# Patent

## Title
Digital video format converter and method therefor

## Abstract

A digital video format converter and method therefor includes a first and second converter. The first converter receives a digital video signal and converts, in the frequency domain, inter-coded pictures in the digital video signal into intra-coded pictures. The first converter performs motion compensation in the frequency domain on the inter-coded pictures in the digital video signal to convert the inter-coded pictures into intra-coded pictures. The second converter down converts output of the first converter. Further signal processing then completes placing the digital video signal output by the second converter in a format acceptable to standard lower definition digital decoders.

## Background
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a method and apparatus for converting the format of a digital video signal; and more particularly, to a method and apparatus for down-converting a digital video signal.
2. Description of the Related Art
Digital video signal processing is an area of science and engineering that has developed rapidly over the past decade. The maturity of the Moving Picture Expert Group (MPEG) video coding standard represents a very important achievement for the video industry and provides strong support for digital transmission of video signals. With advancements in digital compression and other techniques such as digital modulation and packetization, as well as VLSI technology, the fundamentals of television have been reinvented for the digital age.
The first U.S. digital television transmission standard developed for broadcast of high and low definition television by a Grand Alliance of companies has been proposed to the Federal Communications Commission (FCC). High definition digital television broadcasts are typically referred to as HDTV, while low definition digital television broadcasts are generally referred to as SDTV. These terms will be used throughout this application, but are not tied to a particular format or standard. Instead, these terms are used to cover the high and low definition digital television of any coding standard (e.g., such as for VTRs and television).
In 1994 SDTV broadcasts became a reality when the first digital television services, broadcasted via satellite, went on the air. The Digital Satellite Service (DSS) units developed by Thomson Consumer Electronics, etc. have been distributed to more than 1 million homes. The highly sophisticated methods of transmitting and receiving digital television not only produce higher-quality television broadcasts, but also create new services, such as movies on demand, interactive programming, multimedia applications as well as telephone and computer services through the television.
Soon, HDTV will become a reality and join SDTV. Accordingly, in the near future, expect advanced television (ATV) broadcasts which include co-existent broadcasts of HDTV and SDTV. The problem, however, arises in that the HDTV signal can not be decoded by current SDTV decoders or NTSC decoders. (NTSC is the current analog broadcast standard in the U.S.) Several efforts have been reported to address this problem. U.S. Pat. No. 5,262,854 to S. Ng proposes a hierarchical decoder structure to provide a hierarchical representation of a decoded video sequence such that multiple video resolution can be reconstructed from the compressed MPEG bit stream. A modified version of this structure has been proposed by H. Sun, an inventor of the subject application, “Hierarchical Decoder for MPEG Compressed Video Data,” IEEE Trans. on Consumer Electronics, August, 1993, Vol. 39, Number 3, pp. 559-562. Furthermore, a detailed investigation of an all-format ATV decoder has been reported by Jill Boyce, John Henderson and Larry Pearlstein, “An SDTV Decoder with HDTV Capability: An All-format ATV Decoder,” SMPTE Fall Conference, New Orleans, 1995.
Conventionally, the spatial domain down-sizing in converting from HDTV to SDTV was achieved by cutting DCT domain coefficients and performing the inverse DCT with the remaining coefficients. First, the image is filtered by an anti-aliasing low pass filter. The filtered image is then down-sampled by a desired factor in each dimension. For the DCT based coded image (such as MPEG video coding) the conventional method requires converting the compressed image to the spatial domain by inverse DCT and then filtering and down-sampling or down-converting the inverse DCT. Alternatively, both the filtering and down-sampling operations are combined in the DCT domain by cutting frequencies and taking the inverse DCT with a lesser number of DCT coefficients.
When performing, for example, MPEG video encoding of HDTV, image blocks of 8×8 pixels in the spatial domain are converted into 8×8 DCT (discrete cosine transform) blocks of DCT coefficients in the DCT or frequency domain. Specifically, in most coding formats such as MPEG, the HDTV signal is divided into a luminance component (Y) and two chroma components (U) and (V). Furthermore, instead of U and V chroma blocks, some standards use color difference signal chroma blocks. For the purposes of discussion only, U and V chroma blocks will be used. Most formats such as MPEG specify different encoding sequences. In each encoding sequence a sequence header identifies the encoding sequence. Furthermore, in each encoding sequence, macro blocks of 8×8 DCT blocks of DCT coefficients are formed.
Encoding sequences for HDTV typically include the 4:2:0 encoding sequence, the 4:2:2 encoding sequence, and the 4:4:4 encoding sequence. In the 4:2:0 encoding sequence a macro block consists of four 8×8 luminance DCT blocks, one 8×8 U chroma DCT block, and one 8×8 V chroma DCT block. In the 4:2:2 encoding sequence a macro block consists of four 8×8 luminance DCT blocks, two 8×8 U chroma DCT blocks, and two 8×8 V chroma DCT blocks. Finally, in the 4:4:4 encoding sequence a macro block consists of four 8×8 luminance DCT blocks, four 8×8 U chroma DCT blocks, and four 8×8 V chroma DCT blocks. SDTV includes similar coding sequences, but the DCT blocks are 4×4 DCT blocks.
Unfortunately, the conventional conversion techniques produce SDTV signals having a significant amount of block edge effect and other distortions. Additionally, these conventional techniques do not offer high processing speeds, and fail to produce an SDTV signal acceptable to conventional SDTV digital video decoders. Furthermore, when motion compensation must be performed, the conventional down converters require storing two complete anchor pictures in the spatial domain. Consequently, the memory requirements for conventional down converters are quite large. Because memory requirements of down converters account for a large part of the overall device cost, the large memory requirements of conventional down converters adversely impacts the cost of such devices.
SUMMARY OF THE INVENTION
An object of the present invention is to provide a method and apparatus for down-converting an HDTV signal into an SDTV or NTSC signal which performs filtering and down-sampling in the DCT domain.
Another object of the present invention is to provide a method and apparatus for down-converting an HDTV signal into an SDTV or NTSC signal which performs filtering and down-sampling in the DCT domain and reduces block edge effects and other distortions.
A further object of the present invention is to provide a method and apparatus for down-converting an HDTV signal into an SDTV or NTSC signal which performs filtering and down-sampling in the DCT domain and provides high processing speeds.
An additional objective of the present invention is to provide a method and apparatus for down converting a digital video signal which produces a down converted digital video signal acceptable to standard lower definition digital decoders.
A still further objective of the present invention is to provide a method and apparatus for down converting a digital video signal which reduces the frequency and inverse frequency conversion required to down convert the digital video signal.
Another objective of the present invention is to provide a method and apparatus for down converting a digital video signal having significantly lower memory capacity requirements.
These and other objects are achieved by a method for down-converting a digital video signal which includes the steps of receiving the digital video signal including at least first and second DCT blocks of DCT coefficients; synthesizing the first and second DCT blocks into a single synthesized DCT block having dimensions equal to the first and second DCT blocks; and converting the synthesized DCT block from the DCT domain to the spatial domain to produce and output digital video signal.
These and other objects are further achieved by an apparatus for down-converting a digital video signal which includes synthesizing means for receiving the digital video signal including at least first and second DCT blocks of DCT coefficients, and for synthesizing the first and second DCT blocks into a single synthesized DCT block having dimensions equal to the first and second DCT blocks; and converting means for converting the synthesized DCT block from the DCT domain to the spatial domain to produce an output digital video signal.
The objectives of the present invention are also achieved by providing an apparatus for down-converting a digital video signal, comprising: synthesizing means for receiving a digital video signal including at least first and second DCT blocks of DCT coefficients, and for synthesizing said first and second DCT blocks into a single synthesized DCT block having dimensions equal to said first and second DCT blocks; a quantizer quantizing output of said converting means; and a variable length coder variable length coding output of said quantizer.
The objectives of the present invention are additionally achieved by providing a method for down-converting a digital video signal, comprising: (a) receiving a digital video signal including at least first and second DCT blocks of DCT coefficients; (b) synthesizing said first and second DCT blocks into a single synthesized DCT block having dimensions equal to said first and second DCT blocks; (c) quantizing output of said step (b); and (d) variable length coding output of said step (c).
The objectives of the present invention are further achieved by providing an apparatus for down converting a digital video signal, comprising: motion compensation means for receiving a digital video signal and performing motion compensation on said digital video signal in the frequency domain; and a down converter down converting said motion compensated digital video signal.
The objectives of the present invention are also achieved by providing a method for down converting a digital video signal, comprising: (a) receiving a digital video signal; (b) performing motion compensation on said digital video signal in the frequency domain; and (c) down converting said digital video signal output by said step (b).
The objectives of the present invention are still further achieved by providing an apparatus for down converting a digital video signal, comprising: a first converter receiving a digital video signal and converting, in the frequency domain, inter-coded pictures in said digital video signal into intra-coded pictures; and a second converter down converting output of said first converter.
The objectives of the present invention are additionally achieved by providing a method for down converting a digital video signal, comprising: (a) receiving a digital video signal; (b) converting, in the frequency domain, inter-coded pictures in said digital video signal into intra-coded pictures; and (c) down converting output of said step (b).
As discussed above, the method and apparatus of the present invention does not utilize the conventional frequency cutting technique. Instead, the method and apparatus of the present invention employs a frequency synthesis technique which results in reduced block edge effects and other distortions, as well as, high processing speeds.
Other objects, features, and characteristics of the present invention; methods, operation, and functions of the related elements of the structure; combination of parts; and economies of manufacture will become apparent from the following detailed description of the preferred embodiments and accompanying drawings, all of which form a part of this specification, wherein like reference numerals designate corresponding parts in the various figures.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates a conventional apparatus for performing HDTV down-converting wherein filtering and down-sampling are performed in the DCT domain;
FIG. 2 illustrates another conventional apparatus for performing HDTV down-converting wherein filtering and down-sampling are performed in the DCT domain;
FIGS. 3A and 3B illustrate apparatuses, according to the present invention, for performing HDTV down-converting wherein filtering and down-sampling are performed in the DCT domain;
FIGS. 4A and 4B illustrate other apparatuses, according to the present invention, for performing HDTV down-converting wherein filtering and down-sampling are performed in the DCT domain;
FIG. 5 illustrates another embodiment of an HDTV to SDTV converter according to the present invention;
FIG. 6 illustrates a further embodiment of an HDTV to SDTV converter according to the present invention;
FIG. 7 illustrates yet another embodiment of an HDTV to SDTV converter according to the present invention;
FIG. 8 illustrates an embodiment of the memory in the I-picture converter;
FIGS. 9A-9D illustrate the formation of a composite macro block in the spatial domain; and
FIG. 10 illustrates a composite macro block with respect to anchor frame macro blocks.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
More particularly, with respect to the prior art, FIG. 1 illustrates a conventional apparatus for performing HDTV down-converting according to the conventional techniques discussed above wherein the filtering and down-sampling procedures are performed in the DCT domain. As shown in FIG. 1, a variable length decoder (VLD) and dequantizer 10 receives the HDTV bit stream generated, for example, according to MPEG video coding standards. The VLD and dequantizer 10 variable length decodes and dequantizes the HDTV signal in the well known manner. A down-converter 12, including a mask 14 and DCT inverter (IDCT) 16, down-converts the HDTV signal output by the VLD and dequantizer 10. The mask 14 masks the 8×8 blocks of DCT coefficients output by the VLD and dequantizer 10 to produce 4×4 blocks of DCT coefficients, thus, frequency cutting the 8×8 DCT blocks. The IDCT 16 takes the inverse DCT of the 4×4 DCT blocks and converts the 4×4 DCT blocks output by the mask 14 into 4×4 spatial blocks. Depending on the encoding scheme of the HDTV signal, the output of the IDCT 16 produces a bit stream in a corresponding SDTV sequence. For instance, the 4:2:0, 4:2:2 and 4:4:4 HDTV encoding sequences are down-converted to 4:2:0, 4:2:2 and 4:4:4 SDTV encoding sequences, respectively. An NTSC signal can be obtained from the SDTV signal using any well known conversion technique for that purpose.
As alluded to above, whether the output of the IDCT 16 produces a bit stream in SDTV format depends on the encoding scheme of the HDTV signal. Besides variable length encoding, many standards such as MPEG provide for intra- and inter-coding. Intra-coding is where a field or frame of the HDTV signal, referred to as a picture, is encoded based on the pixels therein. Several well known techniques exist for intra-coding. An intra-coded picture is typically referred to as an I-picture. Consequently, the apparatus of FIG. 1 is capable of producing an SDTV signal when the HDTV signal includes only I-pictures.
Inter-coding, sometimes referred to as predictive encoding, is where a picture is encoded based on a reference picture, referred to as an anchor picture. In inter-coding, each macro block (i.e., related luminance and chroma blocks) of the picture being encoded is compared with the macro blocks of the anchor picture to find the macro block of the anchor picture providing the greatest correlation therewith. The vector between the two macro blocks is then determined as the motion vector. The inter-coded HDTV signal for the macro block being encoded will then include the motion vector and the differences between the macro block being encoded and the corresponding macro block of the anchor picture providing the greatest correlation.
For example, a series of pictures may have the display order I1B1B2P1B3B4P2B5B6P3B7B8I2 . . . The transmitted HDTV signal, however, will have the pictures arranged in the order of encoding as follows: I1P1B1B2P2B3B4P3B5B6I2B7B8. P-pictures are encoded using the previous I-picture or P-picture as the anchor picture. In the above example, P-pictures P1, P2, and P3 were encoded using I-picture I1, P-picture P1, and P-picture P2, respectively, as the anchor picture.
The B-pictures may be forward encoded, backward encoded, or bi-directionally encoded. For instance, if B-picture B, was encoded using I-picture I1 as the anchor picture, then B-picture B, is backward or back encoded. Alternatively, if B-picture B, was encoded using P-picture P1 as the anchor picture, then B-picture B1 is forward encoded. If B-picture B, was encoded using both I-picture I1 and P-picture P1 (typically a weighted average thereof) as anchor pictures, then B-picture B, is bi-directionally encoded.
The headers in the HDTV signal indicate whether pictures are I, B, or P-pictures and the direction of encoding. These headers also indicate the group of picture (GOP) size N and the distance between anchor pictures M. The GOP size indicates the distance between I-pictures, which in the above example would be N=12. Since I-pictures and P-pictures are anchor pictures, the distance between anchor pictures in the above example would be M=3. Based on the information provided in the headers, the HDTV signal can be properly decoded.
Therefore, if inter-coding was used to encode a picture, the IDCT 16 outputs only the difference between the present picture and a previous picture. To produce a complete picture requires additional structure such as shown in FIG. 2.
In FIG. 2 the same reference numerals as in FIG. 1 have been used to designate like components. Accordingly, these components will not be described in detail. In FIG. 2, the output of the VLD and dequantizer 10 is also received by a motion vector detector 20. The motion vector detector 20 identifies the motion vectors for inter-coded pictures. Due to the down conversion operation of the down-converter 12, however, the motion vector detector 20 scales down the identified motion vectors by a factor of two. The motion vectors are then output to a frame store 22. The frame store 22 stores the two previous anchor pictures (e.g., I or P-pictures). The frame store 22 will then output the macro blocks of an anchor picture pointed to by the motion vectors. These macro blocks are received by an adder 18. The adder 18 also receives the output of the down-converter 12. Consequently, when a B or P-picture is being down converted, a complete picture can be obtained by adding the output of the down-converter 12, which represents the differences between the picture and an anchor picture, with the anchor picture to create a complete picture. When an I-picture is output from the down-converter 12, there is no need to add anchor picture information thereto, consequently the frame store 22 will not send output to the adder 18, and the output of the adder 18 will be the output of the down-converter 12.
As discussed above, when B-pictures are bi-directionally encoded, a weighted average of two anchor frames is used to inter-code these pictures. The weighting is encoded into the motion vectors by the original encoder. When the frame store 22 receives the two motion vectors, the anchor frame macro blocks pointed thereby are averaged by the frame store 22 to form a weighted average macro block. The frame store 22 then outputs the weighted average macro block to the adder 18.
The frame store 22 also monitors the output of the output of the adder 18. When the frame store 22 determines that the adder 18 outputs an I or P-picture, the frame store 22 replaces the oldest stored anchor frame with the I or P-picture output by the adder 18.
The operation of forming I-pictures from B and P-pictures is referred to as motion compensation. When the pictures are frames, frame motion compensation is performed. When the pictures are fields, field motion compensation is performed. During original encodation, inter-coded even and odd fields are encoded independently based on either the even or odd fields of a previous frame. Consequently, four cases of encodation exist for field motion compensation.
Besides the frame store 22, the output of the adder 18 is also received by a reformatter 24. The reformatter 24 is also connected to the frame store 22. Since the transmission, and therefore, reception order of the pictures is not in the proper display order, the reformatter 24 operates to reformat the order of the pictures into the proper display order.
To better understand the operation of the apparatus illustrated in FIG. 2, assume that an HDTV signal such as discussed above is received. The reformatter 24 determines what output should be sent as the SDTV signal according to the following rules: (1) if the picture received is the first anchor picture received then no output will be sent; (2) if the picture received is an anchor picture but not the first anchor picture received, then the previously received anchor picture will be output; and (3) if the picture received is a B-picture, then the B-picture will be immediately output.
Therefore, upon receipt of I-picture I1, the reformatter 24 will not send any output. The next picture received will be P-picture P1. The adder 18 will then receive the output of the down-converter 12 and macro blocks from the I-picture I1 pointed to by the motion vectors detected by motion vector detector 20. Consequently, the adder 18 will generate a complete picture. Since this complete picture is an anchor picture, the frame store 22 will then store the complete picture P1. According to the rules discussed above, the reformatter 24 will then output from the frame store 22 the I-picture I, (i.e., the previous anchor picture).
The next two pictures received are B-pictures B, and B2. Complete pictures will be formed from these B-pictures in the same manner as discussed above with respect to P-picture P1, except that, depending on the direction of encoding, either the I-picture I1 and/or the P-picture P1 will be used as the anchor picture. Since the adder 18 outputs a B-picture, the reformatter 24 will immediately output the B-picture. Consequently, the output from the reformatter 24 will be I1B1B2.
Next, the P-picture P2 is received and processed in the same manner as P-picture P1. When the adder 18 outputs the complete P-picture P2, the frame store 22 will replace the I-picture I1 with the P-picture P2. The reformatter 24, according to the rules discussed above, will then output the P-picture P1. In this manner, the reformatter 24 will output the pictures in the proper display order.
In another technique, the 4×4 IDCT of FIG. 1 is replaced with an 8×8 IDCT. Accordingly, the 4×4 DCT blocks output by the mask 14 are padded with zeros to produce 8×8 DCT blocks. Then, after performing the 8×8 IDCT, each 8×8 spatial block is down-sampled in each dimension according to conventional techniques to produce 4×4 spatial blocks.
The conventional conversion techniques discussed above with respect to FIGS. 1 and 2 produce SDTV signals having a significant amount of block edge effects and other distortion. Additionally, these down conversion techniques do not offer high speed processing, and fail to produce an SDTV signal acceptable to conventional decoders. Furthermore, the down converter of FIG. 2 must store two anchor pictures in the spatial domain to perform motion compensation. Consequently, the memory requirements for the down converter of FIG. 2 are quite large. Because memory requirements of down converters account for a large part of the overall device cost, the large memory requirements of the down converter illustrated in FIG. 2 adversely impacts the cost of such a device.
First Embodiment
FIG. 3A illustrates an apparatus, according to the present invention, for performing HDTV down-converting wherein filtering and down-sampling are performed in the DCT domain. As shown in FIG. 3A, the apparatus includes VLD and dequantizer 10 such as described above which receives an HDTV signal of any particular format. For the purposes of discussion only, it will be assumed that the HDTV signal was encoded according to the 4:2:0 encoding sequence and includes U and V chroma DCT blocks. A separator 30 receives the output of the VLD and dequantizer 10, and separates the luminance DCT blocks from the V chroma DCT blocks and the U chroma DCT blocks. The luminance DCT blocks are output to a synthesizer 32, and the U and V chroma DCT blocks are output to a down-converter 12.
The down-converter 12 operates on the U and V chroma DCT blocks in the same manner as discussed above with respect to FIG. 1. The synthesizer 32 synthesizes the luminance DCT blocks. An IDCT 34 performs inverse DCT and converts the output of the synthesizer from the DCT or frequency domain into the spatial domain. A recombiner 36 receives the bit stream from the IDCT 34 and the down-converter 12, and recombines these two bit streams to obtain a bit stream in SDTV format. The separator 30, the synthesizer 32, the IDCT 34, the down-converter 12, and the recombiner 36 form the down-converter 40 of HDTV converter according to the present invention illustrated in FIG. 3A.
Operation of the First Embodiment
Next, the operation of the HDTV converter according to the present invention illustrated in FIG. 3A will be described. The separator 30 receives the output of the VLD and dequantizer 10, and based on the sequence header, identifies the sequence of the HDTV signal. When the 4:2:0 sequence has been identified, the separator 30 sends the four 8×8 luminance DCT blocks to the synthesizer 32, and sends the 8×8 U chroma DCT block and the 8×8 V chroma DCT block to the down-converter 12. When the 4:2:2 sequence has been identified, the separator 30 sends the four 8×8 luminance DCT blocks to the synthesizer 32, and sends the two 8×8 U chroma DCT blocks and the two 8×8 V chroma DCT blocks to the down-converter 12. When, however, the separator identifies the 4:4:4 sequence, the separator sends the four 8×8 luminance DCT blocks, the four 8×8 U chroma DCT blocks, and the four 8×8 V chroma DCT blocks to the synthesizer 32.
The synthesizer 32 synthesizes four 8×8 DCT blocks into one 8×8 DCT block. Accordingly, when the synthesizer 32 receives four 8×8 luminance DCT blocks, the synthesizer outputs one 8×8 luminance DCT block. To better explain the operation of the synthesizer 32, the operating principal will first be described for a one-dimensional case, and then the two-dimensional approach used in the synthesizer 32 will be described. Two different techniques for performing the synthesis operation will also be described.
In the one-dimensional case, an 8 element array is obtained from two 8 element arrays (e.g., a 16 element array). Assume that ai (i=0, 1, . . . , 15) represents the 16 element array in the spatial domain, and two one-dimensional arrays in the spatial domain are defined as bi=ai and ci=ai+8 (i=0, 1, . . . , 7). Expressions (1) and (2) represent one-dimensional 8 element arrays b and c as 8×1 DCT arrays B and C, respectively. 



  
    
      
        
          
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In this one-dimensional case, the function of the frequency synthesis operation is to produce an 8×1 DCT array A′ from the two one-dimensional 8×1 DCT arrays B and C. Accordingly, A′ represented in the spatial domain as 8 element array a′ can be defined in terms of 8×1 DCT arrays B and C, and/or 8 element arrays b and c. Expression (3) below shows the relationship between A′ and a′. 



  
    
      
        
          
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The first synthesis technique involves an average smoothing and down-sampling operation. According to the average smoothing operation, a, is obtained from b and c using operators Sb and Sc as shown in expression (4).

a′=Sbb+Scc  (4)

For example, expression (4) may take the form shown in expression (5) below; wherein Sb and Sc are both set to 



  
    
      
        
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                        5
                      
                    
                    )
                  
                
              
            
            
              
                
                  0.5
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        6
                      
                      +
                      
                        c
                        7
                      
                    
                    )
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        5
        )
      
    
  




In accordance with expression (5), the transformation of a′ to A′ in expression (3) can be represented as shown in expression (6) below. 



  
    
      
        
          
            
              
                A
                k
                ′
              
              =
              
              ⁢
              
                
                  
                    g
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  2
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      a
                      i
                      ′
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    g
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        3
                      
                      ⁢
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                2
                                ⁢
                                i
                              
                            
                            +
                            
                              b
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                            
                          
                          )
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    i
                    =
                    4
                  
                  7
                
                ⁢
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          2
                          ⁢
                          
                            (
                            
                              i
                              -
                              4
                            
                            )
                          
                        
                      
                      +
                      
                        c
                        
                          
                            2
                            ⁢
                            
                              (
                              
                                i
                                -
                                4
                              
                              )
                            
                          
                          +
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
              )
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    g
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  4
                
                ⁢
                
                  (
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      3
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      
                        (
                        
                          
                            ∑
                            
                              p
                              =
                              0
                            
                            7
                          
                          ⁢
                          
                            
                              
                                g
                                ⁡
                                
                                  (
                                  p
                                  )
                                
                              
                              2
                            
                            ⁢
                            
                              
                                B
                                p
                              
                              (
                              
                                
                                  cos
                                  ⁢
                                  
                                    
                                      
                                        (
                                        
                                          
                                            4
                                            ⁢
                                            i
                                          
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      p
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      π
                                    
                                    16
                                  
                                
                                +
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
                )
              
              +
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    i
                    =
                    4
                  
                  7
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                    (
                    
                      
                        ∑
                        
                          p
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          
                            g
                            ⁡
                            
                              (
                              p
                              )
                            
                          
                          2
                        
                        ⁢
                        
                          
                            C
                            p
                          
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        4
                                        ⁢
                                        
                                          (
                                          
                                            i
                                            -
                                            4
                                          
                                          )
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                            +
                          
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                (
                                
                                  i
                                  -
                                  4
                                
                                )
                              
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
                )
              
              )
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    B
                    p
                  
                  (
                  
                    
                      
                        
                          g
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        ⁢
                        
                          g
                          ⁡
                          
                            (
                            p
                            )
                          
                        
                      
                      
                        8
                        ⁢
                        
                           
                        
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        3
                      
                      ⁢
                      
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        4
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                            +
                          
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
                )
              
              +
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    C
                    p
                  
                  (
                  
                    
                      
                        
                          g
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        ⁢
                        
                          g
                          ⁡
                          
                            (
                            p
                            )
                          
                        
                      
                      8
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          4
                        
                        7
                      
                      ⁢
                      
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        
                          (
                          
                            
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        4
                                        ⁢
                                        
                                          (
                                          
                                            i
                                            -
                                            4
                                          
                                          )
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                            +
                          
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              (
                              
                                i
                                -
                                4
                              
                              )
                            
                          
                          +
                          3
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
                )
              
              )
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      
                        M
                        k
                      
                      ⁡
                      
                        (
                        p
                        )
                      
                    
                    ⁢
                    
                      B
                      p
                    
                  
                
                +
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      
                        N
                        k
                      
                      ⁡
                      
                        (
                        p
                        )
                      
                    
                    ⁢
                    
                      C
                      p
                    
                  
                
              
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




where Mk (p) and Nk (p) are the matrices represented in expressions (7a) and (7b) below. 



  
    
      
        
          
            M
            k
          
          ⁡
          
            (
            p
            )
          
        
        =
        
          
            
              
                g
                ⁡
                
                  (
                  k
                  )
                
              
              ⁢
              
                g
                ⁡
                
                  (
                  p
                  )
                
              
            
            8
          
          ⁢
          
            
              ∑
              
                i
                =
                0
              
              3
            
            ⁢
            
              cos
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        i
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  k
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        
          7
          ⁢
          a
        
        )
      
    
  
  
    
      
        
          
            N
            k
          
          ⁡
          
            (
            p
            )
          
        
        =
        
          
            
              
                g
                ⁡
                
                  (
                  k
                  )
                
              
              ⁢
              
                g
                ⁡
                
                  (
                  p
                  )
                
              
            
            8
          
          ⁢
          
             
          
          ⁢
          
            
              ∑
              
                i
                =
                4
              
              7
            
            ⁢
            
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        i
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  k
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
              ⁢
              
                 
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                        (
                        
                          
                            4
                            ⁢
                            
                              (
                              
                                i
                                -
                                4
                              
                              )
                            
                          
                          +
                          1
                        
                        )
                      
                      16
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              
                                (
                                
                                  i
                                  -
                                  4
                                
                                )
                              
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        
          7
          ⁢
          b
        
        )
      
    
  




As one skilled in the art will readily note, matrices M and N do not include the DCT coefficients, and are thus independent therefrom. As expression (6) demonstrates, the 8×1 DCT array A1 can be synthesized from the two 8×1 DCT arrays B and C. The 8 element array a′ in the spatial domain can then be obtained by an inverse DCT transformation.
Depending on the desired synthesis characteristics, the smoothing an filtering operations can be modified by modifying Sb and Se in expression (4), which results in changes to matrices M and N.
Having demonstrated the one-dimensional case, one skilled in the art can readily determine the expressions (8)-(15d) below to implement the two-dimensional case, wherein All represents the synthesized 8×8 DCT block; all represents the spatial transformation of A′′; B, C, D, and E each represent one of the four 8×8 DCT blocks; and b, c, d, and e represent the spatial transformation of the 8×8 DCT blocks, respectively. 



  
    
      
        
          B
          
            k
            ⁢
            
               
            
            ⁢
            l
          
        
        =
        
          
            
              
                g
                ⁡
                
                  (
                  k
                  )
                
              
              ⁢
              
                g
                ⁡
                
                  (
                  l
                  )
                
              
            
            4
          
          ⁢
          
            
              ∑
              
                i
                =
                0
              
              7
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                7
              
              ⁢
              
                
                  b
                  
                    i
                    ⁢
                    
                       
                    
                    ⁢
                    j
                  
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  
  
    
      
        
          C
          
            k
            ⁢
            
               
            
            ⁢
            l
          
        
        =
        
          
            
              
                g
                ⁡
                
                  (
                  k
                  )
                
              
              ⁢
              
                g
                ⁡
                
                  (
                  l
                  )
                
              
            
            4
          
          ⁢
          
            
              ∑
              
                i
                =
                0
              
              7
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                7
              
              ⁢
              
                
                  c
                  
                    i
                    ⁢
                    
                       
                    
                    ⁢
                    j
                  
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  
  
    
      
        
          D
          
            k
            ⁢
            
               
            
            ⁢
            l
          
        
        =
        
          
            
              
                g
                ⁡
                
                  (
                  k
                  )
                
              
              ⁢
              
                g
                ⁡
                
                  (
                  l
                  )
                
              
            
            4
          
          ⁢
          
            
              ∑
              
                i
                =
                0
              
              7
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                7
              
              ⁢
              
                
                  d
                  
                    i
                    ⁢
                    
                       
                    
                    ⁢
                    j
                  
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  
  
    
      
        
          E
          
            k
            ⁢
            
               
            
            ⁢
            l
          
        
        =
        
          
            
              
                g
                ⁡
                
                  (
                  k
                  )
                
              
              ⁢
              
                g
                ⁡
                
                  (
                  l
                  )
                
              
            
            4
          
          ⁢
          
            
              ∑
              
                i
                =
                0
              
              7
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                7
              
              ⁢
              
                
                  e
                  
                    i
                    ⁢
                    
                       
                    
                    ⁢
                    j
                  
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where k=0,1, . . . , 7; and 



  
    g
    ⁢
    
      (
      k
      )
    
  
  =
  
    
      g
      ⁡
      
        (
        1
        )
      
    
    =
    
      {
      
        
          
            
              
                1
                
                  2
                
              
              ,
            
          
          
            
              
                
                  if
                  ⁢
                  
                     
                  
                  ⁢
                  k
                
                =
                0
              
              ;
            
          
        
        
          
            
              1
              ,
            
          
          
            
              otherwise
              .
            
          
        
      
      }
    
  




a′′=Sbb+Scc+Sdd+See  (12)




  
    
      
        
          a
          
            i
            ,
            j
          
        
        =
        
          
            1
            4
          
          [
          
             
          
          ⁢
          
            
              
                
                  
                    (
                    
                      
                        b
                        
                          0
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          1
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          0
                          ,
                          1
                        
                      
                      +
                      
                        b
                        
                          1
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          b
                          
                            0
                            ,
                            4
                          
                        
                        +
                        
                          b
                          
                            1
                            ,
                            6
                          
                        
                        +
                        
                          b
                          
                            0
                            ,
                            7
                          
                        
                        +
                        
                          b
                          
                            1
                            ,
                            7
                          
                        
                      
                      )
                    
                  
                
              
              
                
                  
                    (
                    
                      
                        c
                        
                          0
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          1
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          1
                        
                      
                      +
                      
                        c
                        
                          1
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          c
                          
                            0
                            ,
                            6
                          
                        
                        +
                        
                          c
                          
                            1
                            ,
                            6
                          
                        
                        +
                        
                          c
                          
                            0
                            ,
                            7
                          
                        
                        +
                        
                          c
                          
                            1
                            ,
                            7
                          
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    (
                    
                      
                        b
                        
                          2
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          3
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          2
                          ,
                          1
                        
                      
                      +
                      
                        b
                        
                          3
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          b
                          
                            2
                            ,
                            6
                          
                        
                        +
                        
                          b
                          
                            3
                            ,
                            6
                          
                        
                        +
                        
                          b
                          
                            2
                            ,
                            7
                          
                        
                        +
                        
                          b
                          
                            3
                            ,
                            7
                          
                        
                      
                      )
                    
                  
                
              
              
                
                  
                    (
                    
                      
                        c
                        
                          2
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          3
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          2
                          ,
                          1
                        
                      
                      +
                      
                        c
                        
                          3
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          c
                          
                            2
                            ,
                            0
                          
                        
                        +
                        
                          c
                          
                            3
                            ,
                            0
                          
                        
                        +
                        
                          c
                          
                            2
                            ,
                            1
                          
                        
                        +
                        
                          c
                          
                            3
                            ,
                            1
                          
                        
                      
                      )
                    
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
            
            
              
                
                  
                    (
                    
                      
                        b
                        
                          4
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          7
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          6
                          ,
                          2
                        
                      
                      +
                      
                        b
                        
                          7
                          ,
                          2
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          b
                          
                            6
                            ,
                            6
                          
                        
                        +
                        
                          b
                          
                            7
                            ,
                            6
                          
                        
                        +
                        
                          b
                          
                            6
                            ,
                            7
                          
                        
                        +
                        
                          b
                          
                            7
                            ,
                            7
                          
                        
                      
                      )
                    
                  
                
              
              
                
                  
                    (
                    
                      
                        c
                        
                          2
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          3
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          2
                          ,
                          1
                        
                      
                      +
                      
                        c
                        
                          3
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          c
                          
                            6
                            ,
                            6
                          
                        
                        +
                        
                          c
                          
                            7
                            ,
                            6
                          
                        
                        +
                        
                          c
                          
                            6
                            ,
                            7
                          
                        
                        +
                        
                          c
                          
                            7
                            ,
                            7
                          
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    (
                    
                      
                        d
                        
                          0
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          1
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          1
                        
                      
                      +
                      
                        d
                        
                          1
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            0
                            ,
                            6
                          
                        
                        +
                        
                          d
                          
                            1
                            ,
                            6
                          
                        
                        +
                        
                          d
                          
                            0
                            ,
                            7
                          
                        
                        +
                        
                          d
                          
                            1
                            ,
                            7
                          
                        
                      
                      )
                    
                  
                
              
              
                
                  
                    (
                    
                      
                        e
                        
                          0
                          ,
                          0
                        
                      
                      +
                      
                        e
                        
                          1
                          ,
                          0
                        
                      
                      +
                      
                        e
                        
                          0
                          ,
                          1
                        
                      
                      +
                      
                        e
                        
                          1
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          e
                          
                            0
                            ,
                            6
                          
                        
                        +
                        
                          e
                          
                            1
                            ,
                            4
                          
                        
                        +
                        
                          e
                          
                            0
                            ,
                            7
                          
                        
                        +
                        
                          e
                          
                            1
                            ,
                            7
                          
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  
                    (
                    
                      
                        d
                        
                          2
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          3
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          2
                          ,
                          1
                        
                      
                      +
                      
                        d
                        
                          3
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            2
                            ,
                            6
                          
                        
                        +
                        
                          d
                          
                            3
                            ,
                            6
                          
                        
                        +
                        
                          d
                          
                            2
                            ,
                            7
                          
                        
                        +
                        
                          d
                          
                            3
                            ,
                            7
                          
                        
                      
                      )
                    
                  
                
              
              
                
                  
                    (
                    
                      
                        e
                        
                          2
                          ,
                          0
                        
                      
                      +
                      
                        e
                        
                          3
                          ,
                          0
                        
                      
                      +
                      
                        e
                        
                          2
                          ,
                          1
                        
                      
                      +
                      
                        e
                        
                          3
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          e
                          
                            2
                            ,
                            0
                          
                        
                        +
                        
                          e
                          
                            3
                            ,
                            0
                          
                        
                        +
                        
                          e
                          
                            2
                            ,
                            1
                          
                        
                        +
                        
                          e
                          
                            3
                            ,
                            1
                          
                        
                      
                      )
                    
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
            
            
              
                
                  
                    (
                    
                      
                        d
                        
                          6
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          7
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          6
                          ,
                          2
                        
                      
                      +
                      
                        d
                        
                          7
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          d
                          
                            6
                            ,
                            6
                          
                        
                        +
                        
                          d
                          
                            7
                            ,
                            6
                          
                        
                        +
                        
                          d
                          
                            6
                            ,
                            7
                          
                        
                        +
                        
                          d
                          
                            7
                            ,
                            7
                          
                        
                      
                      )
                    
                  
                
              
              
                
                  
                    (
                    
                      
                        e
                        
                          6
                          ,
                          0
                        
                      
                      +
                      
                        e
                        
                          7
                          ,
                          0
                        
                      
                      +
                      
                        e
                        
                          6
                          ,
                          2
                        
                      
                      +
                      
                        e
                        
                          7
                          ,
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    ...
                    ⁡
                    
                      (
                      
                        
                          e
                          
                            6
                            ,
                            6
                          
                        
                        +
                        
                          e
                          
                            7
                            ,
                            6
                          
                        
                        +
                        
                          e
                          
                            6
                            ,
                            7
                          
                        
                        +
                        
                          e
                          
                            6
                            ,
                            6
                          
                        
                      
                      )
                    
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        13
        )
      
    
  




where Sb=Sc=Sd=Sc=0.25



  
    
      
        
          
            
              
                A
                
                  k
                  ⁢
                  
                     
                  
                  ⁢
                  l
                
                ′′
              
              =
              
              ⁢
              
                
                  
                    
                      g
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        l
                        )
                      
                    
                  
                  4
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        a
                        
                          i
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        ′′
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        32
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        32
                      
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      
                        g
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          l
                          )
                        
                      
                    
                    4
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      3
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        3
                      
                      ⁢
                      
                        
                          1
                          4
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                ,
                                
                                  2
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              b
                              
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                ,
                                
                                  2
                                  ⁢
                                  j
                                
                              
                            
                            +
                            
                              b
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                ,
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  1
                                
                              
                            
                            +
                            
                              b
                              
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                ,
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  1
                                
                              
                            
                          
                          )
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          32
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          32
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      i
                      =
                      4
                    
                    7
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      3
                    
                    ⁢
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            c
                            
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    i
                                    -
                                    4
                                  
                                  )
                                
                              
                              ,
                              
                                2
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            c
                            
                              
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      i
                                      -
                                      4
                                    
                                    )
                                  
                                
                                +
                                1
                              
                              ,
                              
                                2
                                ⁢
                                j
                              
                            
                          
                          +
                          
                            c
                            
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    i
                                    -
                                    4
                                  
                                  )
                                
                              
                              ,
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                            
                          
                          +
                          
                          ⁢
                          
                            c
                            
                              
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      i
                                      -
                                      4
                                    
                                    )
                                  
                                
                                +
                                1
                              
                              ,
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        32
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        32
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    3
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        4
                      
                      7
                    
                    ⁢
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        (
                        
                          
                            d
                            
                              
                                2
                                ⁢
                                i
                              
                              ,
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    j
                                    -
                                    4
                                  
                                  )
                                
                              
                            
                          
                          +
                          
                            d
                            
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              ,
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    j
                                    -
                                    4
                                  
                                  )
                                
                              
                            
                          
                          +
                          
                            d
                            
                              
                                2
                                ⁢
                                i
                              
                              ,
                              
                                
                                  2
                                  ⁢
                                  
                                    (
                                    
                                      j
                                      -
                                      4
                                    
                                    )
                                  
                                
                                +
                                1
                              
                            
                          
                          +
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    d
                    
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      ,
                      
                        
                          2
                          ⁢
                          
                            (
                            
                              j
                              -
                              4
                            
                            )
                          
                        
                        +
                        1
                      
                    
                  
                  )
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  32
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  32
                
              
              +
              
              ⁢
              
                
                  ∑
                  
                    i
                    =
                    4
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      4
                    
                    7
                  
                  ⁢
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      (
                      
                        
                          e
                          
                            
                              2
                              ⁢
                              
                                (
                                
                                  i
                                  -
                                  4
                                
                                )
                              
                            
                            ,
                            
                              (
                              
                                j
                                -
                                4
                              
                              )
                            
                          
                        
                        +
                        
                          e
                          
                            
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    i
                                    -
                                    4
                                  
                                  )
                                
                              
                              +
                              1
                            
                            ,
                            
                              2
                              ⁢
                              
                                (
                                
                                  j
                                  -
                                  4
                                
                                )
                              
                            
                          
                        
                        +
                        
                          e
                          
                            
                              2
                              ⁢
                              
                                (
                                
                                  i
                                  -
                                  4
                                
                                )
                              
                            
                            ,
                            
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    j
                                    -
                                    4
                                  
                                  )
                                
                              
                              +
                              1
                            
                          
                        
                        +
                        
                          e
                          
                            
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    i
                                    -
                                    4
                                  
                                  )
                                
                              
                              +
                              1
                            
                            ,
                            
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    j
                                    -
                                    4
                                  
                                  )
                                
                              
                              +
                              1
                            
                          
                        
                      
                      )
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        l
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      g
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        l
                        )
                      
                    
                  
                  4
                
                ⁢
                
                  (
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      3
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        3
                      
                      ⁢
                      
                        
                          1
                          4
                        
                        ⁢
                        
                          (
                          
                            
                              ∑
                              
                                p
                                =
                                0
                              
                              7
                            
                            ⁢
                            
                              
                                ∑
                                
                                  q
                                  =
                                  0
                                
                                7
                              
                              ⁢
                              
                                
                                  
                                    
                                      g
                                      ⁡
                                      
                                        (
                                        p
                                        )
                                      
                                    
                                    ⁢
                                    
                                      g
                                      ⁡
                                      
                                        (
                                        q
                                        )
                                      
                                    
                                  
                                  4
                                
                                ⁢
                                
                                  
                                    B
                                    
                                      p
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      q
                                    
                                  
                                  (
                                  
                                    
                                      cos
                                      ⁢
                                      
                                        
                                          
                                            (
                                            
                                              
                                                4
                                                ⁢
                                                i
                                              
                                              +
                                              1
                                            
                                            )
                                          
                                          ⁢
                                          p
                                          ⁢
                                          
                                             
                                          
                                          ⁢
                                          π
                                        
                                        32
                                      
                                      ⁢
                                      cos
                                      ⁢
                                      
                                        
                                          
                                            (
                                            
                                              
                                                4
                                                ⁢
                                                j
                                              
                                              +
                                              1
                                            
                                            )
                                          
                                          ⁢
                                          q
                                          ⁢
                                          
                                             
                                          
                                          ⁢
                                          π
                                        
                                        32
                                      
                                    
                                    +
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              j
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                  +
                  
                    cos
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        (
        14
        )
      
    
  




where 



  
    
      
        
          
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                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              j
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    
                       
                    
                    ⁢
                    cos
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              j
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    
                       
                    
                    ⁢
                    cos
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              j
                            
                            -
                            13
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              j
                            
                            -
                            13
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        
          15
          ⁢
          c
        
        )
      
    
  
  
    
      
        
          
            Z
            
              k
              ⁢
              
                 
              
              ⁢
              l
            
          
          ⁡
          
            (
            
              p
              ,
              q
            
            )
          
        
        =
        
          
            
              
                g
                ⁡
                
                  (
                  k
                  )
                
              
              ⁢
              
                g
                ⁡
                
                  (
                  l
                  )
                
              
              ⁢
              
                g
                ⁡
                
                  (
                  p
                  )
                
              
              ⁢
              
                g
                ⁡
                
                  (
                  q
                  )
                
              
            
            16
          
          ·
          
            
              ∑
              
                i
                =
                4
              
              7
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  4
                
                7
              
              ⁢
              
                [
                
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              j
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            -
                            13
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    
                       
                    
                    ⁢
                    cos
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              j
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    
                       
                    
                    ⁢
                    cos
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              j
                            
                            -
                            13
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                  +
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              i
                            
                            -
                            13
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              4
                              ⁢
                              j
                            
                            -
                            13
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      32
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        
          15
          ⁢
          d
        
        )
      
    
  




In the expressions above for the two-dimensional case, expressions (8)-(11), (12), (13), (14), and (15a)-(15d) correspond to expressions (1)-(2), (4), (5), (6) and (7a)-(7b) for the one-dimensional case. Therefore, as discussed above with the one-dimensional case, matrices W, X, Y and Z are independent of the DCT coefficients, and the smoothing and filtering operation can be modified by modifying Sb, Sc, Sd and Se in expression (12), which results in changes to matrices W, X, Y and Z.
The second technique involves low-pass filtering and smoothing. In this technique, an ideal low-pass filter, serving as an anti-aliasing filter, removes high-frequencies before down-sampling. The process of ideal low-pass filtering is equivalent to the convolution of the input array with a sin function in the spatial domain. Again, using a to designate a 16 element array, the one-dimensional case will be described. A 16×1 DCT array A is obtained from a using expression (16) below. 



  
    
      
        
          
            A
            k
          
          =
          
            
              
                
                  g
                  ⁡
                  
                    (
                    k
                    )
                  
                
                2
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  15
                
                ⁢
                
                  
                    a
                    k
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    32
                  
                  ⁢
                  
                     
                  
                  ⁢
                  k
                
              
            
            =
            0
          
        
        ,
        1
        ,
        ...
        ⁢
        
           
        
        ,
        15.
      
    
    
      
        (
        16
        )
      
    
  




Implementing ideal low-pass filtering, an 8×1 DCT array A′ is obtained according to expression (17) below.

A′k=Ak, k=0,1, . . . ,7  (17) 

Next, to obtain synthesis, A′ must be characterized in terms of B and C. Expression (18) shows the relationship between A′ and 8 element arrays b and c for even coefficients of A′. 



  
    
      
        
          
            
              
                A
                
                  2
                  ⁢
                  
                     
                  
                  ⁢
                  k
                
                ′
              
              =
              
              ⁢
              
                
                  
                    g
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  2
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    15
                  
                  ⁢
                  
                    
                      a
                      i
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            2
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          32
                        
                        )
                      
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    g
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          a
                          i
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            )
                          
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    i
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    a
                    
                      i
                      +
                      8
                    
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            
                              
                                2
                                ⁢
                                
                                  (
                                  
                                    i
                                    +
                                    8
                                  
                                  )
                                
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    g
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          b
                          i
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            )
                          
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    
                      (
                      
                        -
                        1
                      
                      )
                    
                    k
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        c
                        i
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          )
                        
                      
                    
                  
                
                ,
                
                  k
                  =
                  0
                
                ,
                1
                ,
                
                  ...
                  ⁢
                  
                     
                  
                  ⁢
                  3.
                
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




From expressions (18), (1) and (2), expression (19) is obtained.

A′′2k=Bk(−1)kCkk=0,1, . . . 3,  (19) 

Expression (20) shows the relationship between A′ and 8 element arrays b and c for odd coefficients of A′. 



  
    
      
        
          
            
              
                A
                
                  
                    2
                    ⁢
                    k
                  
                  +
                  1
                
                ′
              
              =
              
              ⁢
              
                
                  
                    g
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          b
                          i
                        
                        ⁢
                        
                          cos
                          (
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  k
                                  +
                                  
                                    1
                                    2
                                  
                                
                                )
                              
                              ⁢
                              π
                            
                            16
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    (
                    
                      -
                      1
                    
                    )
                  
                  k
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      c
                      i
                    
                    ⁢
                    
                      cos
                      (
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                k
                                +
                                
                                  1
                                  2
                                
                              
                              )
                            
                            ⁢
                            π
                          
                          16
                        
                        +
                        
                          
                            (
                            
                              k
                              +
                              
                                1
                                2
                              
                            
                            )
                          
                          ⁢
                          π
                        
                      
                      )
                    
                  
                
              
              )
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    g
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          b
                          i
                        
                        ⁢
                        
                          cos
                          (
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  k
                                  +
                                  
                                    1
                                    2
                                  
                                
                                )
                              
                              ⁢
                              π
                            
                            16
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    
                      (
                      
                        -
                        1
                      
                      )
                    
                    k
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        c
                        i
                      
                      ⁢
                      
                        sin
                        (
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                k
                                +
                                
                                  1
                                  2
                                
                              
                              )
                            
                            ⁢
                            π
                          
                          16
                        
                        )
                      
                    
                  
                
                )
              
              ,
              
                k
                =
                0
              
              ,
              1
              ,
              
                ...
                ⁢
                
                   
                
                ⁢
                3.
              
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




Unfortunately, the odd coefficients can not be reduced to a simple expression as with the even coefficients. Accordingly, a matrix operation is required to express the odd coefficients of A′ in terms of 8×1 matrices B and C. Expression (21) and (22a)-(22b) below shows the relationship between the odd coefficients of A′ and 8×1 matrices B and C. 



  
    
      
        
          
            
              
                A
                
                  
                    2
                    ⁢
                    k
                  
                  +
                  1
                
                ′
              
              =
              
              ⁢
              
                
                  
                    g
                    ⁡
                    
                      (
                      k
                      )
                    
                  
                  2
                
                ⁢
                
                  (
                  
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                k
                                +
                                
                                  1
                                  2
                                
                              
                              )
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        
                          (
                          
                            
                              ∑
                              
                                p
                                =
                                0
                              
                              7
                            
                            ⁢
                            
                              
                                
                                  g
                                  ⁡
                                  
                                    (
                                    p
                                    )
                                  
                                
                                2
                              
                              ⁢
                              
                                B
                                p
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                          )
                        
                      
                    
                    +
                    
                       
                    
                    ⁢
                    
                      (
                      21
                      )
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    (
                    
                      -
                      1
                    
                    )
                  
                  k
                
                ⁢
                
                  (
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      sin
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                            (
                            
                              k
                              +
                              
                                1
                                2
                              
                            
                            )
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      
                        (
                        
                          
                            ∑
                            
                              p
                              =
                              0
                            
                            7
                          
                          ⁢
                          
                            
                              
                                g
                                ⁡
                                
                                  (
                                  p
                                  )
                                
                              
                              2
                            
                            ⁢
                            
                              C
                              p
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                        )
                      
                    
                  
                  )
                
              
              )
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      B
                      p
                    
                    (
                    
                      
                        
                          
                            g
                            ⁡
                            
                              (
                              k
                              )
                            
                          
                          ⁢
                          
                            g
                            ⁡
                            
                              (
                              p
                              )
                            
                          
                        
                        4
                      
                      ⁢
                      
                        
                          ∑
                          
                            i
                            =
                            0
                          
                          7
                        
                        ⁢
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              
                                (
                                
                                  k
                                  +
                                  
                                    1
                                    2
                                  
                                
                                )
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    (
                    
                      -
                      1
                    
                    )
                  
                  k
                
                ⁢
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      C
                      p
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              g
                              ⁡
                              
                                (
                                k
                                )
                              
                            
                            ⁢
                            
                              g
                              ⁡
                              
                                (
                                p
                                )
                              
                            
                          
                          4
                        
                        ⁢
                        
                          
                            ∑
                            
                              i
                              =
                              0
                            
                            7
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                  (
                                  
                                    k
                                    +
                                    
                                      1
                                      2
                                    
                                  
                                  )
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    
                      ∑
                      
                        p
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        B
                        p
                      
                      ⁢
                      
                        
                          U
                          k
                        
                        ⁡
                        
                          (
                          p
                          )
                        
                      
                    
                  
                  +
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      k
                    
                    ⁢
                    
                      
                        ∑
                        
                          p
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          C
                          p
                        
                        ⁢
                        
                          
                            V
                            k
                          
                          ⁡
                          
                            (
                            p
                            )
                          
                        
                      
                    
                  
                
              
              ,
              
                k
                =
                0
              
              ,
              1
              ,
              ...
              ⁢
              
                 
              
              ,
              3.
            
          
        
        
          
            
              
              ⁢
              
                where
                ⁢
                
                   
                
                ⁢
                
                  
                    U
                    k
                  
                  ⁡
                  
                    (
                    p
                    )
                  
                
                ⁢
                
                   
                
                ⁢
                and
                ⁢
                
                   
                
                ⁢
                
                  
                    V
                    k
                  
                  ⁡
                  
                    (
                    p
                    )
                  
                
                ⁢
                
                   
                
                ⁢
                are
                ⁢
                
                   
                
                ⁢
                the
                ⁢
                
                   
                
                ⁢
                conversion
                ⁢
                
                   
                
                ⁢
                matries
                ⁢
                
                   
                
                ⁢
                with
                ⁢
                
                  :
                
              
            
          
        
        
          
            
              
                
                  
                    U
                    k
                  
                  ⁡
                  
                    (
                    p
                    )
                  
                
                =
                
                ⁢
                
                  
                    
                      
                        g
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          p
                          )
                        
                      
                    
                    4
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                            (
                            
                              k
                              +
                              
                                1
                                2
                              
                            
                            )
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
              
              ,
              
                 
              
              ⁢
              
                (
                
                  22
                  ⁢
                  a
                
                )
              
            
          
        
        
          
            
              
              ⁢
              
                
                  k
                  =
                  0
                
                ,
                1
                ,
                
                  ...
                  ⁢
                  
                     
                  
                  ⁢
                  3.
                
              
            
          
        
        
          
            
              
                
                  
                    V
                    k
                  
                  ⁡
                  
                    (
                    p
                    )
                  
                
                =
                
                ⁢
                
                  
                    
                      
                        g
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          p
                          )
                        
                      
                    
                    4
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                            (
                            
                              k
                              +
                              
                                1
                                2
                              
                            
                            )
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
              
              ,
              
                 
              
              ⁢
              
                (
                
                  22
                  ⁢
                  b
                
                )
              
            
          
        
        
          
            
              
              ⁢
              
                
                  k
                  =
                  0
                
                ,
                1
                ,
                
                  ...
                  ⁢
                  
                     
                  
                  ⁢
                  3.
                
              
            
          
        
      
    
    
      
         
      
    
  




As with matrices M and N in the first technique matrices U and V are independent from the DCT coefficients.

Based on expressions (19) and (21), the synthesized 8×1 DCT array A′ can be obtained. Then, the spatial domain representation of A′, a′, can be obtained by inverse DCT.
As with the first technique, having demonstrated the one-dimensional case for the second technique, one skilled in the art can readily determine expressions (23)-(26d) below to implement the two-dimensional case of the second technique, A′′ represents the synthesized 8×8 DCT block; a′′ represents the spatial transformation of A′′; B, C, D, and E each represent one of the four 8×8 DCT blocks; and b, c, d, and 3 represent the spatial transformation of the 8×8 DCT blocks, respectively. 



  
    
      
        
          
            
              
                
                  A
                  
                    2
                    ⁢
                    k2l
                  
                  ′′
                
                =
                
                ⁢
                
                  
                    
                      
                        g
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          l
                          )
                        
                      
                    
                    4
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      15
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        15
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          a
                          ij
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            2
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          32
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            2
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          32
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        g
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          l
                          )
                        
                      
                    
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        
                          ∑
                          
                            i
                            =
                            0
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            7
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              a
                              ij
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      +
                      
                        
                          ∑
                          
                            i
                            =
                            0
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            7
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              a
                              
                                
                                  i
                                  +
                                  8
                                
                                ,
                                j
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    8
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      +
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          a
                          
                            i
                            ,
                            
                              j
                              +
                              8
                            
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                8
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                  +
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          a
                          
                            
                              i
                              +
                              8
                            
                            ,
                            
                              j
                              +
                              8
                            
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                8
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                8
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                
                )
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        
                          g
                          ⁡
                          
                            (
                            k
                            )
                          
                        
                        ⁢
                        
                          g
                          ⁡
                          
                            (
                            l
                            )
                          
                        
                      
                      4
                    
                    ⁢
                    
                      
                        (
                        ∑
                      
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          b
                          ij
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                  +
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      k
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            c
                            ij
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                  
                  +
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      l
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            d
                            ij
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                  
                  +
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      
                        k
                        +
                        l
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            e
                            ij
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    B
                    ij
                  
                  +
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      k
                    
                    ⁢
                    
                      C
                      ij
                    
                  
                  +
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      l
                    
                    ⁢
                    
                      D
                      ij
                    
                  
                  +
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      
                        k
                        +
                        l
                      
                    
                    ⁢
                    
                      E
                      ij
                    
                  
                
              
            
          
        
         
      
    
    
      
        (
        23
        )
      
    
  






  
    
      
        
          
            
              
                
                  A
                  
                    
                      
                        2
                        ⁢
                        k
                      
                      +
                      1
                    
                    ,
                    
                      
                        2
                        ⁢
                        l
                      
                      +
                      1
                    
                  
                  ′′
                
                =
                
                ⁢
                
                  
                    
                      
                        g
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          l
                          )
                        
                      
                    
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        
                          ∑
                          
                            i
                            =
                            0
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            7
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              b
                              
                                i
                                ,
                                j
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      +
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                    ⁢
                    
                      
                        
                          
                            (
                            
                              -
                              1
                            
                            )
                          
                          k
                        
                        ⁢
                        
                          
                            ∑
                            
                              i
                              =
                              0
                            
                            7
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              7
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                c
                                ij
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  
                                    (
                                    
                                      k
                                      +
                                      
                                        1
                                        2
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                      
                      +
                      
                        
                          (
                          
                            k
                            +
                            
                              1
                              2
                            
                          
                          )
                        
                        ⁢
                        π
                      
                    
                    )
                  
                  ·
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                l
                                +
                                
                                  1
                                  2
                                
                              
                              )
                            
                            ⁢
                            π
                          
                          16
                        
                        +
                        
                          
                            (
                            
                              l
                              +
                              
                                1
                                2
                              
                            
                            )
                          
                          ⁢
                          π
                        
                      
                      )
                    
                  
                
                +
                
                  
                    
                      (
                      
                        -
                        1
                      
                      )
                    
                    l
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          d
                          ij
                        
                        ⁢
                        
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    
                                      (
                                      
                                        k
                                        +
                                        
                                          1
                                          2
                                        
                                      
                                      )
                                    
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                +
                                
                                  
                                    (
                                    
                                      k
                                      +
                                      
                                        1
                                        2
                                      
                                    
                                    )
                                  
                                  ⁢
                                  π
                                
                              
                              )
                            
                          
                          ·
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                l
                                +
                                
                                  1
                                  2
                                
                              
                              )
                            
                            ⁢
                            π
                          
                          16
                        
                        +
                        
                          
                            (
                            
                              l
                              +
                              
                                1
                                2
                              
                            
                            )
                          
                          ⁢
                          π
                        
                      
                      )
                    
                  
                  +
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      
                        k
                        +
                        l
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            e
                            ij
                          
                          ⁢
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            i
                                          
                                          +
                                          1
                                        
                                        )
                                      
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      
                                        (
                                        
                                          k
                                          +
                                          
                                            1
                                            2
                                          
                                        
                                        )
                                      
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      π
                                    
                                    16
                                  
                                  +
                                  
                                    
                                      (
                                      
                                        k
                                        +
                                        
                                          1
                                          2
                                        
                                      
                                      )
                                    
                                    ⁢
                                    π
                                  
                                
                                )
                              
                            
                            ·
                            cos
                          
                          ⁢
                          
                            (
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  
                                    (
                                    
                                      l
                                      +
                                      
                                        1
                                        2
                                      
                                    
                                    )
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              +
                              
                                
                                  (
                                  
                                    l
                                    +
                                    
                                      1
                                      2
                                    
                                  
                                  )
                                
                                ⁢
                                π
                              
                            
                            )
                          
                        
                      
                    
                  
                
                )
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      
                        g
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          l
                          )
                        
                      
                    
                    4
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        (
                        
                          
                            
                              b
                              ij
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  (
                                  
                                    k
                                    +
                                    
                                      1
                                      2
                                    
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  (
                                  
                                     
                                  
                                  ⁢
                                  
                                    l
                                    +
                                    
                                      1
                                      2
                                    
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          +
                          
                            
                              
                                (
                                
                                  -
                                  1
                                
                                )
                              
                              k
                            
                            ⁢
                            
                              c
                              ij
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  (
                                  
                                    k
                                    +
                                    
                                      1
                                      2
                                    
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  (
                                  
                                     
                                  
                                  ⁢
                                  
                                    l
                                    +
                                    
                                      1
                                      2
                                    
                                  
                                  )
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          +
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      l
                    
                    ⁢
                    
                      d
                      ij
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          (
                          
                            k
                            +
                            
                              1
                              2
                            
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          (
                          
                             
                          
                          ⁢
                          
                            l
                            +
                            
                              1
                              2
                            
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  +
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      
                        k
                        +
                        l
                      
                    
                    ⁢
                    
                      e
                      ij
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          (
                          
                            k
                            +
                            
                              1
                              2
                            
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          (
                          
                             
                          
                          ⁢
                          
                            l
                            +
                            
                              1
                              2
                            
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                
              
            
          
        
         
      
    
    
      
        (
        24
        )
      
    
  






  
    
      
        
          
            
              
                
                  A
                  
                    
                      
                        2
                        ⁢
                        k
                      
                      +
                      1
                    
                    ,
                    
                      
                        2
                        ⁢
                        l
                      
                      +
                      1
                    
                  
                  ′′
                
                =
                
                ⁢
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        1
                        4
                      
                      [
                      
                        
                          
                            
                              Q
                              kl
                            
                            ⁡
                            
                              (
                              
                                p
                                ,
                                q
                              
                              )
                            
                          
                          ⁢
                          
                            B
                            pq
                          
                        
                        +
                        
                          
                            
                              (
                              
                                -
                                1
                              
                              )
                            
                            k
                          
                          ⁢
                          
                            
                              R
                              kl
                            
                            ⁡
                            
                              (
                              
                                p
                                ,
                                q
                              
                              )
                            
                          
                          ⁢
                          
                            C
                            pq
                          
                        
                        +
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      l
                    
                    ⁢
                    
                      
                        S
                        kl
                      
                      ⁡
                      
                        (
                        
                          p
                          ,
                          q
                        
                        )
                      
                    
                    ⁢
                    
                      D
                      pq
                    
                  
                  +
                  
                    
                      
                        (
                        
                          -
                          1
                        
                        )
                      
                      
                        k
                        +
                        l
                      
                    
                    ⁢
                    
                      
                        T
                        kl
                      
                      ⁡
                      
                        (
                        
                          p
                          ,
                          q
                        
                        )
                      
                    
                    ⁢
                    
                      E
                      pq
                    
                  
                
                ]
              
            
          
        
         
      
    
    
      
        (
        25
        )
      
    
  
  
    
      
        where
        ,
      
    
    
      
         
      
    
  
  
    
      
        
          
            
              
                
                  
                    Q
                    kl
                  
                  ⁡
                  
                    (
                    
                      p
                      ,
                      q
                    
                    )
                  
                
                =
                
                ⁢
                
                  
                    
                      
                        g
                        ⁡
                        
                          (
                          k
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          l
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          p
                          )
                        
                      
                      ⁢
                      
                        g
                        ⁡
                        
                          (
                          q
                          )
                        
                      
                    
                    8
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              p
                            
                            16
                          
                          ·
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          k
                          +
                          
                            1
                            2
                          
                        
                        )
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                        (
                        
                          1
                          +
                          
                            1
                            2
                          
                        
                        )
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
          
        
         
      
    
    
      
        (
        
          26
          ⁢
          a
        
        )
      
    
  
  
    
      
        
          
            
              
                
                  R
                  kl
                
                ⁡
                
                  (
                  
                    p
                    ,
                    q
                  
                  )
                
              
              =
              
              ⁢
              
                
                  
                    
                      g
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        l
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        p
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        q
                        )
                      
                    
                  
                  8
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            p
                          
                          16
                        
                        ·
                      
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                sin
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        k
                        +
                        
                          1
                          2
                        
                      
                      )
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                sin
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        1
                        +
                        
                          1
                          2
                        
                      
                      )
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
        
      
    
    
      
        (
        
          26
          ⁢
          b
        
        )
      
    
  
  
    
      
        
          
            
              
                
                  S
                  kl
                
                ⁡
                
                  (
                  
                    p
                    ,
                    q
                  
                  )
                
              
              =
              
              ⁢
              
                
                  
                    
                      g
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        l
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        p
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        q
                        )
                      
                    
                  
                  8
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                l
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            q
                          
                          16
                        
                        ·
                      
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        k
                        +
                        
                          1
                          2
                        
                      
                      )
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        1
                        +
                        
                          1
                          2
                        
                      
                      )
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
        
      
    
    
      
        (
        
          26
          ⁢
          c
        
        )
      
    
  
  
    
      
        
          
            
              
                
                  T
                  kl
                
                ⁡
                
                  (
                  
                    p
                    ,
                    q
                  
                  )
                
              
              =
              
              ⁢
              
                
                  
                    
                      g
                      ⁡
                      
                        (
                        k
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        l
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        p
                        )
                      
                    
                    ⁢
                    
                      g
                      ⁡
                      
                        (
                        q
                        )
                      
                    
                  
                  8
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                l
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            q
                          
                          16
                        
                        ·
                      
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                sin
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        k
                        +
                        
                          1
                          2
                        
                      
                      )
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                sin
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        1
                        +
                        
                          1
                          2
                        
                      
                      )
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
        
      
    
    
      
        (
        
          26
          ⁢
          d
        
        )
      
    
  




In the expressions above for the two-dimensional case, expressions, (23)-(26d) correspond to expressions (19)-(22b) for the one-dimensional case. Therefore, as discussed above with the one-dimensional case, matrices Q, R, S, and T are independent of the DCT coefficients.
Accordingly, using either expressions (14)-(15d) according to the first technique or expressions (21)-(26d) according to the second technique, the synthesizer 32 synthesizes the four 8×8 luminance DCT blocks into a single 8×8 luminance DCT block.
When the separator 30 identifies either the 4:2:0 or the 4:2:2 sequence, the synthesizer 32 synthesizes the four 8×8 luminance DCT blocks into one 8×8 luminance DCT block according to one of the two techniques discussed above. If, however, the separator 30 identifies the 4:4:4 sequence, the synthesizer 32 synthesizes the four 8×8 luminance DCT blocks into one 8×8 luminance DCT block, synthesizes the four 8×8 U chroma DCT blocks into one 8×8 U chroma DCT block, and synthesizes the four 8×8 V chroma DCT blocks into one 8×8 V chroma DCT block. The IDCT 34 converts the DCT blocks output from the synthesizer 32 into spatial blocks.
During the synthesis operation, the down-converter 12 receives the output of the separator 30 as discussed above, and operates in the same manner as described in the Background of the Invention section. The recombiner 36 also identifies the HDTV sequence based on the sequence header, and recombines the spatial luminance blocks output by the IDCT 34 with the spatial chroma blocks output by the down-converter 12 when the 4:2:0 or 4:2:2 sequence have been identified. Otherwise, when the 4:4:4 sequence has been identified, the recombiner 36 merely passes the output of the IDCT 36. The output of the recombiner 36 is an SDTV bit stream unless as discussed above, the HDTV signal was predictive or inter-coding encoded.
Second Embodiment
If inter-coding was used, the output of the recombiner 36 will not represent spatial data in SDTV format. Accordingly, the adder 18, the motion vector detector 20, the frame store 22, and the reformatter 24 will have to be connected to the converting apparatus of FIG. 3A such as shown in FIG. 4A. Specifically, the motion vector detector 20 is connected to receive the output of the VLD and dequantizer 10. The frame store 22 is connected to the output of the motion vector detector 20, and the adder 18 receives and adds the output of the recombiner 36 and the frame store 22. The frame store 22 also receives the output of the adder 18. The reformatter 24 is connected to both the adder 18 and the frame store 22, and outputs either the output from the adder 18 or a picture stored by the frame store 22. Since the operation of these additional components was described with respect to FIG. 2, that description will not be repeated.
As with the conventional art, once a digital signal in SDTV format is obtained, the SDTV signal can be converted to an NTSC signal using well-known techniques.
Computer simulations were run to compare the converter of the present invention illustrated in FIG. 3A with the prior art technique discussed above with respect to FIG. 1. Furthermore, a prior art technique wherein the HDTV decoded pictures are first filtered by a low-pass filter and then down-sampled by one-half in each dimension was used as a benchmark. In the computer simulations, an 1920×1080 interlaced HDTV sequence, “Whale”, was used. The HDTV bit stream was generated by the MPEG standard video coding software with parameters of GOP size N=15, anchor frame distance M=3, bit rate equal to 19 Mbps, and 4:2:0 sequence. The simulation results showed that the prior art technique illustrated in FIG. 1 suffered from a significant amount of block edge effect and distortions due to poor approximation introduced by simply disregarding higher order coefficients (i.e., masking). The present invention, however, achieved significantly improved results over the prior art technique of FIG. 1. Additionally, the present invention provided improved processing speed, even faster than the benchmark.
Third Embodiment
Standard SDTV decoders expect to receive an SDTV signal or bit stream which is both quantized (Q) and variable length encoded (VLC). Accordingly, the signal output by the HDTV converters of FIGS. 3A and 4A cannot be used by a standard SDTV converter. The HDTV converters of FIGS. 3A and 4A, however, can be modified to supply an SDTV signal such as expected by a standard SDTV decoder. Specifically, a DCT converter 100, a quantizer 102, and variable length encoder 104 would have to be connected in series to the output of the HDTV converter of FIGS. 3A and 4A as shown in FIGS. 3B and 4B.
Operation of the Third Embodiment
The DCT converter 100 converts the spatial domain SDTV signal into the frequency domain, the quantizer 102 then quantizes the output of the DCT converter 100, and the variable length encoder 104 variable length encodes the output from the quantizer 102 to produce an SDTV signal as expected by a standard SDTV decoder. Because variable length coding is lossless coding, only the distortion caused by the quantization step needs to be accounted for by the quantizer 102.
The elimination of distortion, however, can be assured by performing quantization as discussed in detail below. Assuming that Q is the quantization scale determined by the original encoder, I is the intra macro block data before quantization and I′ is the data after quantization, then 



  
    
      
        
          I
          ′
        
        =
        
          
            [
            
              I
              Q
            
            ]
          
          ·
          Q
        
      
    
    
      
        (
        27
        )
      
    
  




where the operator [.] is an integer operator, and I′ can be seen as the reconstructed data.

Assume that Qr, is the quantization scale used in the requantization, and I′′ is the data obtained after requantization, then 



  
    
      
        
          I
          ′′
        
        =
        
          
            
              [
              
                
                  I
                  ′
                
                Qr
              
              ]
            
            ·
            Qr
          
          =
          
            
              
                [
                
                  
                    [
                    
                      I
                      Q
                    
                    ]
                  
                  ·
                  
                    Q
                    Qr
                  
                
                ]
              
              ·
              Qr
            
            =
            
              
                [
                
                  I
                  Q
                
                ]
              
              ·
              
                [
                
                  Q
                  Qr
                
                ]
              
              ·
              Qr
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




and, therefore, 



  
    
      
        
          
            I
            ′′
          
          =
          
            
              
                [
                
                  I
                  Q
                
                ]
              
              ·
              Q
            
            =
            
              I
              ′
            
          
        
        ,
        
          
            if
            ⁢
            
               
            
            [
            
              Q
              Qr
            
            ]
          
          =
          integer
        
      
    
    
      
        (
        29
        )
      
    
  




In other words, if the requantization scale is chosen as an integer times the original quantization scale, requantization will not cause any additional distortion.
For uni-directionally inter-coded macro blocks (i.e., macro blocks from P-pictures),

P′=I′+E′,  (30) 

where P′ is the reconstructed data of the inter-coded macro block from the P-picture, I′ represents the reconstructed reference data for motion-compensation (i.e., the composite macro block) and E′ is the reconstructed prediction error (i.e., the macro block from the variable length decoder and dequantizer 58). Let Qi be the quantizer for I′, Qp for P′ and Qe for E′, then expression (30) can be re-written as: 



  
    
      
        
          P
          ′
        
        =
        
          
            
              [
              
                P
                Qp
              
              ]
            
            ·
            Qp
          
          =
          
            
              
                [
                
                  I
                  Qi
                
                ]
              
              ·
              Qi
            
            +
            
              
                [
                
                  E
                  Qe
                
                ]
              
              ·
              Qe
            
          
        
      
    
    
      
        (
        31
        )
      
    
  




After requantization with quantizer scale Qr, 



  
    
      
        
          P
          ′′
        
        =
        
          
            
              [
              
                P
                Qr
              
              ]
            
            ·
            Qr
          
          =
          
            
              [
              
                
                  
                    [
                    
                      I
                      Qi
                    
                    ]
                  
                  ·
                  
                    Qi
                    Qr
                  
                
                +
                
                  
                    [
                    
                      P
                      Qp
                    
                    ]
                  
                  ·
                  
                    Qp
                    Qr
                  
                
              
              ]
            
            ·
            Qr
          
        
      
    
    
      
        (
        32
        )
      
    
  




where P′′ is the requantized reconstructed data. Accordingly, 



  
    
      
        
          
            P
            ′′
          
          =
          
            
              
                
                  [
                  
                    I
                    Qi
                  
                  ]
                
                ·
                Qi
              
              +
              
                
                  [
                  
                    E
                    Qe
                  
                  ]
                
                ·
                Qe
              
            
            =
            
              P
              ′
            
          
        
        ,
      
    
    
      
        (
        33
        )
      
    
  




provided that both, 



  
    
      
        
          
            Qi
            Qr
          
          =
          integer
        
        ,
        
          
            Qe
            Qr
          
          =
          integer
        
      
    
    
      
        (
        34
        )
      
    
  




are satisfied. Therefore, if Qr is chosen according to the following expression, there is no additional distortion caused by the process of requantizing uni-directionally inter-coded macro blocks.

Qr=GCD(Qi,QE),  (35) 

where GCD stands for greatest common division.

Similarly, for bi-directionally inter-coded macro blocks, relationships can also be derived so that the quantizer scale is properly chosen during the process of requantization such that no additional errors are introduced during the process of requantization.
For bi-directionally inter-coded macro blocks (i.e., macro blocks from B-pictures).

B′=I′+P′+E′I+E′p  (36) 

where B′ is the reconstructed data of the inter-coded macro block from the B-picture, EI′ is the reconstructed prediction error with respect to the reconstructed reference data I′, and Ep′ is the reconstructed prediction error with respect to the reconstructed reference data P′. Let QB be the quantizer for B′, QP for P′, QEI for EI′, and QEP for Ep′, then expression (36) can be written as expression (37) below. 



  
    
      
        
          
            
              
                
                  B
                  ′′
                
                =
                
                ⁢
                
                  
                    [
                    
                      B
                      
                        Q
                        B
                      
                    
                    ]
                  
                  ·
                  
                    Q
                    B
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    
                      [
                      
                        I
                        
                          Q
                          I
                        
                      
                      ]
                    
                    ·
                    
                      Q
                      I
                    
                  
                  +
                  
                    
                      [
                      
                        P
                        
                          Q
                          P
                        
                      
                      ]
                    
                    ·
                    
                      Q
                      P
                    
                  
                  +
                  
                    
                      [
                      
                        
                          E
                          I
                        
                        
                          Q
                          
                            E
                            I
                          
                        
                      
                      ]
                    
                    ·
                    
                      Q
                      
                        E
                        I
                      
                    
                  
                  +
                  
                    
                      [
                      
                        
                          E
                          P
                        
                        
                          Q
                          
                            E
                            P
                          
                        
                      
                      ]
                    
                    ·
                    
                      Q
                      
                        E
                        P
                      
                    
                  
                
              
            
          
        
         
      
    
    
      
        (
        37
        )
      
    
  




After requantization with quantizer scale Qr, 



  
    
      
        
          B
          ′′
        
        =
        
          
            [
            
              
                B
                ′
              
              
                Q
                r
              
            
            ]
          
          ·
          
            Q
            r
          
        
      
    
    
      
        (
        38
        )
      
    
  




where B′′ is the requantized reconstructed data for the bi-directionally inter-coded macro block. Accordingly, 



  
    
      
        
          B
          ′′
        
        =
        
          
            
              {
              
                
                  
                    [
                    
                      I
                      
                        Q
                        I
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      Q
                      I
                    
                    
                      Q
                      r
                    
                  
                
                +
                
                  
                    [
                    
                      P
                      
                        Q
                        P
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      Q
                      P
                    
                    
                      Q
                      r
                    
                  
                
                +
                
                  
                    [
                    
                      
                        E
                        I
                      
                      
                        Q
                        
                          E
                          I
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      Q
                      
                        E
                        I
                      
                    
                    
                      Q
                      r
                    
                  
                
                +
                
                  
                    [
                    
                      
                        E
                        P
                      
                      
                        Q
                        
                          E
                          P
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                      Q
                      
                        E
                        P
                      
                    
                    
                      Q
                      r
                    
                  
                
              
              }
            
            ·
            
              Q
              r
            
          
          =
          
            B
            ′
          
        
      
    
    
      
        (
        39
        )
      
    
  




provided the following ratios are integers: 



  
    
      
        
          
            Q
            I
          
          
            Q
            r
          
        
        ,
        
          
            Q
            P
          
          
            Q
            r
          
        
        ,
        
          
            Q
            
              E
              I
            
          
          
            Q
            r
          
        
        ,
        
          
            
              Q
              
                E
                P
              
            
            
              Q
              r
            
          
          ⁢
          
             
          
          ⁢
          or
        
        ,
      
    
    
      
        (
        40
        )
      
    
  
  
    
      
        
          Q
          r
        
        =
        
          GCD
          ⁡
          
            (
            
              
                Q
                T
              
              ,
              
                Q
                P
              
              ,
              
                Q
                
                  E
                  I
                
              
              ,
              
                Q
                
                  E
                  P
                
              
            
            )
          
        
      
    
    
      
        (
        41
        )
      
    
  




Therefore, if expressions (40) or (41) are satisfied, there is no additional distortion caused by requantizing bi-directionally inter-coded macro blocks. Accordingly, taking into account expressions (34) and (35), if the following expression is satisfied, there is no additional distortion caused by the process of requantizing either uni-directionally or bi-directionally inter-coded macro blocks.

Qr=GCD(Qi, Qe, QEp).  (42) 

Additionally, because the HDTV signal has been down-converted, the quantizer 102 must also perform bit rate control so that the output thereof achieves the SDTV bit rate. Any well-known technique of bit rate control may be used. Also, if the distortion eliminating techniques discussed above are used, bit rate control may also become necessary. The bit rate control performed to obtain the SDTV bit rate, however, accounts for any influences on bit rate that use of the distortion eliminating equations may have.
Producing an SDTV signal as expected by a standard SDTV decoder in the manner discussed above with respect to FIGS. 3B and 4B, has two drawbacks. First, an additional DCT converter (i.e. DCT processor) is required. This increases the processing time and expense of the resulting apparatus. Furthermore, when the HDTV signal includes inter-coded pictures, a frame store 22 (see FIG. 4B) capable of storing two complete anchor pictures in the spatial domain is required to reconstruct spatial pictures. Memory requirements directly impact on the complexity and expense of such a device.
Fourth Embodiment
FIG. 5 illustrates another embodiment of an HDTV to SDTV converter. The HDTV converter of FIG. 5 converts the HDTV signal into an SDTV signal expected by a standard SDTV decoder. The embodiment of FIG. 5, however, does not require performing DCT conversion, and also eliminates the need for inverse DCT conversion. Accordingly, this embodiment as compared to the embodiment of FIG. 3B eliminates both an inverse DCT converter and a DCT converter. As one skilled in the art will readily appreciate, the decoding process is therefore greatly simplified, as is the cost of the resulting device.
As shown in FIG. 5, a variable length decoder and dequantizer 10 is connected to a down-converter 52. The output of the down-converter 52 is processed in series by a quantizer 54 and a variable length encoder 56. The quantizer 54 requantizes the output of the down-converter 52 in the same manner discussed above with respect to the quantizer 102. The down-converter 52 includes a separator 30 which receives the output of the variable length decoder and dequantizer 10. A synthesizer 32 and mask 14 are connected to the separator 30, and send their output to a recombiner 50. The output of the recombiner 50 serves as the output of the down-converter 52.
It should be understood that in FIG. 5 the various data paths have been represented in greatly simplified form for clarity. In reality, many separate data paths into and out of the various components are required. Additionally, many data and control lines have been omitted entirely from FIG. 5 for clarity.
Operation of the Fourth Embodiment
During operation, the variable length decoder and dequantizer 10, variable length decodes and dequantizes the HDTV signal. As discussed with respect to FIG. 3A, the separator 30 receives the output of the variable length decoder and dequantizer 10, and based on the sequence header, identifies the sequence of the HDTV signal. The separator 30 then sends the luminance DCT blocks to the synthesizer 32, and sends the chroma DCT blocks to either the synthesizer 32 or the mask 14 depending upon the sequence identified from the sequence header of the HDTV signal. Both the synthesizer 32 and the mask 14 operate as discussed previously in this disclosure. The luminance DCT blocks output by the synthesizer 32 are then recombined with the chroma DCT blocks output from the mask 14 by the recombiner 50. In this manner the recombiner 50 recombines the DCT blocks corresponding to each picture to produce an SDTV signal in the frequency domain in much the same way the recombiner 36 recombined the spatial domain luminance and chroma blocks. The SDTV signal in the frequency domain output by the recombiner 50 is then quantized by the quantizer 54. As discussed above, the quantizer 54 requantizes the output of the down-converter 52 in the same manner as the quantizer 102. Subsequently, the output of the quantizer 54 is variable length encoded by the variable length encoder 56. Accordingly, the variable length encoder 56 produces an SDTV signal such as expected by a standard SDTV decoder.
The HDTV converter illustrated in FIG. 5 operates independently of the SDTV decoder. Consequently, the HDTV converter of this embodiment can be added to existing standard SDTV decoders without modification of standard SDTV decoders. As a result, the HDTV converter of FIG. 5 is very flexible in its application.
Fifth Embodiment
The HDTV converter of FIG. 5 can only handle HDTV signals having intra-coded pictures. The HDTV converter of FIG. 5 does not provide a means for processing inter-coded pictures. FIG. 6, however, illustrates another HDTV to SDTV converter according to the present invention. The HDTV converter of FIG. 6 can down-convert an HDTV signal into an SDTV signal as expected by a standard SDTV converter, even though the HDTV signal includes inter-coded pictures. Furthermore, in contrast to the HDTV converter of FIG. 4, the HDTV converter of FIG. 6 does not require memory capacity sufficient to store two anchor pictures in the spatial domain. Instead, the HDTV converter of FIG. 6 converts inter-coded pictures into I-pictures in the frequency or DCT domain.
Accordingly, the embodiment of FIG. 6 only requires storing two anchor pictures in the frequency domain. Additionally, the embodiment of FIG. 6 stores the variable length encoded and quantized version of these frequency domain anchor pictures. As a result, the memory requirements for storing these anchor pictures as compared to FIG. 4B are considerably reduced. Applicants estimate that the memory requirements of FIG. 6 are at least one-third less than the memory requirements for the embodiment of FIG. 4B. Since the requisite memory capacity for an HDTV converter is a large component of the cost therefore, this reduction in memory capacity results in a significant decrease in the cost of the resulting device.
As illustrated in FIG. 6, the HDTV signal is received by an I-picture converter 76 which processes an HDTV signal having either I-pictures, or both I-pictures and inter-coded pictures. After processing the HDTV signal, the I-picture converter 76 outputs an HDTV signal comprised only of I-pictures. The HDTV signal output by the I-picture converter 76 is received by a down-converter 12 or 40. Connected in series to the output of the down-converter 12,40 is a DCT converter 80, a quantizer 82, and a variable length encoder 84.
The I-picture converter 76 includes a bit stream parser 60 which receives the HDTV signal input to the I-picture converter 76. A variable length decoder and dequantizer 58 receives one of the outputs of the bit stream parser 60. The variable length decoder and dequantizer 58 outputs pictures to an adder 74 and outputs motion vectors to a motion vector address generator 62. As indicated by dashed lines, the variable length decoder and dequantizer 58 outputs pictures to the adder 74 via an optional field-to-frame converter 69. Both the bit stream parser 60 and the motion vector address generator 62 supply output signals to an I and P-picture macro block bits memory 64 (hereinafter referred to as “memory 64”). A variable length decoder and dequantizer 67 is connected to an output of the memory 64, and has an output thereof connected to a compositer 68. Another variable length decoder and dequantizer 66 is connected to an output of the memory 64, and has an output thereof connected to the reformatter 78.
The compositer 68 also receives an output from the motion vector address generator 62. Besides receiving the output of the variable length decoder and dequantizer 58, the adder 74 also receives the output of the compositer 68. The output of the adder 74 is connected to the reformatter 78, and one of the outputs of the reformatter 78 serves as the output of the I-picture converter 76. Another output of the reformatter 78 is received by a quantizer 70 within the I-picture converter 76. A variable length encoder 72 is connected to the output of the quantizer 70, and a bit stream parser 59 is connected to the output of the variable length encoder 72. The bit stream parser 59 supplies another input of the memory 64.
It should be understood that in FIG. 6 the various data paths have been represented in greatly simplified form for clarity. In reality, many separate data paths into and out of the various components are required. Additionally, many data and control lines have been omitted entirely from FIG. 6 for clarity.
Operation of the Fifth Embodiment
The operation of the embodiment illustrated in FIG. 6 will now be described. As discussed above, the I-picture converter 76 receives an HDTV signal, and converts all inter-coded pictures therein into I-pictures. For purposes of discussion, assume that the HDTV signal has the order of encodation as follows: I1P1B1B2P2B3B4P3B5B6I2B7B8, and when displayed, these pictures have the display order of I1B1B2P1B3B4P2B5B6P3B7B8I2.
As discussed above, an HDTV signal including inter-coded pictures includes both picture data and a motion vector corresponding thereto. The bit stream parser 60 receives the HDTV signal, and parses out the header information for each anchor picture. The header information includes the picture type, the picture timing stamp PTS, the slice header, etc. The header information for each anchor picture is output by the bit stream parser 60 to the memory 64. The memory 64 stores the header information on a picture-by-picture basis. The bit stream parser 60 also outputs the HDTV signal to the variable length decoder and dequantizer 58.
The variable length decoder and dequantizer 58 receives the HDTV signal, variable length decodes and dequantizes the HDTV signal, sends the decoded HDTV signal to the adder 74, and sends the motion vectors to the motion vector address generator 62. When the pictures of the HDTV signal could include fields, as opposed to, or in addition to frames, the I-picture converter 76 includes a field-to-frame converter 69 which converts the separate even and odd fields for a frame into a single frame. This is accomplished by simply interlacing the even and odd fields.
As discussed in more detail below, the compositer 68 produces a composite macro block based on the output of the motion vector address generator 62 and the macro blocks output by the memory 64. The adder 74 adds the composite macro block to the macro block of the picture data currently output by the variable length decoder and dequantizer 58, or optionally the field-to-frame converter 69, to reconstruct a complete macro block. In this manner, both P-pictures and B-pictures are converted into I-pictures.
Of course, if an I-picture is currently being output by the variable length decoder and dequantizer 58, then the adder 74 will not receive anything from the compositer 68 to add thereto. The motion vector address generator 62 will have recognized the I-picture as an I-picture and disabled the compositer 68 from sending output to the adder 74.
As discussed in detail below, when the adder 74 outputs I-pictures or I-pictures which were formally P-pictures, the reformatter 78 outputs these anchor pictures to the memory 64 for storage via the quantizer 70, the variable length encoder 72, and the bit stream parser 59. The memory 64 only stores two anchor pictures at a time, and will replace the oldest stored anchor picture with the newly received anchor picture. Specifically, the I and P-pictures (i.e., anchor pictures) in I-picture format output from the reformatter 78 to the memory 64 are quantized by the quantizer 70, and variable length encoded by the variable length encoder 72.
The quantizer 70 performs quantization in the same manner as discussed above with respect to the quantizer 102. The bit rate output from the quantizer 70, however, is that of an HDTV signal. Accordingly, no bit rate control is required if the quantization scale of the original encoder is used. Furthermore, tests have shown that if the quantizer 70 quantizes the anchor frames using the quantization scale determined by the original encoder, no noticeable distortion is introduced. If, however, the distortion eliminating equations are utilized by the quantizer 70, then the quantizer 70 may need to perform bit rate control to obtain the HDTV bit rate.
Using the various headers within the HDTV signal, the bit stream parser 59 identifies each macro block in an anchor picture. The bit stream parser 59 then separates the macro blocks for the anchor picture, and stores them in the memory 64. These anchor frames are stored in the DCT domain and have been variable length encoded and quantized. Consequently, the memory 64 can be much smaller than the memory needed to implement the frame store 22.
One possible embodiment of memory 64 is illustrates in FIG. 8. As shown in FIG. 8, the memory 64 resembles a memory stack. Each location in the memory stack only stores the data for one macro block. Because the macro blocks are variable length encoded and quantized, the macro blocks can have variable lengths. As illustrated, the macro block MB0 takes up only a portion of the first stack location, the same with macro blocks MB1 and MB2. The fourth macro block MB3, however, contains more data than a single stack location. In this instance, the macro block MB3 fills up an entire stack location and a portion of another stack location. The macro block MB4 following macro block MB3 is then stored in the next stack location. Accordingly, each macro block stored in the memory 64 will be stored beginning in a new stack location. In this manner, each macro block can be stored and retrieved independently of the other macro blocks. Preferably the memory 64 includes two such stacks for storing the macro blocks of two anchor frames. Alternatively, a single, large stack could be divided in half. Furthermore, the memory 64 links the macro blocks of an anchor picture to the previously stored header information therefor.
In the embodiment of FIG. 4, the macro blocks of the anchor pictures (I-pictures and P-pictures) used to form a complete picture were stored in the spatial domain. Accordingly, if the motion vectors caused a shift in the block structure of the anchor picture, the data for the new macro blocks could be easily obtained from the original macro block structure for the anchor picture as discussed in “Manipulation and Compositing of MC-DCT compressed Video” by Shih-Fu Chang, IEEE Journal on Selected Areas in Communication, Vol. 13, No. 1, January 1995 (hereinafter “the Chang article”); which is hereby incorporated by reference.
For instance, as discussed in the Chang article, FIG. 9A illustrates a macro block structure B shown by dashed lines and a macro block structure A illustrated by solid lines. Assume that the macro block structure B represents the macro block structure of an anchor picture stored in frame store memory 22 of FIG. 4B, and that the macro block structure A represents the macro block structure as indicated by the motion vector supplied to the frame store memory 22. If the macro block indicated by the motion vector is the macro block indicated as B′ in FIG. 9A, then the macro block is simply formed from the data of the four macro blocks in the macro block structure B such as shown in FIGS. 9B-9D. Since the macro blocks are stored in the spatial domain, that portion of the macro block structure B forming the macro block B′ in the macro block structure A can simply be read out. Think of it as a picture on a display screen and a window is simply moved about that display screen.
By contrast, macro blocks in the frequency domain (i.e., DCT blocks) do not operate in the same fashion. Instead, the DCT coefficients represent different frequency components of the corresponding macro block in the spatial domain. Namely, the new macro block B′ contains contributions from the lower left corner B13 of block B1, the lower right corner B24 of macro block B2, the upper right corner B3, of block B3 and the upper left corner B42 of macro block B4. While in the spatial domain, the four contributions can be supplemented by zeros as shown in FIG. 9C and added according to the equation B′=B13+B24+B31+B42 as shown in FIG. 9D, this same method cannot be applied in the DCT domain. Instead, as discussed in the Chang article, the DCT macro block B′ can be calculated according to the following expression: 



  
    
      
        
          DCT
          ⁡
          
            (
            
              B
              ′
            
            )
          
        
        =
        
          
            ∑
            
              i
              =
              1
            
            4
          
          ⁢
          
             
          
          ⁢
          
            DCT
            ⁢
            
               
            
            ⁢
            
              (
              
                H
                i1
              
              )
            
            ⁢
            
               
            
            ⁢
            DCT
            ⁢
            
               
            
            ⁢
            
              (
              
                B
                i
              
              )
            
            ⁢
            
               
            
            ⁢
            DCT
            ⁢
            
               
            
            ⁢
            
              (
              
                H
                i2
              
              )
            
            ⁢
            
               
            
            ⁢
            where
          
        
      
    
    
      
        (
        43
        )
      
    
  
  
    
      
        
          H
          1
        
        =
        
          
            
              [
              
                
                  0
                  
                    I
                    h
                  
                
                ⁢
                
                  0
                  0
                
              
              ]
            
            ⁢
            
              H
              2
            
          
          =
          
            [
            
              
                0
                0
              
              ⁢
              
                
                  I
                  w
                
                0
              
            
            ]
          
        
      
    
    
      
         
      
    
  




where Ih and Iw are identity matrices with size h×h and w×w, respectively; and h and w are the number of rows and columns extracted.

Formation of Composite Macro Block
In a preferred embodiment, however, a different set of expressions is used to determine the composite macro block. First the formation of a composite macro block to perform frame-motion compensation will be described. Field-motion compensation will be described thereafter.
Frame-Motion Compensation Full Pixel Accuracy
FIG. 10 illustrates macro blocks a, b, c and d from the anchor frame in the spatial domain, and block e the macro block pointed to by the motion vector, albeit in the spatial domain. As shown in FIG. 10, the macro block e has an upper left hand coordinate of (m,n). The motion vector received by the motion vector address generator 62 points to this coordinate (m,n). Based on the motion vector, the motion vector address generator 62 instructs the memory 64 as to which macro blocks of the anchor frame to output to the compositer 68. The motion vector address generator 62 also outputs a signal to the compositer 68 indicating the coordinate (m,n).
Macro blocks a, b, c, d and e have the relationship shown below by expression (44). 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  a
                  
                    
                      i
                      +
                      m
                    
                    ,
                    
                      j
                      +
                      n
                    
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                  ;
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  b
                  
                    
                      i
                      +
                      m
                    
                    ,
                    
                      j
                      +
                      n
                      -
                      8
                    
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                  ;
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  c
                  
                    
                      i
                      +
                      m
                      -
                      8
                    
                    ,
                    
                      j
                      +
                      n
                    
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  d
                  
                    
                      i
                      +
                      m
                      -
                      8
                    
                    ,
                    
                      j
                      +
                      n
                      -
                      8
                    
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
          
          }
        
      
    
    
      
        (
        44
        )
      
    
  




For instance, if (m,n=2,3) then 



  
    e
    ij
  
  =
  
    [
    
      
        
          
            a
            
              2
              ,
              3
            
          
        
        
          
            a
            
              2
              ,
              4
            
          
        
        
          
            a
            
              2
              ,
              5
            
          
        
        
          
            a
            
              2
              ,
              6
            
          
        
        
          
            a
            
              2
              ,
              7
            
          
        
        
          
            b
            
              2
              ,
              0
            
          
        
        
          
            b
            
              2
              ,
              1
            
          
        
        
          
            b
            
              2
              ,
              2
            
          
        
      
      
        
          
            a
            
              3
              ,
              3
            
          
        
        
          
            a
            
              3
              ,
              4
            
          
        
        
          
            a
            
              3
              ,
              5
            
          
        
        
          
            a
            
              3
              ,
              6
            
          
        
        
          
            a
            
              3
              ,
              7
            
          
        
        
          
            b
            
              3
              ,
              0
            
          
        
        
          
            b
            
              3
              ,
              1
            
          
        
        
          
            b
            
              3
              ,
              2
            
          
        
      
      
        
          
            a
            
              4
              ,
              3
            
          
        
        
          
            a
            
              4
              ,
              4
            
          
        
        
          
            a
            
              4
              ,
              5
            
          
        
        
          
            a
            
              4
              ,
              6
            
          
        
        
          
            a
            
              4
              ,
              7
            
          
        
        
          
            b
            
              4
              ,
              0
            
          
        
        
          
            b
            
              4
              ,
              1
            
          
        
        
          
            b
            
              4
              ,
              2
            
          
        
      
      
        
          
            a
            
              5
              ,
              3
            
          
        
        
          
            a
            
              5
              ,
              4
            
          
        
        
          
            a
            
              5
              ,
              5
            
          
        
        
          
            a
            
              5
              ,
              6
            
          
        
        
          
            a
            
              5
              ,
              7
            
          
        
        
          
            b
            
              5
              ,
              0
            
          
        
        
          
            b
            
              5
              ,
              1
            
          
        
        
          
            b
            
              5
              ,
              2
            
          
        
      
      
        
          
            a
            
              6
              ,
              3
            
          
        
        
          
            a
            
              6
              ,
              4
            
          
        
        
          
            a
            
              6
              ,
              5
            
          
        
        
          
            a
            
              6
              ,
              6
            
          
        
        
          
            a
            
              6
              ,
              7
            
          
        
        
          
            b
            
              6
              ,
              0
            
          
        
        
          
            b
            
              6
              ,
              1
            
          
        
        
          
            b
            
              6
              ,
              2
            
          
        
      
      
        
          
            a
            
              7
              ,
              3
            
          
        
        
          
            a
            
              7
              ,
              4
            
          
        
        
          
            a
            
              7
              ,
              5
            
          
        
        
          
            a
            
              7
              ,
              6
            
          
        
        
          
            a
            
              7
              ,
              7
            
          
        
        
          
            b
            
              7
              ,
              0
            
          
        
        
          
            b
            
              7
              ,
              1
            
          
        
        
          
            b
            
              7
              ,
              2
            
          
        
      
      
        
          
            c
            
              0
              ,
              3
            
          
        
        
          
            c
            
              0
              ,
              4
            
          
        
        
          
            c
            
              0
              ,
              5
            
          
        
        
          
            c
            
              0
              ,
              6
            
          
        
        
          
            c
            
              0
              ,
              7
            
          
        
        
          
            d
            
              0
              ,
              0
            
          
        
        
          
            d
            
              0
              ,
              1
            
          
        
        
          
            d
            
              0
              ,
              2
            
          
        
      
    
    ]
  




whereas if (m,n)=(6,5) then 



  
    e
    ij
  
  =
  
    [
    
      
        
          
            a
            
              6
              ,
              5
            
          
        
        
          
            a
            
              6
              ,
              6
            
          
        
        
          
            a
            
              6
              ,
              7
            
          
        
        
          
            b
            
              6
              ,
              0
            
          
        
        
          
            b
            
              6
              ,
              1
            
          
        
        
          
            b
            
              6
              ,
              2
            
          
        
        
          
            b
            
              6
              ,
              3
            
          
        
        
          
            b
            
              6
              ,
              4
            
          
        
      
      
        
          
            a
            
              7
              ,
              5
            
          
        
        
          
            a
            
              7
              ,
              6
            
          
        
        
          
            a
            
              7
              ,
              7
            
          
        
        
          
            b
            
              7
              ,
              0
            
          
        
        
          
            b
            
              7
              ,
              1
            
          
        
        
          
            b
            
              7
              ,
              2
            
          
        
        
          
            b
            
              7
              ,
              3
            
          
        
        
          
            b
            
              7
              ,
              4
            
          
        
      
      
        
          
            c
            
              0
              ,
              5
            
          
        
        
          
            c
            
              0
              ,
              6
            
          
        
        
          
            c
            
              0
              ,
              7
            
          
        
        
          
            d
            
              0
              ,
              0
            
          
        
        
          
            d
            
              0
              ,
              1
            
          
        
        
          
            d
            
              0
              ,
              2
            
          
        
        
          
            d
            
              0
              ,
              3
            
          
        
        
          
            d
            
              0
              ,
              4
            
          
        
      
      
        
          
            c
            
              1
              ,
              5
            
          
        
        
          
            c
            
              1
              ,
              6
            
          
        
        
          
            c
            
              1
              ,
              7
            
          
        
        
          
            d
            
              1
              ,
              0
            
          
        
        
          
            d
            
              1
              ,
              1
            
          
        
        
          
            d
            
              1
              ,
              2
            
          
        
        
          
            d
            
              1
              ,
              3
            
          
        
        
          
            d
            
              1
              ,
              4
            
          
        
      
      
        
          
            c
            
              2
              ,
              5
            
          
        
        
          
            c
            
              2
              ,
              6
            
          
        
        
          
            c
            
              2
              ,
              7
            
          
        
        
          
            d
            
              2
              ,
              0
            
          
        
        
          
            d
            
              2
              ,
              1
            
          
        
        
          
            d
            
              2
              ,
              2
            
          
        
        
          
            d
            
              2
              ,
              3
            
          
        
        
          
            d
            
              2
              ,
              4
            
          
        
      
      
        
          
            c
            
              3
              ,
              5
            
          
        
        
          
            c
            
              3
              ,
              6
            
          
        
        
          
            c
            
              3
              ,
              7
            
          
        
        
          
            d
            
              3
              ,
              0
            
          
        
        
          
            d
            
              3
              ,
              1
            
          
        
        
          
            d
            
              3
              ,
              2
            
          
        
        
          
            d
            
              3
              ,
              3
            
          
        
        
          
            d
            
              3
              ,
              4
            
          
        
      
      
        
          
            c
            
              4
              ,
              5
            
          
        
        
          
            c
            
              4
              ,
              6
            
          
        
        
          
            c
            
              4
              ,
              7
            
          
        
        
          
            d
            
              4
              ,
              0
            
          
        
        
          
            d
            
              4
              ,
              1
            
          
        
        
          
            d
            
              4
              ,
              2
            
          
        
        
          
            d
            
              4
              ,
              3
            
          
        
        
          
            d
            
              4
              ,
              4
            
          
        
      
      
        
          
            c
            
              5
              ,
              5
            
          
        
        
          
            c
            
              5
              ,
              6
            
          
        
        
          
            c
            
              5
              ,
              7
            
          
        
        
          
            d
            
              5
              ,
              0
            
          
        
        
          
            d
            
              5
              ,
              1
            
          
        
        
          
            d
            
              5
              ,
              2
            
          
        
        
          
            d
            
              5
              ,
              3
            
          
        
        
          
            d
            
              5
              ,
              4
            
          
        
      
    
    ]
  




Assume that arrays A, B, C, D and E are the corresponding DCT macro blocks of macro blocks a, b, c, d and e, respectively. Accordingly, E and e have the relationship shown in expression (45) below. 



  
    
      
        
          E
          kl
        
        =
        
          
            ∑
            
              i
              =
              0
            
            7
          
          ⁢
          
            
              ∑
              
                j
                =
                0
              
              7
            
            ⁢
            
              
                
                  e
                  ij
                
                ·
                cos
              
              ⁢
              
                 
              
              ⁢
              
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ·
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




Substituting a, b, c and d in for e in expression (45) based on expression (44) results in expression (46) below. 



  
    
      
        
          
            E
            kl
          
          =
          
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    a
                  
                
                ⁢
                
                  
                    
                      a
                      
                        
                          i
                          +
                          m
                        
                        ,
                        
                          j
                          +
                          n
                        
                      
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                 
                              
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            ⁢
            
               
            
            +
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      n
                    
                  
                  7
                
                ⁢
                
                  
                    
                      b
                      
                        
                          i
                          +
                          m
                        
                        ,
                        
                          j
                          +
                          n
                          -
                          8
                        
                      
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                 
                              
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            ⁢
            
               
            
            +
            
              
                ∑
                
                  i
                  =
                  
                    8
                    -
                    m
                  
                
                7
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    n
                  
                
                ⁢
                
                  
                    
                      c
                      
                        
                          i
                          +
                          m
                          -
                          8
                        
                        ,
                        
                          j
                          +
                          n
                        
                      
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                 
                              
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            ⁢
            
               
            
            +
            
              
                ∑
                
                  i
                  =
                  
                    8
                    -
                    m
                  
                
                7
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      n
                    
                  
                  7
                
                ⁢
                
                  
                    
                      d
                      
                        
                          i
                          +
                          m
                          -
                          8
                        
                        ,
                        
                          j
                          +
                          n
                          -
                          8
                        
                      
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                 
                              
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            ⁢
            
               
            
            +
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    n
                  
                
                ⁢
                
                  
                    
                      a
                      
                        
                          i
                          +
                          m
                        
                        ,
                        
                          j
                          +
                          n
                        
                      
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                 
                              
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            ⁢
            
               
            
            +
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    n
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      b
                      
                        
                          i
                          +
                          m
                        
                        ,
                        
                          j
                          +
                          n
                          -
                          8
                        
                      
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                 
                              
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            
                              (
                              
                                j
                                -
                                n
                              
                              )
                            
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            ⁢
            
               
            
            +
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  m
                  -
                  1
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    n
                  
                
                ⁢
                
                  
                    
                      c
                      
                        
                          i
                          +
                          m
                          -
                          8
                        
                        ,
                        
                          j
                          +
                          n
                        
                      
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                (
                                
                                  i
                                  -
                                  m
                                
                                )
                              
                            
                            +
                            17
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            ⁢
            
               
            
            +
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  m
                  -
                  1
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    n
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      d
                      
                        i
                        ,
                        j
                      
                    
                    ·
                    cos
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                (
                                
                                  i
                                  -
                                  m
                                
                                )
                              
                            
                            +
                            17
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                (
                                
                                  j
                                  -
                                  n
                                
                                )
                              
                            
                            +
                            17
                          
                          )
                        
                        ⁢
                        
                           
                        
                        ⁢
                        l
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                
              
            
          
        
        ⁢
        
           
        
      
    
    
      
        (
        46
        )
      
    
  




Representing a, b, c and d with the corresponding DCT coefficients A, B, C and D, respectively, using expression (45) and substituting A, B, C, or D for E and a, b, c, or d for e results in expression (47) below. 



  
    
      
        
          E
          kl
        
        =
        
          
            
              ∑
              
                i
                =
                0
              
              
                7
                -
                m
              
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                
                  7
                  -
                  n
                
              
              ⁢
              
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ·
                  cos
                
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ·
                  
                    
                      ∑
                      
                        p
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        ∑
                        
                          q
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          A
                          pq
                        
                        ⁢
                        
                           
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    m
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ·
                          cos
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  j
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  n
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                
              
            
          
          +
          
            
              ∑
              
                i
                =
                0
              
              
                7
                -
                m
              
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                
                  n
                  -
                  1
                
              
              ⁢
              
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ·
                  cos
                
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          -
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            n
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ·
                  
                    
                      ∑
                      
                        p
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        ∑
                        
                          q
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          B
                          pq
                        
                        ⁢
                        
                           
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    m
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ·
                          cos
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                
              
            
          
          +
          
            
              ∑
              
                i
                =
                0
              
              
                m
                -
                1
              
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                
                  7
                  -
                  n
                
              
              ⁢
              
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        
                          (
                          
                            i
                            -
                            m
                            +
                            17
                          
                          )
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                    
                    16
                  
                  ·
                  cos
                
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ·
                  
                    
                      ∑
                      
                        p
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        ∑
                        
                          q
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          C
                          pq
                        
                        ⁢
                        
                           
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ·
                          cos
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  j
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  n
                                
                                +
                                17
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                
              
            
          
          +
          
            
              ∑
              
                i
                =
                0
              
              
                m
                -
                1
              
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                
                  n
                  -
                  1
                
              
              ⁢
              
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            
                              (
                              
                                i
                                -
                                m
                              
                              )
                            
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ·
                  cos
                
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            
                              (
                              
                                j
                                -
                                n
                              
                              )
                            
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ·
                  
                    
                      ∑
                      
                        p
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        ∑
                        
                          q
                          =
                          0
                        
                        7
                      
                      ⁢
                      
                        
                          D
                          pq
                        
                        ⁢
                        
                           
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ·
                          cos
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        47
        )
      
    
  




Changing the summing order and rewriting expression (47) results in expression (48) below. 



  
    
      
        
          E
          kl
        
        =
        
          
            
              ∑
              
                p
                =
                0
              
              7
            
            ⁢
            
              
                ∑
                
                  q
                  =
                  0
                
                7
              
              ⁢
              
                
                  A
                  pq
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    
                      7
                      -
                      m
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        7
                        -
                        n
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  i
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  m
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ·
                        cos
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                j
                              
                              +
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                n
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                             
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                             
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                             
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
              
            
          
          ⁢
          
             
          
          +
          
            
              ∑
              
                p
                =
                0
              
              7
            
            ⁢
            
              
                ∑
                
                  q
                  =
                  0
                
                7
              
              ⁢
              
                
                  B
                  pq
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    
                      7
                      -
                      m
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        n
                        -
                        1
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  i
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  m
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ·
                        cos
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                             
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                  (
                                  
                                     
                                  
                                  ⁢
                                  
                                    j
                                    -
                                    n
                                  
                                  )
                                
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          
                             
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
              
            
          
          ⁢
          
             
          
          +
          
            
              ∑
              
                p
                =
                0
              
              7
            
            ⁢
            
              
                ∑
                
                  q
                  =
                  0
                
                7
              
              ⁢
              
                
                  C
                  pq
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    
                      m
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        7
                        -
                        n
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  j
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  n
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ·
                        cos
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  (
                                  
                                    i
                                    -
                                    m
                                  
                                  )
                                
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          
                             
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
              
            
          
          +
          
            
              ∑
              
                p
                =
                0
              
              7
            
            ⁢
            
              
                ∑
                
                  q
                  =
                  0
                
                7
              
              ⁢
              
                
                  D
                  pq
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      0
                    
                    
                      m
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        n
                        -
                        1
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            
                               
                            
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ·
                        cos
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  (
                                  
                                    i
                                    -
                                    m
                                  
                                  )
                                
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  (
                                  
                                    j
                                    -
                                    n
                                  
                                  )
                                
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        48
        )
      
    
  




Expression (48) can be simplified to expression (49) below. 



  
    
      
        
          E
          kl
        
        =
        
          
            ∑
            
              p
              =
              0
            
            7
          
          ⁢
          
            
              ∑
              
                q
                =
                0
              
              7
            
            ⁢
            
               
            
            ⁢
            
              (
              
                
                  
                    A
                    pq
                  
                  ·
                  
                    M
                    A
                  
                
                +
                
                  
                    B
                    pq
                  
                  ·
                  
                    M
                    B
                  
                
                +
                
                  
                    C
                    pq
                  
                  ·
                  
                    M
                    C
                  
                
                +
                
                  
                    D
                    pq
                  
                  ·
                  
                    M
                    D
                  
                
              
              )
            
          
        
      
    
    
      
        (
        49
        )
      
    
  




where MA, MB, MC and MD are the matrices which are independent of the DCT coefficients and only depend on the values of p, q, k, l, m and n. Therefore, matrices MA, MB, MC and MD can be precalculated according to the following expressions for 0≦p, q, k, l<=7: 



  
    
      
        
          M
          A
        
        =
        
          
            ∑
            
              i
              =
              0
            
            
              7
              -
              m
            
          
          ⁢
          
            
              ∑
              
                j
                =
                0
              
              
                7
                -
                n
              
            
            ⁢
            
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        i
                      
                      +
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        m
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  p
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
              ⁢
              
                 
              
              ⁢
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          n
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ·
                cos
              
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        i
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  
                     
                  
                  ⁢
                  k
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
              ⁢
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        j
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  
                     
                  
                  ⁢
                  l
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
            
          
        
      
    
    
      
        (50a)
      
    
  
  
    
      
        
          M
          B
        
        =
        
          
            ∑
            
              i
              =
              0
            
            
              7
              -
              m
            
          
          ⁢
          
            
              ∑
              
                j
                =
                0
              
              
                n
                -
                1
              
            
            ⁢
            
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        i
                      
                      +
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        m
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  p
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
              ⁢
              
                 
              
              ⁢
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ·
                cos
              
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        i
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  
                     
                  
                  ⁢
                  k
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
              ⁢
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        j
                      
                      -
                      
                        2
                        ⁢
                        n
                      
                      +
                      17
                    
                    )
                  
                  ⁢
                  
                     
                  
                  ⁢
                  l
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
            
          
        
      
    
    
      
        (50b)
      
    
  
  
    
      
        
          
            M
            C
          
          =
          
            
              ∑
              
                i
                =
                0
              
              
                7
                -
                m
              
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                
                  n
                  -
                  1
                
              
              ⁢
              
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            n
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ·
                  cos
                
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          m
                        
                        +
                        17
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
        
        ,
        and
      
    
    
      
        (50c)
      
    
  
  
    
      
        
          M
          D
        
        =
        
          
            ∑
            
              i
              =
              0
            
            
              7
              -
              m
            
          
          ⁢
          
            
              ∑
              
                j
                =
                0
              
              
                n
                -
                1
              
            
            ⁢
            
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        i
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  p
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
              ⁢
              
                 
              
              ⁢
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ·
                cos
              
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        i
                      
                      +
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        m
                      
                      +
                      17
                    
                    )
                  
                  ⁢
                  
                     
                  
                  ⁢
                  k
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
              ⁢
              cos
              ⁢
              
                 
              
              ⁢
              
                
                  
                    (
                    
                      
                        2
                        ⁢
                        
                           
                        
                        ⁢
                        j
                      
                      -
                      
                        2
                        ⁢
                        n
                      
                      +
                      17
                    
                    )
                  
                  ⁢
                  
                     
                  
                  ⁢
                  l
                  ⁢
                  
                     
                  
                  ⁢
                  π
                
                16
              
            
          
        
      
    
    
      
        (50d)
      
    
  




Accordingly, using expressions (49) and (50a)-(50d), the DCT coefficients for the composite macro block E can be determined. As one skilled in the art will readily appreciate from the above disclosure, the expressions above provide a composite macro block for full pixel accuracy. In other words, the coordinate (m,n) is the coordinate for a pixel in the composite macro block e. The coordinate (m,n) does not, however, always fall on a pixel in a macro block of an anchor frame. Instead, this coordinate could fall between two pixels in the horizontal direction, between two pixels in the vertical direction, or between two pixels in the diagonal direction.
Frame-Motion Compensation—Half-Pel Accuracy Horizontal Half-Pel Accuracy
If coordinate (m,n) falls between two pixels in the horizontal direction, then motion compensation must be performed with half-pel (i.e., half pixel) accuracy in the horizontal direction.
Expression (51) below illustrates the relationship between the composite macro block e and the macro blocks a-d in the anchor frame in the spatial domain. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        (
                        
                          
                            a
                            
                              
                                i
                                +
                                m
                              
                              ,
                              
                                j
                                +
                                n
                              
                            
                          
                          +
                          
                            a
                            
                              
                                i
                                +
                                m
                              
                              ,
                              
                                j
                                +
                                n
                                +
                                1
                              
                            
                          
                        
                        )
                      
                      ⁢
                      0
                    
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        (
                        
                          
                            b
                            
                              
                                i
                                +
                                m
                              
                              ,
                              
                                j
                                +
                                n
                                -
                                8
                              
                            
                          
                          +
                          
                            b
                            
                              
                                i
                                +
                                m
                              
                              ,
                              
                                j
                                +
                                n
                                -
                                7
                              
                            
                          
                        
                        )
                      
                      ⁢
                      0
                    
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          (
                          
                            
                              c
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  8
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                            +
                            
                              c
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  8
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  +
                                  1
                                
                              
                            
                          
                          )
                        
                        ⁢
                        8
                      
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  8
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                            +
                            
                              d
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  8
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  7
                                
                              
                            
                          
                          )
                        
                        ⁢
                        8
                      
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        (
                        
                          
                            a
                            
                              
                                i
                                +
                                m
                              
                              ,
                              7
                            
                          
                          +
                          
                            b
                            
                              
                                i
                                +
                                m
                              
                              ,
                              0
                            
                          
                        
                        )
                      
                      ⁢
                      0
                    
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        (
                        
                          
                            c
                            
                              
                                i
                                +
                                m
                                -
                                8
                              
                              ,
                              7
                            
                          
                          +
                          
                            d
                            
                              
                                i
                                +
                                m
                              
                              ,
                              
                                -
                                8
                              
                              ,
                              0
                            
                          
                        
                        )
                      
                      ⁢
                      8
                    
                    -
                    m
                  
                  ≤
                  i
                  ≤
                  7
                
              
            
          
          }
        
      
    
    
      
        (
        51
        )
      
    
  




Expressions (52a)-(52d) below are derived in the same manner as expressions (50a)-(50d) above using expression (51). 



  
    
      
        
          M
          A
        
        =
        
          [
          
             
          
          ⁢
          
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    6
                    -
                    n
                  
                
                ⁢
                
                  
                    
                      Y
                      _
                    
                    _
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            +
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                
                  1
                  2
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        
                          2
                          ⁢
                          m
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      15
                      ⁢
                      
                         
                      
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ·
                  cos
                
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
          ]
        
      
    
    
      
        (52a)
      
    
  
  
    
      
        
          M
          B
        
        =
        
          [
          
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      -
                      n
                    
                  
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      Y
                      _
                    
                    _
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            +
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                
                  1
                  2
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        
                          2
                          ⁢
                          m
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ·
                  cos
                
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
          ]
        
      
    
    
      
        (52b)
      
    
  
  
    
      
        
          M
          C
        
        =
        
          [
          
            
              
                ∑
                
                  i
                  =
                  
                    -
                    m
                  
                
                
                  -
                  1
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    6
                    -
                    m
                  
                
                ⁢
                
                  
                    
                      Y
                      _
                    
                    _
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            +
            
              
                ∑
                
                  i
                  =
                  
                    -
                    m
                  
                
                
                  -
                  1
                
              
              ⁢
              
                
                  1
                  2
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        
                          2
                          ⁢
                          m
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      15
                      ⁢
                      
                         
                      
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ·
                  cos
                
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        17
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
          ]
        
      
    
    
      
        (52c)
      
    
  
  
    
      
        
          M
          D
        
        =
        
          [
          
            
              
                ∑
                
                  i
                  =
                  
                    -
                    m
                  
                
                
                  -
                  1
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      -
                      n
                    
                  
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      Y
                      _
                    
                    _
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            +
            
              
                ∑
                
                  i
                  =
                  
                    -
                    m
                  
                
                
                  -
                  1
                
              
              ⁢
              
                
                  1
                  2
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        
                          2
                          ⁢
                          m
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ·
                  cos
                
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          i
                        
                        +
                        17
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
          ]
        
      
    
    
      
        (52d)
      
    
  




where {overscore (Y)} represents expression (53) below 



  
    
      
        
          
            Y
            _
          
          _
        
        =
        
          cos
          ⁢
          
             
          
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    
                       
                    
                    ⁢
                    i
                  
                  +
                  
                    2
                    ⁢
                    m
                  
                  +
                  1
                
                )
              
              ⁢
              
                 
              
              ⁢
              p
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              (
              
                j
                +
                n
                +
                1
              
              )
            
            8
          
          ⁢
          
             
          
          ⁢
          q
          ⁢
          
             
          
          ⁢
          π
        
      
    
    
      
        (
        53
        )
      
    
  




Using expressions (49) and (52a)-(52d), the composite macro block E can be derived with horizontal half-pel accuracy.
Vertical Half-Pel Accuracy
If coordinate (m,n) falls between two pixels in the vertical direction, then motion compensation must be performed with half-pel (i.e., half pixel) accuracy in the vertical direction. Equation (54) below illustrates the relationship between the composite macro block e and the macro blocks a-d of the anchor frame in the spatial domain. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      n
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      m
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
          
          }
        
      
    
    
      
        (
        54
        )
      
    
  




Expressions (55a)-(55d) below are derived in the same manner as expressions (50a)-(50d) using expression 



  
    
      
        
          M
          A
        
        =
        
          [
          
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  6
                  -
                  m
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    n
                  
                
                ⁢
                
                  
                    
                      X
                      _
                    
                    _
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            +
            
              
                ∑
                
                  j
                  =
                  0
                
                
                  7
                  -
                  n
                
              
              ⁢
              
                
                  1
                  2
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    15
                    ⁢
                    
                       
                    
                    ⁢
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        
                          2
                          ⁢
                          n
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          m
                        
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
          ]
        
      
    
    
      
        (55a)
      
    
  
  
    
      
        
          M
          B
        
        =
        
          [
          
             
          
          ⁢
          
            
              
                ∑
                
                  i
                  =
                  0
                
                
                  6
                  -
                  m
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      -
                      n
                    
                  
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      X
                      _
                    
                    _
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            +
            
              
                ∑
                
                  j
                  =
                  
                    -
                    n
                  
                
                
                  -
                  i
                
              
              ⁢
              
                
                  1
                  2
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    15
                    ⁢
                    
                       
                    
                    ⁢
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        
                          2
                          ⁢
                          n
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          m
                        
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        17
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
          ]
        
      
    
    
      
        (55b)
      
    
  
  
    
      
        
          M
          C
        
        =
        
          [
          
            
              
                ∑
                
                  i
                  =
                  
                    -
                    m
                  
                
                
                  -
                  1
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    n
                  
                
                ⁢
                
                  
                    
                      X
                      _
                    
                    _
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            +
            
               
            
            ⁢
            
              
                ∑
                
                  j
                  =
                  0
                
                
                  7
                  -
                  n
                
              
              ⁢
              
                
                  1
                  2
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        
                          2
                          ⁢
                          n
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          m
                        
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
          ]
        
      
    
    
      
        (55c)
      
    
  
  
    
      
        
          M
          D
        
        =
        
          [
          
             
          
          ⁢
          
            
              
                ∑
                
                  i
                  =
                  
                    -
                    m
                  
                
                
                  -
                  1
                
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      -
                      n
                    
                  
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      X
                      _
                    
                    _
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            i
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  
                     
                  
                  ⁢
                  cos
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                               
                            
                            ⁢
                            j
                          
                          +
                          17
                        
                        )
                      
                      ⁢
                      
                         
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
            +
            
              
                ∑
                
                  j
                  =
                  
                    -
                    n
                  
                
                
                  -
                  1
                
              
              ⁢
              
                
                  1
                  2
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                  
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        
                          2
                          ⁢
                          n
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    
                       
                    
                    ⁢
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          m
                        
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                cos
                ⁢
                
                   
                
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          
                             
                          
                          ⁢
                          j
                        
                        +
                        17
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
          
          ]
        
      
    
    
      
        (55d)
      
    
  




where {overscore (X)} represents expression (56) below. 



  
    
      
        
          
            X
            _
          
          _
        
        =
        
          2
          ⁢
          
             
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    
                       
                    
                    ⁢
                    j
                  
                  +
                  
                    2
                    ⁢
                    n
                  
                  +
                  1
                
                )
              
              ⁢
              
                 
              
              ⁢
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              
                (
                
                  7
                  +
                  m
                  +
                  1
                
                )
              
              ⁢
              p
              ⁢
              
                 
              
              ⁢
              π
            
            8
          
          ⁢
          
             
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              p
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
        
      
    
    
      
        (
        56
        )
      
    
  




Using expressions (49) and (55a)-(55d), the composite macro block E can be derived with vertical half-pel accuracy.
Diagonal Half-Pel Accuracy
If coordinate (m,n) falls between two pixels in the diagonal direction, then motion compensation must be performed with half-pel accuracy in the diagonal direction. Expression (57) below illustrates the relationship between the composite macro block e and the macro blocks a-d of the anchor frame in the spatial domain. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            m
                            +
                            1
                          
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      6
                      -
                      n
                    
                  
                  ;
                
              
            
            
              
                
                  
                    
                      a
                      
                        
                          i
                          +
                          m
                          +
                          1
                        
                        ,
                        
                          j
                          +
                          n
                        
                      
                    
                    +
                    
                      a
                      
                        
                          i
                          +
                          m
                          +
                          1
                        
                        ,
                        
                          j
                          +
                          n
                          +
                          1
                        
                      
                    
                  
                  )
                
              
              
                
                  0
                  ≤
                  i
                  ≤
                  
                    6
                    -
                    n
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            n
                          
                          ,
                          
                            j
                            +
                            m
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      6
                      -
                      m
                    
                  
                  ;
                
              
            
            
              
                
                  
                    
                      b
                      
                        
                          i
                          +
                          m
                          +
                          1
                        
                        ,
                        
                          j
                          +
                          n
                          -
                          8
                        
                      
                    
                    +
                    
                      b
                      
                        
                          i
                          +
                          m
                          +
                          1
                        
                        ,
                        
                          j
                          +
                          n
                          -
                          7
                        
                      
                    
                  
                  )
                
              
              
                
                  8
                  ≤
                  n
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    8
                    ≤
                    m
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    
                      c
                      
                        
                          i
                          +
                          m
                          -
                          7
                        
                        ,
                        
                          j
                          +
                          n
                        
                      
                    
                    +
                    
                      c
                      
                        
                          i
                          +
                          m
                          -
                          7
                        
                        ,
                        
                          j
                          +
                          n
                          +
                          1
                        
                      
                    
                  
                  )
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    6
                    -
                    n
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    
                      d
                      
                        
                          i
                          +
                          m
                          -
                          7
                        
                        ,
                        
                          j
                          +
                          n
                          -
                          8
                        
                      
                    
                    +
                    
                      d
                      
                        
                          i
                          +
                          m
                          -
                          7
                        
                        ,
                        
                          j
                          +
                          n
                          -
                          7
                        
                      
                    
                  
                  )
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  i
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          7
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          7
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      m
                    
                  
                  ;
                
              
            
            
              
                
                  
                    
                      c
                      
                        0
                        ,
                        
                          j
                          +
                          n
                        
                      
                    
                    +
                    
                      c
                      
                        0
                        ,
                        
                          j
                          +
                          n
                          +
                          1
                        
                      
                    
                  
                  )
                
              
              
                
                  0
                  ≤
                  i
                  ≤
                  
                    6
                    -
                    n
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          7
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          1
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      m
                    
                  
                  ;
                
              
            
            
              
                
                  
                    
                      d
                      
                        0
                        ,
                        
                          j
                          +
                          n
                          -
                          8
                        
                      
                    
                    +
                    
                      d
                      
                        0
                        ,
                        
                          j
                          +
                          n
                          -
                          7
                        
                      
                    
                  
                  )
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          7
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            1
                          
                          ,
                          7
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      6
                      -
                      m
                    
                  
                  ;
                
              
            
            
              
                
                  
                    
                      b
                      
                        
                          i
                          +
                          m
                        
                        ,
                        0
                      
                    
                    +
                    
                      b
                      
                        
                          i
                          +
                          m
                          +
                          1
                        
                        ,
                        0
                      
                    
                  
                  )
                
              
              
                
                  j
                  =
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          7
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            7
                          
                          ,
                          7
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    
                      d
                      
                        
                          i
                          +
                          m
                          -
                          8
                        
                        ,
                        0
                      
                    
                    +
                    
                      d
                      
                        
                          i
                          +
                          m
                          -
                          7
                        
                        ,
                        0
                      
                    
                  
                  )
                
              
              
                
                  j
                  =
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          7
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          7
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      m
                    
                  
                  ;
                
              
            
            
              
                
                   
                
              
              
                
                  j
                  =
                  
                    7
                    -
                    n
                  
                
              
            
          
          }
        
      
    
    
      
        (
        57
        )
      
    
  




Expressions (58a)-(58d) below are derived in the same manner as expressions (50a)-(50d) using expression (57). 



  
    
      
        
          
            
              
                M
                A
              
              =
              
                [
                
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      
                        6
                        -
                        m
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        
                          6
                          -
                          n
                        
                      
                      ⁢
                      
                        cos
                        ⁢
                        
                          
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                i
                                +
                                m
                                +
                                1
                              
                              )
                            
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          8
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  n
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ·
                          cos
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        6
                        -
                        n
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          15
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              n
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      
                        6
                        -
                        m
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          15
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              m
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
                +
                
                  
                    1
                    4
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      15
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      15
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            m
                          
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            n
                          
                        
                        )
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        
          58
          ⁢
          a
        
        )
      
    
  
  
    
      
        
          
            
              
                M
                B
              
              =
              
                [
                
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      
                        6
                        -
                        m
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          
                            -
                            n
                          
                        
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        cos
                        ⁢
                        
                          
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                i
                                +
                                m
                                -
                                1
                              
                              )
                            
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          8
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  n
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ·
                          cos
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  j
                                
                                +
                                17
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        
                          -
                          n
                        
                      
                      
                        -
                        1
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          15
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              n
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        0
                      
                      
                        6
                        -
                        m
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              m
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              +
                              
                                2
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
                +
                
                  
                    1
                    4
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      15
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            m
                          
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            n
                          
                        
                        )
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        
          58
          ⁢
          b
        
        )
      
    
  
  
    
      
        
          
            
              
                M
                C
              
              =
              
                [
                
                  
                    
                      ∑
                      
                        i
                        =
                        
                          -
                          m
                        
                      
                      
                        -
                        i
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        
                          6
                          -
                          n
                        
                      
                      ⁢
                      
                        cos
                        ⁢
                        
                          
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                i
                                +
                                m
                                +
                                1
                              
                              )
                            
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          8
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  n
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ·
                          cos
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                17
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        6
                        -
                        n
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              n
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        
                          -
                          m
                        
                      
                      
                        -
                        i
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          15
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              i
                              +
                              m
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          P
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
                +
                
                  
                    1
                    4
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      15
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            m
                          
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            n
                          
                        
                        )
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        
          58
          ⁢
          c
        
        )
      
    
  
  
    
      
        
          
            
              
                M
                D
              
              =
              
                [
                
                  
                    
                      ∑
                      
                        i
                        =
                        
                          -
                          m
                        
                      
                      
                        -
                        i
                      
                    
                    ⁢
                    
                      
                        ∑
                        
                          j
                          =
                          
                            -
                            n
                          
                        
                        
                          -
                          i
                        
                      
                      ⁢
                      
                        cos
                        ⁢
                        
                          
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                i
                                +
                                m
                                +
                                1
                              
                              )
                            
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          8
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              
                                (
                                
                                  j
                                  +
                                  n
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ·
                          
                          ⁢
                          cos
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                17
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                17
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                    
                  
                  +
                
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            -
                            n
                          
                        
                      
                      
                        -
                        i
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              n
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        
                          -
                          m
                        
                      
                      
                        -
                        i
                      
                    
                    ⁢
                    
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              i
                              +
                              m
                              +
                              1
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                  
                
                +
                
                  
                    1
                    4
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            m
                          
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            n
                          
                        
                        )
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        
          58
          ⁢
          d
        
        )
      
    
  




Using expressions (49) and (58a)-(58d), the composite macro block E can be derived with diagonal half-pel accuracy.
The compositer 68 uses expressions (49)-(58d) to form composite macro blocks in the DCT domain when the inter-coded pictures have been encoded on a frame-by-frame basis. Accordingly, when the adder 74 adds the composite macro blocks to the macro blocks of P and B-pictures output from the variable length decoder and dequantizer 58, frame-motion compensation is performed.
Field-Motion Compensation
The HDTV signal can, however, also include fields. The even and odd fields of a frame may be inter-coded independently based on the even and odd fields of a previous frame. Namely, the even field may be inter-coded based on either the even or odd field of the previous frame regardless of whether the odd field was inter-coded based on the even or odd field of the previous frame. As discussed above, decoding inter-coded fields is referred to as field-motion compensation.
To perform field-motion compensation, the motion vector address generator 62 receives the odd field motion vector and the even field motion vector from the variable length decoder and dequantizer 58. The odd field motion vector points to a first coordinate (m,n) of a composite odd field macro block e, while the even field motion vector points to a second coordinate (m′,n′) of a composite even field macro block e′. Based on the first and second motion vectors, the motion vector address generator 62 instructs the memory 64 as to which of the anchor frame macro blocks to output to the compositer 68.
For purposes of discussion, assume that the motion vector address generator instructs the memory 64 to output anchor frame macro blocks A-D with respect to the odd field motion vector and output anchor frame macro blocks A′-D′ with respect to the even field motion vector. The motion vector address generator 62 also outputs a signal to the compositer 68 indicating the coordinates (m,n) and (m′,n′). Additionally, the signal output by the motion vector address generator 62 to the compositer 68 indicates whether frame or field motion compensation is to be performed. If field-motion compensation is to be performed, the compositer 68 then forms a frame composite macro block according to expressions (59)-(94) discussed below.
Because of the operation of the field-to-frame converter 69, the adder 74 receives macro blocks of a frame even when the HDTV signal includes fields. Therefore, the compositer 68 has to output frame composite macro blocks. Also, the macro blocks received from the memory 64 via the variable length decoder and dequantizer 67 are frame macro blocks. Accordingly, when performing field-motion compensation, the compositer 68 (1) separates the anchor frame macro blocks into even and odd fields, (2) forms respective even and odd field composite macro blocks, and (3) interlaces the even and odd field composite macro blocks to form a frame composite macro block. Operations (1), (2), and (3) are simultaneously performed by the compositer 68 according to expressions (59)-(94) discussed below.
Full-Pel Accuracy
With field-motion compensation, four possible encoding schemes exist: 



  
    
      
        
          
            
              o
              →
              o
            
          
        
        
          
            
              o
              →
              e
            
          
        
      
    
    
      
        (
        1
        )
      
    
  
  
    
      
        
          
            
              e
              →
              o
            
          
        
        
          
            
              e
              →
              e
            
          
        
      
    
    
      
        (
        2
        )
      
    
  
  
    
      
        
          
            
              o
              →
              o
            
          
        
        
          
            
              e
              →
              e
            
          
        
      
    
    
      
        (
        3
        )
      
    
  
  
    
      
        
          
            
              o
              →
              e
            
          
        
        
          
            
              e
              →
              o
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




where o→e means that the current even field was encoded based on the previous odd field. In other words, “o” stands for odd field and “e” stands for even field.

Case (1) o→o and e→e
Expression (59) below represents the relationship between the frame composite macro block e and the macro blocks a-d and a′-d′ of the anchor frame in the spatial domain when the even and odd fields are encoded based on the odd field of the anchor frame. 



  
    
      
        
          
            e
            
              i
              ,
              j
            
          
          =
          
            {
            
              
                
                  
                    a
                    
                      
                        i
                        +
                        m
                      
                      ,
                      
                        j
                        +
                        n
                      
                    
                  
                
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      6
                      -
                      m
                    
                  
                
                
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      n
                    
                  
                
              
              
                
                  
                    b
                    
                      
                        i
                        +
                        m
                      
                      ,
                      
                        j
                        +
                        n
                        -
                        8
                      
                    
                  
                
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      6
                      -
                      m
                    
                  
                
                
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
              
                
                  
                    c
                    
                      
                        i
                        +
                        m
                        -
                        8
                      
                      ,
                      
                        j
                        +
                        n
                      
                    
                  
                
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                
                
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      n
                    
                  
                
              
              
                
                  
                    d
                    
                      
                        i
                        +
                        m
                        -
                        8
                      
                      ,
                      
                        j
                        +
                        n
                        -
                        8
                      
                    
                  
                
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                
                
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
              
                
                  
                    
                      a
                      ′
                    
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                      
                    
                  
                
                
                  
                    1
                    ,
                    
                      3
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                  
                
                
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
              
                
                  
                    
                      b
                      ′
                    
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        n
                        -
                        
                          8
                          ′
                        
                      
                    
                  
                
                
                  
                    1
                    ,
                    
                      3
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                  
                
                
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
              
                
                  
                    
                      c
                      ′
                    
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        
                          9
                          ′
                        
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                      
                    
                  
                
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                
                
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
              
                
                  
                    
                      d
                      ′
                    
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        
                          9
                          ′
                        
                      
                      ,
                      
                        j
                        +
                        n
                        -
                        
                          8
                          ′
                        
                      
                    
                  
                
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                
                
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            }
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              where
              ⁢
              
                 
              
              ⁢
              m
            
            =
            0
          
          ,
          2
          ,
          4
          ,
          
            
              6
              ⁢
              
                 
              
              ⁢
              and
              ⁢
              
                 
              
              ⁢
              
                m
                ′
              
            
            =
            0
          
          ,
          2
          ,
          4
          ,
          6
        
      
    
    
      
        (
        59
        )
      
    
  




The top half of expression (59) represents the o→o portion of the frame composite macro block e, while the bottom portion of expression (59) represents the o→e portion of the frame composite macro block e.

Using expression (59), expression (60) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
              ⁢
              
                
                  ∑
                  
                    i
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      e
                      ij
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        l
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      (
                      60
                      )
                    
                  
                
              
            
          
        
        
          
            
              
                E
                kl
              
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁢
                      
                        
                          ∑
                          
                            
                              i
                              =
                              0
                            
                            ,
                            2
                          
                          
                            6
                            -
                            m
                          
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      m
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ·
                              F
                            
                          
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        B
                        pq
                      
                      ⁢
                      
                        
                          ∑
                          
                            
                              i
                              =
                              0
                            
                            ,
                            2
                          
                          
                            6
                            -
                            m
                          
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              
                                8
                                -
                                n
                              
                            
                            7
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      m
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      -
                                      15
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ·
                              F
                            
                          
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        C
                        pq
                      
                      ⁢
                      
                        
                          ∑
                          
                            i
                            =
                            
                              8
                              -
                              m
                            
                          
                          6
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      m
                                    
                                    -
                                    15
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ·
                              F
                            
                          
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        D
                        pq
                      
                      ⁢
                      
                        
                          ∑
                          
                            i
                            =
                            
                              8
                              -
                              m
                            
                          
                          6
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              
                                8
                                -
                                n
                              
                            
                            7
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      m
                                    
                                    -
                                    15
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      -
                                      15
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ·
                              F
                            
                          
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        A
                        pq
                        ′
                      
                      ⁢
                      
                        
                          ∑
                          
                            
                              i
                              =
                              1
                            
                            ,
                            3
                          
                          
                            7
                            -
                            
                              m
                              ′
                            
                          
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          n
                                          ′
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ·
                              F
                            
                          
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        B
                        pq
                        ′
                      
                      ⁢
                      
                        
                          ∑
                          
                            
                              i
                              =
                              1
                            
                            ,
                            3
                          
                          
                            7
                            -
                            
                              m
                              ′
                            
                          
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              
                                8
                                -
                                n
                              
                            
                            7
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                    -
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          n
                                          ′
                                        
                                      
                                      -
                                      15
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ·
                              F
                            
                          
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        C
                        pq
                        ′
                      
                      ⁢
                      
                        
                          ∑
                          
                            i
                            =
                            
                              9
                              -
                              m
                            
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                    -
                                    17
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          n
                                          ′
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ·
                              F
                            
                          
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      D
                      pq
                      ′
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          
                            9
                            -
                            m
                          
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              8
                              -
                              
                                n
                                ′
                              
                            
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                  -
                                  17
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    -
                                    15
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ·
                            F
                          
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              where
            
          
        
        
          
            
              F
              =
              
              ⁢
              
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          i
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        
                          2
                          ⁢
                          j
                        
                        +
                        1
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                
                   
                
                ⁢
                
                  (
                  61
                  )
                
              
            
          
        
      
    
    
      
         
      
    
  




The quadruple summations including the terms Apq, Bpq, Cpq, and Dpq in equation (60) represent the o→o portion of the frame composite macro block E, while the quadruple summations including the terms A′pq, B′pq, C′pq, and D′pq represent the o→e portion of the frame composite macro block E.
Using expression (60), the motion compensated frame macro block E can be derived for full-pel accuracy when the even and odd fields are encoded based on the odd field of the anchor frame.
As expressions (59) and (60) show, the frame composite macro block e can be represented in the spatial domain such that the frame composite macro block E can be derived without having to separately perform the following operations: (1) separate the frame anchor macro blocks into even and odd fields, (2) form respective even and odd field composite macro blocks, and (3) interlace the even and odd field composite macro blocks to form a frame composite macro block.
Case (2) e→o and e→e
Expression (62) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the even and odd fields are encoded based on the even field of the anchor frame. 



  
    
      
        
          e
          
            i
            ,
            j
          
        
        =
        
          {
          
            
              
                
                  a
                  
                    
                      i
                      +
                      m
                    
                    ,
                    
                      j
                      +
                      n
                    
                  
                
              
              
                
                  0
                  ,
                  
                    2
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  b
                  
                    
                      i
                      +
                      m
                    
                    ,
                    
                      j
                      +
                      n
                      -
                      8
                    
                  
                
              
              
                
                  0
                  ,
                  
                    2
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  c
                  
                    
                      i
                      +
                      m
                      -
                      8
                    
                    ,
                    
                      j
                      +
                      n
                    
                  
                
              
              
                
                  
                    9
                    -
                    m
                  
                  ≤
                  i
                  ≤
                  6
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  d
                  
                    
                      i
                      +
                      m
                      -
                      8
                    
                    ,
                    
                      j
                      +
                      n
                      -
                      8
                    
                  
                
              
              
                
                  
                    9
                    -
                    m
                  
                  ≤
                  i
                  ≤
                  6
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  a
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      1
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                    
                  
                  ′
                
              
              
                
                  1
                  ,
                  
                    3
                    ≤
                    i
                    ≤
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  b
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      1
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                      -
                      8
                    
                  
                  ′
                
              
              
                
                  1
                  ,
                  
                    3
                    ≤
                    i
                    ≤
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  c
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      9
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                    
                  
                
              
              
                
                  
                    10
                    -
                    
                      m
                      ′
                    
                  
                  ≤
                  i
                  ≤
                  7
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  d
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      9
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                      -
                      8
                    
                  
                
              
              
                
                  
                    9
                    -
                    
                      m
                      ′
                    
                  
                  ≤
                  i
                  ≤
                  7
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
          
          }
        
      
    
    
      
        (
        62
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        
          
            7
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        7
      
    
    
      
         
      
    
  




The top half of expression (62) represents the e→o portion of the frame composite macro block e, while the bottom portion of expression (62) represents the e→e portion of the frame composite macro block e.
Using expression (62), expression (63) below is derived in the same manner as expression (47). 


  
    
      E
      kl
    
    =
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              A
              pq
            
            ⁢
            
              
                ∑
                
                  
                    i
                    =
                    0
                  
                  ,
                  2
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    n
                  
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              n
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              B
              pq
            
            ⁢
            
              
                ∑
                
                  
                    i
                    =
                    0
                  
                  ,
                  2
                
                
                  7
                  -
                  m
                
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      n
                    
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              n
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
         
      
      ⁢
      
        (
        63
        )
      
    
  




  
    
       
    
    ⁢
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              C
              pq
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    9
                    -
                    m
                  
                
                6
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    n
                  
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          -
                          15
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              n
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              D
              pq
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    9
                    -
                    m
                  
                
                6
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      n
                    
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          -
                          15
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              n
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
        

      
      ⁢
      
         
      
      ⁢
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              A
              pq
              ′
            
            ⁢
            
              
                ∑
                
                  
                    i
                    =
                    1
                  
                  ,
                  3
                
                
                  8
                  -
                  
                    m
                    ′
                  
                
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                          -
                          1
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              
                                n
                                ′
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              B
              pq
              ′
            
            ⁢
            
              
                ∑
                
                  
                    i
                    =
                    1
                  
                  ,
                  3
                
                
                  8
                  -
                  
                    m
                    ′
                  
                
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                          -
                          1
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            j
                          
                          +
                          
                            2
                            ⁢
                            
                              n
                              ′
                            
                          
                          -
                          15
                        
                        )
                      
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
          
        
      
      +
      
        

      
      ⁢
      
         
      
      ⁢
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              C
              pq
              ′
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    10
                    -
                    
                      m
                      ′
                    
                  
                
                7
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                          -
                          17
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            j
                          
                          +
                          
                            2
                            ⁢
                            
                              n
                              ′
                            
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
          
        
      
      +
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              D
              pq
              ′
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    10
                    -
                    
                      m
                      ′
                    
                  
                
                7
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          -
                          17
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            j
                          
                          +
                          
                            2
                            ⁢
                            
                              n
                              ′
                            
                          
                          -
                          15
                        
                        )
                      
                      ⁢
                      q
                    
                    16
                  
                
              
            
          
        
      
    
  



The summation terms including Apq, Bpq, Cpq, and Dpq in expression (63) represent the e→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (63) represent the e→e portion of the frame composite macro block E.
Using expression (63), the motion compensated frame macro block E can be derived for full-pel accuracy when the even and odd fields are encoded based on the even field of the anchor frame.
Case (3) o→o and e→e
Expression (64) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the odd and even fields are encoded based on the odd and even fields, respectively, of the anchor frame. 



  
    
      
        
          e
          
            i
            ,
            j
          
        
        =
        
          {
          
            
              
                
                  a
                  
                    
                      i
                      +
                      m
                    
                    ,
                    
                      j
                      +
                      n
                    
                  
                
              
              
                
                  0
                  ≤
                  i
                  ≤
                  
                    6
                    -
                    m
                  
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  b
                  
                    
                      i
                      +
                      m
                    
                    ,
                    
                      j
                      +
                      n
                      -
                      8
                    
                  
                
              
              
                
                  0
                  ≤
                  i
                  ≤
                  
                    6
                    -
                    m
                  
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  c
                  
                    
                      i
                      +
                      m
                      -
                      8
                    
                    ,
                    
                      j
                      +
                      n
                    
                  
                
              
              
                
                  
                    8
                    -
                    m
                  
                  ≤
                  i
                  ≤
                  6
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  d
                  
                    
                      i
                      +
                      m
                      -
                      8
                    
                    ,
                    
                      j
                      +
                      n
                      -
                      8
                    
                  
                
              
              
                
                  
                    8
                    -
                    m
                  
                  ≤
                  i
                  ≤
                  6
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  a
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      1
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                    
                  
                  ′
                
              
              
                
                  1
                  ,
                  
                    3
                    ≤
                    i
                    ≤
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  b
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      1
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                      -
                      8
                    
                  
                  ′
                
              
              
                
                  1
                  ,
                  
                    3
                    ≤
                    i
                    ≤
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  c
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      9
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                    
                  
                
              
              
                
                  
                    10
                    -
                    
                      m
                      ′
                    
                  
                  ≤
                  i
                  ≤
                  7
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  d
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      9
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                      -
                      8
                    
                  
                
              
              
                
                  
                    9
                    -
                    
                      m
                      ′
                    
                  
                  ≤
                  i
                  ≤
                  7
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
          
          }
        
      
    
    
      
        (
        64
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        
          
            6
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        7
      
    
    
      
         
      
    
  




The top half of expression (64) represents the o→o portion of the frame composite macro block e, while the bottom portion of expression (64) represents the e→e portion of the frame composite macro block e.
Using expression (64), expression (65) below is derived in the same manner as expression (47). 


  
    
      E
      kl
    
    =
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              A
              pq
            
            ⁢
            
              
                ∑
                
                  
                    i
                    =
                    0
                  
                  ,
                  2
                
                
                  6
                  -
                  m
                
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    n
                  
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              n
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              B
              pq
            
            ⁢
            
              
                ∑
                
                  
                    i
                    =
                    0
                  
                  ,
                  2
                
                
                  6
                  -
                  m
                
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      n
                    
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              n
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
         
      
      ⁢
      
        (
        65
        )
      
    
  




  
    
       
    
    ⁢
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              C
              pq
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    8
                    -
                    m
                  
                
                6
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    n
                  
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          -
                          15
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              n
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              D
              pq
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    8
                    -
                    m
                  
                
                6
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      n
                    
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          -
                          15
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              n
                            
                            -
                            15
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
        

      
      ⁢
      
         
      
      ⁢
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              A
              pq
              ′
            
            ⁢
            
              
                ∑
                
                  
                    i
                    =
                    1
                  
                  ,
                  3
                
                
                  8
                  -
                  
                    m
                    ′
                  
                
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                          -
                          1
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            
                              2
                              ⁢
                              
                                n
                                ′
                              
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ·
                    F
                  
                
              
            
          
        
      
      +
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              B
              pq
              ′
            
            ⁢
            
              
                ∑
                
                  
                    i
                    =
                    1
                  
                  ,
                  3
                
                
                  8
                  -
                  
                    m
                    ′
                  
                
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                          -
                          1
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            j
                          
                          +
                          
                            2
                            ⁢
                            
                              n
                              ′
                            
                          
                          -
                          15
                        
                        )
                      
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
          
        
      
      +
      
        

      
      ⁢
      
         
      
      ⁢
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              C
              pq
              ′
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    10
                    -
                    
                      m
                      ′
                    
                  
                
                7
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                          -
                          17
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            j
                          
                          +
                          
                            2
                            ⁢
                            
                              n
                              ′
                            
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
              
            
          
        
      
      +
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
           
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
             
          
          ⁢
          
            
              D
              pq
              ′
            
            ⁢
            
              
                ∑
                
                  i
                  =
                  
                    10
                    -
                    
                      m
                      ′
                    
                  
                
                7
              
              ⁢
              
                 
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            i
                          
                          +
                          
                            2
                            ⁢
                            m
                          
                          -
                          17
                        
                        )
                      
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            j
                          
                          +
                          
                            2
                            ⁢
                            
                              n
                              ′
                            
                          
                          -
                          15
                        
                        )
                      
                      ⁢
                      q
                    
                    16
                  
                
              
            
          
        
      
    
  



The summation terms including Apq, Bpq, Cpq, and Dpq in expression (65) represent the o→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (65) represent the e→e portion of the frame composite macro block E.
Using expression (65), the motion compensated frame macro block E can be derived for full-pel accuracy when the odd and even fields are encoded based on the odd and even fields, respectively, of the anchor frame.
Case (4) e→o and o→e
Expression (66) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the odd and even fields are encoded based on the even and odd fields, respectively, of the anchor frame. 



  
    
      
        
          e
          
            i
            ,
            j
          
        
        =
        
          {
          
            
              
                
                  a
                  
                    
                      i
                      +
                      m
                    
                    ,
                    
                      j
                      +
                      n
                    
                  
                
              
              
                
                  0
                  ,
                  
                    2
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  b
                  
                    
                      i
                      +
                      m
                    
                    ,
                    
                      j
                      +
                      n
                      -
                      8
                    
                  
                
              
              
                
                  0
                  ,
                  
                    2
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  c
                  
                    
                      i
                      +
                      m
                      -
                      8
                    
                    ,
                    
                      j
                      +
                      n
                    
                  
                
              
              
                
                  
                    9
                    -
                    m
                  
                  ≤
                  i
                  ≤
                  6
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  d
                  
                    
                      i
                      +
                      m
                      -
                      8
                    
                    ,
                    
                      j
                      +
                      n
                      -
                      8
                    
                  
                
              
              
                
                  
                    9
                    -
                    m
                  
                  ≤
                  i
                  ≤
                  6
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  a
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      1
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                    
                  
                  ′
                
              
              
                
                  1
                  ,
                  
                    3
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      
                        m
                        ′
                      
                    
                  
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  b
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      1
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                      -
                      8
                    
                  
                  ′
                
              
              
                
                  1
                  ,
                  
                    3
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      
                        m
                        ′
                      
                    
                  
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  c
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      9
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                    
                  
                  ′
                
              
              
                
                  
                    9
                    -
                    
                      m
                      ′
                    
                  
                  ≤
                  i
                  ≤
                  7
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  d
                  
                    
                      i
                      +
                      
                        m
                        ′
                      
                      -
                      9
                    
                    ,
                    
                      j
                      +
                      
                        n
                        ′
                      
                      -
                      8
                    
                  
                  ′
                
              
              
                
                  
                    9
                    -
                    
                      m
                      ′
                    
                  
                  ≤
                  i
                  ≤
                  7
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
          
          }
        
      
    
    
      
        (
        66
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        
          
            7
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        6
      
    
    
      
         
      
    
  




The top half of expression (66) represents the e→o portion of the frame composite macro block e, while the bottom portion of expression (66) represents the o→e portion of the frame composite macro block e.
Using expression (66), expression (67) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁢
                      
                        
                          ∑
                          
                            
                              i
                              =
                              0
                            
                            ,
                            2
                          
                          
                            7
                            -
                            m
                          
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      m
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ·
                              F
                            
                          
                        
                      
                    
                  
                
                +
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        B
                        pq
                      
                      ⁢
                      
                        
                          ∑
                          
                            
                              i
                              =
                              0
                            
                            ,
                            2
                          
                          
                            7
                            -
                            m
                          
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              
                                8
                                -
                                n
                              
                            
                            7
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      m
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      -
                                      15
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ·
                              F
                            
                          
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      C
                      pq
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          
                            9
                            -
                            m
                          
                        
                        6
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            7
                            -
                            n
                          
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    m
                                  
                                  -
                                  15
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ·
                            F
                          
                        
                      
                    
                  
                
              
              +
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      D
                      pq
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          
                            9
                            -
                            m
                          
                        
                        6
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              8
                              -
                              n
                            
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    m
                                  
                                  -
                                  15
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    -
                                    15
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ·
                            F
                          
                        
                      
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      A
                      pq
                      ′
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            =
                            1
                          
                          ,
                          3
                        
                        
                          7
                          -
                          
                            m
                            ′
                          
                        
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            7
                            -
                            
                              n
                              ′
                            
                          
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ·
                            F
                          
                        
                      
                    
                  
                
              
              +
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      B
                      pq
                      ′
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            =
                            1
                          
                          ,
                          3
                        
                        
                          7
                          -
                          
                            m
                            ′
                          
                        
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              8
                              -
                              n
                            
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                  -
                                  1
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                  -
                                  15
                                
                                )
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      C
                      pq
                      ′
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          
                            9
                            -
                            m
                          
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            7
                            -
                            
                              n
                              ′
                            
                          
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                  -
                                  17
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ·
                            F
                          
                        
                      
                    
                  
                
              
              +
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      D
                      pq
                      ′
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          
                            9
                            -
                            m
                          
                        
                        7
                      
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              8
                              -
                              
                                n
                                ′
                              
                            
                          
                          7
                        
                        ⁢
                        
                           
                        
                        ⁢
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                  -
                                  17
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                  -
                                  15
                                
                                )
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        67
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (67) represent the e→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (67) represent the o→e portion of the frame composite macro block E.
Using expression (67), the motion compensated frame macro block E can be derived for full-pel accuracy when the odd and even fields are encoded based on the even and odd fields, respectively, of the anchor frame.
Accordingly, using expressions (60), (63), (65), and (67), the DCT coefficients for the composite macro block E can be determined. As one skilled in the art will readily appreciate from the above disclosure, the expressions above provide a composite macro block for full pixel accuracy. In other words, the coordinates (m,n) and (m′,n′) are the coordinates for a pixel in the composite macro block e. The coordinates (m,n) and (m′,n′) do not, however, always fall on a pixel in a macro block of the anchor frame. Instead, these coordinates could fall between two pixels in the horizontal direction, between two pixels in the vertical direction, or between two pixels in the diagonal direction.
Horizontal Half-Pel Accuracy
Case (1) o→o and o→e
Expression (68) below represents the relationship between the frame composite macro block e and the macro blocks a-d and a′-d′ in the anchor frame in the spatial domain when the even and odd fields are encoded based on the odd field of the anchor frame. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        m
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        6
                        -
                        n
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        m
                      
                    
                    ;
                    
                      
                        8
                        -
                        n
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                
              
            
            
              
                
                  
                    d
                    
                      
                        
                          i
                          +
                          m
                          -
                          8
                        
                        ,
                        
                          j
                          +
                          n
                          -
                          7
                        
                      
                      ⁢
                      
                         
                      
                    
                  
                  )
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
            
            
              
                
                  
                    a
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        1
                      
                    
                    ′
                  
                  )
                
              
              
                
                  i
                  =
                  
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
            
            
              
                
                  
                    b
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        7
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    c
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        9
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        +
                        1
                      
                    
                    ′
                  
                  )
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    6
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    d
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        9
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        7
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
          
          }
        
      
    
    
      
        (
        68
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        
          
            7
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        7
      
    
    
      
         
      
    
  




The top half of expression (68) represents the o→o portion of the frame composite macro block e, while the bottom portion of expression (68) represents the o→e portion of the frame composite macro block e.
Using expression (68), expression (69) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  6
                                  -
                                  n
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        B
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  
                                    
                                      -
                                      n
                                    
                                    -
                                    1
                                  
                                
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      C
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  
                                    -
                                    m
                                  
                                  +
                                  2
                                
                              
                              ,
                              
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                6
                                -
                                n
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              Υ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              i
                              =
                              
                                -
                                m
                              
                            
                            
                              6
                              -
                              m
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      m
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                   
                                
                                ⁢
                                
                                  15
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      n
                                    
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        D
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                i
                                =
                                
                                  -
                                  m
                                
                              
                              
                                -
                                2
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  
                                    -
                                    n
                                  
                                
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                i
                                =
                                
                                  -
                                  m
                                
                              
                              
                                -
                                2
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        A
                        pq
                        ′
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  6
                                  -
                                  
                                    n
                                    ′
                                  
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  Υ
                                  ′
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        3
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          m
                                          ′
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        
                                          n
                                          ′
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        B
                        pq
                        ′
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  
                                    -
                                    
                                      n
                                      ′
                                    
                                  
                                
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  Υ
                                  ′
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        3
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          m
                                          ′
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        
                                          n
                                          ′
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      C
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  
                                    m
                                    ′
                                  
                                  +
                                  2
                                
                              
                              ,
                              
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                6
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                Υ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              i
                              =
                              
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                   
                                
                                ⁢
                                
                                  15
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      D
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              i
                              =
                              
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  
                                    n
                                    ′
                                  
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                Υ
                                ′
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              i
                              =
                              
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                   
                                
                                ⁢
                                
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
            
          
        
      
    
    
      
        (
        69
        )
      
    
  




where 



  
    
      
        Υ
        =
        
          cos
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    i
                  
                  +
                  
                    2
                    ⁢
                    m
                  
                  +
                  1
                
                )
              
              ⁢
              p
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
            
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
            
              
                (
                
                  j
                  +
                  n
                  +
                  1
                
                )
              
              ⁢
              q
              ⁢
              
                 
              
              ⁢
              π
            
            8
          
        
      
    
    
      
        (
        
          70
          ⁢
          A
        
        )
      
    
  
  
    
      
        
          Υ
          ′
        
        =
        
          cos
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    i
                  
                  +
                  
                    2
                    ⁢
                    
                      m
                      ′
                    
                  
                  +
                  1
                
                )
              
              ⁢
              p
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
            
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
            
              
                (
                
                  j
                  +
                  
                    n
                    ′
                  
                  +
                  1
                
                )
              
              ⁢
              q
              ⁢
              
                 
              
              ⁢
              π
            
            8
          
        
      
    
    
      
        (
        
          70
          ⁢
          B
        
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (69) represent the o→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (69) represent the o→e portion of the frame composite macro block E.
Using expression (69), the motion compensated frame macro block E can be derived for horizontal half-pel accuracy when the even and odd fields are encoded based on the odd field of the anchor frame.
Case (2) e→o e→e
Expression (71) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the even and odd fields are encoded based on the even field of the anchor frame. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        m
                      
                    
                    ;
                  
                
              
            
            
              
                
                   
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    6
                    -
                    n
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        m
                      
                    
                    ;
                  
                
              
            
            
              
                
                   
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            
                              i
                              +
                              m
                              -
                              8
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              7
                            
                          
                          ⁢
                          
                             
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  1
                  ,
                  
                    
                      (
                      
                        
                          3
                          ≤
                          i
                          ≤
                          
                            8
                            -
                            
                              m
                              ′
                            
                          
                        
                        ;
                      
                      )
                    
                    ′
                  
                
              
            
            
              
                
                  
                    a
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        1
                      
                    
                    ′
                  
                  )
                
              
              
                
                  i
                  =
                  
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        8
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
            
            
              
                
                  
                    b
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        7
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    c
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        9
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        +
                        1
                      
                    
                    ′
                  
                  )
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    6
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    d
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        9
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        7
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        8
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
          
          }
        
      
    
    
      
        
          (
          71
          )
        
        ⁢
        
           
        
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        
          
            6
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        6
      
    
    
      
         
      
    
  




The top half of expression (71) represents the e→o portion of the frame composite macro block e, while the bottom portion of expression (71) represents the e→e portion of the frame composite macro block e.
Using expression (71), expression (72) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  6
                                  -
                                  n
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        B
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  
                                    -
                                    n
                                  
                                
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        C
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                i
                                =
                                
                                  
                                    -
                                    m
                                  
                                  +
                                  1
                                
                              
                              
                                -
                                2
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  6
                                  -
                                  n
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                i
                                =
                                
                                  
                                    -
                                    m
                                  
                                  +
                                  1
                                
                              
                              
                                -
                                2
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    15
                                    ⁢
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        D
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                i
                                =
                                
                                  
                                    -
                                    m
                                  
                                  +
                                  1
                                
                              
                              
                                -
                                2
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  
                                    -
                                    n
                                  
                                
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                i
                                =
                                
                                  
                                    -
                                    m
                                  
                                  +
                                  1
                                
                              
                              
                                -
                                2
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        A
                        pq
                        ′
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  6
                                  -
                                  
                                    n
                                    ′
                                  
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  Υ
                                  ′
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        3
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          m
                                          ′
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        
                                          n
                                          ′
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        B
                        pq
                        ′
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  
                                    -
                                    
                                      n
                                      ′
                                    
                                  
                                
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  Υ
                                  ′
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        3
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          m
                                          ′
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        
                                          n
                                          ′
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      C
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              i
                              =
                              
                                
                                  m
                                  ′
                                
                                +
                                1
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                6
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                Υ
                                ′
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              i
                              =
                              
                                
                                  -
                                  
                                    m
                                    ′
                                  
                                
                                +
                                1
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                   
                                
                                ⁢
                                
                                  (
                                  
                                    15
                                    ⁢
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  )
                                
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      D
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              i
                              =
                              
                                
                                  -
                                  
                                    m
                                    ′
                                  
                                
                                +
                                1
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  
                                    n
                                    ′
                                  
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                Υ
                                ′
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              i
                              =
                              
                                
                                  -
                                  
                                    m
                                    ′
                                  
                                
                                +
                                1
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                   
                                
                                ⁢
                                
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
            
          
        
      
    
    
      
        (
        72
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (72) represent the e→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (72) represent the e→e portion of the frame composite macro block E.
Using expression (72), the motion compensated frame macro block E can be derived for horizontal half-pel accuracy when the even and odd fields are encoded based on the even field of the anchor frame.
Case (3) o→o and e→e
Expression (73) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the odd and even fields are encoded based on the odd and even fields, respectively, of the anchor frame. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        m
                      
                    
                    ;
                  
                
              
            
            
              
                
                   
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    6
                    -
                    n
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        m
                      
                    
                    ;
                  
                
              
            
            
              
                
                   
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            
                              i
                              +
                              m
                              -
                              8
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              7
                            
                          
                          ⁢
                          
                             
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        8
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
            
            
              
                
                  
                    a
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        1
                      
                    
                    ′
                  
                  )
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    6
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        8
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
            
            
              
                
                  
                    b
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        7
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    c
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        9
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        +
                        1
                      
                    
                    ′
                  
                  )
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    6
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    d
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        9
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        7
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        8
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
          
          }
        
      
    
    
      
        (
        73
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        
          
            6
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        7
      
    
    
      
         
      
    
  




The top half of expression (73) represents the o→o portion of the frame composite macro block e, while the bottom portion of expression (73) represents the e→e portion of the frame composite macro block e.
Using expression (73), expression (74) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  6
                                  -
                                  n
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        B
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  
                                    
                                      -
                                      n
                                    
                                    -
                                    1
                                  
                                
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                6
                                -
                                m
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      C
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  
                                    -
                                    m
                                  
                                  +
                                  2
                                
                              
                              ,
                              
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                6
                                -
                                n
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              Υ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              i
                              =
                              
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      m
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                   
                                
                                ⁢
                                
                                  15
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      n
                                    
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        D
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                i
                                =
                                
                                  -
                                  m
                                
                              
                              
                                -
                                2
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  
                                    -
                                    n
                                  
                                
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                i
                                =
                                
                                  -
                                  m
                                
                              
                              
                                -
                                2
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        n
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        A
                        pq
                        ′
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  6
                                  -
                                  
                                    n
                                    ′
                                  
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  Υ
                                  ′
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        3
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          m
                                          ′
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        
                                          n
                                          ′
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        B
                        pq
                        ′
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  
                                    -
                                    
                                      n
                                      ′
                                    
                                  
                                
                                
                                  -
                                  1
                                
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  Υ
                                  ′
                                
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        3
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        17
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        
                                          m
                                          ′
                                        
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                     
                                  
                                  ⁢
                                  
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        
                                          n
                                          ′
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      C
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              i
                              =
                              
                                
                                  m
                                  ′
                                
                                +
                                1
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                6
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                Υ
                                ′
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              i
                              =
                              
                                
                                  -
                                  
                                    m
                                    ′
                                  
                                
                                +
                                1
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                   
                                
                                ⁢
                                
                                  (
                                  
                                    15
                                    ⁢
                                    q
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  )
                                
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                 
              
              ⁢
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                   
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                     
                  
                  ⁢
                  
                    
                      D
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              i
                              =
                              
                                
                                  -
                                  
                                    m
                                    ′
                                  
                                
                                +
                                1
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  
                                    n
                                    ′
                                  
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                Υ
                                ′
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              i
                              =
                              
                                
                                  -
                                  
                                    m
                                    ′
                                  
                                
                                +
                                1
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                   
                                
                                ⁢
                                
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
            
          
        
      
    
    
      
        (
        74
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq expression (74) represent the o→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (74) represent the e→e portion of the frame composite macro block E.
Using expression (74) the motion compensated frame macro block E can be derived for horizontal half-pel accuracy when the odd and even fields are encoded based on the odd and even fields, respectively, of the anchor frame.
Case (4) e→o and o→e
Expression (75) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the odd and even fields are encoded based on the even and odd fields, respectively, of the anchor frame. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        m
                      
                    
                    ;
                  
                
              
            
            
              
                
                   
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    6
                    -
                    n
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        m
                      
                    
                    ;
                  
                
              
            
            
              
                
                   
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            
                              i
                              +
                              m
                              -
                              8
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              7
                            
                          
                          ⁢
                          
                             
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    0
                    ≤
                    i
                    ≤
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
            
            
              
                
                  
                    a
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        1
                      
                    
                    ′
                  
                  )
                
              
              
                
                  i
                  =
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
            
            
              
                
                  
                    b
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        7
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    c
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        9
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        +
                        1
                      
                    
                    ′
                  
                  )
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    6
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    d
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        9
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        7
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
          
          }
        
      
    
    
      
        (
        75
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        
          
            7
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        6
      
    
    
      
         
      
    
  




The top half of expression (75) represents the e→o portion of the frame composite macro block e, while the bottom portion of expression (75) represents the o→e portion of the frame composite macro block e.
Using expression (75), expression (76) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                m
                              
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  6
                                  -
                                  n
                                
                              
                              ⁢
                              
                                Υ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                7
                                -
                                m
                              
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        m
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
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                              16
                            
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                                  (
                                  
                                    
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                                    (
                                    
                                      
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                                16
                              
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                              16
                            
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                                15
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                    =
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                  7
                
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                      [
                      
                        
                          
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                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
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                  7
                
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                                16
                              
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                                  (
                                  
                                    15
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                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
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                    =
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                  7
                
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                      [
                      
                        
                          
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                                    )
                                  
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                                16
                              
                              ⁢
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                                    (
                                    
                                      
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                              16
                            
                            ⁢
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                                15
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                              16
                            
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                                  (
                                  
                                    
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                                  )
                                
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                              16
                            
                            ⁢
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
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                                      2
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                                  )
                                
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                                ⁢
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                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
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                7
              
              ⁢
              
                
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                    q
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                  7
                
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                    pq
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                  ⁡
                  
                    [
                    
                      
                        
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                        ⁢
                        
                          
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                              j
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                                  )
                                
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                              16
                            
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                      +
                      
                        
                          ∑
                          
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                              -
                              
                                m
                                ′
                              
                            
                          
                          
                            -
                            2
                          
                        
                        ⁢
                        
                          
                            1
                            2
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  19
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
      
    
    
      
        (
        76
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq expression (76) represent the e→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (76) represent the o→e portion of the frame composite macro block E.
Using expression (76), the motion compensated frame macro block E can be derived for horizontal half-pel accuracy when the odd and even fields are encoded based on the even and odd fields, respectively, of the anchor frame.
Vertical Half-Pel Accuracy
Case (1) o→o and o→e
Expression (77) below represents the relationship between the frame composite macro block e and the macro blocks a-d and a′-d′ in the anchor frame in the spatial domain when the even and odd fields are encoded based on the odd field of the anchor frame. 



  
    
      
        
          
            
              
                e
                ij
              
              =
              
                {
                
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              a
                              
                                
                                  i
                                  +
                                  m
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                            +
                            
                              a
                              
                                
                                  i
                                  +
                                  m
                                  +
                                  2
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        0
                        ,
                        
                          
                            2
                            ≤
                            i
                            ≤
                            
                              4
                              -
                              m
                            
                          
                          ;
                        
                      
                    
                  
                  
                    
                      
                         
                      
                    
                    
                      
                        0
                        ≤
                        j
                        ≤
                        
                          7
                          -
                          n
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                
                                  i
                                  +
                                  m
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                            +
                          
                        
                      
                    
                    
                      
                        0
                        ,
                        
                          
                            2
                            ≤
                            i
                            ≤
                            
                              4
                              -
                              m
                            
                          
                          ;
                        
                      
                    
                  
                  
                    
                      
                        
                          b
                          
                            
                              i
                              +
                              m
                              +
                              2
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              8
                            
                          
                        
                        )
                      
                    
                    
                      
                        
                          8
                          -
                          n
                        
                        ≤
                        j
                        ≤
                        7
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              c
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  8
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                            +
                            
                              c
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  6
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          
                            8
                            -
                            m
                          
                          ≤
                          i
                          ≤
                          6
                        
                        ;
                        
                          0
                          ≤
                          j
                          ≤
                          
                            7
                            -
                            n
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  8
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                            +
                          
                        
                      
                    
                    
                      
                        
                          
                            8
                            -
                            m
                          
                          ≤
                          i
                          ≤
                          6
                        
                        ;
                      
                    
                  
                  
                    
                      
                        
                          d
                          
                            
                              i
                              +
                              m
                              -
                              6
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              8
                            
                          
                        
                        )
                      
                    
                    
                      
                        
                          8
                          -
                          n
                        
                        ≤
                        j
                        ≤
                        7
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              a
                              
                                
                                  i
                                  +
                                  m
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                            +
                            
                              c
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  6
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          i
                          =
                          
                            6
                            -
                            m
                          
                        
                        ;
                        
                          0
                          ≤
                          j
                          ≤
                          
                            7
                            -
                            n
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                
                                  i
                                  +
                                  m
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                            +
                            
                              d
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  6
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          8
                          =
                          
                            6
                            -
                            m
                          
                        
                        ;
                        
                          
                            8
                            -
                            n
                          
                          ≤
                          j
                          ≤
                          7
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              a
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  1
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                              ′
                            
                            +
                          
                        
                      
                    
                    
                      
                        1
                        ,
                        
                          
                            3
                            ≤
                            i
                            ≤
                            
                              5
                              -
                              
                                m
                                ′
                              
                            
                          
                          ;
                        
                      
                    
                  
                  
                    
                      
                        
                          a
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              +
                              1
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                            
                          
                          ′
                        
                        )
                      
                    
                    
                      
                        0
                        ≤
                        j
                        ≤
                        
                          7
                          -
                          
                            n
                            ′
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  1
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                  -
                                  8
                                
                              
                              ′
                            
                            +
                          
                        
                      
                    
                    
                      
                        1
                        ,
                        
                          
                            3
                            ≤
                            i
                            ≤
                            
                              5
                              -
                              
                                m
                                ′
                              
                            
                          
                          ;
                        
                      
                    
                  
                  
                    
                      
                        
                          b
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              +
                              1
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                              -
                              8
                            
                          
                          ′
                        
                        )
                      
                    
                    
                      
                        
                          8
                          -
                          
                            n
                            ′
                          
                        
                        ≤
                        j
                        ≤
                        7
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              c
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  9
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                
                              
                              ′
                            
                            +
                            
                              c
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  7
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                
                              
                              ′
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          
                            9
                            -
                            
                              m
                              ′
                            
                          
                          ≤
                          i
                          ≤
                          7
                        
                        ;
                      
                    
                  
                  
                    
                      
                         
                      
                    
                    
                      
                        0
                        ≤
                        j
                        ≤
                        
                          7
                          -
                          
                            n
                            ′
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  9
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                  -
                                  8
                                
                              
                              ′
                            
                            +
                          
                        
                      
                    
                    
                      
                        
                          
                            9
                            -
                            
                              m
                              ′
                            
                          
                          ≤
                          i
                          ≤
                          7
                        
                        ;
                      
                    
                  
                  
                    
                      
                        
                          d
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              7
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                              -
                              8
                            
                          
                          ′
                        
                        )
                      
                    
                    
                      
                        
                          8
                          -
                          
                            n
                            ′
                          
                        
                        ≤
                        j
                        ≤
                        7
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              a
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  1
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                
                              
                              ′
                            
                            +
                            
                              c
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  7
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                
                              
                              ′
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          i
                          =
                          
                            7
                            -
                            
                              m
                              ′
                            
                          
                        
                        ;
                        
                          0
                          ≤
                          j
                          ≤
                          
                            7
                            -
                            
                              n
                              ′
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  1
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                  -
                                  8
                                
                              
                              ′
                            
                            +
                          
                        
                      
                    
                    
                      
                        
                          i
                          =
                          
                            7
                            -
                            
                              m
                              ′
                            
                          
                        
                        ;
                        
                          
                            8
                            -
                            
                              n
                              ′
                            
                          
                          ≤
                          j
                          ≤
                          7
                        
                      
                    
                  
                  
                    
                      
                        
                          d
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              7
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                              -
                              8
                            
                          
                          ′
                        
                        )
                      
                    
                    
                      
                         
                      
                    
                  
                
                }
              
            
          
        
        
          
            
              
                
                  where
                  ⁢
                  
                     
                  
                  ⁢
                  m
                
                =
                0
              
              ,
              2
              ,
              4
              ,
              
                
                  6
                  ⁢
                  
                     
                  
                  ⁢
                  and
                  ⁢
                  
                     
                  
                  ⁢
                  
                    m
                    ′
                  
                
                =
                0
              
              ,
              2
              ,
              4
              ,
              6
            
          
        
      
    
    
      
        (
        78
        )
      
    
  




The top half of expression (77) represents the o→o portion of the frame composite macro block e, while the bottom portion of expression (77) represents the o→e portion of the frame composite macro block e.
Using expression (77), expression (78) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                4
                                -
                                m
                              
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  7
                                  -
                                  n
                                
                              
                              ⁢
                              
                                χ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                n
                              
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  13
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      13
                                      -
                                      
                                        2
                                        ⁢
                                        m
                                      
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      B
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              4
                              -
                              m
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  n
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                n
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                13
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    13
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      C
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  2
                                  -
                                  m
                                
                              
                              ,
                              
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                n
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    13
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      D
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  2
                                  -
                                  m
                                
                              
                              ,
                              
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  n
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    13
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      A
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              4
                              -
                              
                                m
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                13
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      B
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              4
                              -
                              
                                m
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  
                                    n
                                    ′
                                  
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                n
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                13
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      C
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  2
                                  -
                                  
                                    m
                                    ′
                                  
                                
                              
                              ,
                              
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    D
                    pq
                    ′
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∑
                          
                            
                              i
                              =
                              
                                2
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ,
                            
                              -
                              
                                m
                                ′
                              
                            
                          
                          
                            -
                            2
                          
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                1
                              
                            
                            
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              χ
                              ′
                            
                            ⁢
                            
                               
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      +
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            7
                            -
                            
                              n
                              ′
                            
                          
                        
                        ⁢
                        
                          
                            1
                            2
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  17
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
      
    
    
      
        (
        78
        )
      
    
  




where 



  
    
      
        χ
        =
        
          cos
          ⁢
          
             
          
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    i
                  
                  +
                  
                    2
                    ⁢
                    m
                  
                  +
                  3
                
                )
              
              ⁢
              p
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              p
              ⁢
              
                 
              
              ⁢
              π
            
            8
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    j
                  
                  +
                  
                    2
                    ⁢
                    n
                  
                  +
                  1
                
                )
              
              ⁢
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
        
      
    
    
      
        (
        
          79
          ⁢
          A
        
        )
      
    
  
  
    
      
        
          χ
          ′
        
        =
        
          cos
          ⁢
          
             
          
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    i
                  
                  +
                  
                    2
                    ⁢
                    
                      m
                      ′
                    
                  
                  +
                  3
                
                )
              
              ⁢
              p
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              p
              ⁢
              
                 
              
              ⁢
              π
            
            8
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    j
                  
                  +
                  
                    2
                    ⁢
                    
                      n
                      ′
                    
                  
                  +
                  1
                
                )
              
              ⁢
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
        
      
    
    
      
        (
        
          79
          ⁢
          B
        
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (78) represent the o→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (78) represent the o→e portion of the frame composite macro block E.
Using expression (78), the motion compensated frame macro block E can be derived for vertical half-pel accuracy when the even and odd fields are encoded based on the odd field of the anchor frame.
Case (2) e→o and e→e
Expression (80) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the even and odd fields are encoded based on the even field of the anchor frame. 



  
    
      
        
          
            
              
                e
                ij
              
              =
              
                {
                
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              a
                              
                                
                                  i
                                  +
                                  m
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                            +
                            
                              a
                              
                                
                                  i
                                  +
                                  m
                                  +
                                  2
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        0
                        ,
                        
                          
                            2
                            ≤
                            i
                            ≤
                            
                              5
                              -
                              m
                            
                          
                          ;
                        
                      
                    
                  
                  
                    
                      
                         
                      
                    
                    
                      
                        0
                        ≤
                        j
                        ≤
                        
                          7
                          -
                          n
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                
                                  i
                                  +
                                  m
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                            +
                            
                              b
                              
                                
                                  i
                                  +
                                  m
                                  +
                                  2
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        0
                        ,
                        
                          
                            2
                            ≤
                            i
                            ≤
                            
                              5
                              -
                              m
                            
                          
                          ;
                        
                      
                    
                  
                  
                    
                      
                         
                      
                    
                    
                      
                        
                          8
                          -
                          n
                        
                        ≤
                        j
                        ≤
                        7
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              c
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  8
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                            +
                            
                              c
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  6
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          
                            9
                            -
                            m
                          
                          ≤
                          i
                          ≤
                          6
                        
                        ;
                        
                          0
                          ≤
                          j
                          ≤
                          
                            7
                            -
                            n
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  8
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                            +
                            
                              d
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  6
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          
                            9
                            -
                            m
                          
                          ≤
                          i
                          ≤
                          6
                        
                        ;
                        
                          
                            8
                            -
                            n
                          
                          ≤
                          j
                          ≤
                          7
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              a
                              
                                
                                  i
                                  +
                                  m
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                            +
                            
                              c
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  6
                                
                                ,
                                
                                  j
                                  +
                                  n
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          i
                          =
                          
                            7
                            -
                            m
                          
                        
                        ;
                        
                          0
                          ≤
                          j
                          ≤
                          
                            7
                            -
                            n
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                
                                  i
                                  +
                                  m
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                            +
                            
                              d
                              
                                
                                  i
                                  +
                                  m
                                  -
                                  6
                                
                                ,
                                
                                  j
                                  +
                                  n
                                  -
                                  8
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          i
                          =
                          
                            7
                            -
                            m
                          
                        
                        ;
                        
                          
                            8
                            -
                            n
                          
                          ≤
                          j
                          ≤
                          7
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              a
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  1
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                
                              
                              ′
                            
                            +
                          
                        
                      
                    
                    
                      
                        1
                        ,
                        
                          
                            3
                            ≤
                            i
                            ≤
                            
                              6
                              -
                              
                                m
                                ′
                              
                            
                          
                          ;
                        
                      
                    
                  
                  
                    
                      
                        
                          a
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              +
                              1
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                            
                          
                          ′
                        
                        )
                      
                    
                    
                      
                        0
                        ≤
                        j
                        ≤
                        
                          7
                          -
                          
                            n
                            ′
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  1
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                  -
                                  8
                                
                              
                              ′
                            
                            +
                          
                        
                      
                    
                    
                      
                        1
                        ,
                        
                          
                            3
                            ≤
                            i
                            ≤
                            
                              6
                              -
                              
                                m
                                ′
                              
                            
                          
                          ;
                        
                      
                    
                  
                  
                    
                      
                        
                          b
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              +
                              1
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                              -
                              8
                            
                          
                          ′
                        
                        )
                      
                    
                    
                      
                        
                          8
                          -
                          
                            n
                            ′
                          
                        
                        ≤
                        j
                        ≤
                        7
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              c
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  9
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                
                              
                              ′
                            
                            +
                          
                        
                      
                    
                    
                      
                        
                          
                            10
                            -
                            
                              m
                              ′
                            
                          
                          ≤
                          i
                          ≤
                          7
                        
                        ;
                      
                    
                  
                  
                    
                      
                        
                          c
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              7
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                            
                          
                          ′
                        
                        )
                      
                    
                    
                      
                        0
                        ≤
                        j
                        ≤
                        
                          7
                          -
                          
                            n
                            ′
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              d
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  9
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                  -
                                  8
                                
                              
                              ′
                            
                            +
                          
                        
                      
                    
                    
                      
                        
                          
                            10
                            -
                            
                              m
                              ′
                            
                          
                          ≤
                          i
                          ≤
                          7
                        
                        ;
                      
                    
                  
                  
                    
                      
                        
                          d
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              7
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                              -
                              8
                            
                          
                          ′
                        
                        )
                      
                    
                    
                      
                        
                          8
                          -
                          
                            n
                            ′
                          
                        
                        ≤
                        j
                        ≤
                        7
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              a
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  1
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                
                              
                              ′
                            
                            +
                            
                              c
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  7
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                
                              
                              ′
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          i
                          =
                          
                            8
                            -
                            
                              m
                              ′
                            
                          
                        
                        ;
                        
                          0
                          ≤
                          j
                          ≤
                          
                            7
                            -
                            
                              n
                              ′
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          (
                          
                            
                              b
                              
                                
                                  i
                                  +
                                  
                                    m
                                    ′
                                  
                                  -
                                  1
                                
                                ,
                                
                                  j
                                  +
                                  
                                    n
                                    ′
                                  
                                  -
                                  8
                                
                              
                              ′
                            
                            +
                          
                        
                      
                    
                    
                      
                        
                          i
                          =
                          
                            8
                            -
                            
                              m
                              ′
                            
                          
                        
                        ;
                      
                    
                  
                  
                    
                      
                        
                          d
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              7
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                              -
                              8
                            
                          
                          ′
                        
                        )
                      
                    
                    
                      
                        
                          8
                          -
                          
                            n
                            ′
                          
                        
                        ≤
                        j
                        ≤
                        7
                      
                    
                  
                
                }
              
            
          
        
        
          
            
              
                
                  where
                  ⁢
                  
                     
                  
                  ⁢
                  m
                
                =
                1
              
              ,
              3
              ,
              5
              ,
              
                
                  7
                  ⁢
                  
                     
                  
                  ⁢
                  and
                  ⁢
                  
                     
                  
                  ⁢
                  
                    m
                    ′
                  
                
                =
                1
              
              ,
              3
              ,
              5
              ,
              7
            
          
        
      
    
    
      
        (
        80
        )
      
    
  




The top half of expression (80) represents the e→o portion of the frame composite macro block e, while the bottom portion of expression (80) represents the e→e portion of the frame composite macro block e.
Using expression (80), expression (81) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                5
                                -
                                m
                              
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  7
                                  -
                                  n
                                
                              
                              ⁢
                              
                                χ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                n
                              
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        m
                                      
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      B
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              5
                              -
                              m
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  n
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                n
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                15
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      C
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  3
                                  -
                                  m
                                
                              
                              ,
                              
                                1
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                n
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                3
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      D
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  3
                                  -
                                  m
                                
                              
                              ,
                              
                                1
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  n
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                3
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      A
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              5
                              -
                              
                                m
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                15
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      B
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              5
                              -
                              
                                m
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  n
                                  ′
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                15
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      C
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  3
                                  -
                                  
                                    m
                                    ′
                                  
                                
                              
                              ,
                              
                                1
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                3
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    D
                    pq
                    ′
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∑
                          
                            
                              i
                              =
                              
                                3
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ,
                            
                              1
                              -
                              
                                m
                                ′
                              
                            
                          
                          
                            -
                            2
                          
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              χ
                              ′
                            
                            ⁢
                            
                               
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      +
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            7
                            -
                            
                              n
                              ′
                            
                          
                        
                        ⁢
                        
                          
                            1
                            2
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              3
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  17
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  17
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
      
    
    
      
        (
        81
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (81) represent the e→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (81) represent the e→e portion of the frame composite macro block E.
Using expression (81), the motion compensated frame macro block E can be derived for vertical half-pel accuracy when the even and odd fields are encoded based on the even field of the anchor frame.
Case (3) o→o and e→e
Expression (82) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the odd and even fields are encoded based on the odd and even fields, respectively, of the anchor frame. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        4
                        -
                        m
                      
                    
                    ;
                  
                
              
            
            
              
                
                   
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    n
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        4
                        -
                        m
                      
                    
                    ;
                  
                
              
            
            
              
                
                   
                
              
              
                
                  
                    8
                    -
                    n
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    8
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    8
                    =
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
            
            
              
                
                   
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
            
            
              
                
                  
                    b
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        +
                        1
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        8
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    c
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        7
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                      
                    
                    ′
                  
                  )
                
              
              
                
                  0
                  ≤
                  j
                  ≤
                  
                    7
                    -
                    
                      n
                      ′
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                
              
            
            
              
                
                  
                    d
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        7
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        8
                      
                    
                    ′
                  
                  )
                
              
              
                
                  
                    8
                    -
                    
                      n
                      ′
                    
                  
                  ≤
                  j
                  ≤
                  7
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                    
                  
                
              
              
                
                  
                    i
                    =
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    d
                    
                      
                        i
                        +
                        
                          m
                          ′
                        
                        -
                        7
                      
                      ,
                      
                        j
                        +
                        
                          n
                          ′
                        
                        -
                        8
                      
                    
                    ′
                  
                  )
                
              
              
                
                   
                
              
            
          
          }
        
      
    
    
      
        (
        82
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        
          
            6
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        7
      
    
    
      
         
      
    
  




The top half of expression (82) represents the o→o portion of the frame composite macro block e, while the bottom portion of expression (82) represents the e→e portion of the frame composite macro block e.
Using expression (82), expression (83) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                4
                                -
                                m
                              
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  7
                                  -
                                  n
                                
                              
                              ⁢
                              
                                χ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                n
                              
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  13
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      13
                                      -
                                      
                                        2
                                        ⁢
                                        m
                                      
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      B
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              4
                              -
                              m
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  n
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                n
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                13
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    13
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      C
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  2
                                  -
                                  m
                                
                              
                              ,
                              
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                n
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    13
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      D
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  2
                                  -
                                  m
                                
                              
                              ,
                              
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  n
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    13
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      A
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              5
                              -
                              
                                m
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                15
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      B
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              5
                              -
                              
                                m
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  
                                    n
                                    ′
                                  
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                15
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      C
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  3
                                  -
                                  
                                    m
                                    ′
                                  
                                
                              
                              ,
                              
                                1
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                3
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    D
                    pq
                    ′
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∑
                          
                            
                              i
                              =
                              
                                3
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ,
                            
                              1
                              -
                              m
                            
                          
                          
                            -
                            2
                          
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              χ
                              ′
                            
                            ⁢
                            
                               
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      +
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            7
                            -
                            
                              n
                              ′
                            
                          
                        
                        ⁢
                        
                          
                            1
                            2
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              3
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  17
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
      
    
    
      
        (
        83
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (83) represent the o→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (83) represent the e→e portion of the frame composite macro block E.
Using expression (83), the motion compensated frame macro block E can be derived for vertical half-pel accuracy when the odd and even fields are encoded based on the odd and even fields, respectively, of the anchor frame.
Case (4) e→o and o→e
Expression (84) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the odd and even fields are encoded based on the even and odd fields, respectively, of the anchor frame. 



  
    
      
        
          
            e
            ij
          
          =
          
            {
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          a
                          
                            
                              i
                              +
                              m
                            
                            ,
                            
                              j
                              +
                              n
                            
                          
                        
                        +
                        
                          a
                          
                            
                              i
                              +
                              m
                              +
                              2
                            
                            ,
                            
                              j
                              +
                              n
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    0
                    ,
                    
                      
                        2
                        ≤
                        i
                        ≤
                        
                          5
                          -
                          m
                        
                      
                      ;
                    
                  
                
              
              
                
                  
                     
                  
                
                
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      n
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          b
                          
                            
                              i
                              +
                              m
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              8
                            
                          
                        
                        +
                        
                          b
                          
                            
                              i
                              +
                              m
                              +
                              2
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              8
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    0
                    ,
                    
                      
                        2
                        ≤
                        i
                        ≤
                        
                          5
                          -
                          m
                        
                      
                      ;
                    
                  
                
              
              
                
                  
                     
                  
                
                
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          c
                          
                            
                              i
                              +
                              m
                              -
                              8
                            
                            ,
                            
                              j
                              +
                              n
                            
                          
                        
                        +
                        
                          c
                          
                            
                              i
                              +
                              m
                              -
                              6
                            
                            ,
                            
                              j
                              +
                              n
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      
                        9
                        -
                        m
                      
                      ≤
                      i
                      ≤
                      6
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        7
                        -
                        n
                      
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          d
                          
                            
                              i
                              +
                              m
                              -
                              8
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              8
                            
                          
                        
                        +
                        
                          d
                          
                            
                              i
                              +
                              m
                              -
                              6
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              8
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      
                        9
                        -
                        m
                      
                      ≤
                      i
                      ≤
                      6
                    
                    ;
                    
                      
                        8
                        -
                        n
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          a
                          
                            
                              i
                              +
                              m
                            
                            ,
                            
                              j
                              +
                              n
                            
                          
                        
                        +
                        
                          c
                          
                            
                              i
                              +
                              m
                              -
                              6
                            
                            ,
                            
                              j
                              +
                              n
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      i
                      =
                      
                        7
                        -
                        m
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        7
                        -
                        n
                      
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          b
                          
                            
                              i
                              +
                              m
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              8
                            
                          
                        
                        +
                        
                          d
                          
                            
                              i
                              +
                              m
                              -
                              6
                            
                            ,
                            
                              j
                              +
                              n
                              -
                              8
                            
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      i
                      =
                      
                        7
                        -
                        m
                      
                    
                    ;
                    
                      
                        8
                        -
                        n
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          a
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              1
                            
                            ,
                            
                              j
                              +
                              n
                            
                          
                          ′
                        
                        +
                      
                    
                  
                
                
                  
                    1
                    ,
                    
                      
                        3
                        ≤
                        i
                        ≤
                        
                          5
                          -
                          
                            m
                            ′
                          
                        
                      
                      ;
                    
                  
                
              
              
                
                  
                    
                      a
                      
                        
                          i
                          +
                          
                            m
                            ′
                          
                          +
                          1
                        
                        ,
                        
                          j
                          +
                          
                            n
                            ′
                          
                        
                      
                      ′
                    
                    )
                  
                
                
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          b
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              1
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                              -
                              8
                            
                          
                          ′
                        
                        +
                      
                    
                  
                
                
                  
                    1
                    ,
                    
                      
                        3
                        ≤
                        i
                        ≤
                        
                          5
                          -
                          
                            m
                            ′
                          
                        
                      
                      ;
                    
                  
                
              
              
                
                  
                    
                      b
                      
                        
                          i
                          +
                          
                            m
                            ′
                          
                          +
                          1
                        
                        ,
                        
                          j
                          +
                          
                            n
                            ′
                          
                          -
                          8
                        
                      
                      ′
                    
                    )
                  
                
                
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          c
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              9
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                            
                          
                          ′
                        
                        +
                      
                    
                  
                
                
                  
                    
                      
                        9
                        -
                        
                          m
                          ′
                        
                      
                      ≤
                      i
                      ≤
                      7
                    
                    ;
                  
                
              
              
                
                  
                    
                      c
                      
                        
                          i
                          +
                          
                            m
                            ′
                          
                          -
                          7
                        
                        ,
                        
                          j
                          +
                          
                            n
                            ′
                          
                        
                      
                      ′
                    
                    )
                  
                
                
                  
                    0
                    ≤
                    j
                    ≤
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          d
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              9
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                              -
                              8
                            
                          
                          ′
                        
                        +
                      
                    
                  
                
                
                  
                    
                      
                        9
                        -
                        
                          m
                          ′
                        
                      
                      ≤
                      i
                      ≤
                      7
                    
                    ;
                  
                
              
              
                
                  
                    
                      d
                      
                        
                          i
                          +
                          
                            m
                            ′
                          
                          -
                          7
                        
                        ,
                        
                          j
                          +
                          
                            n
                            ′
                          
                          -
                          8
                        
                      
                      ′
                    
                    )
                  
                
                
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          a
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              1
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                            
                          
                          ′
                        
                        +
                        
                          c
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              7
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                            
                          
                          ′
                        
                      
                      )
                    
                  
                
                
                  
                    
                      i
                      =
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      (
                      
                        
                          b
                          
                            
                              i
                              +
                              
                                m
                                ′
                              
                              -
                              1
                            
                            ,
                            
                              j
                              +
                              
                                n
                                ′
                              
                              -
                              8
                            
                          
                          ′
                        
                        +
                      
                    
                  
                
                
                  
                    
                      i
                      =
                      
                        7
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                  
                
              
              
                
                  
                    
                      d
                      
                        
                          i
                          +
                          
                            m
                            ′
                          
                          -
                          7
                        
                        ,
                        
                          j
                          +
                          
                            n
                            ′
                          
                          -
                          8
                        
                      
                      ′
                    
                    )
                  
                
                
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            }
          
        
        ⁢
        
           
        
      
    
    
      
        (
        84
        )
      
    
  
  
    
      
        
           
        
        ⁢
        
          
            
              where
              ⁢
              
                 
              
              ⁢
              m
            
            =
            1
          
          ,
          3
          ,
          5
          ,
          
            
              7
              ⁢
              
                 
              
              ⁢
              and
              ⁢
              
                 
              
              ⁢
              
                m
                ′
              
            
            =
            0
          
          ,
          2
          ,
          4
          ,
          6
        
      
    
    
      
         
      
    
  




The top half of expression (84) represents the e→o portion of the frame composite macro block e, while the bottom portion of expression (84) represents the o→e portion of the frame composite macro block e.
Using expression (84), expression (85) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                kl
              
              =
              
                
                  
                    ∑
                    
                      p
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      ∑
                      
                        q
                        =
                        0
                      
                      7
                    
                    ⁢
                    
                      
                        A
                        pq
                      
                      ⁡
                      
                        [
                        
                          
                            
                              ∑
                              
                                
                                  i
                                  =
                                  0
                                
                                ,
                                2
                              
                              
                                5
                                -
                                m
                              
                            
                            ⁢
                            
                              
                                ∑
                                
                                  j
                                  =
                                  0
                                
                                
                                  7
                                  -
                                  n
                                
                              
                              ⁢
                              
                                χ
                                ⁢
                                
                                   
                                
                                ⁢
                                cos
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          i
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    k
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                                ⁢
                                cos
                                ⁢
                                
                                   
                                
                                ⁢
                                
                                  
                                    
                                      (
                                      
                                        
                                          2
                                          ⁢
                                          j
                                        
                                        +
                                        1
                                      
                                      )
                                    
                                    ⁢
                                    l
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    π
                                  
                                  16
                                
                              
                            
                          
                          +
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                n
                              
                            
                            ⁢
                            
                              
                                1
                                2
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      
                                        2
                                        ⁢
                                        n
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        m
                                      
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      B
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              5
                              -
                              m
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  n
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                n
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                15
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      C
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  3
                                  -
                                  m
                                
                              
                              ,
                              
                                1
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                n
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                3
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      D
                      pq
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  3
                                  -
                                  m
                                
                              
                              ,
                              
                                1
                                -
                                m
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  -
                                  n
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              χ
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              n
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                3
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      A
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              4
                              -
                              
                                m
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                13
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      B
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                0
                              
                              ,
                              2
                            
                            
                              4
                              -
                              
                                m
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                
                                  n
                                  ′
                                
                              
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      3
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      17
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                13
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      C
                      pq
                      ′
                    
                    ⁡
                    
                      [
                      
                        
                          
                            ∑
                            
                              
                                i
                                =
                                
                                  3
                                  -
                                  
                                    m
                                    ′
                                  
                                
                              
                              ,
                              
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            
                              -
                              2
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                j
                                =
                                0
                              
                              
                                7
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            ⁢
                            
                              
                                χ
                                ′
                              
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      19
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                          
                        
                        +
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              7
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              1
                              2
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        n
                                        ′
                                      
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      ]
                    
                  
                
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    D
                    pq
                    ′
                  
                  ⁡
                  
                    [
                    
                      
                        
                          ∑
                          
                            
                              i
                              =
                              
                                3
                                -
                                
                                  m
                                  ′
                                
                              
                            
                            ,
                            
                              1
                              -
                              m
                            
                          
                          
                            -
                            2
                          
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              
                                -
                                
                                  n
                                  ′
                                
                              
                            
                            
                              -
                              
                                n
                                ′
                              
                            
                          
                          ⁢
                          
                            
                              χ
                              ′
                            
                            ⁢
                            
                               
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                        
                      
                      +
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            7
                            -
                            
                              n
                              ′
                            
                          
                        
                        ⁢
                        
                          
                            1
                            2
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  17
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
      
    
    
      
        (
        85
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq expression (85) represent the e→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (85) represent the o→e portion of the frame composite macro block E.
Using expression (85), the motion compensated frame macro block E can be derived for vertical half-pel accuracy when the odd and even fields are encoded based on the even and odd fields, respectively, of the anchor frame.
Diagonal Half-Pel Accuracy
Case (1) o→o and o→e
Expression (86) below represents the relationship between the frame composite macro block e and the macro blocks a-d and a′-d′ in the anchor frame in the spatial domain when the even and odd fields are encoded based on the odd field of the anchor frame. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        4
                        -
                        m
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        6
                        -
                        n
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        4
                        -
                        m
                      
                    
                    ;
                    
                      
                        8
                        -
                        n
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          7
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                  ;
                  
                    0
                    ≤
                    i
                    ≤
                    
                      4
                      -
                      m
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          7
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                  ;
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          6
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        6
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      
                        8
                        -
                        
                          n
                          ′
                        
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          6
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        c
                        
                          0
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          0
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
          
          }
        
      
    
    
      
        (
        86
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        
          
            6
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        6
      
    
    
      
         
      
    
  




The top half of expression (86) represents the o→o portion of the frame composite macro block e, while the bottom portion of expression (86) represents the o→e portion of the frame composite macro block e.
Using expression (86), expression (87) below is derived in the same manner as expression (47). 



  
    
      
        
          E
          kl
        
        =
        
          
            ∑
            
              p
              =
              0
            
            7
          
          ⁢
          
            
              ∑
              
                q
                =
                0
              
              7
            
            ⁢
            
              
                A
                pg
              
              [
              
                
                  
                    ∑
                    
                      
                        i
                        =
                        0
                      
                      ,
                      2
                    
                    
                      4
                      -
                      m
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        6
                        -
                        n
                      
                    
                    ⁢
                    
                      (
                      
                        W
                        ⁢
                        
                           
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                +
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      
                        6
                        -
                        n
                      
                    
                    ⁢
                    
                      (
                      
                        cos
                        ⁢
                        
                          
                            13
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                j
                                +
                                n
                                +
                                1
                              
                              )
                            
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          8
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                13
                                -
                                
                                  2
                                  ⁢
                                  m
                                
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  j
                                
                                +
                                1
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
      
    
    
      
        (
        87
        )
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                
                  ∑
                  
                    
                      i
                      =
                      0
                    
                    ,
                    2
                  
                  
                    4
                    -
                    m
                  
                
                ⁢
                
                  (
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            
                              2
                              ⁢
                              m
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      8
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        15
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            15
                            -
                            
                              2
                              ⁢
                              n
                            
                          
                          )
                        
                        ⁢
                        l
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
              
            
            +
            
              (
              
                
                  1
                  4
                
                ⁢
                cos
                ⁢
                
                  
                    13
                    ⁢
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    15
                    ⁢
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        13
                        -
                        
                          2
                          ⁢
                          m
                        
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
              )
            
          
          ]
        
        +
      
    
    
      
         
      
    
  
  
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
            
              B
              pg
            
            [
            
              
                
                  ∑
                  
                    
                      i
                      =
                      0
                    
                    ,
                    2
                  
                  
                    4
                    -
                    m
                  
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      
                        -
                        n
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    (
                    
                      W
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      
                        -
                        n
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                        
                          13
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              n
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              13
                              -
                              
                                2
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
            
          
        
      
    
    
      
         
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                
                  ∑
                  
                    
                      i
                      =
                      0
                    
                    ,
                    2
                  
                  
                    4
                    -
                    m
                  
                
                ⁢
                
                  (
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            
                              2
                              ⁢
                              m
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      8
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            1
                          
                          )
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            15
                            -
                            
                              2
                              ⁢
                              n
                            
                          
                          )
                        
                        ⁢
                        l
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
              
            
            +
            
              
                1
                4
              
              ⁢
              
                (
                
                  cos
                  ⁢
                  
                    
                      13
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          13
                          -
                          
                            2
                            ⁢
                            m
                          
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            n
                          
                        
                        )
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
                )
              
            
          
          ]
        
        +
      
    
    
      
         
      
    
  
  
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
            
              C
              pg
            
            [
            
              
                
                  ∑
                  
                    i
                    =
                    
                      -
                      m
                    
                  
                  
                    -
                    2
                  
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      0
                    
                    
                      6
                      -
                      n
                    
                  
                  ⁢
                  
                    (
                    
                      W
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      0
                    
                    
                      6
                      -
                      n
                    
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              n
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              13
                              -
                              
                                2
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
            
          
        
      
    
    
      
         
      
    
  
  
    
      
        
          
            
              
                
                  1
                  2
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      
                        -
                        m
                      
                    
                    
                      -
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              
                                2
                                ⁢
                                m
                              
                              +
                              3
                            
                            )
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          15
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
              
                
                  1
                  4
                
                ⁢
                cos
                ⁢
                
                  
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    15
                    ⁢
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        13
                        -
                        
                          2
                          ⁢
                          m
                        
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
            
            )
          
          ]
        
        +
      
    
    
      
         
      
    
  
  
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
            
              D
              pg
            
            [
            
              
                
                  ∑
                  
                    i
                    =
                    
                      -
                      m
                    
                  
                  
                    -
                    2
                  
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      
                        -
                        n
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    (
                    
                      W
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      
                        -
                        n
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              n
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              13
                              -
                              
                                2
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
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                          ⁢
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                        16
                      
                      ⁢
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                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
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                              +
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                            )
                          
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                          ⁢
                          
                             
                          
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                        16
                      
                    
                    )
                  
                
              
              +
            
          
        
      
    
    
      
         
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                
                  ∑
                  
                    i
                    =
                    
                      -
                      m
                    
                  
                  
                    -
                    2
                  
                
                ⁢
                
                  (
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
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                            +
                            
                              2
                              ⁢
                              m
                            
                            +
                            3
                          
                          )
                        
                        ⁢
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                        ⁢
                        
                           
                        
                        ⁢
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                      16
                    
                    ⁢
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                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      8
                    
                    ⁢
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                    ⁢
                    
                      
                        q
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                        ⁢
                        π
                      
                      16
                    
                    ⁢
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                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            17
                          
                          )
                        
                        ⁢
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                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            15
                            -
                            
                              2
                              ⁢
                              n
                            
                          
                          )
                        
                        ⁢
                        l
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
              
            
            +
            
              (
              
                
                  1
                  4
                
                ⁢
                cos
                ⁢
                
                  
                    p
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    q
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        13
                        -
                        
                          2
                          ⁢
                          m
                        
                      
                      )
                    
                    ⁢
                    k
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
                ⁢
                cos
                ⁢
                
                  
                    
                      (
                      
                        15
                        -
                        
                          2
                          ⁢
                          n
                        
                      
                      )
                    
                    ⁢
                    l
                    ⁢
                    
                       
                    
                    ⁢
                    π
                  
                  16
                
              
              )
            
          
          ]
        
        +
      
    
    
      
         
      
    
  
  
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
            
              A
              pg
              ′
            
            [
            
              
                
                  ∑
                  
                    i
                    =
                    0
                  
                  
                    4
                    -
                    
                      m
                      ′
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      0
                    
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                  ⁢
                  
                    (
                    
                      
                        W
                        ′
                      
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              3
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
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                            )
                          
                          ⁢
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                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      0
                    
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                        
                          13
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              
                                n
                                ′
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                  m
                                  ′
                                
                              
                            
                            )
                          
                          ⁢
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                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
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                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
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                              +
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                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
            
          
        
      
    
    
      
         
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                
                  ∑
                  
                    
                      j
                      =
                      0
                    
                    ,
                    2
                  
                  
                    4
                    -
                    
                      m
                      ′
                    
                  
                
                ⁢
                
                  (
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            
                              2
                              ⁢
                              
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                                ′
                              
                            
                            +
                            3
                          
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                        ⁢
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                        ⁢
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                      16
                    
                    ⁢
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                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        p
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                        ⁢
                        π
                      
                      8
                    
                    ⁢
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                    ⁢
                    
                      
                        15
                        ⁢
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                        ⁢
                        
                           
                        
                        ⁢
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                      16
                    
                    ⁢
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                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
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                          )
                        
                        ⁢
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                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
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                    ⁢
                    
                      
                        
                          (
                          
                            15
                            -
                            
                              2
                              ⁢
                              
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                                ′
                              
                            
                          
                          )
                        
                        ⁢
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                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
              
            
            +
            
              
                1
                4
              
              ⁢
              
                (
                
                  cos
                  ⁢
                  
                    
                      13
                      ⁢
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                      ⁢
                      
                         
                      
                      ⁢
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                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      15
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                        
                        )
                      
                      ⁢
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                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
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                  ⁢
                  
                    
                      
                        (
                        
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                          -
                          
                            2
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                              ′
                            
                          
                        
                        )
                      
                      ⁢
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                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
                )
              
            
          
          ]
        
        +
      
    
    
      
         
      
    
  
  
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
            
              B
              pg
              ′
            
            [
            
              
                
                  ∑
                  
                    
                      i
                      =
                      0
                    
                    ,
                    2
                  
                  
                    4
                    -
                    
                      m
                      ′
                    
                  
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      
                        -
                        
                          n
                          ′
                        
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    (
                    
                      
                        W
                        ′
                      
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              3
                            
                            )
                          
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                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
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                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
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                              +
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                            )
                          
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                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      
                        -
                        
                          n
                          ′
                        
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                        
                          13
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                        16
                      
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                          q
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                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
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                                n
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                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                  m
                                  ′
                                
                              
                            
                            )
                          
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                        16
                      
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                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
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                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
            
          
        
      
    
    
      
         
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                
                  ∑
                  
                    
                      i
                      =
                      0
                    
                    ,
                    2
                  
                  
                    4
                    -
                    
                      m
                      ′
                    
                  
                
                ⁢
                
                  (
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            
                              2
                              ⁢
                              
                                m
                                ′
                              
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      8
                    
                    ⁢
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                    ⁢
                    
                      
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              j
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            15
                            -
                            
                              2
                              ⁢
                              
                                n
                                ′
                              
                            
                          
                          )
                        
                        ⁢
                        l
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
              
            
            +
            
              
                1
                4
              
              ⁢
              
                (
                
                  cos
                  ⁢
                  
                    
                      13
                      ⁢
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            
                              n
                              ′
                            
                          
                        
                        )
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
                )
              
            
          
          ]
        
        +
      
    
    
      
         
      
    
  
  
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
            
              C
              pg
              ′
            
            [
            
              
                
                  ∑
                  
                    i
                    =
                    
                      -
                      
                        m
                        ′
                      
                    
                  
                  
                    -
                    2
                  
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      0
                    
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                  ⁢
                  
                    (
                    
                      
                        W
                        ′
                      
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              19
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      0
                    
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              
                                n
                                ′
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                  m
                                  ′
                                
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
            
          
        
      
    
    
      
         
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                
                  ∑
                  
                    i
                    =
                    
                      -
                      
                        m
                        ′
                      
                    
                  
                  
                    -
                    2
                  
                
                ⁢
                
                  (
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            
                              2
                              ⁢
                              
                                m
                                ′
                              
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      8
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        15
                        ⁢
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            19
                          
                          )
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            15
                            -
                            
                              2
                              ⁢
                              
                                n
                                ′
                              
                            
                          
                          )
                        
                        ⁢
                        l
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
              
            
            +
            
              
                1
                4
              
              ⁢
              
                (
                
                  cos
                  ⁢
                  
                    
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      15
                      ⁢
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            
                              n
                              ′
                            
                          
                        
                        )
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
                )
              
            
          
          ]
        
        +
      
    
    
      
         
      
    
  
  
    
      
        
          ∑
          
            p
            =
            0
          
          7
        
        ⁢
        
          
            ∑
            
              q
              =
              0
            
            7
          
          ⁢
          
            
              D
              pg
              ′
            
            [
            
              
                
                  ∑
                  
                    i
                    =
                    
                      -
                      
                        m
                        ′
                      
                    
                  
                  
                    -
                    2
                  
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      
                        -
                        
                          n
                          ′
                        
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    (
                    
                      
                        W
                        ′
                      
                      ⁢
                      
                         
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                i
                              
                              +
                              19
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  
                    ∑
                    
                      j
                      =
                      
                        -
                        
                          n
                          ′
                        
                      
                    
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              j
                              +
                              
                                n
                                ′
                              
                              +
                              1
                            
                            )
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        8
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                  m
                                  ′
                                
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              
                                2
                                ⁢
                                j
                              
                              +
                              17
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              +
            
          
        
      
    
    
      
         
      
    
  
  
    
      
        
          
            
              
                1
                2
              
              ⁢
              
                
                  ∑
                  
                    i
                    =
                    
                      -
                      
                        m
                        ′
                      
                    
                  
                  
                    -
                    2
                  
                
                ⁢
                
                  (
                  
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            
                              2
                              ⁢
                              
                                m
                                ′
                              
                            
                            +
                            3
                          
                          )
                        
                        ⁢
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                       
                    
                    ⁢
                    
                      
                        p
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      8
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        q
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            
                              2
                              ⁢
                              i
                            
                            +
                            19
                          
                          )
                        
                        ⁢
                        k
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                        
                          (
                          
                            15
                            -
                            
                              2
                              ⁢
                              
                                n
                                ′
                              
                            
                          
                          )
                        
                        ⁢
                        l
                        ⁢
                        
                           
                        
                        ⁢
                        π
                      
                      16
                    
                  
                  )
                
              
            
            +
            
              
                1
                4
              
              ⁢
              
                (
                
                  cos
                  ⁢
                  
                    
                      p
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      q
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            
                              m
                              ′
                            
                          
                        
                        )
                      
                      ⁢
                      k
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                      
                        (
                        
                          15
                          -
                          
                            2
                            ⁢
                            
                              n
                              ′
                            
                          
                        
                        )
                      
                      ⁢
                      l
                      ⁢
                      
                         
                      
                      ⁢
                      π
                    
                    16
                  
                
                )
              
            
          
          ]
        
        ⁢
        
           
        
        ⁢
        where
      
    
    
      
         
      
    
  
  
    
      
        W
        =
        
          cos
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    i
                  
                  +
                  
                    2
                    ⁢
                    m
                  
                  +
                  3
                
                )
              
              ⁢
              p
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              p
              ⁢
              
                 
              
              ⁢
              π
            
            8
          
          ⁢
          cos
          ⁢
          
            
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
            
              
                (
                
                  i
                  +
                  n
                  +
                  1
                
                )
              
              ⁢
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
        
      
    
    
      
        (
        
          88
          ⁢
          
            a
          
        
        )
      
    
  
  
    
      
        
          W
          ′
        
        =
        
          cos
          ⁢
          
            
              
                (
                
                  
                    2
                    ⁢
                    i
                  
                  +
                  
                    2
                    ⁢
                    
                      m
                      ′
                    
                  
                  +
                  3
                
                )
              
              ⁢
              p
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
             
          
          ⁢
          
            
              p
              ⁢
              
                 
              
              ⁢
              π
            
            8
          
          ⁢
          cos
          ⁢
          
            
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
          ⁢
          cos
          ⁢
          
            
              
                (
                
                  i
                  +
                  
                    n
                    ′
                  
                  +
                  1
                
                )
              
              ⁢
              q
              ⁢
              
                 
              
              ⁢
              π
            
            16
          
        
      
    
    
      
        (88b)
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (87) represent the o→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (87) represent the o→e portion of the frame composite macro block E.
Using expression (87), the motion compensated frame macro block E can be derived for diagonal half-pel accuracy when the even and odd fields are encoded based on the odd field of the anchor frame.
Case (2) e→o e→e
Expression (89) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the even and odd fields are encoded based on the even field of the anchor frame. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        m
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        6
                        -
                        n
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        m
                      
                    
                    ;
                    
                      
                        8
                        -
                        n
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          7
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      5
                      -
                      m
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          7
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      n
                    
                  
                  ;
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          6
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        6
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            -
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      
                        8
                        -
                        
                          n
                          ′
                        
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          7
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          7
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          1
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          1
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          7
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          7
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          1
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          1
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          7
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          7
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        c
                        
                          1
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          1
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
          
          ⁢
          
             
          
          }
        
      
    
    
      
        (
        89
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        
          
            7
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        7
      
    
    
      
         
      
    
  




The top half of expression (89) represents the e→o portion of the frame composite macro block e, while the bottom portion of expression (89) represents the e→e portion of the frame composite macro block e.
Using expression (89), expression (90) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                k1
              
              =
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      A
                      pq
                    
                    [
                    
                      
                        
                          ∑
                          
                            
                              i
                              =
                              0
                            
                            ,
                            2
                          
                          
                            5
                            -
                            m
                          
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              6
                              -
                              n
                            
                          
                          ⁢
                          
                            (
                            
                              W
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        k
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              6
                              -
                              n
                            
                          
                          ⁢
                          
                            (
                            
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      j
                                      +
                                      n
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                8
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        
                                           
                                        
                                        ⁢
                                        m
                                      
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                            )
                          
                        
                      
                      +
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          m
                        
                      
                      ⁢
                      
                        (
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    m
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              15
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    n
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    (
                    
                      
                        1
                        4
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          15
                          ⁢
                          
                             
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          15
                          ⁢
                          
                             
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    B
                    pq
                  
                  [
                  
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          m
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              n
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            W
                            ⁢
                            
                               
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              n
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                15
                                ⁢
                                
                                   
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
                                    +
                                    n
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              8
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          m
                        
                      
                      ⁢
                      
                        (
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    m
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    n
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    (
                    
                      
                        1
                        4
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          15
                          ⁢
                          
                             
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    C
                    pq
                  
                  [
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
                              m
                            
                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            6
                            -
                            n
                          
                        
                        ⁢
                        
                          (
                          
                            W
                            ⁢
                            
                               
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      i
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            6
                            -
                            n
                          
                        
                        ⁢
                        
                          (
                          
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                3
                                ⁢
                                
                                   
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
                                    +
                                    n
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              8
                            
                            ⁢
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
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                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
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                              16
                            
                            ⁢
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
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                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
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                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  1
                  2
                
                ⁢
                
                  
                    ∑
                    
                      i
                      =
                      
                        
                          -
                          m
                        
                        +
                        1
                      
                    
                    
                      -
                      2
                    
                  
                  ⁢
                  
                    (
                    
                      
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                              (
                              
                                
                                  2
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                                +
                                
                                  2
                                  ⁢
                                  m
                                
                                +
                                3
                              
                              )
                            
                            ⁢
                            p
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                            ⁢
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                          16
                        
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                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            p
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                            ⁢
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                          16
                        
                        ⁢
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                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            15
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                            ⁢
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                            ⁢
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                          16
                        
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                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
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                                +
                                17
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
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                              (
                              
                                15
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                                  2
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                              )
                            
                            ⁢
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                          16
                        
                      
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                          1
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                        ⁢
                        
                          (
                          
                            cos
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                            ⁢
                            
                              
                                3
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                                ⁢
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                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                15
                                ⁢
                                
                                   
                                
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                                ⁢
                                
                                   
                                
                                ⁢
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                              16
                            
                            ⁢
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
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                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
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                    ]
                  
                
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    D
                    pq
                  
                  [
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
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                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              n
                            
                          
                          
                            -
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                        ⁢
                        
                          (
                          
                            W
                            ⁢
                            
                               
                            
                            ⁢
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
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                                    17
                                  
                                  )
                                
                                ⁢
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                                ⁢
                                
                                   
                                
                                ⁢
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                              16
                            
                            ⁢
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                                  (
                                  
                                    
                                      2
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                                  )
                                
                                ⁢
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                              16
                            
                          
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                          ∑
                          
                            j
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                              -
                              n
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
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                            ⁢
                            
                              
                                3
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                              16
                            
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                            ⁢
                            
                              
                                q
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                              16
                            
                            ⁢
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
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                                  )
                                
                                ⁢
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                              8
                            
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                            ⁢
                            
                              
                                
                                  (
                                  
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                              16
                            
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    
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                                    17
                                  
                                  )
                                
                                ⁢
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                                ⁢
                                π
                              
                              16
                            
                          
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                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
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                        ∑
                        
                          i
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                          -
                          2
                        
                      
                      ⁢
                      
                        (
                        
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                                (
                                
                                  
                                    2
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                            16
                          
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                          ⁢
                          
                            
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                              ⁢
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                            8
                          
                          ⁢
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                          ⁢
                          
                            
                              q
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                              ⁢
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                            16
                          
                          ⁢
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                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
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                                  +
                                  17
                                
                                )
                              
                              ⁢
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                              ⁢
                              
                                 
                              
                              ⁢
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                            16
                          
                          ⁢
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                          ⁢
                          
                            
                              
                                (
                                
                                  15
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                                    2
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                                    ⁢
                                    n
                                  
                                
                                )
                              
                              ⁢
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                              ⁢
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                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    (
                    
                      
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                      ⁢
                      
                        
                          3
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                          ⁢
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                        16
                      
                      ⁢
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                      ⁢
                      
                        
                          q
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                          ⁢
                          π
                        
                        16
                      
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                      ⁢
                      
                        
                          
                            (
                            
                              15
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                                2
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                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
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                          ⁢
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                        16
                      
                      ⁢
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                      ⁢
                      
                        
                          
                            (
                            
                              15
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                                2
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                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
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                  7
                
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                    A
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                  [
                  
                    
                      
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                          5
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                      ⁢
                      
                        
                          ∑
                          
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                            6
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                          (
                          
                            
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                                  )
                                
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                              16
                            
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                              16
                            
                          
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                    +
                    
                      
                        1
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                              16
                            
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                              16
                            
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                            ⁢
                            
                              
                                
                                  (
                                  
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                              16
                            
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                            ⁢
                            
                              
                                
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                                  )
                                
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                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
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                        ∑
                        
                          
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                          5
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                            16
                          
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                          ⁢
                          
                            
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                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              15
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                                  +
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                                )
                              
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                            16
                          
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                          ⁢
                          
                            
                              
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                            13
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                          16
                        
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                        ⁢
                        
                          
                            15
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                        ⁢
                        
                          
                            
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                              )
                            
                            ⁢
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                            ⁢
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                          16
                        
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                        ⁢
                        
                          
                            
                              (
                              
                                15
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                          16
                        
                      
                      )
                    
                  
                
                ]
              
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                  p
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                  ∑
                  
                    q
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                  [
                  
                    
                      
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                          5
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                            m
                            ′
                          
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
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                                    +
                                    3
                                  
                                  )
                                
                                ⁢
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                                ⁢
                                
                                   
                                
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                              16
                            
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                            ⁢
                            
                              
                                
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                                  )
                                
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                              16
                            
                          
                          )
                        
                      
                    
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                        1
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                          ∑
                          
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                          (
                          
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                                15
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                              16
                            
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                              16
                            
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                            ⁢
                            
                              
                                
                                  (
                                  
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                                      2
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                                  )
                                
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                              16
                            
                            ⁢
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                            ⁢
                            
                              
                                
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                                  )
                                
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                                ⁢
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                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          
                            m
                            ′
                          
                        
                      
                      ⁢
                      
                        (
                        
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                                    2
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                                  +
                                  
                                    2
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                                      m
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                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      (
                      
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            13
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                17
                                -
                                
                                  2
                                  ⁢
                                  
                                    m
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
                                  ⁢
                                  
                                    n
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    C
                    pq
                    ′
                  
                  [
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
                              
                                m
                                ′
                              
                            
                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            6
                            -
                            
                              n
                              ′
                            
                          
                        
                        ⁢
                        
                          (
                          
                            
                              W
                              ′
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            6
                            -
                            
                              n
                              ′
                            
                          
                        
                        ⁢
                        
                          (
                          
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                3
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
                                    +
                                    
                                      n
                                      ′
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              8
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
                              
                                m
                                ′
                              
                            
                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        (
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              15
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  19
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      (
                      
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            15
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                17
                                -
                                
                                  2
                                  ⁢
                                  
                                    m
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
                                  ⁢
                                  
                                    n
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    D
                    pq
                    ′
                  
                  [
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
                              
                                m
                                ′
                              
                            
                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              
                                n
                                ′
                              
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            
                              W
                              ′
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              
                                n
                                ′
                              
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                3
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
                                    +
                                    
                                      n
                                      ′
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              8
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        
                          
                            -
                            
                              m
                              ′
                            
                          
                          +
                          1
                        
                      
                      
                        -
                        2
                      
                    
                    ⁢
                    
                      (
                      
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    m
                                    ′
                                  
                                
                                +
                                3
                              
                              )
                            
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          8
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                19
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
                                  ⁢
                                  
                                    n
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                +
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              17
                              -
                              
                                2
                                ⁢
                                
                                  m
                                  ′
                                
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                  n
                                  ′
                                
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        90
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (90) represent the e→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (90) represent the e→e portion of the frame composite macro block E.
Using expression (90), the motion compensated frame macro block E can be derived for diagonal half-pel accuracy when the even and odd fields are encoded based on the even field of the anchor frame.
Case (3) o→o and e→e
Expression (91) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the odd and even fields are encoded based on the odd and even fields, respectively, of the anchor frame. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        a
                        
                          
                            j
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        4
                        -
                        m
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        6
                        -
                        n
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        4
                        -
                        m
                      
                    
                    ;
                    
                      
                        8
                        -
                        n
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          7
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      4
                      -
                      m
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          7
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      n
                    
                  
                  ;
                  
                    
                      8
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          6
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      6
                      -
                      m
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        6
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            -
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      
                        8
                        -
                        
                          n
                          ′
                        
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          7
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          7
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          1
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          1
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          7
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          7
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          1
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          1
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        6
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      10
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          7
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          7
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        c
                        
                          1
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          1
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      8
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
          
          ⁢
          
             
          
          }
        
      
    
    
      
        (
        91
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        
          
            6
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        6
      
    
    
      
         
      
    
  




The top half of expression (91) represents the o→o portion of the frame composite macro block e, while the bottom portion of expression (91) represents the e→e portion of the frame composite macro block e.
Using expression (91), expression (92) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                k1
              
              =
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      A
                      pq
                    
                    [
                    
                      
                        
                          ∑
                          
                            
                              i
                              =
                              0
                            
                            ,
                            2
                          
                          
                            4
                            -
                            m
                          
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              6
                              -
                              n
                            
                          
                          ⁢
                          
                            (
                            
                              W
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        k
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              6
                              -
                              n
                            
                          
                          ⁢
                          
                            (
                            
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  13
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      j
                                      +
                                      n
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                8
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      13
                                      -
                                      
                                        2
                                        ⁢
                                        
                                           
                                        
                                        ⁢
                                        m
                                      
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                            )
                          
                        
                      
                      +
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          4
                          -
                          m
                        
                      
                      ⁢
                      
                        (
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    m
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              15
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    n
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    (
                    
                      
                        1
                        4
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          13
                          ⁢
                          
                             
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          15
                          ⁢
                          
                             
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              13
                              -
                              
                                2
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                            )
                          
                          ⁢
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                            (
                            
                              15
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                            )
                          
                          ⁢
                          l
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                          ⁢
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                    )
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
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                  7
                
                ⁢
                
                  
                    B
                    pq
                  
                  [
                  
                    
                      
                        ∑
                        
                          
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                          ,
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                          4
                          -
                          m
                        
                      
                      ⁢
                      
                        
                          ∑
                          
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                            -
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
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                                  )
                                
                                ⁢
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                              16
                            
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                              16
                            
                          
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                          ∑
                          
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                          (
                          
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                                13
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                              16
                            
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                            ⁢
                            
                              
                                q
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                                ⁢
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                              16
                            
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                                  (
                                  
                                    j
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                                    +
                                    1
                                  
                                  )
                                
                                ⁢
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                              8
                            
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                                  (
                                  
                                    13
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                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
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                                ⁢
                                
                                   
                                
                                ⁢
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                              16
                            
                            ⁢
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                                  (
                                  
                                    
                                      2
                                      ⁢
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                                    17
                                  
                                  )
                                
                                ⁢
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                                ⁢
                                
                                   
                                
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                              16
                            
                          
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                      1
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                        ∑
                        
                          
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                          ,
                          2
                        
                        
                          4
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                        (
                        
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                                (
                                
                                  
                                    2
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                                  +
                                  
                                    2
                                    ⁢
                                    m
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
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                            16
                          
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                          ⁢
                          
                            
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                              ⁢
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                            8
                          
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                          ⁢
                          
                            
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                            16
                          
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                                (
                                
                                  
                                    2
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                                  +
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                              ⁢
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                              ⁢
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                            16
                          
                          ⁢
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                                (
                                
                                  15
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                                    2
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                                    ⁢
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                                )
                              
                              ⁢
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                            16
                          
                        
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                  +
                  
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                      ⁢
                      
                        
                          13
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                        16
                      
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                      ⁢
                      
                        
                          q
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                      ⁢
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                      ⁢
                      
                        
                          
                            (
                            
                              13
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                                2
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                            )
                          
                          ⁢
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                          ⁢
                          
                             
                          
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                      ⁢
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                            (
                            
                              15
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                                2
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                            )
                          
                          ⁢
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                          ⁢
                          
                             
                          
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              +
            
          
        
        
          
            
              
                ∑
                
                  p
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                7
              
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                  ∑
                  
                    q
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                  7
                
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                    pq
                  
                  [
                  
                    
                      
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                          i
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                                  )
                                
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                              16
                            
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                              16
                            
                          
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                            6
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                                  )
                                
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                              16
                            
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                                  (
                                  
                                    
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                                  )
                                
                                ⁢
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                              16
                            
                          
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                      1
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                        ∑
                        
                          i
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                          -
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                        (
                        
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                            16
                          
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                            16
                          
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                          ⁢
                          
                            
                              15
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                            16
                          
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                                (
                                
                                  
                                    2
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                                  +
                                  17
                                
                                )
                              
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                            16
                          
                          ⁢
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                          ⁢
                          
                            
                              
                                (
                                
                                  15
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                                    2
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                                )
                              
                              ⁢
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                            16
                          
                        
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                  +
                  
                    
                      1
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                          16
                        
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                        ⁢
                        
                          
                            15
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                          16
                        
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                              (
                              
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                                  ⁢
                                  m
                                
                              
                              )
                            
                            ⁢
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                            ⁢
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                          16
                        
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                        ⁢
                        
                          
                            
                              (
                              
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                          16
                        
                      
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                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    D
                    pq
                  
                  [
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            -
                            m
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
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                            -
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                          (
                          
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                                    17
                                  
                                  )
                                
                                ⁢
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                                ⁢
                                
                                   
                                
                                ⁢
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                              16
                            
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                            ⁢
                            
                              
                                
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                              16
                            
                          
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                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
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                            -
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                        ⁢
                        
                          (
                          
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                              16
                            
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                              16
                            
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                            ⁢
                            
                              
                                
                                  (
                                  
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                                  )
                                
                                ⁢
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                              8
                            
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                            ⁢
                            
                              
                                
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                              16
                            
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
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                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
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                            -
                            m
                          
                        
                        
                          -
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                        (
                        
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                                (
                                
                                  
                                    2
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                                    2
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                                    m
                                  
                                  +
                                  3
                                
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                              ⁢
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                              ⁢
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                            16
                          
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                              ⁢
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                            16
                          
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                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
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                                    i
                                  
                                  +
                                  17
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
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                            16
                          
                          ⁢
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                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
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                                    ⁢
                                    n
                                  
                                
                                )
                              
                              ⁢
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                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    (
                    
                      
                        1
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                      ⁢
                      
                        
                          p
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                          ⁢
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                        16
                      
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                      ⁢
                      
                        
                          q
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                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              13
                              -
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              15
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                                2
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    A
                    pq
                    ′
                  
                  [
                  
                    
                      
                        ∑
                        
                          
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                            =
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                          5
                          -
                          
                            m
                            ′
                          
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
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                          (
                          
                            
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    
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                                    3
                                  
                                  )
                                
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                                ⁢
                                
                                   
                                
                                ⁢
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                              16
                            
                            ⁢
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    
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                                  )
                                
                                ⁢
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                                ⁢
                                
                                   
                                
                                ⁢
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                              16
                            
                          
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                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
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                            6
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                          (
                          
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
                                    +
                                    
                                      n
                                      ′
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              8
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            j
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          
                            m
                            ′
                          
                        
                      
                      ⁢
                      
                        (
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      m
                                      ′
                                    
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              15
                              ⁢
                              
                                 
                              
                              ⁢
                              q
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                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      (
                      
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                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            13
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            15
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                17
                                -
                                
                                  2
                                  ⁢
                                  
                                    m
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
                                  ⁢
                                  
                                    n
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    B
                    pq
                    ′
                  
                  [
                  
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          
                            m
                            ′
                          
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              
                                n
                                ′
                              
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            
                              W
                              ′
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    3
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
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                              16
                            
                            ⁢
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
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                                    +
                                    17
                                  
                                  )
                                
                                ⁢
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                              16
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        1
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                      ⁢
                      
                        
                          ∑
                          
                            i
                            =
                            
                              -
                              
                                n
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                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                15
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
                                    +
                                    
                                      n
                                      ′
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
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                                ⁢
                                
                                   
                                
                                ⁢
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                              8
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          
                            m
                            ′
                          
                        
                      
                      ⁢
                      
                        (
                        
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                      m
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                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
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                          ⁢
                          
                            
                              q
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                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    j
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      (
                      
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            13
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                17
                                -
                                
                                  2
                                  ⁢
                                  
                                    m
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
                                  ⁢
                                  
                                    n
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    C
                    pq
                    ′
                  
                  [
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
                              
                                m
                                ′
                              
                            
                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            6
                            -
                            
                              n
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                        ⁢
                        
                          (
                          
                            
                              W
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                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            6
                            -
                            
                              n
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                        ⁢
                        
                          (
                          
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                3
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                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
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                                      n
                                      ′
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
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                              8
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
                              
                                m
                                ′
                              
                            
                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        (
                        
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                                (
                                
                                  
                                    2
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                                    i
                                  
                                  +
                                  
                                    2
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                                      m
                                      ′
                                    
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
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                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              15
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                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  19
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                      n
                                      ′
                                    
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      (
                      
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            15
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                17
                                -
                                
                                  2
                                  ⁢
                                  
                                    m
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
                                  ⁢
                                  
                                    n
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    D
                    pq
                    ′
                  
                  [
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
                              
                                m
                                ′
                              
                            
                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              
                                n
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                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            
                              W
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                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    19
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              
                                n
                                ′
                              
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                3
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
                                    +
                                    
                                      n
                                      ′
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
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                              8
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    17
                                    -
                                    
                                      2
                                      ⁢
                                      
                                        m
                                        ′
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    
                      ∑
                      
                        i
                        =
                        
                          
                            -
                            
                              m
                              ′
                            
                          
                          +
                          1
                        
                      
                      
                        -
                        2
                      
                    
                    ⁢
                    
                      (
                      
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                
                                  2
                                  ⁢
                                  
                                    m
                                    ′
                                  
                                
                                +
                                3
                              
                              )
                            
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          8
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                
                                  2
                                  ⁢
                                  i
                                
                                +
                                19
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
                                  ⁢
                                  
                                    n
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                +
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              17
                              -
                              
                                2
                                ⁢
                                
                                  m
                                  ′
                                
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                  n
                                  ′
                                
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        92
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (92) represent the o→o portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (92) represent the e→e portion of the frame composite macro block E.
Using expression (92), the motion compensated frame macro block E can be derived for diagonal half-pel accuracy when the odd and even fields are encoded based on the odd and even fields, respectively, of the anchor frame.
Case (4) e→o and o→e
Expression (93) below represents the relationship between frame composite macro block e and the macro blocks a-d and a′-d′ in the spatial domain when the odd and even fields are encoded based on the even and odd fields, respectively, of the anchor frame. 



  
    
      
        
          e
          ij
        
        =
        
          {
          
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        a
                        
                          
                            j
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        m
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        6
                        -
                        n
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  0
                  ,
                  
                    
                      2
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        m
                      
                    
                    ;
                    
                      
                        8
                        -
                        n
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            n
                          
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            -
                            8
                          
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            n
                            -
                            7
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    
                      8
                      -
                      n
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            m
                          
                          ,
                          7
                        
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                          
                          ,
                          0
                        
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            m
                            +
                            2
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      5
                      -
                      m
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          7
                        
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            8
                          
                          ,
                          0
                        
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            m
                            -
                            6
                          
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      n
                    
                  
                  ;
                  
                    
                      9
                      -
                      m
                    
                    ≤
                    i
                    ≤
                    6
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          7
                        
                      
                      +
                      
                        b
                        
                          6
                          ,
                          0
                        
                      
                      +
                      
                        c
                        
                          0
                          ,
                          7
                        
                      
                      +
                      
                        d
                        
                          0
                          ,
                          0
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      m
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      n
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      0
                      ≤
                      j
                      ≤
                      
                        6
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            -
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      
                        8
                        -
                        
                          n
                          ′
                        
                      
                      ≤
                      j
                      ≤
                      7
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    
                      9
                      -
                      
                        m
                        ′
                      
                    
                    ≤
                    i
                    ≤
                    7
                  
                  ;
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        a
                        
                          6
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                          
                        
                        ′
                      
                      +
                      
                        c
                        
                          0
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            +
                            1
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    0
                    ≤
                    j
                    ≤
                    
                      6
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        b
                        
                          6
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            8
                          
                        
                        ′
                      
                      +
                      
                        d
                        
                          0
                          ,
                          
                            j
                            +
                            
                              n
                              ′
                            
                            -
                            7
                          
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    
                      8
                      -
                      
                        n
                        ′
                      
                    
                    ≤
                    j
                    ≤
                    7
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        a
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            1
                          
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        b
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            +
                            1
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        c
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            9
                          
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        d
                        
                          
                            i
                            +
                            
                              m
                              ′
                            
                            -
                            7
                          
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  1
                  ,
                  
                    
                      3
                      ≤
                      i
                      ≤
                      
                        5
                        -
                        
                          m
                          ′
                        
                      
                    
                    ;
                    
                      j
                      =
                      
                        7
                        -
                        
                          n
                          ′
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        
                          6
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        b
                        
                          6
                          ,
                          0
                        
                        ′
                      
                      +
                      
                        c
                        
                          0
                          ,
                          7
                        
                        ′
                      
                      +
                      
                        d
                        
                          0
                          ,
                          0
                        
                        ′
                      
                    
                    )
                  
                
              
              
                
                  
                    i
                    =
                    
                      7
                      -
                      
                        m
                        ′
                      
                    
                  
                  ;
                  
                    j
                    =
                    
                      7
                      -
                      
                        n
                        ′
                      
                    
                  
                
              
            
          
          ⁢
          
             
          
          }
        
      
    
    
      
        (
        93
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
               
            
            ⁢
            m
          
          =
          1
        
        ,
        3
        ,
        5
        ,
        
          
            7
            ⁢
            
               
            
            ⁢
            and
            ⁢
            
               
            
            ⁢
            
              m
              ′
            
          
          =
          0
        
        ,
        2
        ,
        4
        ,
        6
      
    
    
      
         
      
    
  




The top half of expression (93) represents the e→o portion of the frame composite macro block e, while the bottom portion of expression (93) represents the o→e portion of the frame composite macro block e.
Using expression (93), expression (94) below is derived in the same manner as expression (47). 



  
    
      
        
          
            
              
                E
                k1
              
              =
              
                
                  ∑
                  
                    p
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    ∑
                    
                      q
                      =
                      0
                    
                    7
                  
                  ⁢
                  
                    
                      A
                      pq
                    
                    [
                    
                      
                        
                          ∑
                          
                            
                              i
                              =
                              0
                            
                            ,
                            2
                          
                          
                            5
                            -
                            m
                          
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              6
                              -
                              n
                            
                          
                          ⁢
                          
                            (
                            
                              W
                              ⁢
                              
                                 
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        i
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        k
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          
                            ∑
                            
                              j
                              =
                              0
                            
                            
                              6
                              -
                              n
                            
                          
                          ⁢
                          
                            (
                            
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  15
                                  ⁢
                                  p
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      j
                                      +
                                      n
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  q
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                8
                              
                              ⁢
                              cos
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      15
                                      -
                                      
                                        2
                                        ⁢
                                        
                                           
                                        
                                        ⁢
                                        m
                                      
                                    
                                    )
                                  
                                  ⁢
                                  k
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                              ⁢
                              cos
                              ⁢
                              
                                 
                              
                              ⁢
                              
                                
                                  
                                    (
                                    
                                      
                                        2
                                        ⁢
                                        j
                                      
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  l
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  π
                                
                                16
                              
                            
                            )
                          
                        
                      
                      +
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          m
                        
                      
                      ⁢
                      
                        (
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    m
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              15
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    n
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    (
                    
                      
                        1
                        4
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          15
                          ⁢
                          
                             
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          15
                          ⁢
                          
                             
                          
                          ⁢
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    B
                    pq
                  
                  [
                  
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          m
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              n
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            W
                            ⁢
                            
                               
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      i
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            
                              -
                              n
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          (
                          
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                15
                                ⁢
                                
                                   
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
                                    +
                                    n
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              8
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
                            =
                            0
                          
                          ,
                          2
                        
                        
                          5
                          -
                          m
                        
                      
                      ⁢
                      
                        (
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    m
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  1
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    n
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    (
                    
                      
                        1
                        4
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          15
                          ⁢
                          
                             
                          
                          ⁢
                          p
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                n
                              
                            
                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    C
                    pq
                  
                  [
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
                              m
                            
                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            6
                            -
                            n
                          
                        
                        ⁢
                        
                          (
                          
                            W
                            ⁢
                            
                               
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      
                                         
                                      
                                      ⁢
                                      i
                                    
                                    +
                                    17
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          
                            6
                            -
                            n
                          
                        
                        ⁢
                        
                          (
                          
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                3
                                ⁢
                                
                                   
                                
                                ⁢
                                p
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
                                    +
                                    n
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                q
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              8
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
                                    -
                                    
                                      2
                                      ⁢
                                      m
                                    
                                  
                                  )
                                
                                ⁢
                                k
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
                                      ⁢
                                      j
                                    
                                    +
                                    1
                                  
                                  )
                                
                                ⁢
                                l
                                ⁢
                                
                                   
                                
                                ⁢
                                π
                              
                              16
                            
                          
                          )
                        
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          i
                          =
                          
                            
                              -
                              m
                            
                            +
                            1
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        (
                        
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  
                                    2
                                    ⁢
                                    m
                                  
                                  +
                                  3
                                
                                )
                              
                              ⁢
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              p
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              15
                              ⁢
                              
                                 
                              
                              ⁢
                              q
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  17
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    n
                                  
                                
                                )
                              
                              ⁢
                              l
                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      (
                      
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            3
                            ⁢
                            p
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            15
                            ⁢
                            
                               
                            
                            ⁢
                            q
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
                                  ⁢
                                  
                                     
                                  
                                  ⁢
                                  m
                                
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
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                          16
                        
                        ⁢
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                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
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                                  2
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                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
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                          16
                        
                      
                      )
                    
                  
                
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              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
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                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    D
                    pq
                  
                  [
                  
                    
                      
                        ∑
                        
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                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
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                            =
                            
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                            -
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
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                                  )
                                
                                ⁢
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                              16
                            
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                                  (
                                  
                                    
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                              16
                            
                          
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                          ∑
                          
                            j
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                            -
                            1
                          
                        
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                          (
                          
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                                3
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                              16
                            
                            ⁢
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                            ⁢
                            
                              
                                q
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                                ⁢
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                              16
                            
                            ⁢
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
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                                    n
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                                    1
                                  
                                  )
                                
                                ⁢
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                              8
                            
                            ⁢
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                            ⁢
                            
                               
                            
                            ⁢
                            
                              
                                
                                  (
                                  
                                    15
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                                  )
                                
                                ⁢
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                              16
                            
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    
                                      2
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                                    17
                                  
                                  )
                                
                                ⁢
                                l
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                              16
                            
                          
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                      1
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                        ∑
                        
                          i
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                          -
                          2
                        
                      
                      ⁢
                      
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                                (
                                
                                  
                                    2
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                                  +
                                  
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                                    m
                                  
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                                  3
                                
                                )
                              
                              ⁢
                              p
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                              ⁢
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                            16
                          
                          ⁢
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                          ⁢
                          
                            
                              p
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                              ⁢
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                            8
                          
                          ⁢
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                          ⁢
                          
                            
                              q
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                              ⁢
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                            16
                          
                          ⁢
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                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
                                    ⁢
                                    i
                                  
                                  +
                                  17
                                
                                )
                              
                              ⁢
                              k
                              ⁢
                              
                                 
                              
                              ⁢
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                            16
                          
                          ⁢
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                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
                                    ⁢
                                    
                                       
                                    
                                    ⁢
                                    n
                                  
                                
                                )
                              
                              ⁢
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                              ⁢
                              
                                 
                              
                              ⁢
                              π
                            
                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    (
                    
                      
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                      ⁢
                      
                        
                          3
                          ⁢
                          
                             
                          
                          ⁢
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                          ⁢
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                        16
                      
                      ⁢
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                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          q
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
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                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              15
                              -
                              
                                2
                                ⁢
                                
                                   
                                
                                ⁢
                                m
                              
                            
                            )
                          
                          ⁢
                          k
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                      ⁢
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                      ⁢
                      
                         
                      
                      ⁢
                      
                        
                          
                            (
                            
                              15
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                                2
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                            )
                          
                          ⁢
                          l
                          ⁢
                          
                             
                          
                          ⁢
                          π
                        
                        16
                      
                    
                    )
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
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                    A
                    pq
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                  [
                  
                    
                      
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                          2
                        
                        
                          4
                          -
                          
                            m
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                      ⁢
                      
                        
                          ∑
                          
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                            0
                          
                          
                            6
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                          (
                          
                            
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                                  (
                                  
                                    
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                                  )
                                
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                              16
                            
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                                  (
                                  
                                    
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                                  )
                                
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                              16
                            
                          
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                        1
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                          ∑
                          
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                            6
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                                13
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                              16
                            
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                            ⁢
                            
                              
                                q
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                              16
                            
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    j
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                              8
                            
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                            ⁢
                            
                              
                                
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                              16
                            
                            ⁢
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                            ⁢
                            
                              
                                
                                  (
                                  
                                    
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                                  )
                                
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                              16
                            
                          
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                      1
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                        ∑
                        
                          
                            j
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                          4
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                        (
                        
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                            16
                          
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                          ⁢
                          
                            
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                            8
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              15
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                            16
                          
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                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
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                                  +
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                                )
                              
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                            16
                          
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                          ⁢
                          
                            
                              
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                                  15
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                            13
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                          16
                        
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                        ⁢
                        
                          
                            15
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                          16
                        
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                                15
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                                  2
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                              )
                            
                            ⁢
                            k
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                          16
                        
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                        ⁢
                        
                          
                            
                              (
                              
                                15
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                                  2
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                              )
                            
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                          16
                        
                      
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                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
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                7
              
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                  ∑
                  
                    q
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                  7
                
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                    B
                    pq
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                  [
                  
                    
                      
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                          4
                          -
                          
                            m
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                      ⁢
                      
                        
                          ∑
                          
                            j
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                                  (
                                  
                                    
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                                    +
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                                  )
                                
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                              16
                            
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                                  (
                                  
                                    
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                                  )
                                
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                        1
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                          ∑
                          
                            i
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                            -
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                          (
                          
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                                13
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                              16
                            
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                            ⁢
                            
                              
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                                ⁢
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                              16
                            
                            ⁢
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                            ⁢
                            
                              
                                
                                  (
                                  
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                                  )
                                
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                              16
                            
                            ⁢
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                            ⁢
                            
                              
                                
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                                  )
                                
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                              16
                            
                          
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                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      1
                      2
                    
                    ⁢
                    
                      
                        ∑
                        
                          
                            i
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                          ,
                          2
                        
                        
                          4
                          -
                          
                            m
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                      ⁢
                      
                        (
                        
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                                    2
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                                    2
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                            16
                          
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                          ⁢
                          
                            
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                              ⁢
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                            16
                          
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                          ⁢
                          
                            
                              
                                (
                                
                                  
                                    2
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                                  +
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                                )
                              
                              ⁢
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                              ⁢
                              
                                 
                              
                              ⁢
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                            16
                          
                          ⁢
                          cos
                          ⁢
                          
                             
                          
                          ⁢
                          
                            
                              
                                (
                                
                                  15
                                  -
                                  
                                    2
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                                      n
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                                )
                              
                              ⁢
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                              ⁢
                              
                                 
                              
                              ⁢
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                            16
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      (
                      
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                        ⁢
                        
                          
                            13
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                            ⁢
                            π
                          
                          16
                        
                        ⁢
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                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            q
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                            ⁢
                            π
                          
                          16
                        
                        ⁢
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                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
                                  ⁢
                                  
                                    m
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            k
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                        ⁢
                        cos
                        ⁢
                        
                           
                        
                        ⁢
                        
                          
                            
                              (
                              
                                15
                                -
                                
                                  2
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                                    n
                                    ′
                                  
                                
                              
                              )
                            
                            ⁢
                            l
                            ⁢
                            
                               
                            
                            ⁢
                            π
                          
                          16
                        
                      
                      )
                    
                  
                
                ]
              
              +
            
          
        
        
          
            
              
                ∑
                
                  p
                  =
                  0
                
                7
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    0
                  
                  7
                
                ⁢
                
                  
                    C
                    pq
                    ′
                  
                  [
                  
                    
                      
                        ∑
                        
                          i
                          =
                          
                            -
                            
                              m
                              ′
                            
                          
                        
                        
                          -
                          2
                        
                      
                      ⁢
                      
                        
                          ∑
                          
                            j
                            =
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                            6
                            -
                            
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                        ⁢
                        
                          (
                          
                            
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                +
                
                  
                    1
                    4
                  
                  ⁢
                  
                    (
                    
                      cos
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                      ⁢
                      
                        
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                      ⁢
                      
                         
                      
                      ⁢
                      
                        
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                      ⁢
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                      ⁢
                      
                        
                          
                            (
                            
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                                  ′
                                
                              
                            
                            )
                          
                          ⁢
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                      ⁢
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                      ⁢
                      
                        
                          
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                        16
                      
                    
                    )
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        94
        )
      
    
  




The summation terms including Apq, Bpq, Cpq, and Dpq in expression (94) represent the ego portion of the frame composite macro block E, while the summation terms including A′pq, B′pq, C′pq, and D′pq in expression (94) represent the o→e portion of the frame composite macro block E.
Using expression (94), the motion compensated frame macro block E can be derived for diagonal half-pel accuracy when the odd and even fields are encoded based on the even and odd fields, respectively, of the anchor frame.
With field motion compensation, both motion vectors do not always have the same accuracy, half or full-pel, in the same direction. For instance, the odd field portion vector could have full-pel accuracy, while the even field motion vector has half-pel accuracy in the horizontal direction. As another example, the odd field could have half-pel accuracy in the horizontal direction, while the even field motion vector has half-pel accuracy in the vertical direction.
The composite macro block expressions for these instances can be readily generated from those portions of the previously discussed expressions corresponding to the motion vectors. For instance, if the odd field is encoded based on the previous odd field and the motion vector therefor has half-pel accuracy in the horizontal direction, then the top half of expression (68) represents the first part of the frame composite macro block eij, while the summation terms Apq, Bpq, Cpq, and Dpq from expression (69) are used to generate the first part of the frame composite macro block Ekl. Next, if the even field is encoded based on the previous even field and the motion vector therefor has half-pel accuracy in the vertical direction, then the bottom half of expression (82) represents the second part of the frame composite macro block eij, while the summation terms A′pq, B′pq, C′pq, and D′pq from expression (83) are used to generate the second part of the frame composite macro block Ekl.
As discussed above, B-pictures may be bi-directionally encoded. Accordingly, two motion vectors are associated with each field or frame of a bi-directionally encoded B-picture. The motion vector address generator 62 identifies and informs the compositer 68 when a picture is a bi-directionally inter-coded picture. In response, the compositer 68 forms composite anchor frame blocks with respect to each anchor frame pointed to by the motion vectors. The compositer 68 then averages the two composite macro blocks to form a weighted average composite macro block, and outputs the weighted average composite macro block to the adder 74.
The adder 74 adds the composite macro block to the macro block of the picture data currently output by the variable length decoder and dequantizer 58, or optionally the field-to-frame converter 69, to reconstruct a complete macro block. In this manner, both P-pictures and B-pictures are converted into I-pictures.
Of course, if an I-picture is currently being output by the variable length decoder and dequantizer 58, then the adder 74 will not receive anything from the compositer 68 to add thereto. The motion vector address generator 62 will have recognized the I-picture as an I-picture and disabled the compositer 68 from sending output to the adder 74.
The output of the adder 74 is received by a reformatter 78. The reformatter 78 is also connected to the memory 64 via the variable length decoder and dequantizer 66. Because all of the pictures in the HDTV signal are converted into I-pictures, if not already I-pictures, the reformatter 78 restores the order of the pictures in the HDTV signal to the display order.
The reformatter 78 generates output according to the following rules: (1) if the picture received is the first anchor picture received, then no output will be sent as the output of the I-picture converter 76 and the I-picture will be output to the memory 64 for storage; (2) if the picture received is an anchor picture but not the first anchor picture received, then the previously received anchor picture will be obtained from the memory 64 via the variable length decoder 66 and output as the output of the I-picture converter 78, and the current anchor frame will be output to the memory 64 for storage; and (3) if the picture received is a B-picture, then the B-picture will be immediately output.
Therefore, upon receipt of I-picture I1, the reformatter 78 will output the I-picture I1 to the memory 64 for storage via the quantizer 70, the variable length coder 72, and the bit stream parser 59. The next picture output by the adder 74 will be P-picture P1. Since this picture is an anchor picture, the memory 64 will then store the P-picture P1, albeit converted into an I-picture, under the control of the bit stream parser 59. And, the reformatter 78 will send the memory 64 a control signal to output the I-picture I1 (i.e., the previous anchor picture) to the reformatter 78, and the reformatter 78 will output the I-picture I1. The reformatter 78 will then receive and immediately output the B-pictures. Consequently, the output order from the reformatter 78 will be I1B1B2.
Next, the P-picture P2 is received and processed in the same manner as P-picture Pa. Consequently, the bit stream parser 59 will replace the I-picture I1 with the P-picture P2 in the memory 64. The reformatter 78, according to the rules discussed above, will then receive P-picture P1 from the memory 64, and output P-picture P1. In this manner, the reformatter 78 will output the pictures in the proper display order, and these pictures will all be I-pictures. Therefore, the I-picture converter 76 outputs the HDTV signal comprised entirely of, I-pictures in the display order.
The HDTV signal output by the I-picture converter is then received by a down-converter 12,40. The down-converter 12,40 can be the down-converter 12 disclosed with respect to FIG. 1, or the down-converter 40 disclosed with respect to FIGS. 3A and 3B. As discussed above with respect to FIGS. 1, 3A, and 3B, the down-converter 12,40 down converts the HDTV signal into an SDTV signal, and outputs the SDTV signal in the spatial domain. Consequently, to place the SDTV signal output by the down-converter 12,40 into the format expected by a standard SDTV decoder, the SDTV signal output by the down-converter 12,40 must be converted back into the DCT domain. Therefore, the DCT converter 80 receives the output of the down-converter 12,40, and converts the output of the down-converter 12,40 into the DCT domain.
The output of the DCT converter 80 is then quantized by the quantizer 82, and variable length encoded by the variable length encoder 84. The quantizer 82 requantizes the output of the down-converter 52 in the same manner discussed above with respect to the quantizer 102. The resulting SDTV signal output by the variable length encoder 84 is of the format expected by a standard SDTV decoder.
Because the HDTV converter of FIG. 6 stores anchor frames which are in the DCT domain and which have been variable length encoded and quantized, the HDTV converter of FIG. 6 significantly reduces the amount of memory required to perform down conversion of HDTV signals including inter-coded pictures. The HDTV converter of FIG. 6, however, still requires an additional DCT converter in order to place the SDTV signal in the format expected by a standard SDTV converter.
FIG. 7 illustrates another HDTV to SDTV converter according to the present invention. The embodiment of FIG. 7 also provides the advantage provided in the embodiment of FIG. 6 in that the amount of memory required to perform down conversion is seriously reduced. The embodiment of FIG. 7, however, further provides the additional benefit of eliminating the need for DCT converters.
In FIG. 7, the HDTV signal is received by the I-picture converter 76. The output of the I-picture converter 76 is received by the down-converter 52, and the output of the down-converter 52 is quantized by a quantizer 54 and variable length encoded by a variable length encoder 56. The operation of each of these elements has been discussed in detail above. Accordingly, that description will not be repeated for the sake of brevity. It is important to note that the I-picture converter 76 outputs an HDTV signal comprised of I-pictures in the frequency domain, and that the down-converter 52 performs down conversion without having to perform an inverse DCT conversion. Consequently, the output of the down-converter 52 is in SDTV format with the pictures in the frequency domain. Thus, the need for a DCT converter at the output of down-converter has been eliminated. The embodiment of FIG. 7 achieves both the cost and complexity reduction benefits from having reduced the memory requirements for handling an HDTV signal with inter-coded pictures as well as the elimination of inverse DCT and DCT converters.
Each of the above described embodiments may have several different forms of implementation. For instance, each embodiment could be implemented using a programmed microprocessor. Alternatively, each embodiment could be implemented using hard-wired logic.
While the present invention has been described with respect to specific coding sequences, it should be understood that the present invention is not limited to the coding sequences described. Additionally, the present invention is not restricted to use with a particular digital format such as MPEG. Instead, the present invention may be used with any format in any application where down-converting of digital data is desired.
Furthermore, while the invention has been described in connection with what is presently considered the most practical and preferred embodiments, it is to be understood that the invention is not limited to the disclosed embodiments, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.

## Claims

1. An apparatus for down-converting a digital video signal, comprising:
receiving means for receiving an encoded digital video signal, said encoded digital video signal having been previously generated by converting spatial video data into the DCT domain, quantizing DCT domain coefficients, and variable length encoding the quantized DCT domain coefficients; 
a variable length decoder for generating variable length decoded DCT data from said received encoded digital video signal; 
a dequantizer for dequantizing the variable length decoded DCT coefficients generated by said variable length decoder to generate at least first and second DCT blocks, said first and second DCT blocks each having first dimensions; 
converting means for receiving said at least first and second DCT blocks, and for converting said first and second DCT blocks into a single DCT block having dimensions equal to said first dimensions; 
a quantizer for re-quantizing the output of said converting means, the quantization scale used by said quantizer for re-quantizing being different than the quantization scale used to generate said encoded digital video signal; and 
a variable length coder variable length coding output of said quantizer.

2. The apparatus of claim 1, wherein said quantizer re-quantizes said output of said converting means such that distortion is eliminated.

3. A method for down-converting a digital video signal, comprising:
(a) receiving an encoded digital video signal, said encoded digital video signal having been previously generated by converting spatial video data into the DCT domain, quantizing DCT domain coefficients, and variable length encoding the quantized DCT domain coefficients; 
(b) variable length decoding said encoded digital video signal to generate variable length decoded DCT data; 
(c) dequantizing said variable length decoded DCT data to generate at least first and second DCT blocks, said first and second DCT blocks each having first dimensions; 
(d) converting said first and second DCT blocks into a single DCT block having dimensions equal to said first dimensions; 
(e) re-quantizing the output of said step (d), the quantization scale used by said quantizer for re-quantizing being different than the quantization scale used to generate said encoded digital video signal; and 
(f) variable length coding the output of said step (e).

4. The method of claim 3 wherein said step (e) re-quantizes said output of step (d) using a quantization scale which does not cause distortion in the re-quantized data.