# Patent

## Title
Method and apparatus for transmitting uplink signals using multi-antenna

## Abstract

A method and apparatus for allowing a UE to transmit uplink signals using a MIMO scheme are disclosed. In order to maintain good Peak power to Average Power Ratio (PAPR) or Cubic Metric (CM) properties when the UE transmits uplink signals using the MIMO scheme, the UE uses a precoding scheme based on a precoding matrix established in a manner that one layer is transmitted to each antenna in specific rank transmission.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
This application is a continuation application of U.S. patent application Ser. No. 14/627,414, filed on Feb. 20, 2015, now U.S. Pat. No. 9,281,886, which is a continuation application of U.S. patent application Ser. No. 12/539,420, filed on Aug. 11, 2009, now U.S. Pat. No. 9,001,802, which claims the benefit of earlier filing date and right of priority to Korean Patent Application Nos. 10-2008-0132994, filed on Dec. 24, 2008 and 10-2009-0073606, filed on Aug. 11, 2009, and also claims the benefit of U.S. Provisional Application Nos. 61/087,990, filed on Aug. 11, 2008, 61/160,711, filed on Mar. 17, 2009, 61/169,726, filed on Apr. 16, 2009, 61/170,106, filed on Apr. 17, 2009 and 61/173,585, filed on Apr. 28, 2009, the contents of which are all incorporated by reference herein in their entirety.


BACKGROUND OF THE INVENTION
Field of the Invention
The present invention relates to a wireless mobile communication system, and more particularly to a communication system based on a Multiple Input Multiple Output (MIMO) scheme.
Discussion of the Related Art
MIMO technology is an abbreviation for Multiple Input Multiple Output technology. MIMO technology uses a plurality of transmission (Tx) antennas and a plurality of reception (Rx) antennas to improve the efficiency of transmission and reception (Tx/Rx) of data. In other words, MIMO technology allows a transmission end or reception end of a wireless communication system to use multiple antennas (hereinafter referred to as a multi-antenna), so that the capacity or performance can be improved. For convenience of description, the term “MIMO” can also be considered to be a multi-antenna technology.
In more detail, MIMO technology is not dependent on a single antenna path to receive a single total message. Instead, the MIMO technology collects a plurality of data fragments received via several antennas, merges the collected data fragments, and completes total data. As a result, MIMO technology can increase a data transfer rate within a predetermined-sized cell region, or can increase system coverage while guaranteeing a specific data transfer rate. Under this situation, MIMO technology can be widely applied to mobile communication terminals, repeaters, or the like. MIMO technology can extend the range of data communication, so that it can overcome the limited amount of transmission (Tx) data of mobile communication systems.
FIG. 1 is a block diagram illustrating a general MIMO communication system.
Referring to FIG. 1, the number of transmission (Tx) antennas in a transmitter is NT, and the number of reception (Rx) antennas in a receiver is NR. In this way, theoretical channel transmission capacity of the MIMO communication system when both the transmitter and the receiver use a plurality of antennas is greater than that of another case in which only the transmitter or the receiver uses several antennas. The theoretical channel transmission capacity of the MIMO communication system increases in proportion to the number of antennas. Therefore, data transfer rate and frequency efficiency are greatly increased. Provided that a maximum data transfer rate acquired when a single antenna is used is set to Ro, a data transfer rate acquired when multiple antennas are used can theoretically increase by a predetermined amount that corresponds to the maximum data transfer rate (Ro) multiplied by a rate of increase Ri. The rate of increase (Ri) can be represented by the following equation 1.

Ri=min(NT,NR)  [Equation 1]

For example, provided that a MIMO system uses four transmission (Tx) antennas and four reception (Rx) antennas, the MIMO system can theoretically acquire a high data transfer rate which is four times higher than that of a single antenna system. After the above-mentioned theoretical capacity increase of the MIMO system was demonstrated in the mid-1990s, many developers began to conduct intensive research into a variety of technologies which can substantially increase a data transfer rate using the theoretical capacity increase. Some of the above technologies have been reflected in a variety of wireless communication standards, for example, a third-generation mobile communication or a next-generation wireless LAN, etc.
The above-mentioned MIMO technology can be classified into a spatial diversity scheme (also called a Transmit Diversity scheme) and a spatial multiplexing scheme. The spatial diversity scheme increases transmission reliability using symbols passing various channel paths. The spatial multiplexing scheme simultaneously transmits a plurality of data symbols via a plurality of transmission (Tx) antennas, so that it increases a transfer rate of data. In addition, the combination of the spatial diversity scheme and the spatial multiplexing scheme has also been recently developed to properly acquire unique advantages of the two schemes.
In association with the MIMO technology, a variety of MIMO-associated technologies have been intensively researched by many companies or developers, for example, research into an information theory associated with a MIMO communication capacity calculation under various channel environments or multiple access environments, research into radio frequency (RF) channel measurement and modeling of the MIMO system, and research into a space-time signal processing technology for increasing transmission reliability and data transfer rate.
In a 3rd Generation Partnership Project Long Term Evolution (3GPP LTE) system, the above-mentioned MIMO scheme is applied to only downlink signal transmission of the 3GPP LTE system. The MIMO technology may also be applied to uplink signal transmission. In this case, a transmitter structure is changed to implement the MIMO technology, so that a Peak power to Average Power Ratio (PAPR) or Cubic Metric (CM) characteristics may be deteriorated. Therefore, there is needed a new technology capable of effectively applying the MIMO scheme to uplink signal transmission.
SUMMARY OF THE INVENTION
Accordingly, the present invention is directed to a method and apparatus for transmitting uplink signals via multiple antennas that substantially obviate one or more problems due to limitations and disadvantages of the related art.
An object of the present invention is to provide a technology for effectively carrying out uplink signal transmission according to a MIMO scheme.
Additional advantages, objects, and features of the invention will be set forth in part in the description which follows and in part will become apparent to those having ordinary skill in the art upon examination of the following or may be learned from practice of the invention. The objectives and other advantages of the invention may be realized and attained by the structure particularly pointed out in the written description and claims hereof as well as the appended drawings.
To achieve these objects and other advantages and in accordance with the purpose of the invention, as embodied and broadly described herein, a method for enabling a user equipment (UE) to transmit uplink signals via multiple antennas includes mapping the uplink signals to a predetermined number of layers, performing Discrete Fourier Transform (DFT) spreading upon each of the predetermined number of layer signals, precoding the DFT-spread layer signals by selecting a specific precoding matrix established in a manner that one layer signal is transmitted to each of the multiple antennas from among a prestored codebook, and performing a predetermined process for constructing a Single Carrier-Frequency Division Multiple Access (SC-FDMA) symbol upon the precoded signals, and transmitting the processed signals to a base station (BS) via the multiple antennas.
The specific precoding matrix may be a precoding matrix established in a manner that the multiple antennas have uniform transmission power therebetween. The specific precoding matrix may be a precoding matrix established in a manner that the predetermined number of layers have uniform transmission power therebetween
The codebook may include a first type precoding matrix, wherein the first type precoding matrix may be configured in a form of




  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          X
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          Y
        
      
    
    ]
  
  ,




as a Rank 2 precoding matrix utilized when the number of the multiple antennas is 4 and a rank is set to 2, and may satisfy a condition of





  X
  ,
  
    Y
    ⁢
    
      
    
    ∈
    
      
        {
        
          1
          ,
          
            
              1
              +
              j
            
            
              2
            
          
          ,
          j
          ,
          
            
              1
              -
              j
            
            
              2
            
          
          ,
          
            -
            1
          
          ,
          
            
              
                -
                1
              
              -
              j
            
            
              2
            
          
          ,
          
            -
            j
          
          ,
          
            
              
                -
                1
              
              +
              j
            
            
              2
            
          
        
        }
      
      .
    
  




The Rank 2 precoding matrix may further include a precoding matrix generated when positions of individual rows of the first type precoding matrix are changed.

The Rank 2 precoding matrix may further include a second type precoding matrix configured in a form of




  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          X
        
        
          0
        
      
      
        
          0
        
        
          Y
        
      
    
    ]
  
  ,




and a third type precoding matrix configured in a form of





  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          Y
        
      
      
        
          X
        
        
          0
        
      
    
    ]
  
  ,




where individual rows of the precoding matrix may respectively correspond to four antennas of the multiple antennas, and individual columns may respectively correspond to layers.

The Rank 2 precoding matrix may further include a precoding matrix generated when positions of individual columns of the first type precoding matrix are changed.
The codebook may include a first type precoding matrix, wherein the first type precoding matrix, serving as a Rank 3 precoding matrix utilized when the number of the multiple antennas is 4 and a rank is set to 3, is configured in a form of




  
    
      [
      
        
          
            1
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            1
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            1
          
        
        
          
            X
          
          
            0
          
          
            0
          
        
      
      ]
    
    .
  
  ,




and satisfies a condition of





  X
  ∈
  
    
      {
      
        1
        ,
        
          
            1
            +
            j
          
          
            2
          
        
        ,
        j
        ,
        
          
            1
            -
            j
          
          
            2
          
        
        ,
        
          -
          1
        
        ,
        
          
            
              -
              1
            
            -
            j
          
          
            2
          
        
        ,
        
          -
          j
        
        ,
        
          
            
              -
              1
            
            +
            j
          
          
            2
          
        
      
      }
    
    .
  




The Rank 3 precoding matrix may further include a precoding matrix generated when positions of individual rows of the first type precoding matrix are changed. The Rank 3 precoding matrix may further include a precoding matrix generated when positions of individual columns of the first type precoding matrix are changed. That is, the codebook may includes a precoding matrix configured to alternatively map a first layer to first and second antennas and second and third layers to third and fourth antennas, respectively, as the precoding matrix used for the case when the number of antennas is 4 and the rank is 3.

When the number of antennas is 4, the Rank is 3, and the number of codewords is 2, one of the codeword is mapped to a single layer, and the other codeword is mapped to two layers. The precoding matrix can be configured so that the total transmission power from the layer perspective may be different in order to enforce uniform transmission power between multiple antennas. In such a case the precoding matrix column which has larger effective transmission power is mapped to the layer which is solely mapped to a single codeword. Thus in case of precoding matrix in the form of




  
    
      [
      
        
          
            1
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            1
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            1
          
        
        
          
            X
          
          
            0
          
          
            0
          
        
      
      ]
    
    .
  
  ,




the first column is mapped to the layer which is solely mapped to a single codeword, and the second and third column is mapped to layers which is mapped to the other codeword.

The codebook may include a different number of precoding matrices for each rank.
Each of the uplink signals may be entered in units of a codeword, and the mapping step of the uplink signals to the predetermined number of layers may periodically change a layer mapped to a specific codeword to another layer. One example of this periodicity can be 1 SC-FDMA symbol.
In another aspect of the present invention, a user equipment (UE) for transmitting uplink signals via multiple antennas includes multiple antennas for transmitting and receiving signals, a memory for storing a codebook having a precoding matrix established in a manner that one layer signal is transmitted to the multiple antennas, and a processor connected to the multiple antennas and the memory so as to process the uplink signal transmission. The processor includes a layer mapper for mapping the uplink signals to a predetermined number of layers corresponding to a specific rank, a Discrete Fourier Transform (DFT) module for performing DFT spreading upon each of the predetermined number of layer signals, a precoder for precoding each of the DFT-spread layer signals received from the DFT module by selecting a specific precoding matrix established in a manner that one layer signal is transmitted to each of the multiple antennas from among a codebook stored in the memory, and a transmission module for performing a predetermined process for constructing a Single Carrier-Frequency Division Multiple Access (SC-FDMA) symbol upon the precoded signals, and transmitting the processed signals to a base station (BS) via the multiple antennas.
In this case, the memory may store the codebook. The processor may perform the antenna shift and/or the layer shift either in a different way from the precoding of a precoder or through row permutation and/or column permutation of a precoding matrix.
It is to be understood that both the foregoing general description and the following detailed description of the present invention are exemplary and explanatory and are intended to provide further explanation of the invention as claimed.



BRIEF DESCRIPTION OF THE DRAWINGS
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this application, illustrate embodiment(s) of the invention and together with the description serve to explain the principle of the invention. In the drawings:
FIG. 1 is a conceptual diagram illustrating a general MIMO communication system.
FIGS. 2 and 3 illustrate a general structure of a transmitter based on a MIMO technology.
FIG. 4 is a conceptual diagram illustrating a method for precoding information of each layer and transmitting the precoded information via an antenna.
FIG. 5 is a conceptual diagram illustrating a general SC-FDMA scheme.
FIG. 6 is a conceptual diagram illustrating a method for mapping a codeword to several layers.
FIG. 7 is a conceptual diagram illustrating a method for performing a DFT upon each layer after performing codeword-to-layer mapping (i.e., codeword-layer mapping) so as to prevent a CM value for each antenna from being increased.
FIG. 8 is a conceptual diagram illustrating a method for performing permutation on the position of a row or column of a precoding matrix.
FIG. 9 is a conceptual diagram illustrating a chordal distance.
FIG. 10 is a block diagram illustrating a general base station (BS) and a general user equipment (UE).
FIGS. 11 and 12 illustrate an SC-FDMA scheme for transmitting an uplink signal in a 3GPP LTE system and an OFDMA scheme for transmitting a downlink signal in the 3GPP LTE system.
FIG. 13 is a block diagram illustrating a processor for enabling a base station (BS) to transmit a downlink signal using a MIMO scheme in a 3GPP LTE system.
FIG. 14 illustrates a processor of a UE according to one embodiment of the present invention.



DETAILED DESCRIPTION OF THE INVENTION
Reference will now be made in detail to the preferred embodiments of the present invention, examples of which are illustrated in the accompanying drawings. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts.
The detailed description, which will be given below with reference to the accompanying drawings, is intended to explain exemplary embodiments of the present invention, rather than to show the only embodiments that can be implemented according to the present invention. The following detailed description includes specific details in order to provide a thorough understanding of the present invention. However, it will be apparent to those skilled in the art that the present invention may be practiced without such specific details. For example, the following description will be given centering on specific terms, but the present invention is not limited thereto and any other terms may be used to represent the same meanings. Also, wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts.
Peak power to Average Power Ratio (PAPR) is a parameter indicating characteristics of a waveform. PAPR is a specific value acquired when a peak amplitude of the waveform is divided by a time-averaged Root Mean Square (RMS) value of the waveform. PAPR is a dimensionless value. In general, a PAPR of a single carrier signal is better than that of a multi-carrier signal.
An LTE-Advanced scheme can implement MIMO technology using Single Carrier-Frequency Division Multiple Access (SC-FDMA) so as to maintain a superior CM property. When using general precoding, a signal including information corresponding to several layers is multiplexed and transmitted via a single antenna, so that the signal transmitted via this antenna may be considered to be a kind of multi-carrier signal. PAPR is associated with a dynamic range that must be supported by a power amplifier of a transmitter, and a CM value is another value capable of being used as a substitute for the PAPR.
FIG. 2 shows a general structure of a transmitter based on a MIMO technology.
In FIG. 2, one or more codewords are mapped to a plurality of layers. In this case, mapping information is mapped to each physical antenna by a precoding process, and is then transmitted via each physical antenna.
FIG. 3 is a detailed block diagram illustrating the MIMO-based transmitter shown in FIG. 2.
The term ‘codeword’ indicates that Cyclic Redundancy Check (CRC) bits are attached to data information and are then encoded by a specific coding method. There are a variety of coding methods, for example, a turbo code, a tail biting convolution code, and the like. Each codeword is mapped to one or more layers (i.e., one or more virtual layers), and a total number of mapped layers is equal to a rank value. In other words, if a transmission rank is 3, a total number of transmission layers is also set to 3. Information mapped to each layer is precoded. In this case, data information mapped to each layer is mapped to a physical layer through a precoding process (where, the term ‘layer’ means a virtual layer as far as it especially designates a physical layer). Information is transmitted to each antenna via each physical layer. Under the condition that no specified explanation is shown in FIG. 3, the precoding is carried out in a frequency domain, and an OFDM information transmission scheme is used for information mapped to the physical layer. The information mapped to the physical layer is mapped to a specific frequency domain, and is then IFFT-processed. After that, a cyclic prefix (CP) is attached to the IFFT result. Thereafter, information is transmitted to each antenna via a radio frequency (RF) chain.
The precoding process may be carried out by matrix multiplication. In each of the matrices, the number of rows is equal to the number of physical layers (i.e., the number of antennas), and the number of columns is equal to a rank value. The rank value is equal to the number of layers, so that the number of columns is equal to the number of layers. Referring to the following equation 2, information mapped to a layer (i.e., a virtual layer) is x1 and x2, each element pij of a (4×2) matrix is a weight used for precoding. y1, y2, y3, and y4 are information mapped to physical layers, and are transmitted via respective antennas using individual OFDM transmission schemes.




  
    
      
        
          [
          
            
              
                
                  y
                  1
                
              
            
            
              
                
                  y
                  2
                
              
            
            
              
                
                  y
                  3
                
              
            
            
              
                
                  y
                  4
                
              
            
          
          ]
        
        =
        
          
            [
            
              
                
                  
                    p
                    11
                  
                
                
                  
                    p
                    21
                  
                
              
              
                
                  
                    p
                    12
                  
                
                
                  
                    p
                    22
                  
                
              
              
                
                  
                    p
                    13
                  
                
                
                  
                    p
                    23
                  
                
              
              
                
                  
                    p
                    14
                  
                
                
                  
                    p
                    24
                  
                
              
            
            ]
          
          ·
          
            [
            
              
                
                  
                    x
                    1
                  
                
              
              
                
                  
                    x
                    2
                  
                
              
            
            ]
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          2
        
        ]
      
    
  




In the following description, a virtual layer will hereinafter be referred to as a layer so long as such use will not lead to confusion. An operation for mapping a virtual layer signal to a physical layer will hereinafter be considered to be an operation for directly mapping a layer to an antenna.
The precoding method can be mainly classified into two methods, i.e., a wideband precoding method and a subband precoding method.
The wideband precoding method is as follows. According to the wideband precoding method, when precoding is carried out in a frequency domain, the same precoding matrix is applied to all information transmitted to the frequency domain.
FIG. 4 is a conceptual diagram illustrating a method for precoding information of each layer and transmitting the precoded information via an antenna.
Referring to FIG. 4, it can be recognized that information corresponding to a plurality of layers is precoded while being classified according to subcarriers of each frequency domain, and the precoded information is transmitted via each antenna. All precoding matrices ‘P’ for use in the wideband precoding method are equal to each other.
The subband precoding method is provided by the extension of the wideband precoding method. The subband precoding method applies a variety of precoding matrices to each subcarrier without applying the same precoding matrix to all subcarriers. In other words, according to the subband precoding method, a precoding matrix ‘P’ is used in a specific subcarrier, and another precoding matrix ‘M’ is used in the remaining subcarriers other than the specific subcarrier. Herein, element values of the precoding matrix ‘P’ are different from those of the other precoding matrix ‘M’.
Uplink signal transmission is relatively sensitive to PAPR or CM properties as compared to downlink signal transmission. The increase of filter costs caused by the increase of PAPR or CM properties may generate more serious problems in a user equipment (UE). Thus, the SC-FDMA scheme is used for uplink signal transmission.
FIG. 5 is a conceptual diagram illustrating a general SC-FDMA scheme.
As shown in FIG. 5, the OFDM scheme and the SC-FDMA scheme are considered to be identical with each other, because they convert a serial signal into parallel signals, map the parallel signals to subcarriers, perform an IDFT or IFFT process on the mapped signals, convert the IDFT- or IFFT-processed signals into a serial signal, attach a cyclic prefix (CP) to the resultant serial signal, and transmit the CP resultant signal via a radio frequency (RF) module. However, in contrast to the OFDM scheme, the SC-FDMA scheme converts parallel signals into a serial signal, and performs DFT spreading upon the serial signal, so that it reduces the influence of a next IDFT or IFFT process and maintains a single signal characteristic of more than a predetermined level.
In the meantime, the reason why the CM value is degraded when a MIMO scheme is applied to uplink signal transmission is as follows. If a plurality of single-carrier signals each having good CM properties is simultaneously overlapped with each other, the overlapped signals may have poor CM properties. Therefore, if the SC-FDMA system multiplexes output information of several layers using a minimum number of single-carrier signals or one single-carrier signal on a single physical antenna, a transmission signal having a good CM can be generated.
A codeword-layer mapping process may be performed before information to be transmitted is precoded. Since the SC-FDMA scheme is generally used for one transmission mode (1Tx), the number of layers is 1. However, if the SC-FDMA scheme supports a MIMO scheme, the number of layers is plural, and a codeword composed of a single transport block may be mapped to a plurality of layers.
FIG. 6 is a conceptual diagram illustrating a method for mapping a codeword to several layers.
Referring to FIG. 6, if the codeword-layer mapping is carried out after a DFT process for the SC-FDMA scheme is performed, a CM value may be increased. That is, because an output signal of a DFT block undergoes other processes before entering an IFFT module, i.e., because the output signal of the DFT block is divided into two layers, a CM value may be increased.
FIG. 7 is a conceptual diagram illustrating a method for performing DFT upon each layer after performing codeword-to-layer mapping (i.e., a codeword-layer mapping) so as to prevent a CM value for each antenna from being increased.
Therefore, if the number of DFT blocks is changed while being classified according to layer numbers based on a rank value, a low CM value can be maintained. That is, the output signal of the DFT block is directly input to the IFFT block without passing through other processes, so that a low CM value can be maintained. In the case of actual implementation, a plurality of layers may share a single DFT block.
If a plurality of layer signals is transmitted via a single antenna by applying the MIMO scheme to uplink signal transmission, a PAPR or a CM property may be deteriorated. In order to overcome the above-mentioned problem, the following embodiments of the present invention will describe a method for designing a codebook based on a precoding matrix by which only one layer signal is transmitted via a single antenna.
For convenience of description and better understanding of the present invention, in a transmission system, it is assumed that a set of signals transferred to a precoding block is set to ‘x’, and a set of precoded signals is set to ‘y’. In this case, if the precoding matrix is ‘P’, the following equation 3 is acquired.

Y=P·x  [Equation 3]

In Equation 3, a dimension of ‘P’ is NT×NL, a dimension of ‘x’ is NL×1, a dimension of ‘y’ is NT×1. In this case, NT is the number of antennas, and NL is the number of layers.
In the following description, a principle of designing a codebook that is capable of being applied to uplink signal transmission using a MIMO scheme by a UE will be firstly described in chapter (I), and a detailed format of the codebook will then be described in chapter (II).
I. Principle of Codebook Design
<2Tx Codebook>
A variety of embodiments according to a structure of a precoding matrix contained in a codebook used in a 2Tx mode will hereinafter be described.
The method according to embodiments of the present invention includes: generating a plurality of streams by mapping a codeword to a plurality of layers; and precoding the generated streams, mapping the precoded streams to a plurality of antennas, and transmitting the mapped result via the antennas. In this case, the codebook may be configured as follows. A precoding matrix used in Rank 1 and the other precoding matrix used in Rank 2 will be described in different ways.
2Tx-Rank 1 Precoding Matrix
In case of 2Tx-Rank 1, Equation 3 can be rewritten as the following equation 4 according to embodiments of the present invention.




  
    
      
        y
        =
        
          
            [
            
              
                
                  
                    y
                    1
                  
                
              
              
                
                  
                    y
                    2
                  
                
              
            
            ]
          
          =
          
            
              P
              ·
              x
            
            =
            
              
                
                  [
                  
                    
                      
                        a
                      
                    
                    
                      
                        b
                      
                    
                  
                  ]
                
                ·
                
                  [
                  
                    x
                    1
                  
                  ]
                
              
              =
              
                [
                
                  
                    
                      
                        ax
                        1
                      
                    
                  
                  
                    
                      
                        bx
                        1
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          4
        
        ]
      
    
  




In general, if it is assumed that a wideband precoding scheme is used, a specific constant value is multiplied by a signal of each layer according to a Rank 1 precoding scheme, a PAPR and CM value of a signal transmitted via each antenna in the 2Tx mode are equal to those in a 1Tx mode. Thus, when using wideband precoding, the PAPR and the CM are not affected by values of a 2Tx-Rank 1 precoding matrix.
Precoding is a method for changing a channel so as to acquire a constructive effect between signals transferred via each channel. Thus, transmission performance of each signal is improved. Accordingly, ‘a’ indicating a first element of the precoding matrix P illustrated in Equation 4 is set to ‘1’, and a second element ‘b’ of the precoding matrix P may be set to an arbitrary value. Signals transferred via respective antennas have the same power, so that all power amplifiers contained in each antenna can be maximally used. For this purpose, the above-mentioned second element ‘b’ may be a complex number having an absolute value of 1. In other words, P shown in Equation 4 may be represented by




  P
  =
  
    
      [
      
        
          
            1
          
        
        
          
            
              e
              
                j
                ⁢
                
                  
                
                ⁢
                θ
              
            
          
        
      
      ]
    
    .
  




There is a limitation in the number of precoding matrices contained in a codebook used for the precoding, because both a transmission end and a reception end must have a codebook and information about a predetermined precoding matrix is communicated between the transmission end and the reception end. As a result, the transmission end and the reception end must use a limited number of precoding matrices. For this operation, a complex number that has an absolute value of 1 and a phase corresponding to any one of +0°, +45°, +90°, +135°, +180°, −135°, −90°, and −45° may be used as each element of the precoding matrix. That is, in the above-mentioned expression,




  
    P
    =
    
      [
      
        
          
            1
          
        
        
          
            
              e
              
                j
                ⁢
                
                  
                
                ⁢
                θ
              
            
          
        
      
      ]
    
  
  ,




θ may be represented by





  θ
  ∈
  
    
      {
      
        0
        ,
        
          π
          4
        
        ,
        
          π
          2
        
        ,
        
          
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          4
        
        ,
        π
        ,
        
          
            5
            ⁢
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            6
            ⁢
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          4
        
        ,
        
          
            7
            ⁢
            π
          
          4
        
      
      }
    
    .
  




In other words, P may be represented by





  P
  ∈
  
    
      {
      
        
          [
          
            
              
                1
              
            
            
              
                1
              
            
          
          ]
        
        ,
        
          [
          
            
              
                1
              
            
            
              
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
              
            
          
          ]
        
        ,
        
          [
          
            
              
                1
              
            
            
              
                j
              
            
          
          ]
        
        ,
        
          [
          
            
              
                1
              
            
            
              
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
              
            
          
          ]
        
        ,
        
          [
          
            
              
                1
              
            
            
              
                
                  -
                  1
                
              
            
          
          ]
        
        ,
        
          [
          
            
              
                1
              
            
            
              
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
              
            
          
          ]
        
        ,
        
          [
          
            
              
                1
              
            
            
              
                
                  -
                  j
                
              
            
          
          ]
        
        ,
        
          [
          
            
              
                1
              
            
            
              
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
            
          
          ]
        
      
      }
    
    .
  




2Tx-Rank 2 Precoding Matrix
In case of a 2Tx-Rank 2, the equation 3 may be rewritten as the following equation 5.




  
    
      
        y
        =
        
          
            [
            
              
                
                  
                    y
                    1
                  
                
              
              
                
                  
                    y
                    2
                  
                
              
            
            ]
          
          =
          
            
              P
              ·
              x
            
            =
            
              
                
                  [
                  
                    
                      
                        
                          p
                          11
                        
                      
                      
                        
                          p
                          12
                        
                      
                    
                    
                      
                        
                          p
                          21
                        
                      
                      
                        
                          p
                          22
                        
                      
                    
                  
                  ]
                
                ·
                
                  [
                  
                    
                      
                        
                          x
                          1
                        
                      
                    
                    
                      
                        
                          x
                          2
                        
                      
                    
                  
                  ]
                
              
              =
              
                [
                
                  
                    
                      
                        
                          
                            p
                            11
                          
                          ⁢
                          
                            x
                            1
                          
                        
                        +
                        
                          
                            p
                            12
                          
                          ⁢
                          
                            x
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            p
                            21
                          
                          ⁢
                          
                            x
                            1
                          
                        
                        +
                        
                          
                            p
                            22
                          
                          ⁢
                          
                            x
                            2
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          5
        
        ]
      
    
  




In Equation 5, the signal yk transferred via each antenna is composed of a combination of several input signals xi, so that a CM value may be increased.
In this case, if each of p12 and p21 is set to zero ‘0’ or if each of p11 and p22 is set to zero ‘0’, only one signal can be transmitted via each antenna. Thus, if it is assumed that a CM value of a signal xi is considered to be good, a CM value of the precoded signal also becomes good. In association with FIG. 7, in the case where a codeword is mapped to each layer, DFT spreading is applied to the resultant signal mapped to each layer, and a precoding process for allowing each antenna to transmit only one layer signal is carried out, the same effect as in an IDFT or IFFT process that is performed as soon as a DFT process was performed can be acquired, and a PAPR or CM property can be maintained at a good status. A detailed description of this will hereinafter be explained in the following description.
In this case, if each of p12 and p21 is zero ‘0’, a signal corresponding to each layer is transmitted via each antenna after being multiplied by a constant complex value. As a result, although the above-mentioned constant complex value is set to 1, performance is not affected by this constant complex value of 1.
Therefore, Equation 5 can be represented by the following equation 6.




  
    
      
        
          y
          =
          
            
              [
              
                
                  
                    
                      y
                      1
                    
                  
                
                
                  
                    
                      y
                      2
                    
                  
                
              
              ]
            
            =
            
              
                P
                ·
                x
              
              =
              
                
                  
                    [
                    
                      
                        
                          
                            p
                            11
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            p
                            22
                          
                        
                      
                    
                    ]
                  
                  ·
                  
                    [
                    
                      
                        
                          
                            x
                            1
                          
                        
                      
                      
                        
                          
                            x
                            2
                          
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    
                      [
                      
                        
                          
                            1
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                        
                      
                      ]
                    
                    ·
                    
                      [
                      
                        
                          
                            
                              x
                              1
                            
                          
                        
                        
                          
                            
                              x
                              2
                            
                          
                        
                      
                      ]
                    
                  
                  =
                  
                    [
                    
                      
                        
                          
                            x
                            1
                          
                        
                      
                      
                        
                          
                            x
                            2
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          
        
        ⁢
        
          P
          ∈
          
            {
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
            }
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          6
        
        ]
      
    
  




<4Tx Codebook>
A variety of embodiments according to a structure of a precoding matrix contained in a codebook used in a 4Tx mode will hereinafter be described.
The method according to embodiments of the present invention includes: generating a plurality of streams by mapping a codeword to a plurality of layers; and precoding the generated streams, mapping the precoded streams to a plurality of antennas, and transmitting the mapped result via the antennas. In this case, the codebook may be configured as follows. Precoding matrices respectively used in Rank 1, Rank 2, Rank 3, and Rank 4 will be described in different ways.
4Tx-Rank 1 Precoding Matrix
In case of 4Tx-Rank 1, Equation 3 can be rewritten as the following equation 7.




  
    
      
        y
        =
        
          
            [
            
              
                
                  
                    y
                    1
                  
                
              
              
                
                  
                    y
                    2
                  
                
              
              
                
                  
                    y
                    3
                  
                
              
              
                
                  
                    y
                    4
                  
                
              
            
            ]
          
          =
          
            
              P
              ·
              x
            
            =
            
              
                
                  [
                  
                    
                      
                        a
                      
                    
                    
                      
                        b
                      
                    
                    
                      
                        c
                      
                    
                    
                      
                        d
                      
                    
                  
                  ]
                
                ·
                
                  [
                  
                    x
                    1
                  
                  ]
                
              
              =
              
                [
                
                  
                    
                      
                        ax
                        1
                      
                    
                  
                  
                    
                      
                        bx
                        1
                      
                    
                  
                  
                    
                      
                        cx
                        1
                      
                    
                  
                  
                    
                      
                        dx
                        1
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          7
        
        ]
      
    
  




In case of using the wideband precoding scheme in the same manner as in the 2Tx-Rank 1 codebook, a CM of a signal transmitted via each antenna by a 4Tx-Rank 1 precoding process is equal to a CM of a signal used in the 1Tx mode. Thus, all kinds of precoding matrices can be freely applied to such a CM without any problems.
4Tx-Rank 2 Precoding Matrix
In case of 4Tx-Rank 2, Equation 3 can be rewritten as the following equation 8.




  
    
      
        y
        =
        
          
            [
            
              
                
                  
                    y
                    1
                  
                
              
              
                
                  
                    y
                    2
                  
                
              
              
                
                  
                    y
                    3
                  
                
              
              
                
                  
                    y
                    4
                  
                
              
            
            ]
          
          =
          
            
              P
              ·
              x
            
            =
            
              
                
                  [
                  
                    
                      
                        
                          p
                          11
                        
                      
                      
                        
                          p
                          12
                        
                      
                    
                    
                      
                        
                          p
                          21
                        
                      
                      
                        
                          p
                          22
                        
                      
                    
                    
                      
                        
                          p
                          31
                        
                      
                      
                        
                          p
                          32
                        
                      
                    
                    
                      
                        
                          p
                          41
                        
                      
                      
                        
                          p
                          42
                        
                      
                    
                  
                  ]
                
                ·
                
                  [
                  
                    
                      
                        
                          x
                          1
                        
                      
                    
                    
                      
                        
                          x
                          2
                        
                      
                    
                  
                  ]
                
              
              =
              
                [
                
                  
                    
                      
                        
                          
                            p
                            11
                          
                          ⁢
                          
                            x
                            1
                          
                        
                        +
                        
                          
                            p
                            12
                          
                          ⁢
                          
                            x
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            p
                            21
                          
                          ⁢
                          
                            x
                            1
                          
                        
                        +
                        
                          
                            p
                            22
                          
                          ⁢
                          
                            x
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            p
                            31
                          
                          ⁢
                          
                            x
                            1
                          
                        
                        +
                        
                          
                            p
                            32
                          
                          ⁢
                          
                            x
                            2
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            p
                            41
                          
                          ⁢
                          
                            x
                            1
                          
                        
                        +
                        
                          
                            p
                            42
                          
                          ⁢
                          
                            x
                            2
                          
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          8
        
        ]
      
    
  




In a 4Tx-Rank 2 codebook, in a similar way as in the 2Tx-Rank 2 codebook, a specific element of a precoding matrix is set to zero ‘0’, so that the overlapping of signals transmitted via respective antennas is minimized and thus a CM can be maintained at a low value.
In Equation 8, if it is assumed that pk1 or pk2 in a signal (pk1x1+pk2x2) transmitted via each antenna is set to zero ‘0’, the signal transmitted via each antenna becomes equal to a signal transmitted from a single layer, and therefore a CM of the signal transmitted via each antenna can be maintained at a low value.
In one embodiment of the present invention, ‘P’ included in Equation 8 may be represented by




  P
  =
  
    
      [
      
        
          
            
              p
              11
            
          
          
            0
          
        
        
          
            
              p
              21
            
          
          
            0
          
        
        
          
            0
          
          
            
              p
              32
            
          
        
        
          
            0
          
          
            
              p
              42
            
          
        
      
      ]
    
    .
  




Equation 8 may be rewritten as the following equation 9.





  
    
      
        y
        =
        
          
            [
            
              
                
                  
                    y
                    1
                  
                
              
              
                
                  
                    y
                    2
                  
                
              
              
                
                  
                    y
                    3
                  
                
              
              
                
                  
                    y
                    4
                  
                
              
            
            ]
          
          =
          
            
              P
              ·
              x
            
            =
            
              
                
                  [
                  
                    
                      
                        
                          p
                          11
                        
                      
                      
                        0
                      
                    
                    
                      
                        
                          p
                          21
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          p
                          32
                        
                      
                    
                    
                      
                        0
                      
                      
                        
                          p
                          42
                        
                      
                    
                  
                  ]
                
                ·
                
                  [
                  
                    
                      
                        
                          x
                          1
                        
                      
                    
                    
                      
                        
                          x
                          2
                        
                      
                    
                  
                  ]
                
              
              =
              
                [
                
                  
                    
                      
                        
                          p
                          11
                        
                        ⁢
                        
                          x
                          1
                        
                      
                    
                  
                  
                    
                      
                        
                          p
                          21
                        
                        ⁢
                        
                          x
                          2
                        
                      
                    
                  
                  
                    
                      
                        
                          p
                          32
                        
                        ⁢
                        
                          x
                          2
                        
                      
                    
                  
                  
                    
                      
                        
                          p
                          42
                        
                        ⁢
                        
                          x
                          2
                        
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          9
        
        ]
      
    
  




Referring to Equation 9, only one layer is mapped to a signal transmitted via each antenna. From the viewpoint of a single layer, it is considered that the 2Tx-Rank 1 precoding is applied to information transmitted via this single layer. Thus, the 4Tx-Rank 2 precoding matrix can be configured using a 2Tx-Rank 2 precoding matrix. In other words, the 4Tx-Rank 2 precoding matrix may be a super matrix of the 2Tx-Rank 1 precoding matrix.
For example, ‘P’ according to one embodiment of the present invention can be represented by Equation 10.




  
    
      
        
          
        
        ⁢
        
          
            P
            =
            
              
                [
                
                  
                    
                      
                        [
                        
                          
                            
                              1
                            
                          
                          
                            
                              X
                            
                          
                        
                        ]
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        [
                        
                          
                            
                              1
                            
                          
                          
                            
                              Y
                            
                          
                        
                        ]
                      
                    
                  
                
                ]
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                  
                
                ]
              
            
          
          ,
          
            

          
          ⁢
          X
          ,
          
            Y
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          10
        
        ]
      
    
  




The above-mentioned 2Tx-Rank1 precoding matrix is used for a method for transmitting information by applying two antennas to a single layer signal. However, if it is assumed that there are four physical antennas, communication performance may be changed according to which combination composed of two antennas is used for data transmission. In this case, the selected combination of antennas may be changed according to a value of the precoding matrix P.
For example, according to one embodiment of the present invention, the precoding matrix P may be configured in various formats. Respective formats may indicate different antenna combinations.




  
    
      
        P
        ∈
        
          {
          
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    X
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                
              
              ]
            
            ,
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    X
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                
              
              ]
            
            ,
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                
                
                  
                    X
                  
                  
                    0
                  
                
              
              ]
            
          
          }
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          11
        
        ]
      
    
  




In Equation 11, if an appropriate value is selected as a precoding matrix P, performance improvement due to precoding can be enhanced. If the precoding matrix is configured as described above, a signal corresponding to each layer uses two antennas among a total of four antennas, channel estimation performances among respective layers become similar to each other, and a CM value for each antenna can be minimized.
Generally, although a constant value is multiplied by a specific column vector of an arbitrary precoding matrix, characteristics of the precoding matrix are not changed. Therefore, although a constant value is multiplied by a specific column of the above-mentioned precoding matrix, characteristics of the precoding matrix are not changed. As a result, the above-mentioned operation for multiplying a constant value by a specific column vector of the precoding matrix does not depart from the scope of the present invention.
In addition, if a predetermined scaling factor is multiplied by the precoding matrix shown in Equation 11, the multiplied result may be represented by the following equation 12.




  
    
      
        
          
        
        ⁢
        
          
            P
            ∈
            
              {
              
                
                  k
                  ·
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                      
                    
                    ]
                  
                
                ,
                
                  k
                  ·
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                      
                    
                    ]
                  
                
                ,
                
                  k
                  ·
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
              }
            
          
          ⁢
          
            

          
          ⁢
          
            X
            ,
            
              Y
              ∈
              
                {
                
                  1
                  ,
                  
                    
                      1
                      +
                      j
                    
                    
                      2
                    
                  
                  ,
                  j
                  ,
                  
                    
                      1
                      -
                      j
                    
                    
                      2
                    
                  
                  ,
                  
                    -
                    1
                  
                  ,
                  
                    
                      
                        -
                        1
                      
                      -
                      j
                    
                    
                      2
                    
                  
                  ,
                  
                    -
                    j
                  
                  ,
                  
                    
                      
                        -
                        1
                      
                      +
                      j
                    
                    
                      2
                    
                  
                
                }
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          12
        
        ]
      
    
  




4Tx-Rank 3 Precoding Matrix (1)
In case of 4Tx-Rank 3, Equation 3 can be rewritten as the following equation 13.




  
    
      
        y
        =
        
          
            [
            
              
                
                  
                    y
                    1
                  
                
              
              
                
                  
                    y
                    2
                  
                
              
              
                
                  
                    y
                    3
                  
                
              
              
                
                  
                    y
                    4
                  
                
              
            
            ]
          
          =
          
            
              P
              ·
              x
            
            =
            
              
                
                  [
                  
                    
                      
                        
                          p
                          11
                        
                      
                      
                        
                          p
                          12
                        
                      
                      
                        
                          p
                          13
                        
                      
                    
                    
                      
                        
                          p
                          21
                        
                      
                      
                        
                          p
                          22
                        
                      
                      
                        
                          p
                          23
                        
                      
                    
                    
                      
                        
                          p
                          31
                        
                      
                      
                        
                          p
                          32
                        
                      
                      
                        
                          p
                          33
                        
                      
                    
                    
                      
                        
                          p
                          41
                        
                      
                      
                        
                          p
                          42
                        
                      
                      
                        
                          p
                          43
                        
                      
                    
                  
                  ]
                
                ·
                
                  [
                  
                    
                      
                        
                          x
                          1
                        
                      
                    
                    
                      
                        
                          x
                          2
                        
                      
                    
                    
                      
                        
                          x
                          3
                        
                      
                    
                  
                  ]
                
              
              =
              
                 
                
                  [
                  
                    
                      
                        
                          
                            
                              p
                              11
                            
                            ⁢
                            
                              x
                              1
                            
                          
                          +
                          
                            
                              p
                              12
                            
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                              p
                              13
                            
                            ⁢
                            
                              x
                              3
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              p
                              21
                            
                            ⁢
                            
                              x
                              1
                            
                          
                          +
                          
                            
                              p
                              22
                            
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                              p
                              23
                            
                            ⁢
                            
                              x
                              3
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              p
                              31
                            
                            ⁢
                            
                              x
                              1
                            
                          
                          +
                          
                            
                              p
                              32
                            
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                              p
                              33
                            
                            ⁢
                            
                              x
                              3
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              p
                              41
                            
                            ⁢
                            
                              x
                              1
                            
                          
                          +
                          
                            
                              p
                              42
                            
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                              p
                              43
                            
                            ⁢
                            
                              x
                              3
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          13
        
        ]
      
    
  




In a 4Tx-Rank 3 precoding matrix in a similar way as in the 4Tx-Rank 2 precoding matrix, a specific element of a precoding matrix is set to zero ‘0’, so that the overlapping of signals transmitted via respective antennas is minimized and thus a CM can be maintained at a low value.
In Equation 13, if it is assumed that pk1, pk2, or pk3 in a signal (pk1x1+pk2x2+pk3x3) transmitted via each antenna is set to zero ‘0’, a CM of the signal transmitted via each antenna can be maintained at a low value.
In one embodiment of the present invention, ‘P’ included in Equation 12 may be represented by




  P
  =
  
    
      [
      
        
          
            
              p
              11
            
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            
              p
              22
            
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            
              p
              33
            
          
        
        
          
            
              p
              41
            
          
          
            
              p
              42
            
          
          
            
              P
              43
            
          
        
      
      ]
    
    .
  




Equation 13 may be rewritten as the following equation 14.





  
    
      
        y
        =
        
          
            [
            
              
                
                  
                    y
                    1
                  
                
              
              
                
                  
                    y
                    2
                  
                
              
              
                
                  
                    y
                    3
                  
                
              
              
                
                  
                    y
                    4
                  
                
              
            
            ]
          
          =
          
            
              P
              ·
              x
            
            =
            
              
                
                  [
                  
                    
                      
                        
                          p
                          11
                        
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          p
                          22
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        
                          p
                          33
                        
                      
                    
                    
                      
                        
                          p
                          41
                        
                      
                      
                        
                          p
                          42
                        
                      
                      
                        
                          P
                          43
                        
                      
                    
                  
                  ]
                
                ·
                
                  [
                  
                    
                      
                        
                          x
                          1
                        
                      
                    
                    
                      
                        
                          x
                          2
                        
                      
                    
                    
                      
                        
                          x
                          3
                        
                      
                    
                  
                  ]
                
              
              =
              
                 
                
                  [
                  
                    
                      
                        
                          
                            p
                            11
                          
                          ⁢
                          
                            x
                            1
                          
                        
                      
                    
                    
                      
                        
                          
                            p
                            22
                          
                          ⁢
                          
                            x
                            2
                          
                        
                      
                    
                    
                      
                        
                          
                            p
                            33
                          
                          ⁢
                          
                            x
                            3
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              p
                              41
                            
                            ⁢
                            
                              x
                              1
                            
                          
                          +
                          
                            
                              p
                              42
                            
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                              p
                              43
                            
                            ⁢
                            
                              x
                              3
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          14
        
        ]
      
    
  




In Rank 3, the number of layers to be transmitted is 3, and the number of physical antennas is 4. In this case, each of the three antennas may be independently mapped to a single layer. Herein, only a signal of a single layer may be mapped to the remaining one antenna, or signals of at least two layers may be mapped to the remaining one antenna. If only a signal of a specific single layer is mapped to the remaining one antenna, a CM of the signal transmitted via this antenna may have good characteristics, but communication performance of the specific single layer may be different from that of another layer. For example, in the case where information of a first layer (Layer 1) is mapped to a first antenna (Antenna 1) and a fourth antenna (Antenna 4), information of a second layer (Layer 2) is mapped to a second antenna (Antenna 2), and information of a third layer (Layer 3) is mapped to a third antenna (Antenna 3), communication performance of the Layer 1 information may be different from that of either the Layer 2 or the Layer 3.
In one embodiment of the present invention, in order to minimize a CM value for each antenna in the precoding process, the precoding matrix P may have any one of the values of P1, P2, and P3 shown in the following equation 15.




  
    
      
        
          
            
              
                P
                1
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                    
                      0
                    
                  
                
                ]
              
            
            ,
            
              
                P
                2
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                    
                      0
                    
                  
                
                ]
              
            
            ,
            
              
                P
                3
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      Z
                    
                  
                
                ]
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          where
        
        ,
        
          

        
        ⁢
        X
        ,
        Y
        ,
        
          
        
        ⁢
        
          Z
          ∈
          
             
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          15
        
        ]
      
    
  




In case of using the above-mentioned precoding matrices P1, P2, and P3, numbers of antennas used for individual layers are different from each other. However, if it is assumed that the precoding matrices P1, P2 and P3 are evenly used to transmit certain information, instead of using any one of the precoding matrices P1, P2 and P3, numbers of antennas used for individual layers may be normalized. Although the precoding matrices P1, P2 and P3 can be alternately used in a frequency domain, a single carrier property of a signal composed of a single carrier is damaged, so that a CM value is unavoidably increased. Therefore, if the precoding matrices P1, P2 and P3 are alternately applied to each SC-FDMA symbol, no additional increase in CM is achieved. In case of transmitting data, information may be decoded in units of one subframe. Thus, if the precoding matrices P1, P2 and P3 are alternately applied to each SC-FDMA symbol, each layer information of the whole information transmitted via a single subframe can be transmitted via the same number of antennas on average.
In another embodiment of the present invention, the position of an antenna used by each layer is changed so that performance can be improved. The changing of the antenna position may be carried out with time. In particular, the antenna position can be changed at each SC-FDMA symbol. A detailed method for changing the antenna position will hereinafter be described in detail.
For example, the position of a value other than ‘O’ in the precoding matrix is changed to another position in the range of a row vector, so that the position of an antenna via which each layer signal is transmitted can be changed to another position. As another example, the above-mentioned method may be implemented by a row/column permutation because position permutation is carried out between rows or columns of a given precoding matrix.
FIG. 8 is a conceptual diagram illustrating a method for performing permutation on the position of a row or column of a precoding matrix.
In more detail, FIG. 8(a) is a conceptual diagram illustrating a method for performing permutation on the position of a row, and FIG. 8(b) is a conceptual diagram illustrating a method for performing permutation on the position of a column.
In the precoding matrix shown in Equation 15, a precoding matrix P1 can be row-permuted and/or column-permuted, so that a precoding matrix P2 or P3 can be generated. Therefore, in the structure such as the precoding matrix P1, P2 or P3, a new unique precoding matrix can be acquired only by row permutation.
The order of rows changed by row permutation available in the 4Tx mode can be represented by the following expression.
{1, 2, 3, 4}, {1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2},
{1, 4, 2, 3}, {1, 4, 3, 2}, {2, 1, 3, 4}, {2, 1, 4, 3},
{2, 3, 1, 4}, {2, 3, 4, 1}, {2, 4, 1, 3}, {2, 4, 3, 1},
{3, 2, 1, 4}, {3, 2, 4, 1}, {3, 1, 2, 4}, {3, 1, 4, 2},
{3, 4, 2, 1}, {3, 4, 1, 2}, {4, 2, 3, 1}, {4, 2, 1, 3},
{4, 3, 2, 1}, {4, 3, 1, 2}, {4, 1, 2, 3}, {4, 1, 3, 2}
In the above-mentioned expression, {w, x, y, z} means that row vectors 1, 2, 3 and 4 of the precoding matrix are rearranged in the order of parenthesized numbers on the condition that a given precoding matrix Pk exists.
By row permutation, signals corresponding to a specific layer are mapped to different antennas. By column permutation, the same effect as in the switching of information of different layers can be acquired. If there is no need to distinguish performance of each layer, and a system for requesting similar performance from each layer need not utilize the column permutation. Thus, the effect corresponding to antenna selection can be acquired using only the row permutation.
In the meantime, in the case where a given scaling factor is multiplied by each precoding matrix shown in Equation 15, the result can be represented by the following equation 16.




  
    
      
        
          
            P
            1
          
          =
          
            
              
                k
                ·
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
              
              ⁢
              
                
              
              ⁢
              
                P
                2
              
            
            =
            
              
                
                  k
                  ·
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
                ⁢
                
                  
                
                ⁢
                
                  P
                  3
                
              
              =
              
                k
                ·
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        Z
                      
                    
                  
                  ]
                
              
            
          
        
        ,
        
          

        
        ⁢
        X
        ,
        Y
        ,
        
          Z
          ∈
          
            {
            
              1
              ,
              
                
                  1
                  +
                  j
                
                
                  2
                
              
              ,
              j
              ,
              
                
                  1
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                1
              
              ,
              
                
                  
                    -
                    1
                  
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                j
              
              ,
              
                
                  
                    -
                    1
                  
                  +
                  j
                
                
                  2
                
              
            
            }
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          16
        
        ]
      
    
  




4Tx-Rank 3 Precoding Matrix (2)
In case of 4Tx-Rank 3, if each antenna transmits information corresponding to only one layer, a CM value of a signal transmitted via each antenna can be maintained at a low value, however information of only one layer is transmitted via only one antenna so that communication performance can be deteriorated. Therefore, in case of 4Tx-Rank 3, there is a need for a codebook to be designed in a manner that a maximum of two layers are multiplexed and transmitted via a single antenna, so that the increment of CM can be minimized and at the same time communication performance can be increased.
In accordance with one embodiment of the present invention, when information corresponding to two layers is transmitted via a single antenna, the precoding matrix P shown in Equation 13 can be represented by P4 of Equation 17 or P5 of Equation 18.




  
    
      
        
          
        
        ⁢
        
          
            
              P
              4
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    Z
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            X
            ≠
            Z
          
          ,
          
            

          
          ⁢
          X
          ,
          Y
          ,
          
            Z
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          17
        
        ]
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              P
              5
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    X
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    Z
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          X
          ,
          Y
          ,
          
            Z
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          18
        
        ]
      
    
  




In Equation 17, in order to satisfy Rank 3, ‘X’ must be different from ‘Z’ in the precoding matrix P4.
A method for using the precoding matrix P4 or P5 has a disadvantage in that only a signal of a single layer is transmitted via the other antenna whereas a signal of two layers is multiplexed and transmitted via a specific antenna.
In one embodiment of the present invention, in order to obviate the above-mentioned problem, the precoding matrix P may have any one of values Pa, P7 and Pa shown in the following equation 19.




  
    
      
        
          
            
          
          ⁢
          
            
              
                
                  P
                  6
                
                =
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        Z
                      
                    
                    
                      
                        X
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        1
                      
                    
                    
                      
                        A
                      
                      
                        0
                      
                      
                        C
                      
                    
                  
                  ]
                
              
              ,
              
                
                  P
                  7
                
                =
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        Z
                      
                    
                    
                      
                        X
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        B
                      
                      
                        C
                      
                    
                  
                  ]
                
              
              ,
              
                

              
              ⁢
              
                
              
              ⁢
              
                
                  P
                  8
                
                =
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        Z
                      
                    
                    
                      
                        X
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        1
                      
                    
                    
                      
                        A
                      
                      
                        B
                      
                      
                        0
                      
                    
                  
                  ]
                
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            ⁢
            
              where
              ,
              
                

              
              ⁢
              X
              ,
              Y
              ,
              Z
              ,
              A
              ,
              
                
              
              ⁢
              B
              ,
              
                
              
              ⁢
              
                C
                ∈
              
            
          
           
        
        ⁢
        
          {
          
            1
            ,
            
              
                1
                +
                j
              
              
                2
              
            
            ,
            j
            ,
            
              
                1
                -
                j
              
              
                2
              
            
            ,
            
              -
              1
            
            ,
            
              
                
                  -
                  1
                
                -
                j
              
              
                2
              
            
            ,
            
              -
              j
            
            ,
            
              
                
                  -
                  1
                
                +
                j
              
              
                2
              
            
          
          }
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          19
        
        ]
      
    
  




In association with the precoding matrix P4, P5, P6, P7 or P8, row permutation and/or column permutation can be carried out on the 4Tx-Rank 3 precoding matrix. Because the row permutation and the column permutation are carried out, an antenna selection function and layer permutation function for enabling a signal of a specific layer to be transmitted via an arbitrary antenna can be implemented by the precoding.
In one embodiment of the present invention, individual column vectors of the precoding matrix may be configured to have orthogonality therebetween.
If individual column vectors of the precoding matrix are configured to have orthogonality therebetween, the precoding matrix is able to satisfy properties of a one side unitary matrix. That is, the precoding matrix P can have a characteristic denoted by the following equation 20.

PHP=α·I≠PPH  [Equation 20]

In one embodiment of the present invention, the precoding matrix of Rank 3 can be configured in the form of the following equation 21. The precoding matrix P for satisfying the following equation 21 is able to satisfy the relationship illustrated in Equation 20.




  
    
      
        
          
        
        ⁢
        
          
            P
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    
                      -
                      X
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          X
          ,
          
            Y
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          21
        
        ]
      
    
  




In Equation 21, since the relationship denoted by




  
    
      P
      H
    
    ⁢
    P
  
  =
  
    
      
        [
        
          
            
              1
            
            
              
                X
                *
              
            
            
              0
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              1
            
            
              
                Y
                *
              
            
          
          
            
              1
            
            
              
                -
                
                  X
                  *
                
              
            
            
              0
            
            
              0
            
          
        
        ]
      
      ·
      
        [
        
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              X
            
            
              0
            
            
              
                -
                X
              
            
          
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              0
            
            
              Y
            
            
              0
            
          
        
        ]
      
    
    =
    
      
        [
        
          
            
              2
            
            
              0
            
            
              0
            
          
          
            
              0
            
            
              2
            
            
              0
            
          
          
            
              0
            
            
              0
            
            
              2
            
          
        
        ]
      
      =
      
        α
        ·
        I
      
    
  




is satisfied, it can be recognized that the matrix P satisfies Equation 20.

4Tx-Rank 4 Precoding Matrix (1)
In case of 4Tx-Rank 4, Equation 3 can be rewritten as the following equation 22.




  
    
      
        y
        =
        
          
            [
            
              
                
                  
                    y
                    1
                  
                
              
              
                
                  
                    y
                    2
                  
                
              
              
                
                  
                    y
                    3
                  
                
              
              
                
                  
                    y
                    4
                  
                
              
            
            ]
          
          =
          
            
              P
              ·
              x
            
            =
            
              
                
                  [
                  
                    
                      
                        
                          p
                          11
                        
                      
                      
                        
                          p
                          12
                        
                      
                      
                        
                          p
                          13
                        
                      
                      
                        
                          p
                          14
                        
                      
                    
                    
                      
                        
                          p
                          21
                        
                      
                      
                        
                          p
                          22
                        
                      
                      
                        
                          p
                          23
                        
                      
                      
                        
                          p
                          24
                        
                      
                    
                    
                      
                        
                          p
                          31
                        
                      
                      
                        
                          p
                          32
                        
                      
                      
                        
                          p
                          33
                        
                      
                      
                        
                          p
                          34
                        
                      
                    
                    
                      
                        
                          p
                          41
                        
                      
                      
                        
                          p
                          42
                        
                      
                      
                        
                          p
                          43
                        
                      
                      
                        
                          p
                          44
                        
                      
                    
                  
                  ]
                
                ·
                
                  [
                  
                    
                      
                        
                          x
                          1
                        
                      
                    
                    
                      
                        
                          x
                          2
                        
                      
                    
                    
                      
                        
                          x
                          3
                        
                      
                    
                    
                      
                        
                          x
                          4
                        
                      
                    
                  
                  ]
                
              
              =
              
                 
                
                  [
                  
                    
                      
                        
                          
                            
                              p
                              11
                            
                            ⁢
                            
                              x
                              1
                            
                          
                          +
                          
                            
                              p
                              12
                            
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                              p
                              13
                            
                            ⁢
                            
                              x
                              3
                            
                          
                          +
                          
                            
                              p
                              14
                            
                            ⁢
                            
                              x
                              4
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              p
                              21
                            
                            ⁢
                            
                              x
                              1
                            
                          
                          +
                          
                            
                              p
                              22
                            
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                              p
                              23
                            
                            ⁢
                            
                              x
                              3
                            
                          
                          +
                          
                            
                              p
                              24
                            
                            ⁢
                            
                              x
                              4
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              p
                              31
                            
                            ⁢
                            
                              x
                              1
                            
                          
                          +
                          
                            
                              p
                              32
                            
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                              p
                              33
                            
                            ⁢
                            
                              x
                              3
                            
                          
                          +
                          
                            
                              p
                              34
                            
                            ⁢
                            
                              x
                              4
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              p
                              41
                            
                            ⁢
                            
                              x
                              1
                            
                          
                          +
                          
                            
                              p
                              42
                            
                            ⁢
                            
                              x
                              2
                            
                          
                          +
                          
                            
                              p
                              43
                            
                            ⁢
                            
                              x
                              3
                            
                          
                          +
                          
                            
                              p
                              44
                            
                            ⁢
                            
                              x
                              4
                            
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          22
        
        ]
      
    
  




In case of 4Tx-Rank 4, signals from four layers are multiplexed and transmitted via respective antennas.
In one embodiment of the present invention, if a precoding matrix is configured in the form of an identity matrix, one antenna is able to transmit only a signal corresponding to a single layer. In this case, Equation 22 can be rewritten as the following equation 23.




  
    
      
        y
        =
        
          
            [
            
              
                
                  
                    y
                    1
                  
                
              
              
                
                  
                    y
                    2
                  
                
              
              
                
                  
                    y
                    3
                  
                
              
              
                
                  
                    y
                    4
                  
                
              
            
            ]
          
          =
          
            
              P
              ·
              x
            
            =
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
                ·
                
                  [
                  
                    
                      
                        
                          x
                          1
                        
                      
                    
                    
                      
                        
                          x
                          2
                        
                      
                    
                    
                      
                        
                          x
                          3
                        
                      
                    
                    
                      
                        
                          x
                          4
                        
                      
                    
                  
                  ]
                
              
              =
              
                [
                
                  
                    
                      
                        x
                        1
                      
                    
                  
                  
                    
                      
                        x
                        2
                      
                    
                  
                  
                    
                      
                        x
                        3
                      
                    
                  
                  
                    
                      
                        x
                        4
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          23
        
        ]
      
    
  




4Tx-Rank 4 Precoding Matrix (2)
In a 4Tx-Rank 4 codebook, if the number of Rank-4 precoding matrices is increased, communication performance can also be increased. As the number of precoding matrices contained in a codebook increases, a precoding matrix closer to an actual channel can be selected. Thus, the greater the number of precoding matrices, the higher the performance. However, the selection of a precoding matrix in a codebook becomes complicated, so that it is preferable that an appropriate number of precoding matrices should be included in such a codebook. However, in case of 4Tx-Rank 4, in order to transmit only a signal corresponding to a single layer via each antenna, the precoding matrix should be an identity matrix, so that a signal corresponding to two or more layers should sometimes be transmitted via a single antenna in case of using several Rank 4 precoding matrices. Therefore, in order to minimize a CM value and increase the number of Rank 4 precoding matrices in a codebook, a specific element of the precoding matrix may be set to zero ‘0’. In Equation 22, if it is assumed that two values of pk1, pk2, pk3 and pk4 in the signal (pk1x1+pk2x2+pk3x3+pk4x4) transmitted via each antenna are respectively set to zero ‘0’, a CM of the signal transmitted via each antenna can be maintained at a low value.
In one embodiment of the present invention, the precoding matrix can be set to P9 of the following equation 24, P10 of the following equation 25, or P11 of the following equation 26.




  
    
      
        
          
            
          
          ⁢
          
            
              
                P
                9
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      A
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      B
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                    
                      C
                    
                  
                  
                    
                      D
                    
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
            
            ,
            
              1
              ≠
              
                A
                ⁢
                
                  
                
                ⁢
                B
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                D
              
            
            ,
            
              

            
            ⁢
            A
            ,
            B
            ,
            C
            ,
            
              D
              ∈
            
          
           
        
        ⁢
        
          {
          
            1
            ,
            
              
                1
                +
                j
              
              
                2
              
            
            ,
            j
            ,
            
              
                1
                -
                j
              
              
                2
              
            
            ,
            
              -
              1
            
            ,
            
              
                
                  -
                  1
                
                -
                j
              
              
                2
              
            
            ,
            
              -
              j
            
            ,
            
              
                
                  -
                  1
                
                +
                j
              
              
                2
              
            
          
          }
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          24
        
        ]
      
    
  
  
    
      
        
          
            
          
          ⁢
          
            
              
                P
                10
              
              =
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        A
                      
                      
                        0
                      
                      
                        C
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        B
                      
                      
                        0
                      
                      
                        D
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                ⁢
                where
              
            
            ,
            
              A
              ≠
              C
            
            ,
            
              B
              ≠
              D
            
            ,
            
              

            
            ⁢
            A
            ,
            B
            ,
            C
            ,
            
              D
              ∈
            
          
           
        
        ⁢
        
          {
          
            1
            ,
            
              
                1
                +
                j
              
              
                2
              
            
            ,
            j
            ,
            
              
                1
                -
                j
              
              
                2
              
            
            ,
            
              -
              1
            
            ,
            
              
                
                  -
                  1
                
                -
                j
              
              
                2
              
            
            ,
            
              -
              j
            
            ,
            
              
                
                  -
                  1
                
                +
                j
              
              
                2
              
            
          
          }
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          25
        
        ]
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              P
              11
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    A
                  
                  
                    0
                  
                  
                    
                      -
                      A
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    B
                  
                  
                    0
                  
                  
                    
                      -
                      B
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          A
          ,
          
            B
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          26
        
        ]
      
    
  




The precoding matrix P9, P10 or P11 is an example of a precoding matrix for transmitting a signal corresponding to a maximum of two layers via each antenna. As described above, the row/column permutation is performed on the precoding matrix P9, P10 or P11, so that signals of different layers can be transmitted via different antennas.
The precoding matrix P11 is a unitary matrix, so that the advantages of the unitary precoding matrix can be utilized.
4Tx-Rank 4 Precoding Matrix (3)
In case of 4Tx-Rank 4, only one element among elements of each row of a precoding matrix can be set to zero ‘0’. In case of using the above method, a signal corresponding to three layers can be multiplexed and transmitted via a single antenna, so that communication performance can be improved. However, in the case of using the above-mentioned method, a CM value further increases, but the increased CM value may be lower than another CM value acquired when all elements of the precoding matrix are each set to any of other values except for zero ‘0’. Thus, the above-mentioned method can be effectively utilized under a good SNR status in which a transmitter need not transmit data or information at a maximum transmission power.
In one embodiment of the present invention, the precoding matrix P can be represented by P12 of the following equation 27, P13 of the following equation 28, P14 of the following equation 29, or P15 of the following equation 30.




  
    
      
        
          
        
        ⁢
        
          
            
              P
              12
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    
                      m
                      12
                    
                  
                  
                    
                      m
                      13
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      m
                      23
                    
                  
                  
                    
                      m
                      24
                    
                  
                
                
                  
                    
                      m
                      31
                    
                  
                  
                    0
                  
                  
                    1
                  
                  
                    
                      m
                      34
                    
                  
                
                
                  
                    
                      m
                      41
                    
                  
                  
                    
                      m
                      42
                    
                  
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              m
              ik
            
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          i
          ,
          
            k
            =
            1
          
          ,
          2
          ,
          3
          ,
          4
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          27
        
        ]
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              P
              13
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    
                      m
                      21
                    
                  
                  
                    0
                  
                  
                    
                      m
                      23
                    
                  
                  
                    
                      m
                      24
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      m
                      33
                    
                  
                  
                    
                      m
                      34
                    
                  
                
                
                  
                    0
                  
                  
                    
                      m
                      42
                    
                  
                  
                    
                      m
                      43
                    
                  
                  
                    
                      m
                      44
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              m
              ik
            
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          i
          ,
          
            k
            =
            1
          
          ,
          2
          ,
          3
          ,
          4
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          28
        
        ]
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              P
              14
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                  
                    1
                  
                
                
                  
                    
                      m
                      21
                    
                  
                  
                    0
                  
                  
                    
                      m
                      23
                    
                  
                  
                    
                      m
                      24
                    
                  
                
                
                  
                    
                      m
                      31
                    
                  
                  
                    0
                  
                  
                    
                      m
                      33
                    
                  
                  
                    
                      m
                      34
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      m
                      43
                    
                  
                  
                    
                      m
                      44
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              m
              ik
            
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          i
          ,
          
            k
            =
            1
          
          ,
          2
          ,
          3
          ,
          4
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          29
        
        ]
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              P
              15
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    c
                  
                  
                    
                      -
                      c
                    
                  
                  
                    c
                  
                
                
                  
                    a
                  
                  
                    0
                  
                  
                    
                      -
                      a
                    
                  
                  
                    
                      -
                      a
                    
                  
                
                
                  
                    b
                  
                  
                    
                      -
                      b
                    
                  
                  
                    0
                  
                  
                    b
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          a
          ,
          b
          ,
          
            c
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          30
        
        ]
      
    
  




The precoding matrix P15 illustrated in Equation 30 is a unitary matrix, so that the advantages of the unitary precoding matrix can be utilized.
A matrix acquired when a constant is multiplied by a specific column of the precoding matrix or another matrix acquired when row/column permutation is performed on the above-mentioned precoding matrix may be used as a part of a codebook.
Elements of the above-mentioned precoding matrices are selected from a complex number that has an absolute value of 1 and a phase corresponding to any one of +0°, +45°, +90°, +135°, +180°, −135°, −90°, and −45°. That is, elements of the precoding matrix are selected from




  
    {
    
      
        e
        jθ
      
      ,
      
        θ
        ∈
        
          {
          
            0
            ,
            
              π
              4
            
            ,
            
              π
              2
            
            ,
            
              
                3
                ⁢
                π
              
              4
            
            ,
            π
            ,
            
              
                5
                ⁢
                π
              
              4
            
            ,
            
              
                6
                ⁢
                π
              
              4
            
            ,
            
              
                7
                ⁢
                π
              
              4
            
          
          }
        
      
    
    }
  
  .




For example, the above-mentioned selection has been disclosed only for illustrative purposes, and the elements of the precoding matrix may be selected from a set of complex numbers that have an absolute value of 1 and different phases. For example, each element of the precoding matrix may be selected from





  {
  
    
      e
      
        jθ
        +
        a
      
    
    ,
    
      θ
      ∈
      
        {
        
          0
          ,
          
            π
            4
          
          ,
          
            π
            2
          
          ,
          
            
              3
              ⁢
              π
            
            4
          
          ,
          π
          ,
          
            
              5
              ⁢
              π
            
            4
          
          ,
          
            
              6
              ⁢
              π
            
            4
          
          ,
          
            
              7
              ⁢
              π
            
            4
          
        
        }
      
    
  
  }




(where a is an arbitrary constant).

Power Balancing
In the meantime, transmission power balancing of respective antennas and/or transmission power balancing of respective layers may be considered to be an important matter in codebook design. If transmission powers of individual antennas are not adjusted for maximal uniformity, there arises a difference in performance between respective transmission antennas. Likewise, if transmission powers of individual layers are not adjusted for maximal uniformity, there arises a difference in performance between respective codewords.
Therefore, one embodiment of the present invention proposes a method for designing a precoding matrix in consideration of antenna power balancing using norms of all elements (i.e., all elements of a specific row of the precoding matrix) corresponding to individual antennas of the precoding matrix. In more detail, the precoding matrix shown in the following equation 31 may be utilized in the form of a antenna power balancing shown in the following equation 32.




  
    
      
        
          
        
        ⁢
        
          P
          =
          
            k
            ·
            
              [
              
                
                  
                    
                      p
                      11
                    
                  
                  
                    ...
                  
                  
                    
                      p
                      
                        1
                        ⁢
                        
                          
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                  
                
                
                  
                    
                      p
                      21
                    
                  
                  
                    ⋱
                  
                  
                    
                      p
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                  
                
                
                  
                    ⋮
                  
                  
                    ⋱
                  
                  
                    ⋮
                  
                
                
                  
                    
                      p
                      
                        
                          N
                          T
                        
                        ⁢
                        1
                      
                    
                  
                  
                    ...
                  
                  
                    
                      p
                      
                        
                          N
                          T
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                  
                
              
              ]
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          31
        
        ]
      
    
  
  
    
      
        P
        =
        
          k
          ·
          
            [
            
              
                
                  
                    
                      p
                      11
                    
                    
                      
                        
                          p
                          11
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              1
                              ⁢
                              
                                
                              
                              ⁢
                              
                                N
                                L
                              
                            
                            2
                          
                        
                      
                    
                  
                
                
                  ...
                
                
                  
                    
                      p
                      
                        1
                        ⁢
                        
                          
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                    
                      
                        
                          p
                          11
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              1
                              ⁢
                              
                                
                              
                              ⁢
                              
                                N
                                L
                              
                            
                            2
                          
                        
                      
                    
                  
                
              
              
                
                  
                    
                      p
                      21
                    
                    
                      
                        
                          p
                          21
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              
                                N
                                L
                              
                            
                            2
                          
                        
                      
                    
                  
                
                
                  ⋱
                
                
                  
                    
                      p
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                    
                      
                        
                          p
                          21
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              
                                N
                                L
                              
                            
                            2
                          
                        
                      
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    
                      p
                      
                        
                          N
                          T
                        
                        ⁢
                        1
                      
                    
                    
                      
                        
                          p
                          
                            
                              N
                              T
                            
                            ⁢
                            1
                          
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              
                                N
                                T
                              
                              ⁢
                              
                                N
                                L
                              
                            
                            2
                          
                        
                      
                    
                  
                
                
                  ...
                
                
                  
                    
                      p
                      
                        
                          N
                          T
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                    
                      
                        
                          p
                          
                            
                              N
                              T
                            
                            ⁢
                            1
                          
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              
                                N
                                T
                              
                              ⁢
                              
                                N
                                L
                              
                            
                            2
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          32
        
        ]
      
    
  




On the other hand, one embodiment of the present invention provides a method for designing a precoding matrix in consideration of layer power balancing using norms of all elements (i.e., all elements of a specific column of the precoding matrix) of individual layers. In more detail, the precoding matrix shown in the following equation 33 may be utilized in the form of layer power balancing shown in the following equation 34.




  
    
      
        
          
        
        ⁢
        
          P
          =
          
            k
            ·
            
              [
              
                
                  
                    
                      p
                      11
                    
                  
                  
                    ...
                  
                  
                    
                      p
                      
                        1
                        ⁢
                        
                          
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                  
                
                
                  
                    
                      p
                      21
                    
                  
                  
                    ⋱
                  
                  
                    
                      p
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                  
                
                
                  
                    ⋮
                  
                  
                    ⋱
                  
                  
                    ⋮
                  
                
                
                  
                    
                      p
                      
                        
                          N
                          T
                        
                        ⁢
                        1
                      
                    
                  
                  
                    ...
                  
                  
                    
                      p
                      
                        
                          N
                          T
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                  
                
              
              ]
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          33
        
        ]
      
    
  
  
    
      
        P
        =
        
          k
          ·
          
            [
            
              
                
                  
                    
                      p
                      11
                    
                    
                      
                        
                          p
                          11
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              
                                N
                                T
                              
                              ⁢
                              1
                            
                            2
                          
                        
                      
                    
                  
                
                
                  ...
                
                
                  
                    
                      p
                      
                        1
                        ⁢
                        
                          
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                    
                      
                        
                          p
                          
                            1
                            ⁢
                            
                              
                            
                            ⁢
                            
                              N
                              L
                            
                          
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              
                                N
                                T
                              
                              ⁢
                              
                                N
                                L
                              
                            
                            2
                          
                        
                      
                    
                  
                
              
              
                
                  
                    
                      p
                      21
                    
                    
                      
                        
                          p
                          11
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              
                                N
                                T
                              
                              ⁢
                              1
                            
                            2
                          
                        
                      
                    
                  
                
                
                  ⋱
                
                
                  
                    
                      p
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                    
                      
                        
                          p
                          
                            1
                            ⁢
                            
                              
                            
                            ⁢
                            
                              N
                              L
                            
                          
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              
                                N
                                T
                              
                              ⁢
                              
                                N
                                L
                              
                            
                            2
                          
                        
                      
                    
                  
                
              
              
                
                  ⋮
                
                
                  ⋱
                
                
                  ⋮
                
              
              
                
                  
                    
                      p
                      
                        
                          N
                          T
                        
                        ⁢
                        1
                      
                    
                    
                      
                        
                          p
                          11
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              
                                N
                                T
                              
                              ⁢
                              1
                            
                            2
                          
                        
                      
                    
                  
                
                
                  ...
                
                
                  
                    
                      p
                      
                        
                          N
                          T
                        
                        ⁢
                        
                          N
                          L
                        
                      
                    
                    
                      
                        
                          p
                          
                            1
                            ⁢
                            
                              
                            
                            ⁢
                            
                              N
                              L
                            
                          
                          2
                        
                        +
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            p
                            
                              
                                N
                                T
                              
                              ⁢
                              
                                N
                                L
                              
                            
                            2
                          
                        
                      
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          34
        
        ]
      
    
  




In this case, differently from a Rank 2 precoding matrix, it can be inappropriate for the number of rows and the number of columns in a 4Tx-Rank 3 precoding matrix to simultaneously perform the antenna power balancing and the power balancing. However, in a specific situation, for example, in a system of using a layer shift that changes a layer used for transmission to another layer according to a specific pattern in a transmission mode, there occurs an effect in which a difference in performance between layers is dispersed, the layer power balancing may be relatively less important than the antenna power balancing. Therefore, one embodiment of the present invention proposes the use of a precoding matrix acquired when the antenna power balancing is firstly carried out on the condition that it is impossible to simultaneously perform the antenna power balancing and the layer power balancing.
In the meantime, the following precoding matrices among the above-mentioned 4Tx-Rank 3 precoding matrices indicate that the antenna power balancing can be carried out because two symbols are transmitted to each layer, as denoted by the following equation 35.




  
    
      
        
          
            P
            0
            ′
          
          =
          
            k
            ·
            
              [
              
                
                  
                    
                      p
                      11
                    
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      p
                      22
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    
                      p
                      33
                    
                  
                
                
                  
                    
                      
                        p
                        41
                      
                      
                        3
                      
                    
                  
                  
                    
                      
                        p
                        42
                      
                      
                        3
                      
                    
                  
                  
                    
                      
                        p
                        43
                      
                      
                        3
                      
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          
        
        ⁢
        
          

        
        ⁢
        
          
            P
            4
            ′
          
          =
          
            k
            ·
            
              [
              
                
                  
                    
                      1
                      
                        2
                      
                    
                  
                  
                    0
                  
                  
                    
                      1
                      
                        2
                      
                    
                  
                
                
                  
                    
                      X
                      
                        2
                      
                    
                  
                  
                    0
                  
                  
                    
                      Z
                      
                        2
                      
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    0
                  
                
              
              ]
            
          
        
        ⁢
        
          
        
        ⁢
        
          

        
        ⁢
        
          
            P
            5
            ′
          
          =
          
            k
            ·
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    
                      X
                      
                        2
                      
                    
                  
                  
                    
                      1
                      
                        2
                      
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      Y
                      
                        2
                      
                    
                  
                  
                    
                      1
                      
                        2
                      
                    
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    Z
                  
                
              
              ]
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          35
        
        ]
      
    
  




Similarly, in case of the following precoding matrices among the 4Tx-Rank 3 precoding matrices, because only one symbol is transmitted to one antenna, only the layer power balancing can be carried out as shown in the following equation 36.




  
    
      
        
          
            P
            1
            ′
          
          =
          
            
              
                k
                ·
                
                  [
                  
                    
                      
                        
                          1
                          
                            2
                          
                        
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        
                          X
                          
                            2
                          
                        
                      
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
              
              ⁢
              
                
              
              ⁢
              
                P
                2
                ′
              
            
            =
            
              k
              ·
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        1
                        
                          2
                        
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      
                        Y
                        
                          2
                        
                      
                    
                    
                      0
                    
                  
                
                ]
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            P
            3
            ′
          
          =
          
            k
            ·
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    
                      1
                      
                        2
                      
                    
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    
                      Z
                      
                        2
                      
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          
        
        ⁢
        
          

        
        ⁢
        
          
            P
            7
            ′
          
          =
          
            k
            ·
            
              [
              
                
                  
                    
                      1
                      
                        3
                      
                    
                  
                  
                    0
                  
                  
                    
                      Z
                      
                        3
                      
                    
                  
                
                
                  
                    
                      X
                      
                        3
                      
                    
                  
                  
                    
                      1
                      
                        2
                      
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      Y
                      
                        2
                      
                    
                  
                  
                    
                      1
                      
                        3
                      
                    
                  
                
                
                  
                    
                      A
                      
                        3
                      
                    
                  
                  
                    0
                  
                  
                    
                      C
                      
                        3
                      
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            P
            8
            ′
          
          =
          
            k
            ·
            
              [
              
                
                  
                    
                      1
                      
                        2
                      
                    
                  
                  
                    0
                  
                  
                    
                      Z
                      
                        3
                      
                    
                  
                
                
                  
                    
                      X
                      
                        2
                      
                    
                  
                  
                    
                      1
                      
                        3
                      
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      Y
                      
                        3
                      
                    
                  
                  
                    
                      1
                      
                        3
                      
                    
                  
                
                
                  
                    0
                  
                  
                    
                      B
                      
                        3
                      
                    
                  
                  
                    
                      C
                      
                        3
                      
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            P
            9
            ′
          
          =
          
            k
            ·
            
              [
              
                
                  
                    
                      1
                      
                        3
                      
                    
                  
                  
                    0
                  
                  
                    
                      Z
                      
                        2
                      
                    
                  
                
                
                  
                    
                      X
                      
                        3
                      
                    
                  
                  
                    
                      1
                      
                        3
                      
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      Y
                      
                        3
                      
                    
                  
                  
                    
                      1
                      
                        2
                      
                    
                  
                
                
                  
                    
                      A
                      
                        3
                      
                    
                  
                  
                    
                      B
                      
                        3
                      
                    
                  
                  
                    0
                  
                
              
              ]
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          36
        
        ]
      
    
  




In the meantime, in accordance with another embodiment of the present invention, from the viewpoint of simultaneous execution of both the antenna power balancing and the layer power balancing, the present invention proposes the 4Tx-Rank 3 precoding matrix including the following precoding matrices denoted by Equation 37.




  
    
      
        
          
            P
            0
            
              
                (
                1
                )
              
              ′
            
          
          =
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                  
                
                ]
              
              ⁢
              
                
              
              ⁢
              
                P
                0
                
                  
                    (
                    2
                    )
                  
                  ′
                
              
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            P
            0
            
              
                (
                3
                )
              
              ′
            
          
          =
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
              ⁢
              
                
              
              ⁢
              
                P
                0
                
                  
                    (
                    4
                    )
                  
                  ′
                
              
            
            =
            
              [
              
                
                  
                    0
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    1
                  
                  
                    0
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    0
                  
                  
                    1
                  
                
              
              ]
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          37
        
        ]
      
    
  




In other words, Equation 37 shows precoding matrices used as the 4Tx-Rank 3 precoding matrices, and each precoding matrix of Equation 37 is established to transmit no signal to a single specific antenna.
In the meantime, examples of the precoding matrix acquired when the layer power balancing is carried out on the 4Tx-Rank 4 precoding matrix can be represented by the following equation 38.




  
    
      
        
          
            P
            13
            ′
          
          =
          
            k
            ·
            
              [
              
                
                  
                    
                      1
                      
                        2
                      
                    
                  
                  
                    0
                  
                  
                    
                      1
                      2
                    
                  
                  
                    
                      1
                      2
                    
                  
                
                
                  
                    
                      
                        m
                        21
                      
                      
                        2
                      
                    
                  
                  
                    0
                  
                  
                    
                      
                        m
                        23
                      
                      2
                    
                  
                  
                    
                      
                        m
                        24
                      
                      2
                    
                  
                
                
                  
                    0
                  
                  
                    
                      1
                      
                        2
                      
                    
                  
                  
                    
                      
                        m
                        33
                      
                      2
                    
                  
                  
                    
                      
                        m
                        34
                      
                      2
                    
                  
                
                
                  
                    0
                  
                  
                    
                      
                        m
                        42
                      
                      
                        2
                      
                    
                  
                  
                    
                      
                        m
                        43
                      
                      2
                    
                  
                  
                    
                      
                        m
                        44
                      
                      2
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            P
            14
            ′
          
          =
          
            k
            ·
            
              [
              
                
                  
                    
                      1
                      
                        3
                      
                    
                  
                  
                    0
                  
                  
                    
                      1
                      2
                    
                  
                  
                    
                      1
                      2
                    
                  
                
                
                  
                    
                      
                        m
                        21
                      
                      
                        3
                      
                    
                  
                  
                    0
                  
                  
                    
                      
                        m
                        23
                      
                      2
                    
                  
                  
                    
                      
                        m
                        24
                      
                      2
                    
                  
                
                
                  
                    
                      
                        m
                        31
                      
                      
                        3
                      
                    
                  
                  
                    0
                  
                  
                    
                      
                        m
                        33
                      
                      2
                    
                  
                  
                    
                      
                        m
                        34
                      
                      2
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      
                        m
                        43
                      
                      2
                    
                  
                  
                    
                      
                        m
                        44
                      
                      2
                    
                  
                
              
              ]
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          38
        
        ]
      
    
  




<Codebook Pruning>
In a 4Tx system, precoding matrices corresponding to Rank 1, Rank 2, Rank 3, and Rank 4 can be used as elements of a codebook used in transmission and reception ends. However, in the case of using all precoding matrices, the size of a codebook is excessively increased, so that it is necessary to reduce the number of precoding matrices simultaneously while maintaining performance at an appropriate level. Embodiments capable of reducing the number of precoding matrices will hereinafter be described in detail. Methods for restricting the following precoding matrix can be independently or simultaneously utilized.
Codebook Element Alphabet Restriction
Each element of the above-mentioned precoding matrices is selected from a complex number that has an absolute value of 1 and a phase corresponding to any one of +0°, +45°, +90°, +135°, +180°, −135°, −90°, and −45°.
In one embodiment of the present invention, in order to reduce the number of precoding matrices, each element of the precoding matrices may be selected from a complex number that has an absolute value of 1 and a phase corresponding to any one of +0°, +90°, +180°, and −90°. That is, each element of the precoding matrix may be selected from {1, j, −1, −j}.
Otherwise, each element of the precoding matrix may be extracted from a subset composed of N alphabetical letters among 8 alphabets which are spaced apart from each other by an angle of 45′.
Restriction to Unitary Precoding Matrix
In the case where individual column vectors contained in a precoding matrix have orthogonality therebetween, the precoding matrix may be a unitary matrix or a partially unitary matrix. If the precoding matrix has the above-mentioned characteristics, an additional gain can be obtained.
Thus, in accordance with one embodiment of the present invention, unitary matrices or partially unitary matrices among all the aforementioned precoding matrices are collected so that a codebook can be formed.
For example, the row/column permutation is carried out on the precoding matrices shown in the following equation 39 and the precoding matrices shown in the following equation 40 so as to obtain a few matrices, and the obtained matrices are combined, so that a codebook can be generated.




  
    
      
        
          
        
        ⁢
        
          
            
              P
              
                (
                1
                )
              
            
            =
            
              [
              
                
                  
                    1
                  
                
                
                  
                    a
                  
                
                
                  
                    b
                  
                
                
                  
                    c
                  
                
              
              ]
            
          
          ,
          
            
              P
              
                (
                2
                )
              
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    a
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    b
                  
                
              
              ]
            
          
          ,
          
            
              P
              
                (
                3
                )
              
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    a
                  
                  
                    0
                  
                  
                    
                      -
                      a
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    b
                  
                  
                    0
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              P
              1
              
                (
                4
                )
              
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    a
                  
                  
                    0
                  
                  
                    
                      -
                      a
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    b
                  
                  
                    0
                  
                  
                    
                      -
                      b
                    
                  
                
              
              ]
            
          
          ,
          
            
              P
              2
              
                (
                4
                )
              
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    1
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    c
                  
                  
                    
                      -
                      c
                    
                  
                  
                    c
                  
                
                
                  
                    a
                  
                  
                    0
                  
                  
                    
                      -
                      a
                    
                  
                  
                    
                      -
                      a
                    
                  
                
                
                  
                    b
                  
                  
                    
                      -
                      b
                    
                  
                  
                    0
                  
                  
                    b
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          where
          ,
          
            

          
          ⁢
          a
          ,
          b
          ,
          
            c
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          39
        
        ]
      
    
  




Restriction to Nested Structure
When constructing the precoding matrices of Rank 1, Rank 2, Rank 3 and Rank 4, in the case where the precoding matrix of Rank 2 or Rank 3 can be constructed with column vectors of the Rank 4 precoding matrix, the constructed precoding matrix is called a precoding matrix with a nested structure. If a specific Rank 4 precoding matrix is used as a part of a precoding codebook, the Rank 3 precoding matrix should be configured with column vectors of the specific Rank 4 precoding matrix, such that there occurs a limitation in the construction of the precoding matrix. Thus, the codebook size can be limited according to the aforementioned norm or standard.
In one embodiment of the present invention, the precoding matrix of Rank 1, Rank 2, Rank 3, or Rank 4 may have a nested structure.
For example, a codebook can be constructed with a combination of matrices acquired by performing the row/column permutation on the precoding matrices shown in the following equation 40.




  
    
      
        
          
        
        ⁢
        
          
            
              
                P
                
                  (
                  1
                  )
                
              
              =
              
                [
                
                  
                    
                      1
                    
                  
                  
                    
                      a
                    
                  
                  
                    
                      b
                    
                  
                  
                    
                      c
                    
                  
                
                ]
              
            
            ,
            
              
                P
                
                  (
                  2
                  )
                
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      a
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      b
                    
                  
                
                ]
              
            
            ,
            
              
                P
                
                  (
                  3
                  )
                
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      a
                    
                    
                      0
                    
                    
                      
                        -
                        a
                      
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      b
                    
                    
                      0
                    
                  
                
                ]
              
            
            ,
            
              

            
            ⁢
            
              
            
            ⁢
            
              
                P
                1
                
                  (
                  4
                  )
                
              
              =
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      a
                    
                    
                      0
                    
                    
                      
                        -
                        a
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      b
                    
                    
                      0
                    
                    
                      
                        -
                        b
                      
                    
                  
                
                ]
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            where
            ,
            
              

            
            ⁢
            a
            ,
            b
            ,
            
              c
              ∈
              
                {
                
                  1
                  ,
                  
                    
                      1
                      +
                      j
                    
                    
                      2
                    
                  
                  ,
                  j
                  ,
                  
                    
                      1
                      -
                      j
                    
                    
                      2
                    
                  
                  ,
                  
                    -
                    1
                  
                  ,
                  
                    
                      
                        -
                        1
                      
                      -
                      j
                    
                    
                      2
                    
                  
                  ,
                  
                    -
                    j
                  
                  ,
                  
                    
                      
                        -
                        1
                      
                      +
                      j
                    
                    
                      2
                    
                  
                
                }
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          40
        
        ]
      
    
  




In addition to the matrices shown in the above-mentioned equations, other applicable matrices may also exist. It can be easily understood that the applicable matrices can be obtained by performing the row permutation and/or the column permutation on the above-mentioned matrices. In the present invention, because the precoding matrix has elements each having a value of 0, a certain antenna may not be mapped to a specific input stream. This operation may be recognized as an antenna selection function.
II. Detailed Format of Codebook
Hereinafter, in the case where a codebook is designed to satisfy the above-mentioned codebook design rule, a method for deciding a precoding matrix for each rank contained in the codebook in consideration of a chordal distance will be described in detail.
FIG. 9 is a conceptual diagram illustrating a chordal distance.
A chordal distance is well known as one of norms (or standards) for comparing performances of various codebook sets. Herein, the term “chordal” indicates a straight line between two points located at the circumference. Therefore, given a two-dimensional (2D) case, a chordal distance indicates a distance between two points located at the circumference of a circle (e.g., a unit circle) as shown in FIG. 9.
There is a need for the 4Tx-codebook to consider a four-dimensional chordal distance, so that the following equation 41 can be used as a chordal distance for selecting a codebook set.




  
    
      
        
          
            d
            c
          
          ⁡
          
            (
            
              P
              ,
              Q
            
            )
          
        
        =
        
          
            1
            
              2
            
          
          ⁢
          
            
              
              
                
                  PP
                  H
                
                -
                
                  QQ
                  H
                
              
              
            
            F
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          41
        
        ]
      
    
  




In Equation 41, P is P=[v1 v2 . . . vN], and Q is Q=[u1 u2 . . . uN], where vi and ui (i=1, 2, . . . N, N=4 in the case of 4Tx antennas) are principal vectors of the matrices P and Q, respectively. In addition,




  
    
      
      A
      
    
    F
  
  =
  
    
      
        
          ∑
          
            i
            =
            1
          
          m
        
        ⁢
        
          
        
        ⁢
        
          
            ∑
            
              j
              =
              1
            
            n
          
          ⁢
          
            
          
          ⁢
          
            
              
              
                a
                ij
              
              
            
            2
          
        
      
    
    =
    
      
        trace
        ⁡
        
          (
          
            AA
            H
          
          )
        
      
    
  




is the Frobenius norm of the matrix. The above-mentioned chordal distance can also be measured by the following equation 42.





  
    
      
        
          
            
              
                
                  d
                  c
                
                ⁡
                
                  (
                  
                    P
                    ,
                    Q
                  
                  )
                
              
              =
              
              ⁢
              
                
                  1
                  
                    2
                  
                
                ⁢
                
                  
                    
                    
                      
                        PP
                        H
                      
                      -
                      
                        QQ
                        H
                      
                    
                    
                  
                  F
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  n
                  -
                  
                    trace
                    ⁡
                    
                      (
                      
                        
                          AA
                          H
                        
                        ⁢
                        
                          BB
                          H
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          42
        
        ]
      
    
  




where A and B are orthonormal generation magtrices for P and Q respectively
The above-mentioned codebook design for the 4Tx system based on four transmission antennas will be described using the above-mentioned chordal distance concept. For convenience of description and better understanding of the present invention, a factor related to the power balancing will be omitted from the following expressions.
Rank 2
Firstly, it is assumed that the following codebooks of three groups capable of maintaining good CM performance about the 4Tx-Rank 2 system are used.




  
    
      
        
          Group
          ⁢
          
            
          
          ⁢
          1
          ⁢
          
            (
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        Y
                      
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      
                        -
                        X
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      
                        -
                        X
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        Y
                      
                    
                  
                
                ]
              
            
            )
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          X
          ,
          
            Y
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          43
        
        ]
      
    
  
  
    
      
        
          Group
          ⁢
          
            
          
          ⁢
          2
          ⁢
          
            (
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        Y
                      
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        X
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      
                        -
                        X
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        Y
                      
                    
                  
                
                ]
              
            
            )
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          X
          ,
          
            Y
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
              
              }
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          Group
          ⁢
          
            
          
          ⁢
          3
          ⁢
          
            (
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        Y
                      
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                  
                  
                    
                      
                        -
                        X
                      
                    
                    
                      0
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      
                        -
                        Y
                      
                    
                  
                  
                    
                      
                        -
                        X
                      
                    
                    
                      0
                    
                  
                
                ]
              
            
            )
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          X
          ,
          
            Y
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
              
              }
            
          
        
      
    
    
      
        
      
    
  




While the number of precoding matrices satisfying the above-mentioned formats may be a considerably high number, it is preferable that a codebook for including a predetermined number of precoding matrices be designed according to a reasonable norm. The following description proposes a method for limiting the number of precoding matrices for each rank to a predetermined number or less using the following norms.
First Norm (Norm 1): Chordal Distance
Second Norm (Norm 2): Reference indicating whether the precoding matrices are uniformly selected from individual groups. If the number of precoding matrices/vectors in a codebook is not divided by the number of groups, the precoding matrices are most uniformly selected in consideration of the first norm (Norm 1).
The above-mentioned norm can be equally applied not only to Rank 3 but also to Rank 4 to be described later.
In more detail, one embodiment of the present invention proposes a method for selecting the set of precoding matrices from a codebook about a specific rank using the Norm 1. In a first step, a chordal distance about all precoding matrix pairs contained in a single codebook is calculated using Equation 42. For example, if four codebook sets exist, four minimum chordal distance values can be represented by the following expression.

dc,min1=1, dc,min2=0.56, dc,min3=0.71 and dc,min4=1  [Expression]

In the above expression, the higher the value of dc,mini (where i is a codebook set number), the higher the system performance. Thus, it is preferable that first and fourth codebooks go to a next selection step.
In a second step, in order to support a variety of wireless channel environments, the present invention proposes a method for most uniformly selecting the precoding matrices for each group. For example, in accordance with the proposed method of the present invention, if there are three codebook groups and 16 precoding matrices are needed as the Rank-2 codebook, 5 precoding matrices are selected from two groups, and 6 precoding matrices are selected from the remaining one group. For example, in accordance with the proposed method of the present invention, 5 precoding matrices are selected from first two groups, and 6 precoding matrices are selected from the last one group. One embodiment of the present invention may consider a method for limiting alphabets of each precoding matrix as described above, in which, for example, an alphabet ‘X’ may be limited to X=1, j, −1, or −j. The following description illustrates exemplary 4Tx Rank-2 codebooks capable of being configured by the above steps.







TABLE 1







Rank-2 codebook set 1-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
   






 







Rank-2 codebook set 2-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   






 







Rank-2 codebook set 3-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
   






 







Rank-2 codebook set 4-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
   






 







Rank-2 codebook set 5-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   






 







Rank-2 codebook set 6-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   






 







Rank-2 codebook set 7-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
   






 







Rank-2 codebook set 8-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   






 







Rank-2 codebook set 9-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
   






 







Rank-2 codebook set 10-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
   






 







Rank-2 codebook set 11-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   






 







Rank-2 codebook set 12-1


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   













The above-mentioned codebooks shown in Table 1 are disclosed only for illustrative purposes, and row permutation and/or column permutation may be applied to all or some of the precoding matrices.
If the 4Tx Rank-2 codebook includes 15 precoding matrices, one precoding matrix may be removed from a group of selecting the largest number of precoding matrices among individual precoding matrix groups. The following description shows exemplary 4Tx Rank-2 codebooks configured by the above-mentioned schemes.







TABLE 2







Rank-2 codebook set 1-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
   







 







Rank-2 codebook set 2-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   







 







Rank-2 codebook set 3-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
   







 







Rank-2 codebook set 4-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
   







 







Rank-2 codebook set 5-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   







 







Rank-2 codebook set 6-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   







 







Rank-2 codebook set 7-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
   







 







Rank-2 codebook set 8-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   







 







Rank-2 codebook set 9-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
   







 







Rank-2 codebook set 10-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
   







 







Rank-2 codebook set 11-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   







 







Rank-2 codebook set 12-2


















  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
      
      
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
      
      
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
      
      
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   













The codebooks shown in Table 2 are also disclosed only for illustrative purposes, the row permutation and/or column permutation may be carried out on all or some of precoding matrices of the codebooks.
Rank 3—First Embodiment
In order to design the 4Tx Rank-3 codebook so as to maintain good CM properties, it is assumed that the following three precoding matrix groups are used. For convenience of description, a factor related to power balancing will be omitted herein.



  
    
      
    
    ⁢
    
      [
      
        Equation
        ⁢
        
          
        
        ⁢
        44
      
      ]
    
  




  
    
      
    
    ⁢
    
      Group
      ⁢
      
        
      
      ⁢
      1
    
  




  
    (
    
      
        [
        
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              X
            
            
              0
            
            
              
                -
                X
              
            
          
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              0
            
            
              Y
            
            
              0
            
          
        
        ]
      
      ,
      
        [
        
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              X
            
            
              0
            
            
              
                -
                X
              
            
          
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              0
            
            
              
                -
                Y
              
            
            
              0
            
          
        
        ]
      
      ,
      
        [
        
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              0
            
            
              X
            
            
              0
            
          
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              Y
            
            
              0
            
            
              
                -
                Y
              
            
          
        
        ]
      
      ,
      
        [
        
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              0
            
            
              
                -
                X
              
            
            
              0
            
          
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              Y
            
            
              0
            
            
              
                -
                Y
              
            
          
        
        ]
      
    
    )
  




  
    
      
    
    ⁢
    
      X
      ,
      
        Y
        ∈
        
          {
          
            1
            ,
            
              
                1
                +
                j
              
              
                2
              
            
            ,
            j
            ,
            
              
                1
                -
                j
              
              
                2
              
            
          
          }
        
      
    
  




  
    
      
    
    ⁢
    
      Group
      ⁢
      
        
      
      ⁢
      2
    
  




  
    (
    
      
        [
        
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              X
            
            
              0
            
            
              
                -
                X
              
            
          
          
            
              0
            
            
              Y
            
            
              0
            
          
        
        ]
      
      ,
      
        [
        
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              X
            
            
              0
            
            
              
                -
                X
              
            
          
          
            
              0
            
            
              
                -
                Y
              
            
            
              0
            
          
        
        ]
      
      ,
      
        [
        
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              0
            
            
              X
            
            
              0
            
          
          
            
              Y
            
            
              0
            
            
              
                -
                Y
              
            
          
        
        ]
      
      ,
      
        [
        
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              0
            
            
              
                -
                X
              
            
            
              0
            
          
          
            
              Y
            
            
              0
            
            
              
                -
                Y
              
            
          
        
        ]
      
    
    )
  




  
    
      
    
    ⁢
    
      X
      ,
      
        Y
        ∈
        
          {
          
            1
            ,
            
              
                1
                +
                j
              
              
                2
              
            
            ,
            j
            ,
            
              
                1
                -
                j
              
              
                2
              
            
          
          }
        
      
    
  




  
    
      
    
    ⁢
    
      Group
      ⁢
      
        
      
      ⁢
      3
    
  




  
    (
    
      
        [
        
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              0
            
            
              Y
            
            
              0
            
          
          
            
              X
            
            
              0
            
            
              
                -
                X
              
            
          
        
        ]
      
      ,
      
        [
        
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              0
            
            
              
                -
                Y
              
            
            
              0
            
          
          
            
              X
            
            
              0
            
            
              
                -
                X
              
            
          
        
        ]
      
      ,
      
        [
        
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              Y
            
            
              0
            
            
              
                -
                Y
              
            
          
          
            
              0
            
            
              X
            
            
              0
            
          
        
        ]
      
      ,
      
        [
        
          
            
              0
            
            
              1
            
            
              0
            
          
          
            
              1
            
            
              0
            
            
              1
            
          
          
            
              Y
            
            
              0
            
            
              
                -
                Y
              
            
          
          
            
              0
            
            
              
                -
                X
              
            
            
              0
            
          
        
        ]
      
    
    )
  




  
    
      
    
    ⁢
    
      X
      ,
      
        Y
        ∈
        
          {
          
            1
            ,
            
              
                1
                +
                j
              
              
                2
              
            
            ,
            j
            ,
            
              
                1
                -
                j
              
              
                2
              
            
          
          }
        
      
    
  



In case of Rank 3, the present invention proposes a method for constructing a codebook according to the above-mentioned Norm 1 and Norm 2 in the same manner as in Rank 2. In more detail, a chordal distance about all precoding matrix combinations available in a codebook is calculated using Equation 42, and then a minimum number of sets each having a maximum chordal distance can be selected. In addition, the present invention proposes a method for most uniformly selecting the precoding matrix from each group (Group 1, 2, or 3). If the letter denoted by a precoding matrix component of each group is restricted to (1, j, −1, −j), the following codebook capable of satisfying a minimum chordal distance dc, =0.707 can be obtained.







TABLE 3







Rank-3 codebook set 1-1
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
   








 







Rank-3 codebook set 2-1
















  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
   








 







Rank-3 codebook set 3-1
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
   








 







Rank-3 codebook set 4-1
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   








 







Rank-3 codebook set 5-1
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   








 







Rank-3 codebook set 6-1
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
   








 







Rank-3 codebook set 7-1
















  
    
      [
      
        
          
            0
          
          
            1
          
          
            0
          
        
        
          
            0
          
          
            
              -
              1
            
          
          
            0
          
        
        
          
            1
          
          
            0
          
          
            1
          
        
        
          
            j
          
          
            0
          
          
            
              -
              j
            
          
        
      
      ]
    
    ,
  
  ⁢
  
    
  








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   








 







Rank-3 codebook set 8-1
















  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  [
  
    
      
        1
      
      
        0
      
      
        1
      
    
    
      
        0
      
      
        1
      
      
        0
      
    
    
      
        0
      
      
        j
      
      
        0
      
    
    
      
        j
      
      
        0
      
      
        
          -
          j
        
      
    
  
  ]













It should be noted that the row permutation and/or the column permutation may be carried out on all or some of precoding matrices of the above codebooks shown in Table 3.
If only 15 precoding matrices are included in the Rank-3 codebook, one precoding matrix of a group for selecting the largest number of precoding matrices among individual groups is removed from the codebooks shown in Table 3, so that the removed result may be configured as shown in the following Table 4.







TABLE 4







Rank-3 codebook set 1-2
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
   









 







Rank-3 codebook set 2-2
















  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








   
  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  









 







Rank-3 codebook set 3-2
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
   









 







Rank-3 codebook set 4-2
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   









 







Rank-3 codebook set 5-2
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   









 







Rank-3 codebook set 6-2
















  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
   









 







Rank-3 codebook set 7-2
















  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








   
  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  









 







Rank-3 codebook set 8-2
















  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
  ,






 






  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          
            -
            1
          
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   













It should be noted that the row permutation and/or the column permutation may be carried out on all or some of the above precoding matrices shown in Table 4.
Rank 3—Second Embodiment
In one embodiment of the present invention, a method for constructing a codebook using 6 precoding matrix groups capable of maintaining good CM properties will hereinafter be described. The six 4Tx Rank-3 precoding matrix groups for maintaining good CM properties can be represented by the following equation 45.




  
    
      
        
          
        
        ⁢
        
          
            
              Group
              ⁢
              
                
              
              ⁢
              
                1
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
              
            
            ,
            
              Group
              ⁢
              
                
              
              ⁢
              
                2
                ⁡
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
              
            
            ,
            
              

            
            ⁢
            
              
            
            ⁢
            
              Group
              ⁢
              
                
              
              ⁢
              
                3
                ⁡
                
                  [
                  
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
              
            
            ,
            
              Group
              ⁢
              
                
              
              ⁢
              
                4
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        0
                      
                    
                  
                  ]
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              Group
              ⁢
              
                
              
              ⁢
              
                5
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                  
                  ]
                
              
            
            ,
            
              Group
              ⁢
              
                
              
              ⁢
              
                6
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
              
            
            ,
            
              

            
            ⁢
            
              
            
            ⁢
            where
          
          ⁢
          
            

          
          ⁢
          
            X
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    
                      2
                      ⁢
                      
                        
                      
                    
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          45
        
        ]
      
    
  




An example of the Rank-3 codebook including 24 precoding matrixes from 6 groups shown in Equation 45 is shown in the following table 5. In order to reduce complexity, in the example shown in Table 5, letters denoted by precoding matrix elements are restricted to 1, −j, −1, and −j.










TABLE 5











  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  
   













For another example, the present invention proposes a method for utilizing the remaining groups other than a fourth group (Group 4) generated by applying column permutation to a first group (Group 1) among all groups shown in Equation 45. Generally, if three column vectors are represented by [c1, c2, c3], 5 column permutation matrices such as [c1, c3, c2], [c2, c1, c3], [c2, c3, c1], [c3, c2, c1], and [c3, c1, c2] can be generated, thus 6 matrices can be achieved.
The reason why the specific vector permutation matrix is not used as described above is that an encoded sequence is mapped to a specific column vector (or a specific layer) of the precoding matrix. It is assumed that two independently-encoded codewords in the above-mentioned precoding matrix groups are mapped to different layers as described below.
(1) A first codeword is mapped to a first layer.
(2) A second codeword is evenly distributed and mapped to second and third layers.
On the assumption that the above codeword-layer mapping is used, a specific column permutation does not generate a difference in average SINR between different codewords. For example, permutation from a column vector [c1, c2, c3] to another column vector [c1, c3, c2] may indicate that only a layer of a second codeword is swapped. In this way, the swapping between two layers to which the same second codeword is evenly distributed and mapped does not cause a variation in performance. For systems utilizing SIC receivers, correct decoding of a codeword given transmission of plurality of codewords leads to performance enhancements. This is because once a codeword is correctly decoded. So the correctly decoded codeword information can be used to cancel out spatial layer interference. In the case that transmission power of multiple antennas is uniformly normalized, some column vectors of the precoding matrix may have larger transmission power. In the case there is no layer shifting/permutation between all transmission layers, a specific layer corresponding to the column vector of the precoding matrix which column vectors has larger transmission power may have better performance. In case there is no layer shifting/permutation across all transmitted layers, in order to fully utilized SIC receivers the first layer, which the first codeword is solely mapped to is mapped to the precoding matrix column vector which has larger transmission power, and the second codeword which is mapped to second and third layer is mapped to precoding vector columns which has relatively smaller transmission power compared to the first layer. In the case above codeword-layer mapping is used, Precoding matrices as shown in [Equation 46] can be used to further enhance performance in case Successive Interference Cancellation (SIC) receiver algorithm is used.




  
    
      
        
          
            Group
            ⁢
            
              
            
            ⁢
            
              1
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                    
                      0
                    
                  
                
                ]
              
            
          
          ,
          
            Group
            ⁢
            
              
            
            ⁢
            
              2
              ⁡
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                    
                      0
                    
                  
                
                ]
              
            
          
          ,
          
            

          
          ⁢
          
            Group
            ⁢
            
              
            
            ⁢
            
              3
              ⁡
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                    
                      0
                    
                  
                
                ]
              
            
          
          ,
          
            Group
            ⁢
            
              
            
            ⁢
            
              4
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                
                ]
              
            
          
          ,
          
            

          
          ⁢
          
            Group
            ⁢
            
              
            
            ⁢
            
              5
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
            
          
          ,
          
            

          
          ⁢
          where
        
        ⁢
        
          

        
        ⁢
        X
        ∈
        
          {
          
            1
            ,
            
              
                1
                +
                j
              
              
                2
              
            
            ,
            j
            ,
            
              
                1
                -
                j
              
              
                2
              
            
            ,
            
              -
              1
            
            ,
            
              
                
                  -
                  1
                
                -
                j
              
              
                2
              
            
            ,
            
              -
              j
            
            ,
            
              
                
                  -
                  1
                
                +
                j
              
              
                
                  2
                  ⁢
                  
                    
                  
                
              
            
          
          }
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          46
        
        ]
      
    
  




The following codebooks are exemplary 4Tx Rank-3 codebooks, each of which restricts letters contained in each of the above precoding matrices groups to 1, j, −1, and −j, and includes 20 precoding matrices.










TABLE 6











  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  
  ,








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  
   













In the meantime, in accordance with another embodiment of the present invention, the number of precoding matrices required for acquiring optimum performance from a high rank is less than the number of precoding matrices required for acquiring optimum performance from a low rank, so that the present invention can restrict the Rank-3 codebook to have below 24 precoding matrices. In this case, the present invention may evenly select the precoding matrices from 6 precoding matrix groups using the Norm 2.










TABLE 7











  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          1
        
        
          0
        
        
          0
        
      
    
    ]
  
   








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            1
          
        
        
          0
        
        
          0
        
      
    
    ]
  
   








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          j
        
        
          0
        
        
          0
        
      
    
    ]
  
   








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          
            -
            j
          
        
        
          0
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
    
    ]
  
   








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            1
          
        
        
          0
        
      
    
    ]
  
   








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          j
        
        
          0
        
      
    
    ]
  
   








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          
            -
            j
          
        
        
          0
        
      
    
    ]
  
   






 






  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  
   








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          
            -
            1
          
        
      
    
    ]
  
   








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          j
        
      
    
    ]
  
   








  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          
            -
            j
          
        
      
    
    ]
  
   













As can be seen from the example of Table 7, if e−jθ is multiplied by a specific column vector, column permutation in a precoding matrix has no influence upon improvement of performance, so that the number of precoding matrices contained in a codebook is limited to 12. Meanwhile, in accordance with one embodiment of the present invention, antenna permutation may be carried out to obtain antenna selection gain. This antenna permutation may also be implemented by row permutation of a precoding matrix contained in the above-mentioned codebook.
Rank 3—Third Embodiment
In the third embodiment of the present invention, it is assumed that the following 6 precoding matrix groups are considered as precoding matrices capable of maintaining good CM performance.




  
    
      
        
          
            
              
                Group
                ⁢
                
                  
                
                ⁢
                
                  1
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
              ,
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      X
                    
                    
                      0
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      X
                    
                  
                
                ]
              
            
            ⁢
            
              

            
            ⁢
            
              
                Group
                ⁢
                
                  
                
                ⁢
                
                  2
                  ⁡
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      X
                    
                    
                      0
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      X
                    
                  
                
                ]
              
            
            ⁢
            
              

            
            ⁢
            
              
                Group
                ⁢
                
                  
                
                ⁢
                
                  3
                  ⁡
                  
                    [
                    
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
              ,
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      X
                    
                    
                      0
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      X
                    
                  
                
                ]
              
            
            ⁢
            
              

            
            ⁢
            
              
                Group
                ⁢
                
                  
                
                ⁢
                
                  4
                  ⁡
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
              ,
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      X
                    
                  
                
                ]
              
            
            ⁢
            
              

            
            ⁢
            
              
                Group
                ⁢
                
                  
                
                ⁢
                
                  5
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
              ,
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      X
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                
                ]
              
              ,
              
                [
                
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      X
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                
                ]
              
            
            ⁢
            
              

            
            ⁢
            Group
            ⁢
            
              
            
            ⁢
            
              
                6
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        0
                      
                      
                        1
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      X
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      1
                    
                  
                
                ]
              
            
          
          ,
          
            [
            
              
                
                  0
                
                
                  0
                
                
                  1
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  X
                
              
              
                
                  1
                
                
                  0
                
                
                  0
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        
          X
          ∈
          
            {
            
              1
              ,
              
                
                  1
                  +
                  j
                
                
                  2
                
              
              ,
              j
              ,
              
                
                  1
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                1
              
              ,
              
                
                  
                    -
                    1
                  
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                j
              
              ,
              
                
                  
                    -
                    1
                  
                  +
                  j
                
                
                  
                    2
                    ⁢
                    
                      
                    
                  
                
              
            
            }
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          47
        
        ]
      
    
  




In case of a first group (Group 1) in Equation 47, it can be recognized that three permutation matrices are selected from [c1, c3, c2], [c2, c1, c3], [c2, c3, c1], [c3, c2, c1], and [c3, c1, c2]. In case of a fourth group (Group 4), it can be recognized that one constituent precoding matrix is excluded, because the excluded precoding matrix has already been included in a first group (Group 1). It is preferable that the third embodiment be utilized when a layer shift operation is not carried out. The third embodiment can implement a layer shift using a codebook including a precoding matrix set upon which column permutation is carried out. Thus, an information sequence may be mapped to all layers, so that an SINR difference between layers can be normalized.
The third embodiment can select a precoding matrix using the first norm (Norm 1) and the second norm (Norm 2).
Rank 3—Fourth Embodiment
The fourth embodiment considers the following three groups as precoding matrix groups for maintaining good CM properties.




  
    
      
        
          
            
              G
              ⁢
              
                
              
              ⁢
              1
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    a
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    b
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    c
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    d
                  
                
              
              ]
            
          
          ,
          
            
              G
              ⁢
              
                
              
              ⁢
              2
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    
                      a
                      ′
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      b
                      ′
                    
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    
                      c
                      ′
                    
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    
                      d
                      ′
                    
                  
                
              
              ]
            
          
          ,
          
            
              G
              ⁢
              
                
              
              ⁢
              3
            
            =
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    
                      a
                      ′′
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      b
                      ′′
                    
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    
                      c
                      ′′
                    
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    
                      d
                      ′′
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          where
          ⁢
          
            

          
          ⁢
          X
          ,
          
            Y
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          48
        
        ]
      
    
  




The last vectors




  
    [
    
      
        
          a
        
      
      
        
          b
        
      
      
        
          c
        
      
      
        
          d
        
      
    
    ]
  
  ,
  
    [
    
      
        
          
            a
            ′
          
        
      
      
        
          
            b
            ′
          
        
      
      
        
          
            c
            ′
          
        
      
      
        
          
            d
            ′
          
        
      
    
    ]
  
  ,
  
    [
    
      
        
          
            a
            ′′
          
        
      
      
        
          
            b
            ′′
          
        
      
      
        
          
            c
            ′′
          
        
      
      
        
          
            d
            ′′
          
        
      
    
    ]
  




among precoding matrix groups shown in Equation 48 may be different precoding matrices such as DFT-based precoding vectors/matrices or household-based precoding vectors/matrices. For example, an example of the last vectors may be a Rank-1 codebook of the 3GPP LTE system (Release 8 system). Preferably, in order to maintain orthogonal/partial unitary characteristics of the matrix





   
  
    
      [
      
        
          
            1
          
          
            0
          
          
            a
          
        
        
          
            X
          
          
            0
          
          
            b
          
        
        
          
            0
          
          
            1
          
          
            c
          
        
        
          
            0
          
          
            Y
          
          
            d
          
        
      
      ]
    
    ,
  




the matrices





  
    [
    
      
        
          1
        
        
          a
        
      
      
        
          X
        
        
          b
        
      
    
    ]
  
  ⁢
  
    
  
  ⁢
  
    and
    ⁢
    
      
    
    [
    
      
        
          1
        
        
          c
        
      
      
        
          Y
        
        
          d
        
      
    
    ]
  




must satisfy unitary characteristics. Similarly, matrices





  
    [
    
      
        
          1
        
        
          
            a
            ′
          
        
      
      
        
          X
        
        
          
            c
            ′
          
        
      
    
    ]
  
  ⁢
  
    
  
  ⁢
  
    and
    ⁢
    
      
    
    [
    
      
        
          1
        
        
          
            b
            ′
          
        
      
      
        
          Y
        
        
          
            d
            ′
          
        
      
    
    ]
  




of the matrix





   
  
    [
    
      
        
          1
        
        
          0
        
        
          
            a
            ′
          
        
      
      
        
          0
        
        
          1
        
        
          
            b
            ′
          
        
      
      
        
          X
        
        
          0
        
        
          
            c
            ′
          
        
      
      
        
          0
        
        
          Y
        
        
          
            d
            ′
          
        
      
    
    ]
  




and matrices





  
    [
    
      
        
          1
        
        
          
            a
            ′′
          
        
      
      
        
          X
        
        
          
            d
            ′′
          
        
      
    
    ]
  
  ⁢
  
    
  
  ⁢
  
    and
    ⁢
    
      
    
    [
    
      
        
          1
        
        
          
            b
            ′′
          
        
      
      
        
          Y
        
        
          
            c
            ′′
          
        
      
    
    ]
  




of the matrix





   
  
    [
    
      
        
          1
        
        
          0
        
        
          
            a
            ′′
          
        
      
      
        
          0
        
        
          1
        
        
          
            b
            ′′
          
        
      
      
        
          0
        
        
          Y
        
        
          
            c
            ′′
          
        
      
      
        
          X
        
        
          0
        
        
          
            d
            ′′
          
        
      
    
    ]
  




must satisfy unitary characteristics. This means that parameters must satisfy the following relationship.

In Group 1: a=1, b=−X, and c=−d·Y* 

In Group 2: a′=1, b′=−X, and c′=−d′·Y* 

In Group 3: a′′=1, b′′=−X, and c′′=−d′′·Y*  [Equation 49]

In this case, although a certain complex constant is multiplied by each column vector of a specific precoding matrix, this means that the multiplied results indicate the same precoding matrix, so that it is assumed that a, a′, or a′′ is set to 1.
Preferably, the fourth embodiment may be applied to a case when layer permutation is executed. The layer permutation operation indicates that a specific information sequence is cyclically mapped and transmitted to all layers so that SINR performance differences of individual layers are normalized. If the same power is used in different layers, a data sequence of the last layer corresponding to the last column having no value of 0 has the highest power from the viewpoint of a precoding output signal.
In case layer permutation is not used and enhanced SIC receiver algorithm is used, layer which the first codeword is mapped to should preferably correspond to the precoding vector column which the transmission power is relatively larger than other precoding vector columns. In case of [Equation 48] the third column may have larger transmission power than others. For cases which the first column is mapped to the first layer, the second column is mapped to the second layer, and the third column is mapped to the third layer, [Equation 48a] may be used instead of [Equation 48]. This precoding matrix structure will allow enhanced performance in case no layer permutation is used and SIC receiver is used, due to the increased correct decoding probability of a entire codeword given a plurality of codeword transmission.




  
    
      
        
          
        
        ⁢
        
          
            
              
                G
                ⁢
                
                  
                
                ⁢
                1
              
              =
              
                [
                
                  
                    
                      a
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      b
                    
                    
                      0
                    
                    
                      X
                    
                  
                  
                    
                      c
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      d
                    
                    
                      Y
                    
                    
                      0
                    
                  
                
                ]
              
            
            ,
            
              
                G
                ⁢
                
                  
                
                ⁢
                2
              
              =
              
                [
                
                  
                    
                      
                        a
                        ′
                      
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      
                        b
                        ′
                      
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      
                        c
                        ′
                      
                    
                    
                      0
                    
                    
                      X
                    
                  
                  
                    
                      
                        d
                        ′
                      
                    
                    
                      Y
                    
                    
                      0
                    
                  
                
                ]
              
            
            ,
            
              

            
            ⁢
            
              
            
            ⁢
            
              
                G
                ⁢
                
                  
                
                ⁢
                3
              
              =
              
                [
                
                  
                    
                      
                        a
                        ′′
                      
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      
                        b
                        ′′
                      
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      
                        c
                        ′′
                      
                    
                    
                      0
                    
                    
                      X
                    
                  
                  
                    
                      
                        d
                        ′′
                      
                    
                    
                      Y
                    
                    
                      0
                    
                  
                
                ]
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          where
          ⁢
          
            

          
          ⁢
          
            X
            ,
            
              Y
              ∈
              
                {
                
                  1
                  ,
                  
                    
                      1
                      +
                      j
                    
                    
                      2
                    
                  
                  ,
                  j
                  ,
                  
                    
                      1
                      -
                      j
                    
                    
                      2
                    
                  
                  ,
                  
                    -
                    1
                  
                  ,
                  
                    
                      
                        -
                        1
                      
                      -
                      j
                    
                    
                      2
                    
                  
                  ,
                  
                    -
                    j
                  
                  ,
                  
                    
                      
                        -
                        1
                      
                      +
                      j
                    
                    
                      
                        2
                        ⁢
                        
                          
                        
                      
                    
                  
                
                }
              
            
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          48
          ⁢
          a
        
        ]
      
    
  




Rank 3—Fifth Embodiment
In the fifth embodiment, it is assumed that the following groups shown in Equation 50 are used as precoding matrix groups for maintaining good CM performance.




  
    
      
        
          
        
        ⁢
        
          
            [
            
              Equation
              ⁢
              
                
              
              ⁢
              50
            
            ]
          
          ⁢
          
            

          
          ⁢
          
            
              G
              ⁢
              
                
              
              ⁢
              1
            
            =
            
              (
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          a
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          b
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          a
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          b
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          c
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          d
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        a
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        b
                      
                      
                        0
                      
                      
                        X
                      
                    
                    
                      
                        c
                      
                      
                        1
                      
                      
                        0
                      
                    
                    
                      
                        d
                      
                      
                        Y
                      
                      
                        0
                      
                    
                  
                  ]
                
              
              )
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          G
          ⁢
          
            
          
          ⁢
          2
        
        =
        
          (
          
            
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      
                        a
                        ′
                      
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      
                        b
                        ′
                      
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                    
                      
                        c
                        ′
                      
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                    
                      
                        d
                        ′
                      
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                    
                      
                        a
                        ′
                      
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                    
                      
                        b
                        ′
                      
                    
                  
                  
                    
                      0
                    
                    
                      X
                    
                    
                      
                        c
                        ′
                      
                    
                  
                  
                    
                      Y
                    
                    
                      0
                    
                    
                      
                        d
                        ′
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      a
                      ′
                    
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    
                      b
                      ′
                    
                  
                  
                    1
                  
                  
                    0
                  
                
                
                  
                    
                      c
                      ′
                    
                  
                  
                    0
                  
                  
                    X
                  
                
                
                  
                    
                      d
                      ′
                    
                  
                  
                    Y
                  
                  
                    0
                  
                
              
              ]
            
          
          )
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            G
            ⁢
            
              
            
            ⁢
            3
          
          =
          
            (
            
              
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        
                          a
                          ′′
                        
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        
                          b
                          ′′
                        
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        
                          c
                          ′′
                        
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        
                          d
                          ′′
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        0
                      
                      
                        1
                      
                      
                        
                          a
                          ′′
                        
                      
                    
                    
                      
                        1
                      
                      
                        0
                      
                      
                        
                          b
                          ′′
                        
                      
                    
                    
                      
                        Y
                      
                      
                        0
                      
                      
                        
                          c
                          ′′
                        
                      
                    
                    
                      
                        0
                      
                      
                        X
                      
                      
                        
                          d
                          ′′
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        a
                        ′′
                      
                    
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      
                        b
                        ′′
                      
                    
                    
                      1
                    
                    
                      0
                    
                  
                  
                    
                      
                        c
                        ′′
                      
                    
                    
                      Y
                    
                    
                      0
                    
                  
                  
                    
                      
                        d
                        ′′
                      
                    
                    
                      0
                    
                    
                      X
                    
                  
                
                ]
              
            
            )
          
        
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        X
        ,
        
          Y
          ∈
          
            {
            
              1
              ,
              
                
                  1
                  +
                  j
                
                
                  2
                
              
              ,
              j
              ,
              
                
                  1
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                1
              
              ,
              
                
                  
                    -
                    1
                  
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                j
              
              ,
              
                
                  
                    -
                    1
                  
                  +
                  j
                
                
                  2
                
              
            
            }
          
        
      
    
    
      
        
      
    
  




The precoding matrix groups shown in Equation 50 are composed of a plurality of precoding matrices acquired when row permutation or column permutation is carried out on the structure of the fourth embodiment. The column vectors




  
    [
    
      
        
          a
        
      
      
        
          b
        
      
      
        
          c
        
      
      
        
          d
        
      
    
    ]
  
  ,
  
    [
    
      
        
          
            a
            ′
          
        
      
      
        
          
            b
            ′
          
        
      
      
        
          
            c
            ′
          
        
      
      
        
          
            d
            ′
          
        
      
    
    ]
  
  ,
  
    [
    
      
        
          
            a
            ′′
          
        
      
      
        
          
            b
            ′′
          
        
      
      
        
          
            c
            ′′
          
        
      
      
        
          
            d
            ′′
          
        
      
    
    ]
  




in the precoding matrix groups shown in Equation 50 may be different precoding matrices such as DFT-based precoding vectors/matrices or household-based precoding vectors/matrices. For example, an example of the above column vectors may be a Rank-1 codebook of the 3GPP LTE system (Release 8 system).

Similar to the fourth embodiment, in the fifth embodiment, it is preferable that precoding matrix vectors be orthogonal to each other and elements other than a first value of 0 in all column vectors of each precoding matrix group be set to 1.
A codebook according to the fifth embodiment includes a precoding matrix generated when column permutation is carried out on the precoding matrices of the fourth embodiment. As described above, the precoding matrix having a column vector [c1, c2, c3] may have 6 column permutation precoding matrices such as [c1, c3, c2], [c2, c1, c3], [c2, c3, c1], [c3, c2, c1], [c3, c1, c2] and [c3, c1, c2].
The reason why a specific column permutation is not included is that second and third column permutations of the precoding matrix in a system in which a first codeword is mapped to a first layer and a second codeword is distributed and mapped to second and third layers do not cause a difference in performance.
Rank 3—Sixth Embodiment
A precoding matrix according to the sixth embodiment is configured in a format acquired when row permutation is carried out on a precoding matrix of the codebook shown in the fourth embodiment, because the precoding matrix of the sixth embodiment can be acquired by antenna switching.
The precoding matrix groups according to the sixth embodiment can be represented by the following equation 51.



  
    
      
    
    ⁢
    
      
    
    ⁢
    
      [
      
        Equation
        ⁢
        
          
        
        ⁢
        51
      
      ]
    
  




  
    
      
        
          
            G
            ⁢
            
              
            
            ⁢
            1
          
          =
          
            (
            
              
                
                  
                    
                      
                        
                          [
                          
                            
                              
                                1
                              
                              
                                0
                              
                              
                                a
                              
                            
                            
                              
                                X
                              
                              
                                0
                              
                              
                                b
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                c
                              
                            
                            
                              
                                0
                              
                              
                                Y
                              
                              
                                d
                              
                            
                          
                          ]
                        
                        ⁡
                        
                          [
                          
                            
                              
                                X
                              
                              
                                0
                              
                              
                                b
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                a
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                c
                              
                            
                            
                              
                                0
                              
                              
                                Y
                              
                              
                                d
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              0
                            
                            
                              1
                            
                            
                              c
                            
                          
                          
                            
                              X
                            
                            
                              0
                            
                            
                              b
                            
                          
                          
                            
                              1
                            
                            
                              0
                            
                            
                              a
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              d
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            0
                          
                          
                            Y
                          
                          
                            d
                          
                        
                        
                          
                            X
                          
                          
                            0
                          
                          
                            b
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                          
                            c
                          
                        
                        
                          
                            1
                          
                          
                            0
                          
                          
                            a
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          a
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          b
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        a
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        d
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        c
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        b
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      a
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                    
                      b
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                    
                      d
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      c
                    
                  
                
                ]
              
            
            )
          
        
      
    
    
      
        
          
            G
            ⁢
            
              
            
            ⁢
            2
          
          =
          
            (
            
              
                
                  
                    
                      
                        
                          [
                          
                            
                              
                                1
                              
                              
                                0
                              
                              
                                
                                  a
                                  ′
                                
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                
                                  b
                                  ′
                                
                              
                            
                            
                              
                                X
                              
                              
                                0
                              
                              
                                
                                  c
                                  ′
                                
                              
                            
                            
                              
                                0
                              
                              
                                Y
                              
                              
                                
                                  d
                                  ′
                                
                              
                            
                          
                          ]
                        
                        ⁡
                        
                          [
                          
                            
                              
                                0
                              
                              
                                1
                              
                              
                                
                                  b
                                  ′
                                
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                
                                  a
                                  ′
                                
                              
                            
                            
                              
                                X
                              
                              
                                0
                              
                              
                                
                                  c
                                  ′
                                
                              
                            
                            
                              
                                0
                              
                              
                                Y
                              
                              
                                
                                  d
                                  ′
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                c
                                ′
                              
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                b
                                ′
                              
                            
                          
                          
                            
                              1
                            
                            
                              0
                            
                            
                              
                                a
                                ′
                              
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              
                                d
                                ′
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            0
                          
                          
                            Y
                          
                          
                            
                              d
                              ′
                            
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                          
                            
                              b
                              ′
                            
                          
                        
                        
                          
                            X
                          
                          
                            0
                          
                          
                            
                              c
                              ′
                            
                          
                        
                        
                          
                            1
                          
                          
                            0
                          
                          
                            
                              a
                              ′
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          
                            a
                            ′
                          
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          
                            c
                            ′
                          
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          
                            b
                            ′
                          
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          
                            d
                            ′
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        
                          a
                          ′
                        
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        
                          d
                          ′
                        
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        
                          c
                          ′
                        
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        
                          b
                          ′
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      1
                    
                    
                      0
                    
                    
                      
                        a
                        ′
                      
                    
                  
                  
                    
                      0
                    
                    
                      1
                    
                    
                      
                        b
                        ′
                      
                    
                  
                  
                    
                      0
                    
                    
                      Y
                    
                    
                      
                        d
                        ′
                      
                    
                  
                  
                    
                      X
                    
                    
                      0
                    
                    
                      
                        c
                        ′
                      
                    
                  
                
                ]
              
            
            )
          
        
      
    
    
      
        
          
            
              G
              ⁢
              
                
              
              ⁢
              3
            
            =
            
              (
              
                
                  
                    
                      
                        
                          
                            [
                            
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  
                                    a
                                    ′′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  1
                                
                                
                                  
                                    b
                                    ′′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  Y
                                
                                
                                  
                                    c
                                    ′′
                                  
                                
                              
                              
                                
                                  X
                                
                                
                                  0
                                
                                
                                  
                                    d
                                    ′′
                                  
                                
                              
                            
                            ]
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  0
                                
                                
                                  1
                                
                                
                                  
                                    b
                                    ′′
                                  
                                
                              
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  
                                    a
                                    ′′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  Y
                                
                                
                                  
                                    c
                                    ′′
                                  
                                
                              
                              
                                
                                  X
                                
                                
                                  0
                                
                                
                                  
                                    d
                                    ′′
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                0
                              
                              
                                Y
                              
                              
                                
                                  c
                                  ′′
                                
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                
                                  b
                                  ′′
                                
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                
                                  a
                                  ′′
                                
                              
                            
                            
                              
                                X
                              
                              
                                0
                              
                              
                                
                                  d
                                  ′′
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                d
                                ′′
                              
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                b
                                ′′
                              
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              
                                c
                                ′′
                              
                            
                          
                          
                            
                              1
                            
                            
                              0
                            
                            
                              
                                a
                                ′′
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            1
                          
                          
                            0
                          
                          
                            
                              a
                              ′′
                            
                          
                        
                        
                          
                            0
                          
                          
                            Y
                          
                          
                            
                              c
                              ′′
                            
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                          
                            
                              b
                              ′′
                            
                          
                        
                        
                          
                            X
                          
                          
                            0
                          
                          
                            
                              d
                              ′′
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          
                            a
                            ′′
                          
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          
                            d
                            ′′
                          
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          
                            c
                            ′′
                          
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          
                            b
                            ′′
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        
                          a
                          ′′
                        
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        
                          b
                          ′′
                        
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        
                          d
                          ′′
                        
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        
                          c
                          ′′
                        
                      
                    
                  
                  ]
                
              
              )
            
          
          ⁢
          
            

          
          ⁢
          X
          ,
          
            Y
            ∈
            
              {
              
                1
                ,
                
                  
                    1
                    +
                    j
                  
                  
                    2
                  
                
                ,
                j
                ,
                
                  
                    1
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  1
                
                ,
                
                  
                    
                      -
                      1
                    
                    -
                    j
                  
                  
                    2
                  
                
                ,
                
                  -
                  j
                
                ,
                
                  
                    
                      -
                      1
                    
                    +
                    j
                  
                  
                    2
                  
                
              
              }
            
          
        
      
    
  



[Equation 51]
The column vectors




  
    [
    
      
        
          a
        
      
      
        
          b
        
      
      
        
          c
        
      
      
        
          d
        
      
    
    ]
  
  ,
  
    [
    
      
        
          
            a
            ′
          
        
      
      
        
          
            b
            ′
          
        
      
      
        
          
            c
            ′
          
        
      
      
        
          
            d
            ′
          
        
      
    
    ]
  
  ,
  
    [
    
      
        
          
            a
            ′′
          
        
      
      
        
          
            b
            ′′
          
        
      
      
        
          
            c
            ′′
          
        
      
      
        
          
            d
            ′′
          
        
      
    
    ]
  




or their row permutation formats may be different precoding matrices such as DFT-based precoding vectors/matrices or household-based precoding vectors/matrices. For example, an example of the above column vectors may be a Rank-1 codebook of the 3GPP LTE system (Release 8 system).

Similar to the fourth embodiment, in the sixth embodiment, it is preferable that column vectors of the precoding matrix be orthogonal to each other and elements a, a′, or a′′ are set to 1. An example of the codebook according to the sixth embodiment can be represented by the following equation 52.




  
    
      
        
          
            G
            ⁢
            
              
            
            ⁢
            
              1
              ′
            
          
          =
          
            
              
                
                  
                    
                      
                        [
                        
                          
                            
                              1
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                -
                                X
                              
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                            
                              c
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              d
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                -
                                X
                              
                            
                          
                          
                            
                              1
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                            
                              c
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              d
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            0
                          
                          
                            1
                          
                          
                            c
                          
                        
                        
                          
                            X
                          
                          
                            0
                          
                          
                            
                              -
                              X
                            
                          
                        
                        
                          
                            1
                          
                          
                            0
                          
                          
                            1
                          
                        
                        
                          
                            0
                          
                          
                            Y
                          
                          
                            d
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          
                            -
                            X
                          
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          1
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        c
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        
                          -
                          X
                        
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        d
                      
                    
                  
                  ]
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    d
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    c
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    
                      -
                      X
                    
                  
                
              
              ]
            
            ⁡
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    
                      -
                      X
                    
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    d
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    c
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            G
            ⁢
            
              
            
            ⁢
            
              2
              ′
            
          
          =
          
            
              
                
                  
                    
                      
                        [
                        
                          
                            
                              1
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                b
                                ′
                              
                            
                          
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                -
                                X
                              
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              
                                d
                                ′
                              
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                b
                                ′
                              
                            
                          
                          
                            
                              1
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                -
                                X
                              
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              
                                d
                                ′
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            X
                          
                          
                            0
                          
                          
                            
                              -
                              X
                            
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                          
                            
                              b
                              ′
                            
                          
                        
                        
                          
                            1
                          
                          
                            0
                          
                          
                            1
                          
                        
                        
                          
                            0
                          
                          
                            Y
                          
                          
                            
                              d
                              ′
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          
                            d
                            ′
                          
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          
                            b
                            ′
                          
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          
                            -
                            X
                          
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          1
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        
                          -
                          X
                        
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        
                          b
                          ′
                        
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        
                          d
                          ′
                        
                      
                    
                  
                  ]
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    
                      d
                      ′
                    
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    
                      -
                      X
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      b
                      ′
                    
                  
                
              
              ]
            
            ⁡
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      b
                      ′
                    
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    
                      d
                      ′
                    
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    
                      -
                      X
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            G
            ⁢
            
              
            
            ⁢
            
              3
              ′
            
          
          =
          
            
              
                
                  
                    
                      
                        [
                        
                          
                            
                              1
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                b
                                ′′
                              
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              
                                c
                                ′′
                              
                            
                          
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                -
                                X
                              
                            
                          
                        
                        ]
                      
                      ⁡
                      
                        [
                        
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                b
                                ′′
                              
                            
                          
                          
                            
                              1
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              
                                c
                                ′′
                              
                            
                          
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                -
                                X
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            0
                          
                          
                            Y
                          
                          
                            
                              c
                              ′′
                            
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                          
                            
                              b
                              ′′
                            
                          
                        
                        
                          
                            1
                          
                          
                            0
                          
                          
                            1
                          
                        
                        
                          
                            X
                          
                          
                            0
                          
                          
                            
                              -
                              X
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          X
                        
                        
                          0
                        
                        
                          
                            -
                            X
                          
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          
                            b
                            ′′
                          
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          
                            c
                            ′′
                          
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          1
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        1
                      
                      
                        0
                      
                      
                        1
                      
                    
                    
                      
                        0
                      
                      
                        Y
                      
                      
                        
                          c
                          ′′
                        
                      
                    
                    
                      
                        0
                      
                      
                        1
                      
                      
                        
                          b
                          ′′
                        
                      
                    
                    
                      
                        X
                      
                      
                        0
                      
                      
                        
                          -
                          X
                        
                      
                    
                  
                  ]
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    
                      -
                      X
                    
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    
                      c
                      ′′
                    
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      b
                      ′′
                    
                  
                
              
              ]
            
            ⁡
            
              [
              
                
                  
                    1
                  
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    0
                  
                  
                    1
                  
                  
                    
                      b
                      ′′
                    
                  
                
                
                  
                    X
                  
                  
                    0
                  
                  
                    
                      -
                      X
                    
                  
                
                
                  
                    0
                  
                  
                    Y
                  
                  
                    
                      c
                      ′′
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        X
        ,
        
          Y
          ∈
          
            {
            
              1
              ,
              
                
                  1
                  +
                  j
                
                
                  2
                
              
              ,
              j
              ,
              
                
                  1
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                1
              
              ,
              
                
                  
                    -
                    1
                  
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                j
              
              ,
              
                
                  
                    -
                    1
                  
                  +
                  j
                
                
                  2
                
              
            
            }
          
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          52
        
        ]
      
    
  




Rank 3—Seventh Embodiment
A codebook according to the seventh embodiment is configured in a format of row permutation of the codebook shown in the fifth embodiment. An example of the codebook according to the seventh embodiment can be represented by the following equation 53.




  
    
      
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          53
        
        ]
      
    
  
  
    
      
        
          G
          ⁢
          
            
          
          ⁢
          1
        
        =
        
          (
          
            
              
                
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          a
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          b
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          X
                        
                        
                          0
                        
                        
                          b
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          a
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          b
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          a
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          b
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          a
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          a
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          b
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          a
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          b
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          a
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          b
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                    
                    ]
                  
                
              
            
            
              
                
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          a
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          b
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          c
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          d
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          0
                        
                        
                          X
                        
                        
                          b
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          a
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          c
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          d
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          c
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          b
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          a
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          d
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          d
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          b
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          c
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          a
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          a
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          c
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          b
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          d
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          a
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          d
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          c
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          b
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          a
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          b
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          d
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          c
                        
                      
                    
                    ]
                  
                
              
            
            
              
                
                  
                    [
                    
                      
                        
                          a
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          b
                        
                        
                          0
                        
                        
                          X
                        
                      
                      
                        
                          c
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          d
                        
                        
                          Y
                        
                        
                          0
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          b
                        
                        
                          0
                        
                        
                          X
                        
                      
                      
                        
                          a
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          c
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          d
                        
                        
                          Y
                        
                        
                          0
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          c
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          b
                        
                        
                          0
                        
                        
                          X
                        
                      
                      
                        
                          a
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          d
                        
                        
                          Y
                        
                        
                          0
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          d
                        
                        
                          Y
                        
                        
                          0
                        
                      
                      
                        
                          b
                        
                        
                          0
                        
                        
                          X
                        
                      
                      
                        
                          c
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          a
                        
                        
                          0
                        
                        
                          1
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          a
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          c
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          b
                        
                        
                          0
                        
                        
                          X
                        
                      
                      
                        
                          d
                        
                        
                          Y
                        
                        
                          0
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          a
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          d
                        
                        
                          Y
                        
                        
                          0
                        
                      
                      
                        
                          c
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          b
                        
                        
                          0
                        
                        
                          X
                        
                      
                    
                    ]
                  
                  ,
                  
                    [
                    
                      
                        
                          a
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          b
                        
                        
                          0
                        
                        
                          X
                        
                      
                      
                        
                          d
                        
                        
                          Y
                        
                        
                          0
                        
                      
                      
                        
                          c
                        
                        
                          1
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
            
          
          )
        
      
    
    
      
        
      
    
  
  
    
      
        
          G
          ⁢
          
            
          
          ⁢
          2
        
        =
        
          (
          
            
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    
                                      a
                                      ′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    
                                      b
                                      ′
                                    
                                  
                                
                                
                                  
                                    X
                                  
                                  
                                    0
                                  
                                  
                                    
                                      c
                                      ′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    Y
                                  
                                  
                                    
                                      d
                                      ′
                                    
                                  
                                
                              
                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    
                                      b
                                      ′
                                    
                                  
                                
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    
                                      a
                                      ′
                                    
                                  
                                
                                
                                  
                                    X
                                  
                                  
                                    0
                                  
                                  
                                    
                                      c
                                      ′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    Y
                                  
                                  
                                    
                                      d
                                      ′
                                    
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  X
                                
                                
                                  0
                                
                                
                                  
                                    c
                                    ′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  1
                                
                                
                                  
                                    b
                                    ′
                                  
                                
                              
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  
                                    a
                                    ′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  Y
                                
                                
                                  
                                    d
                                    ′
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                0
                              
                              
                                Y
                              
                              
                                
                                  d
                                  ′
                                
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                
                                  b
                                  ′
                                
                              
                            
                            
                              
                                X
                              
                              
                                0
                              
                              
                                
                                  c
                                  ′
                                
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                
                                  a
                                  ′
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              1
                            
                            
                              0
                            
                            
                              
                                a
                                ′
                              
                            
                          
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                c
                                ′
                              
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                b
                                ′
                              
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              
                                d
                                ′
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            1
                          
                          
                            0
                          
                          
                            
                              a
                              ′
                            
                          
                        
                        
                          
                            0
                          
                          
                            Y
                          
                          
                            
                              d
                              ′
                            
                          
                        
                        
                          
                            X
                          
                          
                            0
                          
                          
                            
                              c
                              ′
                            
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                          
                            
                              b
                              ′
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          
                            a
                            ′
                          
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          
                            b
                            ′
                          
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          
                            d
                            ′
                          
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          
                            c
                            ′
                          
                        
                      
                    
                    ]
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    
                                      a
                                      ′
                                    
                                  
                                
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    
                                      b
                                      ′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    X
                                  
                                  
                                    
                                      c
                                      ′
                                    
                                  
                                
                                
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                  
                                    
                                      d
                                      ′
                                    
                                  
                                
                              
                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    
                                      b
                                      ′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    
                                      a
                                      ′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    X
                                  
                                  
                                    
                                      c
                                      ′
                                    
                                  
                                
                                
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                  
                                    
                                      d
                                      ′
                                    
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  0
                                
                                
                                  X
                                
                                
                                  
                                    c
                                    ′
                                  
                                
                              
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  
                                    b
                                    ′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  1
                                
                                
                                  
                                    a
                                    ′
                                  
                                
                              
                              
                                
                                  Y
                                
                                
                                  0
                                
                                
                                  
                                    d
                                    ′
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                Y
                              
                              
                                0
                              
                              
                                
                                  d
                                  ′
                                
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                
                                  b
                                  ′
                                
                              
                            
                            
                              
                                0
                              
                              
                                X
                              
                              
                                
                                  c
                                  ′
                                
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                
                                  a
                                  ′
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                a
                                ′
                              
                            
                          
                          
                            
                              0
                            
                            
                              X
                            
                            
                              
                                c
                                ′
                              
                            
                          
                          
                            
                              1
                            
                            
                              0
                            
                            
                              
                                b
                                ′
                              
                            
                          
                          
                            
                              Y
                            
                            
                              0
                            
                            
                              
                                d
                                ′
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            0
                          
                          
                            1
                          
                          
                            
                              a
                              ′
                            
                          
                        
                        
                          
                            Y
                          
                          
                            0
                          
                          
                            
                              d
                              ′
                            
                          
                        
                        
                          
                            0
                          
                          
                            X
                          
                          
                            
                              c
                              ′
                            
                          
                        
                        
                          
                            1
                          
                          
                            0
                          
                          
                            
                              b
                              ′
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          
                            a
                            ′
                          
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          
                            b
                            ′
                          
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          
                            d
                            ′
                          
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          
                            c
                            ′
                          
                        
                      
                    
                    ]
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    
                                      a
                                      ′
                                    
                                  
                                  
                                    0
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    
                                      b
                                      ′
                                    
                                  
                                  
                                    1
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    
                                      c
                                      ′
                                    
                                  
                                  
                                    0
                                  
                                  
                                    X
                                  
                                
                                
                                  
                                    
                                      d
                                      ′
                                    
                                  
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                
                              
                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    
                                      b
                                      ′
                                    
                                  
                                  
                                    1
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    
                                      a
                                      ′
                                    
                                  
                                  
                                    0
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    
                                      c
                                      ′
                                    
                                  
                                  
                                    0
                                  
                                  
                                    X
                                  
                                
                                
                                  
                                    
                                      d
                                      ′
                                    
                                  
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    c
                                    ′
                                  
                                
                                
                                  0
                                
                                
                                  X
                                
                              
                              
                                
                                  
                                    b
                                    ′
                                  
                                
                                
                                  1
                                
                                
                                  0
                                
                              
                              
                                
                                  
                                    a
                                    ′
                                  
                                
                                
                                  0
                                
                                
                                  1
                                
                              
                              
                                
                                  
                                    d
                                    ′
                                  
                                
                                
                                  Y
                                
                                
                                  0
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  d
                                  ′
                                
                              
                              
                                Y
                              
                              
                                0
                              
                            
                            
                              
                                
                                  b
                                  ′
                                
                              
                              
                                1
                              
                              
                                0
                              
                            
                            
                              
                                
                                  c
                                  ′
                                
                              
                              
                                0
                              
                              
                                X
                              
                            
                            
                              
                                
                                  a
                                  ′
                                
                              
                              
                                0
                              
                              
                                1
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                a
                                ′
                              
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              
                                c
                                ′
                              
                            
                            
                              0
                            
                            
                              X
                            
                          
                          
                            
                              
                                b
                                ′
                              
                            
                            
                              1
                            
                            
                              0
                            
                          
                          
                            
                              
                                d
                                ′
                              
                            
                            
                              Y
                            
                            
                              0
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              a
                              ′
                            
                          
                          
                            0
                          
                          
                            1
                          
                        
                        
                          
                            
                              d
                              ′
                            
                          
                          
                            Y
                          
                          
                            0
                          
                        
                        
                          
                            
                              c
                              ′
                            
                          
                          
                            0
                          
                          
                            X
                          
                        
                        
                          
                            
                              b
                              ′
                            
                          
                          
                            1
                          
                          
                            0
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            a
                            ′
                          
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          
                            b
                            ′
                          
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          
                            d
                            ′
                          
                        
                        
                          Y
                        
                        
                          0
                        
                      
                      
                        
                          
                            c
                            ′
                          
                        
                        
                          0
                        
                        
                          X
                        
                      
                    
                    ]
                  
                
              
            
          
          )
        
      
    
    
      
        
      
    
  
  
    
      
        
          G
          ⁢
          
            
          
          ⁢
          3
        
        =
        
          (
          
            
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    
                                      a
                                      ′′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    
                                      b
                                      ′′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    Y
                                  
                                  
                                    
                                      c
                                      ′′
                                    
                                  
                                
                                
                                  
                                    X
                                  
                                  
                                    0
                                  
                                  
                                    
                                      d
                                      ′′
                                    
                                  
                                
                              
                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    
                                      b
                                      ′′
                                    
                                  
                                
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    
                                      a
                                      ′′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    Y
                                  
                                  
                                    
                                      c
                                      ′′
                                    
                                  
                                
                                
                                  
                                    X
                                  
                                  
                                    0
                                  
                                  
                                    
                                      d
                                      ′′
                                    
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  0
                                
                                
                                  Y
                                
                                
                                  
                                    c
                                    ′′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  1
                                
                                
                                  
                                    b
                                    ′′
                                  
                                
                              
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  
                                    a
                                    ′′
                                  
                                
                              
                              
                                
                                  X
                                
                                
                                  0
                                
                                
                                  
                                    d
                                    ′′
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                X
                              
                              
                                0
                              
                              
                                
                                  d
                                  ′′
                                
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                
                                  b
                                  ′′
                                
                              
                            
                            
                              
                                0
                              
                              
                                Y
                              
                              
                                
                                  c
                                  ′′
                                
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                
                                  a
                                  ′′
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              1
                            
                            
                              0
                            
                            
                              
                                a
                                ′′
                              
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              
                                c
                                ′′
                              
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                b
                                ′′
                              
                            
                          
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                d
                                ′′
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            1
                          
                          
                            0
                          
                          
                            
                              a
                              ′′
                            
                          
                        
                        
                          
                            X
                          
                          
                            0
                          
                          
                            
                              d
                              ′′
                            
                          
                        
                        
                          
                            0
                          
                          
                            Y
                          
                          
                            
                              c
                              ′′
                            
                          
                        
                        
                          
                            0
                          
                          
                            1
                          
                          
                            
                              b
                              ′′
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          
                            a
                            ′′
                          
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          
                            b
                            ′′
                          
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          
                            d
                            ′′
                          
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          
                            c
                            ′′
                          
                        
                      
                    
                    ]
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    
                                      a
                                      ′′
                                    
                                  
                                
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    
                                      b
                                      ′′
                                    
                                  
                                
                                
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                  
                                    
                                      c
                                      ′′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    X
                                  
                                  
                                    
                                      d
                                      ′′
                                    
                                  
                                
                              
                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    
                                      b
                                      ′′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    
                                      a
                                      ′′
                                    
                                  
                                
                                
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                  
                                    
                                      c
                                      ′′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    X
                                  
                                  
                                    
                                      d
                                      ′′
                                    
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  Y
                                
                                
                                  0
                                
                                
                                  
                                    c
                                    ′′
                                  
                                
                              
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  
                                    b
                                    ′′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  1
                                
                                
                                  
                                    a
                                    ′′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  X
                                
                                
                                  
                                    d
                                    ′′
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                0
                              
                              
                                X
                              
                              
                                
                                  d
                                  ′′
                                
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                
                                  b
                                  ′′
                                
                              
                            
                            
                              
                                Y
                              
                              
                                0
                              
                              
                                
                                  c
                                  ′′
                                
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                
                                  a
                                  ′′
                                
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              0
                            
                            
                              1
                            
                            
                              
                                a
                                ′′
                              
                            
                          
                          
                            
                              Y
                            
                            
                              0
                            
                            
                              
                                c
                                ′′
                              
                            
                          
                          
                            
                              1
                            
                            
                              0
                            
                            
                              
                                b
                                ′′
                              
                            
                          
                          
                            
                              0
                            
                            
                              X
                            
                            
                              
                                d
                                ′′
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            0
                          
                          
                            1
                          
                          
                            
                              a
                              ′′
                            
                          
                        
                        
                          
                            0
                          
                          
                            X
                          
                          
                            
                              d
                              ′′
                            
                          
                        
                        
                          
                            Y
                          
                          
                            0
                          
                          
                            
                              c
                              ′′
                            
                          
                        
                        
                          
                            1
                          
                          
                            0
                          
                          
                            
                              b
                              ′′
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          
                            a
                            ′′
                          
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          
                            b
                            ′′
                          
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          
                            d
                            ′′
                          
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          
                            c
                            ′′
                          
                        
                      
                    
                    ]
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    
                                      a
                                      ′′
                                    
                                  
                                  
                                    0
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    
                                      b
                                      ′′
                                    
                                  
                                  
                                    1
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    
                                      c
                                      ′′
                                    
                                  
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    
                                      d
                                      ′′
                                    
                                  
                                  
                                    0
                                  
                                  
                                    X
                                  
                                
                              
                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    
                                      b
                                      ′′
                                    
                                  
                                  
                                    1
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    
                                      a
                                      ′′
                                    
                                  
                                  
                                    0
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    
                                      c
                                      ′′
                                    
                                  
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    
                                      d
                                      ′′
                                    
                                  
                                  
                                    0
                                  
                                  
                                    X
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    c
                                    ′′
                                  
                                
                                
                                  Y
                                
                                
                                  0
                                
                              
                              
                                
                                  
                                    b
                                    ′′
                                  
                                
                                
                                  1
                                
                                
                                  0
                                
                              
                              
                                
                                  
                                    a
                                    ′′
                                  
                                
                                
                                  0
                                
                                
                                  1
                                
                              
                              
                                
                                  
                                    d
                                    ′′
                                  
                                
                                
                                  0
                                
                                
                                  X
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  d
                                  ′′
                                
                              
                              
                                0
                              
                              
                                X
                              
                            
                            
                              
                                
                                  b
                                  ′′
                                
                              
                              
                                1
                              
                              
                                0
                              
                            
                            
                              
                                
                                  c
                                  ′′
                                
                              
                              
                                Y
                              
                              
                                0
                              
                            
                            
                              
                                
                                  a
                                  ′′
                                
                              
                              
                                0
                              
                              
                                1
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              
                                a
                                ′′
                              
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              
                                c
                                ′′
                              
                            
                            
                              Y
                            
                            
                              0
                            
                          
                          
                            
                              
                                b
                                ′′
                              
                            
                            
                              1
                            
                            
                              0
                            
                          
                          
                            
                              
                                d
                                ′′
                              
                            
                            
                              0
                            
                            
                              X
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              a
                              ′′
                            
                          
                          
                            0
                          
                          
                            1
                          
                        
                        
                          
                            
                              d
                              ′′
                            
                          
                          
                            0
                          
                          
                            X
                          
                        
                        
                          
                            
                              c
                              ′′
                            
                          
                          
                            Y
                          
                          
                            0
                          
                        
                        
                          
                            
                              b
                              ′′
                            
                          
                          
                            1
                          
                          
                            0
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            a
                            ′′
                          
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          
                            b
                            ′′
                          
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          
                            d
                            ′′
                          
                        
                        
                          0
                        
                        
                          X
                        
                      
                      
                        
                          
                            c
                            ′′
                          
                        
                        
                          Y
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
            
          
          )
        
      
    
    
      
        
      
    
  
  
    
      
        where
        ⁢
        
          

        
        ⁢
        X
        ,
        
          Y
          ∈
          
            {
            
              1
              ,
              
                
                  1
                  +
                  j
                
                
                  2
                
              
              ,
              j
              ,
              
                
                  1
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                1
              
              ,
              
                
                  
                    -
                    1
                  
                  -
                  j
                
                
                  2
                
              
              ,
              
                -
                j
              
              ,
              
                
                  
                    -
                    1
                  
                  +
                  j
                
                
                  2
                
              
            
            }
          
        
      
    
    
      
        
      
    
  




The column vectors




  
    [
    
      
        
          a
        
      
      
        
          b
        
      
      
        
          c
        
      
      
        
          d
        
      
    
    ]
  
  ,
  
    [
    
      
        
          
            a
            ′
          
        
      
      
        
          
            b
            ′
          
        
      
      
        
          
            c
            ′
          
        
      
      
        
          
            d
            ′
          
        
      
    
    ]
  
  ,
  
    [
    
      
        
          
            a
            ′′
          
        
      
      
        
          
            b
            ′′
          
        
      
      
        
          
            c
            ′′
          
        
      
      
        
          
            d
            ′′
          
        
      
    
    ]
  




or their row permutation formats may be different precoding matrices such as DFT-based precoding vectors/matrices or household-based precoding vectors/matrices. For example, an example of the above column vectors may be a Rank-1 codebook of the 3GPP LTE system (Release 8 system).

Similar to the fourth embodiment, in the seventh embodiment, it is preferable that column vectors of the precoding matrix be orthogonal to each other and elements a, a′, or a′′ are set to 1. It is preferable that the codebook according to this embodiment be used when antenna permutation is not carried out, because the antenna permutation effect can be achieved by the precoding matrix to which row permutation is carried out when using the codebook of the seventh embodiment.
An example of the codebook according to the seventh embodiment can be represented by the following equation 54.




  
    
      
        
          
            
          
          ⁢
          
            [
            
              Equation
              ⁢
              
                
              
              ⁢
              54
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          G
          ⁢
          
            
          
          ⁢
          1
        
        =
        
          (
          
            
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    X
                                  
                                  
                                    0
                                  
                                  
                                    
                                      -
                                      X
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    c
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    Y
                                  
                                  
                                    d
                                  
                                
                              
                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    X
                                  
                                  
                                    0
                                  
                                  
                                    
                                      -
                                      X
                                    
                                  
                                
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    c
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    Y
                                  
                                  
                                    d
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  0
                                
                                
                                  1
                                
                                
                                  c
                                
                              
                              
                                
                                  X
                                
                                
                                  0
                                
                                
                                  
                                    -
                                    X
                                  
                                
                              
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  1
                                
                              
                              
                                
                                  0
                                
                                
                                  Y
                                
                                
                                  d
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                0
                              
                              
                                Y
                              
                              
                                d
                              
                            
                            
                              
                                X
                              
                              
                                0
                              
                              
                                
                                  -
                                  X
                                
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                c
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                1
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              1
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              0
                            
                            
                              1
                            
                            
                              c
                            
                          
                          
                            
                              X
                            
                            
                              0
                            
                            
                              
                                -
                                X
                              
                            
                          
                          
                            
                              0
                            
                            
                              Y
                            
                            
                              d
                            
                          
                        
                        ]
                      
                    
                    ⁢
                    
                      
                    
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          
                            -
                            X
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          X
                        
                        
                          0
                        
                        
                          
                            -
                            X
                          
                        
                      
                      
                        
                          0
                        
                        
                          Y
                        
                        
                          d
                        
                      
                      
                        
                          0
                        
                        
                          1
                        
                        
                          c
                        
                      
                    
                    ]
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    X
                                  
                                  
                                    
                                      -
                                      X
                                    
                                  
                                
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    c
                                  
                                
                                
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                  
                                    d
                                  
                                
                              
                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    0
                                  
                                  
                                    X
                                  
                                  
                                    
                                      -
                                      X
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    c
                                  
                                
                                
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                  
                                    d
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  c
                                
                              
                              
                                
                                  0
                                
                                
                                  X
                                
                                
                                  
                                    -
                                    X
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  1
                                
                                
                                  1
                                
                              
                              
                                
                                  Y
                                
                                
                                  0
                                
                                
                                  d
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                Y
                              
                              
                                0
                              
                              
                                d
                              
                            
                            
                              
                                0
                              
                              
                                X
                              
                              
                                
                                  -
                                  X
                                
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                c
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                1
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
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                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    
                                      b
                                      ′′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    1
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                  
                                    
                                      c
                                      ′′
                                    
                                  
                                
                                
                                  
                                    0
                                  
                                  
                                    X
                                  
                                  
                                    
                                      -
                                      X
                                    
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  Y
                                
                                
                                  0
                                
                                
                                  
                                    c
                                    ′′
                                  
                                
                              
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  
                                    b
                                    ′′
                                  
                                
                              
                              
                                
                                  0
                                
                                
                                  1
                                
                                
                                  1
                                
                              
                              
                                
                                  0
                                
                                
                                  X
                                
                                
                                  
                                    -
                                    X
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                0
                              
                              
                                X
                              
                              
                                
                                  d
                                  ′′
                                
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                
                                  b
                                  ′′
                                
                              
                            
                            
                              
                                Y
                              
                              
                                0
                              
                              
                                
                                  c
                                  ′′
                                
                              
                            
                            
                              
                                0
                              
                              
                                1
                              
                              
                                1
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              0
                            
                            
                              1
                            
                            
                              1
                            
                          
                          
                            
                              Y
                            
                            
                              0
                            
                            
                              
                                c
                                ′′
                              
                            
                          
                          
                            
                              1
                            
                            
                              0
                            
                            
                              
                                b
                                ′′
                              
                            
                          
                          
                            
                              0
                            
                            
                              X
                            
                            
                              
                                -
                                X
                              
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            0
                          
                          
                            1
                          
                          
                            1
                          
                        
                        
                          
                            0
                          
                          
                            X
                          
                          
                            
                              -
                              X
                            
                          
                        
                        
                          
                            Y
                          
                          
                            0
                          
                          
                            
                              c
                              ′′
                            
                          
                        
                        
                          
                            1
                          
                          
                            0
                          
                          
                            
                              b
                              ′′
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          0
                        
                        
                          1
                        
                        
                          1
                        
                      
                      
                        
                          1
                        
                        
                          0
                        
                        
                          
                            b
                            ′′
                          
                        
                      
                      
                        
                          0
                        
                        
                          X
                        
                        
                          
                            -
                            X
                          
                        
                      
                      
                        
                          Y
                        
                        
                          0
                        
                        
                          
                            c
                            ′′
                          
                        
                      
                    
                    ]
                  
                
              
            
            
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    
                                      b
                                      ′′
                                    
                                  
                                  
                                    1
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    
                                      c
                                      ′′
                                    
                                  
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    
                                      -
                                      X
                                    
                                  
                                  
                                    0
                                  
                                  
                                    X
                                  
                                
                              
                              ]
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    
                                      b
                                      ′′
                                    
                                  
                                  
                                    1
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    1
                                  
                                  
                                    0
                                  
                                  
                                    1
                                  
                                
                                
                                  
                                    
                                      c
                                      ′′
                                    
                                  
                                  
                                    Y
                                  
                                  
                                    0
                                  
                                
                                
                                  
                                    
                                      -
                                      X
                                    
                                  
                                  
                                    0
                                  
                                  
                                    X
                                  
                                
                              
                              ]
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    c
                                    ′′
                                  
                                
                                
                                  Y
                                
                                
                                  0
                                
                              
                              
                                
                                  
                                    b
                                    ′′
                                  
                                
                                
                                  1
                                
                                
                                  0
                                
                              
                              
                                
                                  1
                                
                                
                                  0
                                
                                
                                  1
                                
                              
                              
                                
                                  
                                    -
                                    X
                                  
                                
                                
                                  0
                                
                                
                                  X
                                
                              
                            
                            ]
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  -
                                  X
                                
                              
                              
                                0
                              
                              
                                X
                              
                            
                            
                              
                                
                                  b
                                  ′′
                                
                              
                              
                                1
                              
                              
                                0
                              
                            
                            
                              
                                
                                  c
                                  ′′
                                
                              
                              
                                Y
                              
                              
                                0
                              
                            
                            
                              
                                1
                              
                              
                                0
                              
                              
                                1
                              
                            
                          
                          ]
                        
                      
                      ⁡
                      
                        [
                        
                          
                            
                              1
                            
                            
                              0
                            
                            
                              1
                            
                          
                          
                            
                              
                                c
                                ′′
                              
                            
                            
                              Y
                            
                            
                              0
                            
                          
                          
                            
                              
                                b
                                ′′
                              
                            
                            
                              1
                            
                            
                              0
                            
                          
                          
                            
                              
                                -
                                X
                              
                            
                            
                              0
                            
                            
                              X
                            
                          
                        
                        ]
                      
                    
                    ⁡
                    
                      [
                      
                        
                          
                            1
                          
                          
                            0
                          
                          
                            1
                          
                        
                        
                          
                            
                              -
                              X
                            
                          
                          
                            0
                          
                          
                            X
                          
                        
                        
                          
                            
                              c
                              ′′
                            
                          
                          
                            Y
                          
                          
                            0
                          
                        
                        
                          
                            
                              b
                              ′′
                            
                          
                          
                            1
                          
                          
                            0
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          1
                        
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          
                            b
                            ′′
                          
                        
                        
                          1
                        
                        
                          0
                        
                      
                      
                        
                          
                            -
                            X
                          
                        
                        
                          0
                        
                        
                          X
                        
                      
                      
                        
                          
                            c
                            ′′
                          
                        
                        
                          Y
                        
                        
                          0
                        
                      
                    
                    ]
                  
                
              
            
          
          )
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          where
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            X
            ,
            
              Y
              ∈
              
                {
                
                  1
                  ,
                  
                    
                      1
                      +
                      j
                    
                    
                      2
                    
                  
                  ,
                  j
                  ,
                  
                    
                      1
                      -
                      j
                    
                    
                      2
                    
                  
                  ,
                  
                    -
                    1
                  
                  ,
                  
                    
                      
                        -
                        1
                      
                      -
                      j
                    
                    
                      2
                    
                  
                  ,
                  
                    -
                    j
                  
                  ,
                  
                    
                      
                        -
                        1
                      
                      +
                      j
                    
                    
                      2
                    
                  
                
                }
              
            
          
        
      
    
    
      
        
      
    
  




Reference for Selecting Additional Precoding Matrix
In addition to the Norm 1 and the Norm 2, this embodiment is designed to consider another norm. In this norm, elements denoted by letters contained in each precoding matrix group are not selected from among eight values, but are limited to 1, j, −1 and −j, thus reducing the number of precoding matrices contained in a codebook.
In accordance with this embodiment, a codebook set including 16 precoding matrices is considered. For example, Rank 1 DFT vectors about 4Tx antennas can be represented as follows.
N×N DFT matrix (or Fourier Matrix) FN based on a given component, such as FN=e−j·2π/N normalized to 1/√{square root over (N)} can be represented by the following equation 55.




  
    
      
        
          F
          N
        
        =
        
          [
          
            
              
                1
              
              
                1
              
              
                1
              
              
                ...
              
              
                1
              
            
            
              
                1
              
              
                
                  F
                  N
                  1
                
              
              
                
                  F
                  N
                  2
                
              
              
                ...
              
              
                
                  F
                  N
                  
                    N
                    -
                    1
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
              
                ...
              
              
                ⋮
              
            
            
              
                1
              
              
                
                  F
                  N
                  
                    (
                    
                      N
                      -
                      1
                    
                    )
                  
                
              
              
                
                  F
                  N
                  
                    2
                    ·
                    
                      (
                      
                        N
                        -
                        1
                      
                      )
                    
                  
                
              
              
                ...
              
              
                
                  F
                  N
                  
                    
                      (
                      
                        N
                        -
                        1
                      
                      )
                    
                    ·
                    
                      (
                      
                        N
                        -
                        1
                      
                      )
                    
                  
                
              
            
          
          ]
        
      
    
    
      
        [
        
          Equation
          ⁢
          
            
          
          ⁢
          55
        
        ]
      
    
  




Rank 1 DFT vectors about the 4Tx antennas composed of 16 column vectors located at the first four rows of Equation 55.







TABLE 8











   
  
    
      
        
          
            
              
                
                  
                    
                      
                        [
                        
                          
                            
                              1
                            
                          
                          
                            
                              1
                            
                          
                          
                            
                              1
                            
                          
                          
                            
                              1
                            
                          
                        
                        ]
                      
                      ⁢
                      
                        
                      
                      [
                      
                        
                          
                            1
                          
                        
                        
                          
                            
                              e
                              
                                
                                  -
                                  j
                                
                                ⁢
                                
                                  1
                                  8
                                
                                ⁢
                                π
                              
                            
                          
                        
                        
                          
                            
                              e
                              
                                
                                  -
                                  j
                                
                                ⁢
                                
                                  1
                                  4
                                
                                ⁢
                                π
                              
                            
                          
                        
                        
                          
                            
                              e
                              
                                
                                  -
                                  j
                                
                                ⁢
                                
                                  3
                                  8
                                
                                ⁢
                                π
                              
                            
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      
                    
                    [
                    
                      
                        
                          1
                        
                      
                      
                        
                          
                            e
                            
                              
                                -
                                j
                              
                              ⁢
                              
                                1
                                4
                              
                              ⁢
                              π
                            
                          
                        
                      
                      
                        
                          
                            e
                            
                              
                                -
                                j
                              
                              ⁢
                              
                                1
                                2
                              
                              ⁢
                              π
                            
                          
                        
                      
                      
                        
                          
                            e
                            
                              
                                -
                                j
                              
                              ⁢
                              
                                3
                                4
                              
                              ⁢
                              π
                            
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                  
                  [
                  
                    
                      
                        1
                      
                    
                    
                      
                        
                          e
                          
                            
                              -
                              j
                            
                            ⁢
                            
                              3
                              8
                            
                            ⁢
                            π
                          
                        
                      
                    
                    
                      
                        
                          e
                          
                            
                              -
                              j
                            
                            ⁢
                            
                              3
                              4
                            
                            ⁢
                            π
                          
                        
                      
                    
                    
                      
                        
                          e
                          
                            
                              -
                              j
                            
                            ⁢
                            
                              9
                              8
                            
                            ⁢
                            π
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                [
                
                  
                    
                      1
                    
                  
                  
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            1
                            2
                          
                          ⁢
                          π
                        
                      
                    
                  
                  
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                      
                    
                  
                  
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            3
                            2
                          
                          ⁢
                          π
                        
                      
                    
                  
                
                ]
              
              ⁢
              
                
              
              [
              
                
                  
                    1
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          5
                          8
                        
                        ⁢
                        π
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          5
                          4
                        
                        ⁢
                        π
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          15
                          8
                        
                        ⁢
                        π
                      
                    
                  
                
              
              ]
            
            ⁢
            
              
            
            [
            
              
                
                  1
                
              
              
                
                  
                    e
                    
                      
                        -
                        j
                      
                      ⁢
                      
                        3
                        4
                      
                      ⁢
                      π
                    
                  
                
              
              
                
                  
                    e
                    
                      
                        -
                        j
                      
                      ⁢
                      
                        3
                        2
                      
                      ⁢
                      π
                    
                  
                
              
              
                
                  
                    e
                    
                      
                        -
                        j
                      
                      ⁢
                      
                        9
                        4
                      
                      ⁢
                      π
                    
                  
                
              
            
            ]
          
          ⁢
          
            
          
          [
          
            
              
                1
              
            
            
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      7
                      8
                    
                    ⁢
                    π
                  
                
              
            
            
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      7
                      4
                    
                    ⁢
                    π
                  
                
              
            
            
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      5
                      8
                    
                    ⁢
                    π
                  
                
              
            
          
          ]
        
      
    
    
      
        
          
            
              
                
                  
                    
                      [
                      
                        
                          
                            1
                          
                        
                        
                          
                            
                              e
                              
                                
                                  -
                                  j
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                            
                          
                        
                        
                          
                            1
                          
                        
                        
                          
                            
                              e
                              
                                
                                  -
                                  j
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                            
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      
                    
                    [
                    
                      
                        
                          1
                        
                      
                      
                        
                          
                            e
                            
                              
                                -
                                j
                              
                              ⁢
                              
                                9
                                8
                              
                              ⁢
                              π
                            
                          
                        
                      
                      
                        
                          
                            e
                            
                              
                                -
                                j
                              
                              ⁢
                              
                                1
                                4
                              
                              ⁢
                              π
                            
                          
                        
                      
                      
                        
                          
                            e
                            
                              
                                -
                                j
                              
                              ⁢
                              
                                11
                                8
                              
                              ⁢
                              π
                            
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                  
                  [
                  
                    
                      
                        1
                      
                    
                    
                      
                        
                          e
                          
                            
                              -
                              j
                            
                            ⁢
                            
                              5
                              4
                            
                            ⁢
                            π
                          
                        
                      
                    
                    
                      
                        
                          e
                          
                            
                              -
                              j
                            
                            ⁢
                            
                              5
                              2
                            
                            ⁢
                            π
                          
                        
                      
                    
                    
                      
                        
                          e
                          
                            
                              -
                              j
                            
                            ⁢
                            
                              7
                              4
                            
                            ⁢
                            π
                          
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                [
                
                  
                    
                      1
                    
                  
                  
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            11
                            8
                          
                          ⁢
                          π
                        
                      
                    
                  
                  
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            11
                            4
                          
                          ⁢
                          π
                        
                      
                    
                  
                  
                    
                      
                        e
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            1
                            8
                          
                          ⁢
                          π
                        
                      
                    
                  
                
                ]
              
              ⁢
              
                
              
              [
              
                
                  
                    1
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          3
                          2
                        
                        ⁢
                        π
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                    
                  
                
                
                  
                    
                      e
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          1
                          2
                        
                        ⁢
                        π
                      
                    
                  
                
              
              ]
            
            ⁢
            
              
            
            [
            
              
                
                  1
                
              
              
                
                  
                    e
                    
                      
                        -
                        j
                      
                      ⁢
                      
                        13
                        8
                      
                      ⁢
                      π
                    
                  
                
              
              
                
                  
                    e
                    
                      
                        -
                        j
                      
                      ⁢
                      
                        5
                        4
                      
                      ⁢
                      π
                    
                  
                
              
              
                
                  
                    e
                    
                      
                        -
                        j
                      
                      ⁢
                      
                        7
                        8
                      
                      ⁢
                      π
                    
                  
                
              
            
            ]
          
          ⁢
          
            
          
          [
          
            
              
                1
              
            
            
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      7
                      4
                    
                    ⁢
                    π
                  
                
              
            
            
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      3
                      2
                    
                    ⁢
                    π
                  
                
              
            
            
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      5
                      4
                    
                    ⁢
                    π
                  
                
              
            
          
          ]
        
      
    
    
      
        
          [
          
            
              
                1
              
            
            
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      15
                      8
                    
                    ⁢
                    π
                  
                
              
            
            
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      7
                      4
                    
                    ⁢
                    π
                  
                
              
            
            
              
                
                  e
                  
                    
                      -
                      j
                    
                    ⁢
                    
                      13
                      8
                    
                    ⁢
                    π
                  
                
              
            
          
          ]
        
      
    
  













Next, 4Tx Rank 1 house hold vector (HH vector) may be represented by the following Table 9.







TABLE 9











   
  
    
      
        
          
            
              
                
                  
                    
                      
                        [
                        
                          
                            
                              1
                            
                          
                          
                            
                              1
                            
                          
                          
                            
                              1
                            
                          
                          
                            
                              1
                            
                          
                        
                        ]
                      
                      ⁢
                      
                        
                      
                      [
                      
                        
                          
                            1
                          
                        
                        
                          
                            
                              -
                              j
                            
                          
                        
                        
                          
                            
                              -
                              1
                            
                          
                        
                        
                          
                            j
                          
                        
                      
                      ]
                    
                    ⁢
                    
                      
                    
                    [
                    
                      
                        
                          1
                        
                      
                      
                        
                          
                            -
                            1
                          
                        
                      
                      
                        
                          1
                        
                      
                      
                        
                          
                            -
                            1
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                  
                  [
                  
                    
                      
                        1
                      
                    
                    
                      
                        j
                      
                    
                    
                      
                        
                          -
                          1
                        
                      
                    
                    
                      
                        
                          -
                          j
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  
                
                [
                
                  
                    
                      1
                    
                  
                  
                    
                      
                        
                          1
                          -
                          j
                        
                        
                          2
                        
                      
                    
                  
                  
                    
                      
                        -
                        j
                      
                    
                  
                  
                    
                      
                        
                          
                            -
                            1
                          
                          -
                          j
                        
                        
                          2
                        
                      
                    
                  
                
                ]
              
              ⁢
              
                
              
              [
              
                
                  
                    1
                  
                
                
                  
                    
                      
                        
                          -
                          1
                        
                        -
                        j
                      
                      
                        2
                      
                    
                  
                
                
                  
                    j
                  
                
                
                  
                    
                      
                        1
                        -
                        j
                      
                      
                        2
                      
                    
                  
                
              
              ]
            
            ⁢
            
              
            
            [
            
              
                
                  1
                
              
              
                
                  
                    
                      
                        -
                        1
                      
                      +
                      j
                    
                    
                      2
                    
                  
                
              
              
                
                  
                    -
                    j
                  
                
              
              
                
                  
                    
                      1
                      +
                      j
                    
                    
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Codebook Size Restriction
At least one of the first to third norms (Norm 1, Norm 2 and Norm 3) may be used to limit the number of precoding matrices contained in a codebook. In this embodiment, codebook size restriction for each rank, especially, size restriction in a Rank 1 codebook, will be described in detail.
Presently, a downlink 4Tx codebook for the 3GPP LTE system has prescribed that respective ranks have the same number of vectors/matrices (i.e., 16 vectors/matrices). However, it is well known in the art that the number of precoding matrices required to acquire optimum performance from a high rank is less than the number of precoding matrices required to acquire optimum performance from a low rank. For this purpose, this embodiment of the present invention proposes a new codebook format in which the number of precoding matrices of a low rank is higher than that of a high rank so that individual ranks have different numbers of precoding matrices.
In the meantime, a mobile communication system can support a variety of transmission modes. It is assumed that an X-th transmission mode is effectively used for a UE located at a cell edge so that the UE can support a closed loop operation using a Rank 1 Precoding Matrix Indicator (PMI). In this case, a Rank 1 PMI vector may be selected from the Rank 1 precoding matrices contained in an overall codebook composed of a plurality of precoding matrices of all ranks supporting a Y-th transmission mode such as an open loop MIMO or closed loop MIMO. In this case, it is assumed that the X-th transmission mode is different from the Y-th transmission mode. For the Y-th transmission mode, the size of the Rank 1 codebook need not be configured as a power of 2. In addition, although the Rank 1 codebook size is configured as a power of 2, only the codebook size can be increased without higher performance improvement. Thus, this embodiment proposes a method for rationally restricting the codebook size simultaneously while having appropriate performance so that the codebook can be expressed with a smaller amount of feedback information.
Firstly, it is assumed that numbers of precoding matrices of individual ranks supporting the Y-th transmission mode are set to A—Rank 1, B—Rank 2, C—Rank 3, and D-Rank 4 (where D≦C≦B≦A). In this case, the size of an overall codebook is equal to the sum of A, B, C, and D. In order to support the above codebook size, m-bit signaling for satisfying the following condition shown in Equation 56 may be needed.

A+B+C+D≦2m  [Equation 56]

If a UE is configured to use the X-th transmission mode, a UE is able to use Rank 1 PMI information. It is preferable that 2n Rank 1 PMIs (where n<m) be newly defined to reduce the number of bits required for signaling. A variety of methods (1), (2), (3), (4), (5) and (6) may be used to reduce the number of signaling bits.
(1) Method 1
If possible, an even-th index is selected.
(2) Method 2
If possible, an odd-th index is selected.
(3) Method 3
Initial 2n indexes are selected.
(4) Method 4
Last 2n indexes are selected.
(5) Method 5
Indexes are arbitrarily selected.
(6) Method 6
Construction is achieved by signaling.
For example, for the Y-th transmission mode, 33 precoding matrices may be given for Rank 1, 15 precoding matrices may be given for Rank 2, 15 precoding matrices may be given for Rank 3, and 4 precoding matrices may be given for Rank 4.
In this case, a variety of methods (1), (2), (3), (4), (5) and (6) for constructing the Rank 1 codebook used for indicating only 16 precoding matrices can be used.
(1) Method 1
If possible, an even-th index is selected.
(2) Method 2
If possible, an odd-th index is selected.
(3) Method 3
Initial 16 indexes are selected.
(4) Method 4
Last 16 indexes are selected.
(5) Method 5
Indexes are arbitrarily selected.
(6) Method 6
Construction is achieved by signaling.
In the meantime, a variety of methods (1), (2), (3) and (4) for constructing the Rank 1 codebook used for indicating only 32 precoding matrices can be used.
(1) Method 1
Initial 32 indexes are selected.
(2) Method 2
Last 32 indexes are selected.
(3) Method 3
Indexes are arbitrarily selected.
(4) Method 4
Construction is achieved by signaling.
If 16 downlink Rank 1 vectors are contained in the Rank 1 codebook including 32 precoding matrices, the following restriction methods (I) and (II) can be used.
The restriction method (I) corresponds to a case for constructing the 16-sized Rank 1 codebook, and a detailed description thereof will hereinafter be described in detail.
A) 16 downlink Rank 1 vectors are selected.
B) The 16-sized Rank 1 codebook is selected regardless of downlink Rank 1 vectors.
(1) Initial 16 indexes are selected.
(2) Last 16 indexes are selected.
(3) Indexes are arbitrarily selected.
(4) Construction is achieved by signaling.
The other restriction method (II) corresponds to another case for constructing the 32-sized Rank 1 codebook, and a detailed description thereof will hereinafter be described in detail.
A) Selection of 16 downlink Rank 1 vectors+additional vectors.
(1) Initial 16 indexes are selected.
(2) Last 16 indexes are selected.
(3) Indexes are arbitrarily selected.
(4) Construction is achieved by signaling.
B) Selection of 32-sized Rank 1 codebook regardless of downlink Rank 1 vectors.
(1) Initial 32 indexes are selected.
(2) Last 32 indexes are selected.
(3) Indexes are arbitrarily selected.
(4) Construction is achieved by signaling.
The number of codebooks for each rank can be effectively constructed according to the above-mentioned schemes.
III. Apparatus Configuration
Chapter III will hereinafter disclose an improved structure to be contained in a UE, wherein the improved structure can maintain good PAPR or CM properties simultaneously while applying the MIMO scheme to uplink signal transmission.
FIG. 10 is a block diagram illustrating a general base station (BS) and a general user equipment (UE).
Referring to FIG. 10, a base station (BS) 10 includes a processor 11, a memory 12, and a Radio Frequency (RF) module 13. The RF module 13 is used as a transmission/reception module for receiving an uplink signal and transmitting a downlink signal. The processor 11 may control downlink signal transmission using downlink signal transmission information (for example, a specific precoding matrix contained in a codebook for downlink signal transmission) stored in the memory 12. Otherwise, as an inverse process of the precoding process, the processor 11 may control a signal reception process by multiplying uplink signal reception information (e.g., an uplink signal) stored in the memory 12 by a Hermitian matrix of the same precoding matrix as a precoding matrix used in the UE 20.
The UE 20 may include a processor 21, a memory 22, and an RF module 23 used as a transmission/reception module for transmitting an uplink signal and receiving a downlink signal. The processor 21 may control uplink signal transmission using uplink signal transmission information (for example, a specific precoding matrix contained in the above-mentioned codebook for uplink signal transmission) stored in the memory 22. Otherwise, as an inverse process of the precoding process, the processor 21 may control a signal reception process by multiplying downlink signal reception information (e.g., a downlink signal) stored in the memory 22 by a Hermitian matrix of the same precoding matrix as a precoding matrix used in the UE 20.
In the meantime, a detailed description about a processor of the UE 20 (or the BS 10), particularly, a structure for transmitting a signal using the SC-FDMA scheme, will hereinafter be described. A processor for transmitting a signal based on the SC-FDMA scheme in the 3GPP LTE system and a processor for transmitting a signal based on an OFDM scheme in the 3GPP LTE system will hereinafter be described, and a processor for enabling a UE to transmit an uplink signal using the SC-FDMA scheme as well as the MIMO scheme will then be described below.
FIGS. 11 and 12 illustrate an SC-FDMA scheme for transmitting an uplink signal in the 3GPP LTE system and an OFDMA scheme for transmitting a downlink signal in the 3GPP LTE system.
Referring to FIG. 11, not only a UE for transmitting an uplink signal but also a base station (BS) for transmitting a downlink signal includes a Serial-to-Parallel converter 401, a subcarrier mapper 403, an M-point IDFT module 404, a Parallel-to-Serial converter 405, and the like. However, a UE for transmitting a signal using the SC-FDMA scheme further includes an N-point DFT module 402, and compensates for a predetermined part of the IDFT processing influence of the M-point IDFT module 404 so that a transmission signal can have single carrier characteristics.
FIG. 12 shows the relationship between a block diagram for an uplink signal process prescribed in TS 36.211 including the 3GPP LTE system specification and a processor for transmitting a signal using the SC-FDMA scheme. In accordance with the TS 36.211, each UE scrambles a transmission signal using a specific scrambling sequence so as to transmit an uplink signal, and the scrambled signal is modulated so that complex symbols are generated. After that, transform precoding for performing a DFT spreading process on complex symbols is carried out. That is, a transform precoder prescribed in TS 36.211 may correspond to an N-point DFT module. Thereafter, the DFT-spread signal may be mapped to a specific resource element according to a resource block (RB)-based mapping rule by a resource element mapper, and it can be recognized that this operation corresponds to the subcarrier mapper shown in FIG. 11. The signal mapped to the resource element is M-point IDFT or IFFT-processed by the SC-FDMA signal generator, parallel-to-serial conversion is performed on the IDFT or IFFT processed result, and then a cyclic prefix (CP) is added to the P/S conversion result.
In the meantime, FIG. 12 further shows a processor of a base station (BS) that is used to receive a signal which has been received in the base station through the above-mentioned processes.
In this way, the processor for SC-FDMA transmission in the 3GPP LTE system does not include a structure for utilizing the MIMO scheme. Therefore, the BS processor for MIMO transmission in the 3GPP LTE system will be described first, and a processor for transmitting an uplink signal by combining the SC-FDMA scheme with the MIMO scheme using the above BS processor will then be described.
FIG. 13 is a block diagram illustrating a processor for enabling the base station (BS) to transmit a downlink signal using the MIMO scheme in the 3GPP LTE system.
A base station (BS) in the 3GPP LTE system can transmit one or more codewords via a downlink. Therefore, one or more codewords may be processed as complex symbols by the scrambling module 301 and the modulation mapper 302 in the same manner as in the uplink operation shown in FIG. 12. Thereafter, the complex symbols are mapped to a plurality of layers by the layer mapper 303, and each layer is multiplied by a predetermined precoding matrix selected according to the channel status and is then allocated to each transmission antenna by the precoding module 304. The processed transmission signals of individual antennas are mapped to time-frequency resource elements to be used for data transmission by the resource element mapper 305. Thereafter, the mapped result may be transmitted via each antenna after passing through the OFDMA signal generator 306.
However, if a downlink signal scheme shown in FIG. 13 is used in the 3GPP LTE system, PAPR or CM properties may be degraded. Thus, it is necessary for a UE to effectively combine the SC-FDMA scheme for maintaining good PAPR and CM properties described in FIGS. 11 and 12 with the MIMO scheme shown in FIG. 13, and a UE for performing precoding using the precoding matrix capable of maintaining good PAPR and CM properties described in the above embodiment must be constructed.
In accordance with one embodiment of the present invention, it is assumed that a UE for transmitting an uplink signal via multiple antennas (multi-antenna) includes multiple antennas (not shown) for transmitting and receiving signals. Referring to FIG. 10, the UE 20 includes a memory 22 for storing a codebook, and a processor 21 that are connected to multiple antennas (not shown) and the memory 22 so as to process uplink signal transmission. In this case, the codebook stored in the memory 22 includes precoding matrices established in a manner that a single layer signal is transmitted to each of the multiple antennas. The processor 21 of the UE configured as described above will hereinafter be described in detail.
FIG. 14 illustrates a processor of the UE according to one embodiment of the present invention.
Referring to FIG. 14, the processor of the UE 20 according to one embodiment of the present invention includes a codeword to layer mapper 1401 for mapping uplink signals to a predetermined number of layers corresponding to a specific rank, a predetermined number of DFT modules 1402 for performing Discrete Fourier Transform (DFT) spreading on each of the predetermined number of layer signals, and a precoder 1403. The precoder 1403 selects a specific precoding matrix established in a manner that one layer signal is transmitted to each antenna 1405 so as to precode a DFT-spread resultant layer signal received from the DFT module 1402. Particularly, in this embodiment of the present invention, each DFT module 1402 performs spreading of each layer signal, this DFT module 1402 for spreading each layer signal is located just before the precoder 1403. When the precoder 1403 performs precoding, the precoder 1403 is configured such that each layer signal is mapped to one antenna and then transmitted via the mapped antenna, so that single carrier characteristics of each layer signal are maintained and good PAPR and CM properties are also maintained. In the meantime, the UE 20 further includes a transmission module. The transmission module performs a process constructing an SC-FDMA symbol upon the precoded signal, and transmits the resultant precoded signal to the base station (BS) via multiple antennas 1405.
In the meantime, the precoder 1403 selects a precoding matrix to be used for signal transmission from among the codebook stored in the memory 22, and performs precoding on the selected precoding matrix. Preferably, these precoding matrices may be precoding matrices established for equalizing transmission powers of multiple antennas and/or transmission powers of respective layers.
The number of multiple antennas 1405 may be 2 or 4. The processor of the UE according to one embodiment of the present invention may further perform not only a layer shift function for periodically or aperiodically changing a layer mapped to a specific codeword but also an antenna shift function for periodically or aperiodically changing an antenna via which a specific layer signal is transmitted. The layer shift function may be performed by the layer mapper 1401 separately from the precoding of the precoder 1403, or may also be performed through column permutation of the precoding matrix when the precoder 1403 performs precoding. In addition, the antenna shift function may also be carried out separately from the precoding of the precoder 1403, or may also be performed through row permutation of the precoding matrix.
The exemplary embodiments described hereinabove are combinations of elements and features of the present invention. The elements or features may be considered selective unless otherwise mentioned. Each element or feature may be practiced without being combined with other elements or features. Further, the embodiments of the present invention may be constructed by combining parts of the elements and/or features. Operation orders described in the embodiments of the present invention may be rearranged. Some constructions or characteristics of any one embodiment may be included in another embodiment and may be replaced with corresponding constructions or characteristics of another embodiment. It is apparent that the present invention may be embodied by a combination of claims which do not have an explicit cited relation in the appended claims or may include new claims by amendment after application.
The embodiments of the present invention may be achieved by various means, for example, hardware, firmware, software, or a combination thereof. In a hardware configuration, the embodiments of the present invention may be implemented by one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, microcontrollers, microprocessors, etc.
In a firmware or software configuration, the embodiments of the present invention may be achieved by a module, a procedure, a function, etc. performing the above-described functions or operations. Software code may be stored in a memory unit and driven by a processor. The memory unit may be located at the interior or exterior of the processor and may transmit data to and receive data from the processor via various known means.
It will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit or scope of the invention. Therefore, the above-mentioned detailed description must be considered only for illustrative purposes instead of restrictive purposes. The scope of the present invention must be decided by a rational analysis of claims, and all modifications within equivalent ranges of the present invention are contained in the scope of the present invention. It is apparent that the present invention may be embodied by a combination of claims which do not have an explicit cited relation in the appended claims or may include new claims by amendment after application.
As apparent from the above description, the present invention can maintain PAPR or CM properties while transmitting uplink signals using a MIMO scheme.
In addition, the present invention uniformly controls or adjusts antenna/layer transmission power, minimizes an amount of signaling overhead required for precoding matrix information, and acquires a maximum diversity gain.
The present invention is applicable to a wideband wireless mobile communication system.
It will be apparent to those skilled in the art that various modifications and variations can be made in the present invention without departing from the spirit or scope of the inventions. Thus, it is intended that the present invention covers the modifications and variations of this invention provided they come within the scope of the appended claims and their equivalents.

## Claims

1. A method for controlling a user equipment (UE) to transmit uplink signals via 4 antennas, the method comprising:
mapping the uplink signals to 3 layers;
performing Discrete Fourier Transform (DFT) spreading upon each of the 3 layers;
precoding each of the DFT-spread layers by using a specific precoding matrix selected from among a prestored codebook for rank 3, wherein every precoding matrix in the prestored codebook for rank 3 is established in a manner that each one of the 4 antennas transmits only one layer of the 3 layers, and a first column has 2 non-zero elements while each of a second and a third columns respectively has one non-zero element; and
transmitting the precoded signals to a base station (BS) via the 4 antennas by performing a predetermined process for constructing a Single Carrier-Frequency Division Multiple Access (SC-FDMA) symbol upon the precoded signals.

2. The method according to claim 1, wherein every precoding matrix in the prestored codebook for rank 3 is established in a manner that the 4 antennas have uniform transmission power therebetween.

3. The method according to claim 1, wherein the uplink signals comprise 2 codewords, and
wherein a first codeword of the 2 codewords is mapped to a first layer of the 3 layers while a second codeword of the 2 codewords is alternately mapped to a second and a third layers of the 3 layers.

4. The method according to claim 1, wherein the prestored codebook for rank 3 includes 6 types of precoding matrixes,
wherein one of the 6 types of precoding matrix is configured in a form of





  
    
      [
      
        
          
            1
          
          
            0
          
          
            0
          
        
        
          
            0
          
          
            1
          
          
            0
          
        
        
          
            0
          
          
            0
          
          
            1
          
        
        
          
            X
          
          
            0
          
          
            0
          
        
      
      ]
    
    .
  
  ,




and satisfying a condition of




  
    X
    ∈
    
      {
      
        1
        ,
        
          
            1
            +
            j
          
          
            2
          
        
        ,
        j
        ,
        
          
            1
            -
            j
          
          
            2
          
        
        ,
        
          -
          1
        
        ,
        
          
            
              -
              1
            
            -
            j
          
          
            2
          
        
        ,
        
          -
          j
        
        ,
        
          
            
              -
              1
            
            +
            j
          
          
            2
          
        
      
      }
    
  
  ,




and
wherein individual rows of the precoding matrix respectively correspond to four antennas of the 4 antennas, and individual columns of the precoding matrix respectively correspond to layers.

5. A user equipment (UE) for transmitting uplink signals, the UE comprising:
4 antennas for transmitting and receiving signals;
a memory for storing a codebook for rank 3, wherein every precoding matrix in the prestored codebook for rank 3 is established in a manner that each one of the 4 antennas transmits only one layer of 3 layers, and a first column has 2 non-zero elements while each of a second and a third columns respectively has one non-zero element; and
a processor connected to the 4 antennas and the memory so as to process transmission of the uplink signals,
wherein the processor includes:
a layer mapper for mapping the uplink signals to the 3 layers;
a Discrete Fourier Transform (DFT) module for performing DFT spreading upon each of the 3 layers;
a precoder for precoding each of the DFT-spread layer signals received from the DFT module by selecting a specific precoding matrix from among the codebook for rank 3 stored in the memory; and
a transmission module for performing a predetermined process for constructing a Single Carrier-Frequency Division Multiple Access (SC-FDMA) symbol upon the precoded signals, and transmitting the processed signals to a base station (BS) via the 4 antennas.

6. The UE according to claim 5, wherein every precoding matrix in the codebook for rank 3 is established in a manner that the 4 antennas have uniform transmission power therebetween.

7. The UE according to claim 5, wherein the uplink signals comprise 2 codewords, and
wherein a first codeword of the 2 codewords is mapped to a first layer of the 3 layers while a second codeword of the 2 codewords is alternately mapped to a second and a third layers of the 3 layers.

8. The UE according to claim 5, wherein the prestored codebook for rank 3 includes 6 types of precoding matrixes,
wherein one of the 6 types of precoding matrix is configured in a form of





  
    
      [
      
        
          
            1
          
          
            0
          
          
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            0
          
          
            1
          
        
        
          
            X
          
          
            0
          
          
            0
          
        
      
      ]
    
    .
  
  ,




and satisfying a condition of




  
    X
    ∈
    
      {
      
        1
        ,
        
          
            1
            +
            j
          
          
            2
          
        
        ,
        j
        ,
        
          
            1
            -
            j
          
          
            2
          
        
        ,
        
          -
          1
        
        ,
        
          
            
              -
              1
            
            -
            j
          
          
            2
          
        
        ,
        
          -
          j
        
        ,
        
          
            
              -
              1
            
            +
            j
          
          
            2
          
        
      
      }
    
  
  ,




and
wherein individual rows of the precoding matrix respectively correspond to four antennas of the 4 antennas, and individual columns of the precoding matrix respectively correspond to layers.