Patent Publication Number: US-2012039402-A1

Title: Multiple input multiple output communication system using at least two codebooks

Description:
PRIORITY 
     This application claims the benefit under 35 U.S.C. §119(e) of U.S. provisional application No. 61/372,170, filed on Aug. 10, 2010 in the U.S. Patent and Trademark Office, and the benefit under 35 U.S.C. §119(a) of Korean patent application No. 10-2011-0011945, filed on Feb. 10, 2011, in the Korean Intellectual Property Office, the entire disclosure of each of which is hereby incorporated by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to an apparatus and method for controlling a communication system. More particularly, the present invention relates to a Multiple Input Multiple Output (MIMO) communication system to share channel information using a codebook. 
     2. Description of Related Art 
     In a Multiple Input Multiple Output (MIMO) communication system, a transmitter and a receiver may use a codebook to share channel information. The channel information may include channel direction information and channel quality information. 
     The codebook may include a plurality of codewords. The receiver may select a single codeword from the plurality of codewords, and may feed back, to the transmitter, a precoding matrix indicator indicating an index of the selected codeword. The precoding matrix indicator may be one example of the channel direction information. 
     The transmitter may verify a codeword selected by the receiver, based on the precoding matrix indicator, and may generate a precoding matrix based on the codeword selected by the receiver. The transmitter may precode data based on the precoding matrix and transmit the precoded data via a plurality of transmit antennas. A number of transmit antennas may be variously determined, for example, 2, 4, 8, and the like. 
     SUMMARY OF THE INVENTION 
     Aspects of the present invention are to address at least the above-mentioned problems and/or disadvantages and to provide at least the advantages described below. 
     In accordance with an aspect of the present invention, a communication method of a receiver in a communication system including a transmitter and the receiver is provided. The method includes accessing a memory storing a first codebook including first codewords and a second codebook including second codewords, generating a first precoding matrix indicator corresponding to one of the first codewords and a second precoding matrix indicator corresponding to one of the second codewords, and transmitting, to the transmitter, the first precoding matrix indicator and the second precoding matrix indicator. 
     When the transmitter includes four transmit antennas, the second codewords may include one of 
     
       
         
           
             
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               and 
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             , 
           
         
       
     
     and θ may correspond to a real number. 
     In accordance with another aspect of the present invention, a communication method of a receiver in a communication system including a transmitter and the receiver is provided. The method includes accessing a memory storing a first codebook including first codewords and a second codebook including second codewords, generating a first precoding matrix indicator corresponding to one of the first codewords and a second precoding matrix indicator corresponding to one of the second codewords, and transmitting, to the transmitter, the first precoding matrix indicator and the second precoding matrix indicator. The first precoding matrix indicator may indicate a preferred T matrix that is defined by applying a diagonal operation to at least one column vector included in one of the first codewords. At least one column vector included in each of the second codewords may include diagonal elements present in at least one matrix of 
     
       
         
           
             
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             , 
           
         
       
     
     and θ may correspond to a real number. 
     In accordance with still another aspect of the present invention, a communication method of a transmitter in a communication system including the transmitter and a receiver is provided. The method includes receiving a first precoding matrix indicator and a second precoding matrix indicator, accessing a memory storing a first codebook including first codewords and a second codebook including second codewords, and generating a precoding matrix based on a first codeword indicated by the first precoding matrix indicator and a second codeword indicated by the second precoding matrix indicator. Each of the second codewords may correspond to a diagonal matrix. 
     In accordance with yet another aspect of the present invention, a communication method of a transmitter in a communication system including the transmitter and a receiver is provided. The method includes receiving a first precoding matrix indicator and a second precoding matrix indicator, accessing a memory storing a first codebook including first codewords and a second codebook including second codewords, and generating a precoding matrix based on a first codeword indicated by the first precoding matrix indicator and a second codeword indicated by the second precoding matrix indicator. The precoding matrix indicator may indicate a preferred T matrix that is defined by applying a diagonal operation to at least one column vector included in one of the first codewords. At least one column vector included in each of the second codewords may include diagonal elements present in at least one matrix of 
     
       
         
           
             
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               and 
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               ] 
             
             , 
           
         
       
     
     and θ may correspond to a real number. 
     Other aspects, advantages, and salient features of the invention will become apparent to those skilled in the art from the following detailed description, which, taken in conjunction with the annexed drawings, discloses exemplary embodiments of the invention. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects, features, and advantages of certain exemplary embodiments of the present invention will be more apparent from the following description taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a diagram illustrating a Multiple Input Multiple Output (MIMO) communication system according to an exemplary embodiment of the present invention. 
         FIG. 2  is a diagram illustrating a communication method of a receiver and a transmitter to share channel information using a single codebook according to an exemplary embodiment of the present invention. 
         FIG. 3  is a diagram illustrating a conceptual relationship between two codebooks and a precoding matrix according to an exemplary embodiment of the present invention. 
         FIG. 4  is a diagram illustrating a communication method of a receiver and a transmitter to share channel information using two codebooks according to an exemplary embodiment of the present invention. 
         FIG. 5  is a diagram illustrating a communication apparatus according to an exemplary embodiment of the present invention. 
     
    
    
     Throughout the drawings, like reference numerals will be understood to refer to like parts, components, and structures. 
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     The following description with reference to the accompanying drawings is provided to assist in a comprehensive understanding of exemplary embodiments of the invention as defined by the claims and their equivalents. It includes various specific details to assist in that understanding but these are to be regarded as merely exemplary. Accordingly, those of ordinary skill in the art will recognize that various changes and modifications of the embodiments described herein can be made without departing from the scope and spirit of the invention. In addition, description of well-known functions and constructions may be omitted for i clarity and conciseness. 
     The terms and words used in the following description and claims are not limited to the bibliographical meanings, but, are merely used by the inventor to enable a clear and consistent understanding of the invention. Accordingly, it should be apparent to those skilled in the art that the following description of exemplary embodiments of the present invention is provided for illustration purpose only and not for the purpose of limiting the invention as defined by the appended claims and their equivalents. 
     It is to be understood that the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. Thus, for example, reference to “a component surface” includes reference to one or more of such surfaces. 
       FIG. 1  illustrates a Multiple Input Multiple Output (MIMO) communication system according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 1 , the MIMO communication system may include a transmitter  110  and a plurality of receivers  120 ,  130 , and  140 . 
     N t  transmit antennas may be installed in the transmitter  110 . The transmitter  110  may function as a base station in a downlink, and may function as a terminal in an uplink. N r  receive antennas may be installed in the receivers  120 ,  130 , and  140 . Each of the receivers  120 ,  130 , and  140  may function as a terminal in the downlink, and may function as a base station in the uplink. Hereinafter, exemplary embodiments will be described based on an operation of the transmitter  110  and the receivers  120 ,  130 , and  140  in the downlink. The exemplary embodiments may be applicable to the uplink. 
     Channels may be formed between the transmitter  110  and the receivers  120 ,  130 , and  140 . Data may be transmitted from the transmitter  110  to the receivers  120 ,  130 , and  140  via the channels. The transmitter  110  may precode at least one data stream using a precoding matrix, enhancing a performance of the MIMO communication system. A data stream may also be referred to as data. 
     The transmitter  110  may generate a more accurate precoding matrix by verifying information associated with channel direction and information associated with channel quality. Information associated with the channel direction and information associated with the channel quality may be one example of channel information. Information associated with the channel direction may include a precoding matrix indicator. 
     For example, the transmitter  110  and the receivers  120 ,  130 , and  140  may share the precoding matrix indicator using a codebook. The codebook may include a plurality of codewords. Each of the plurality of codewords may correspond to a vector or a matrix. A size of the codebook may correspond to a number of codewords. For example, a 3-bit codebook may include eight codewords, and a 4-bit codebook may include 16 codewords. 
     Each of the receivers  120 ,  130 , and  140  may select a single codeword from the plurality of codewords, and may generate an indicator of the selected codeword as a precoding matrix indicator. The precoding matrix indicator may be fed back to the transmitter  110 . The transmitter  110  may verify a codeword indicated by the precoding matrix indicator, using the codebook. The transmitter  110  may generate an optimal precoding matrix based on the codeword corresponding to the precoding matrix indicator. 
     A dimension of a precoding matrix may be dependent on a rank of the transmitter  110 . The rank of the transmitter  110  may correspond to a number of data streams desired to be transmitted or a number of layers of the transmitter  110 . 
       FIG. 2  illustrates a communication method of a receiver and a transmitter to share channel information using a single codebook according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 2 , at step  210 , the transmitter may transmit a well-known signal to the receiver. The well-known signal may be a pilot signal. 
     At step  220 , the receiver may estimate a channel formed from the transmitter to the receiver based on the well-known signal. 
     At step  230 , the receiver may select, from a codebook, a codeword suitable for the estimated channel and generate a precoding matrix indicator including an index of the selected codeword. In this example, the same codebook may be stored in both the transmitter and the receiver. 
     At step  240 , the receiver may feed back a precoding matrix indicator to the transmitter. The receiver may also feed back channel quality information and a rank indicator. 
     At step  250 , the transmitter may generate an optimal precoding matrix based on the fed back precoding matrix indicator. At step  260 , the transmitter may transmit data using the precoding matrix. 
     The communication method of the transmitter and the receiver when the transmitter and the receiver use the same single codebook is described above with reference to  FIG. 2 . According to exemplary embodiments, two codebooks may be used for the receiver and the transmitter to share two precoding matrix indicators. 
     Hereinafter, it is assumed that a first codebook C 1  and a second codebook C 2  are present, and two codebooks are stored in the receiver and the transmitter, respectively. It is also assumed that a precoding matrix W is finally recommended by the receiver and is used by the transmitter. 
     Rotation-Based Differential Codebook W 2 W 1    
     Exemplary embodiments may define a precoding matrix W according to Equation 1. 
         W=W   2   W   1   Equation 1
 
     In Equation 1, W 1  corresponds to a preferred first precoding matrix of a receiver corresponding to a first precoding matrix indicator selected by the receiver from the first codebook C 1 , and W 2  corresponds to a preferred second precoding matrix of a receiver corresponding to a second precoding matrix indicator selected by the receiver from the second codebook C 2 . The first codebook C 1  or the first precoding matrix indicator may be used to indicate a property of a channel in a wideband including a plurality of subbands or to indicate a long-term property of the channel. The second codebook C 2  or the second precoding matrix indicator may be used to indicate a property of a channel in a subband or to indicate a short-term property of the channel. The first codebook C 1  may be well defined in various standards, for example, a 3 rd  Generation Partnership Project (3GPP) Long Term Evolution (LTE) Release 8 standard, and the like. For example, the first codebook C 1  may correspond to a codebook defined with respect to four transmit antennas in the 3GPP LTE Release 8 standard. 
     Regardless of the definition of first codewords included in the first codebook C 1 , exemplary embodiments may provide various candidates of the second codebook C 2 . Each of second codewords included in the second codebook C 2  may be a diagonal matrix. When the transmitter has four transmit antennas, each of the second codewords may be defined through the following functions. Since one of the second codewords is selected as W 2 , a structure of the second codewords may be defined by defining a probable structure of W 2  according to Equation 2. 
     
       
         
           
             
               
                 
                   
                     
                       Function 
                        
                       
                           
                       
                        
                       S 
                        
                       
                         : 
                       
                        
                       
                           
                       
                        
                       
                         W 
                         2 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
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                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
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                             0 
                           
                           
                             
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                       ] 
                     
                   
                    
                   
                     
 
                   
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                       D 
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                       1 
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                         : 
                       
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                         W 
                         2 
                       
                     
                     = 
                     
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                       D 
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                         : 
                       
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                         W 
                         2 
                       
                     
                     = 
                     
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                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
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                             0 
                           
                           
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                   Equation 
                    
                   
                       
                   
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                   2 
                 
               
             
           
         
       
     
     In Equation 2, θ corresponds to a real number and may have a variety of values. For example, θ may correspond to one of 0, π, π/2, −π/2, −π/8, π/8, −π/16, π/16, −3π/16, 3π/16, −π/32, π/32, −3π/32, and 3π/32. 
     Referring to Table 1, second codewords of the second codebook C 2  may be defined by applying a corresponding function to entry θ within the table. For example, the following Table 1 may be assumed. 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 θ 
                 0 
                 π 
                 π/2 
                 −π/2 
                 −π/8 
                 π/8 
                 −π/16 
                 π/16 
                 −3π/16 
                 3π/16 
                 −π/32 
                 π/32 
                 −3π/32 
                 3π/32 
               
               
                   
               
             
            
               
                 C2 
                 D1 
                 D1 
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                   
               
            
           
         
       
     
     In this example, the second codebook C 2  may include eight second codewords. For example, D1 may be applied to θ=0, π, π/2, and −π/2, and S may be applied to −π/32, π/32, −3π/32, and 3π/32. When k denotes an index of each of the eight second codewords (k=1, 2, . . . , 8), first through fourth second codewords in the second codebook C 2  may be defined as follows: 
     
       
         
           
             
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                   ] 
                 
               
               } 
             
           
         
       
     
     Similarly, fifth through eighth second codewords in the second codebook C 2  may be defined as follows: 
     
       
         
           
             
               C 
               
                 2 
                 , 
                 
                   5 
                    
                   
                       
                   
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                       , 
                     
                   
                   
                     
                       
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                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   j2π 
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   j3π 
                                   / 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j3π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j6π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j9π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                
                               
                                 j3π 
                                 / 
                                 32 
                               
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                
                               
                                 j6π 
                                 / 
                                 32 
                               
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                
                               
                                 j9π 
                                 / 
                                 32 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               } 
             
           
         
       
     
     In this example, a final precoding matrix W may be defined based on the first precoding matrix indicator and the second precoding matrix indicator. More specifically, the final precoding matrix W may be expressed as a multiplication between one of C 2,k  and one of the first codewords included in the first codebook C 1 . 
     Probable candidates of the second codebook C 2  may be various. For example, the probable candidates of the second codebook C 2  may be defined according to Table 2. 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 θ 
                 1 
                 −1 
                 j 
                 −j 
                 −π/4 
                 π/4 
                 −π/8 
                 π/8 
                 −π/16 
                 π/16 
                 −3π/16 
                 3π/16 
                 −π/32 
                 π/32 
                 −3π/32 
                 3π/32 
                 −π/20 
                 π/20 
                 −π/10 
                 π/10 
               
               
                   
               
             
            
               
                 C 2   
                 D1 
                 D1 
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                   
                   
               
               
                 C 2   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S, 
                 S, 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 D2 
                 D2 
               
               
                 C 2   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 S 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S, D2 
                 S, D2 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D2 
                 D1D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                   
                 D1 
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                   
                 D1 
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1, D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1 
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S, 
                 S, 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D1 
                 D1 
                 D1 
                 D1 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D2 
                 D1, 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1, 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D2 
                 D2 
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D2 
                 D1, 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                   
                 D1 
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                   
                 D1 
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D2 
                 D2 
                 D1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1, 
                 D2 
                   
                   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1 
                   
                 D2 
                 D2 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                   
                 D1 
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D1 
                 D1 
                 D1 
                 D1 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                 D1 
                 D1 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D2 
                 D2 
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                 D2 
                 D2 
                 D2 
                 D2 
               
               
                 C 2   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S, 
                 S, 
                 D2 
                 D2 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                   
                   
                   
                   
                   
                   
                 S, 
                 S, 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
                 D2 
                 D2 
               
               
                 C 2   
                   
                   
                   
                   
                 D2 
                 D2 
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D2 
                 D2 
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D2 
                   
                 D2 
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                 D1 
                 D1 
                 D1 
                 D1 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                 D2 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                 S 
                 S 
                 S 
                 S 
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                 S 
                 S 
                 S 
                 S 
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                 S 
                 S 
                 S 
                 S 
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                   
                 D1 
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D2 
                 D1, 
                 S 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                 D2 
               
               
                 C 2   
                 D1 
                   
                 D2 
                 D2 
                   
                 D1 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                 S 
                 S 
                 S 
                 S 
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1 
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                   
                   
                 D1, 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                 D2 
               
               
                 C 2   
                 D1 
                   
                 D2 
                 D2 
                 D1 
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                 S 
                 S 
                 S, 
                 S, 
               
               
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                 S, 
                 S, 
                 S 
                 S 
               
               
                   
                   
                   
                   
                   
                   
                   
                 D2 
                 D2 
               
               
                 C 2   
                 D1 
                   
                 D1 
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                   
                 D1, 
                 D2 
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                   
                   
                   
                 D2 
               
               
                 C 2   
                 D1 
                 D1 
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                 C 2   
                 D1 
                 D1 
                   
                   
                   
                   
                 D2 
                 D2 
                 S 
                 S 
                 S 
                 S 
               
               
                   
               
            
           
         
       
     
     Another probable candidate of the second codebook C 2  may be defined based on Table 3 as well as Table 2. 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                 θ 
                 1 
                 −1 
                 j 
                 −j 
                 −π/4 
                 π/4 
                 −π/8 
                 π/8 
                 −π/16 
                 π/16 
                 −3π/16 
                 3π/16 
                 −π/32 
                 π/32 
                 −3π/32 
                 3π/32 
                 −π/20 
                 π/20 
                 −π/10 
                 π/10 
               
               
                   
               
             
            
               
                 C 2   
                 D1 
                   
                 D1 
                   
                   
                   
                 D2 
                 D2 
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
                   
                   
                   
                   
               
               
                   
               
            
           
         
       
     
     First through eighth second codewords of the second codebook C 2  defined based on Table 3 may be expressed as follows: 
     
       
         
           
             
               C 
               
                 2 
                 , 
                 
                   1 
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   4 
                 
               
             
             = 
             
               { 
               
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         j 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         j 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             
                               - 
                               jπ 
                             
                             / 
                             8 
                           
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             
                               - 
                               jπ 
                             
                             / 
                             8 
                           
                         
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             jπ 
                             / 
                             8 
                           
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             jπ 
                             / 
                             8 
                           
                         
                       
                     
                   
                   ] 
                 
               
               } 
             
           
         
       
       
         
           
             
               C 
               
                 2 
                 , 
                 
                   5 
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   8 
                 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     jπ 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j2π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j3π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   jπ 
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   j2π 
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   j3π 
                                   / 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j3π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j6π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j9π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                
                               
                                 j3π 
                                 / 
                                 32 
                               
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                
                               
                                 j6π 
                                 / 
                                 32 
                               
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                
                               
                                 j9π 
                                 / 
                                 32 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               } 
             
           
         
       
     
     The aforementioned candidates of the second codebook C 2  may provide a 3-bit codebook. More particularly, predetermined subsets of the second codebook C 2  may be used for rank 2 or more. 
     Transformation-Based Differential Codebook 
     A precoding matrix W finally used by the transmitter may be defined by an inner product between matrices T 1  and W 2 . 
     
       
      
       W=T 
       1 
       W 
       2  
      
     
     The T matrix may be generated based on W 1  corresponding to the first precoding matrix indicator among codewords included in the first codebook C 1 . 
     (1) Rank 1 Case: 
         T   1 =diag( W   1 ) 
     The T matrix T 1  may be defined by applying a diagonal operation to W 1 . When X is a matrix, the diagonal operation diag(X) may transform the matrix X to a column vector including diagonal elements of X. When X is a vector, that is, a matrix including a single column, the diagonal operation diag(X) may transform the vector X to a diagonal matrix including, as diagonal elements, elements included in the vector X. 
     Since W 1  corresponds to a column vector in rank 1 case, T 1  may correspond to the diagonal matrix including, as diagonal elements, elements included in W 1 . 
     (2) Rank 2 Case: 
     In rank 2 case, each of first codewords included in the first codebook may include two column vectors. Each of the first codewords may be divided into two parts. For example, W 1  may be divided into two column vectors w 1  and w 2 . 
     
       
         
           
             
               W 
               1 
             
             = 
             
               [ 
               
                 
                   
                     | 
                   
                   
                     | 
                   
                 
                 
                   
                     
                       w 
                       1 
                     
                   
                   
                     
                       w 
                       2 
                     
                   
                 
                 
                   
                     | 
                   
                   
                     | 
                   
                 
               
               ] 
             
           
         
       
     
     When the receiver selects a first codeword W 1  from the first codebook, the T matrix may be defined by applying the diagonal operation to each of the column vectors included in W 1  as follows: 
         T   1 =[diag( w   1 )diag( w   2 )] 
     where the T matrix corresponds to an N t ×(2N t ) matrix. 
     As described above, W 1  may be selected from the first codebook C 1  based on a wideband channel property or a long-term channel property, and W 2  may be selected from the second codebook C 2  based on a subband channel property or a short-term channel property. More particularly, the first codebook C 1  may be well defined in various standards, for example, a 3GPP LTE Release 8 standard. The second codebook C 2  may be described as follows. 
     C 2,r,k  may be assumed as a k th  codeword of the second codebook C 2  for rank r. 
     For rank 1, C 2,1,k  may be defined to establish C 2,1,k =diag(C 2,k ). In this example, C 2,k  may be assumed to be defined based on Table 1 through Table 3. 
     For rank 2, C 2,2,k  may be defined to establish 
     
       
         
           
             
               C 
               
                 2 
                 , 
                 2 
                 , 
                 k 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         C 
                         
                           2 
                           , 
                           1 
                           , 
                           k 
                         
                       
                     
                     
                       
                         0 
                         
                           4 
                           × 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         0 
                         
                           4 
                           × 
                           1 
                         
                       
                     
                     
                       
                         C 
                         
                           2 
                           , 
                           1 
                           , 
                           k 
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     For example, C 2,k  may be assumed as follows: 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 θ 
                 0 
                 π 
                 π/2 
                 −π/2 
                 −π/8 
                 π/8 
                 −π/16 
                 π/16 
                 −3π/16 
                 3π/16 
                 −π/32 
                 π/32 
                 −3π/32 
                 3π/32 
               
               
                   
               
             
            
               
                   
                 D1 
                 D1 
                 D1 
                 D1 
                   
                   
                   
                   
                   
                   
                 S 
                 S 
                 S 
                 S 
               
               
                   
               
            
           
         
       
     
     For rank 1, the second codebook C 2  may be obtained by applying the diagonal operation to C 2,k . Accordingly, 
     
       
         
           
             
               C 
               
                 2 
                 , 
                 
                   1 
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   4 
                 
               
             
             = 
             
               { 
               
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                           - 
                           1 
                         
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         j 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         j 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                           - 
                           j 
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                       
                         
                           - 
                           j 
                         
                       
                     
                   
                   ] 
                 
               
               } 
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               C 
               
                 2 
                 , 
                 
                   5 
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   8 
                 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     jπ 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j2π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j3π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   jπ 
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   j2π 
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   j3π 
                                   / 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                 
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j3π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j6π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                  
                                 
                                   
                                     - 
                                     j9π 
                                   
                                   / 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                
                               
                                 j3π 
                                 / 
                                 32 
                               
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                
                               
                                 j6π 
                                 / 
                                 32 
                               
                             
                           
                           
                             
                                 
                             
                           
                         
                         
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                 
                             
                           
                           
                             
                                
                               
                                 j9π 
                                 / 
                                 32 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               } 
             
           
         
       
       
         
           
             
               may 
                
               
                   
               
                
               be 
                
               
                   
               
                
               transformed 
                
               
                   
               
                
               to 
                
               
                   
               
                
               
                 C 
                 
                   2 
                   , 
                   1 
                   , 
                   
                     1 
                      
                     
                         
                     
                      
                     … 
                      
                     
                         
                     
                      
                     4 
                   
                 
               
             
             = 
             
               { 
               
                 
                   [ 
                   
                     
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         
                           - 
                           1 
                         
                       
                     
                     
                       
                         
                           - 
                           1 
                         
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         j 
                       
                     
                     
                       
                         j 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         
                           - 
                           j 
                         
                       
                     
                     
                       
                         
                           - 
                           j 
                         
                       
                     
                   
                   ] 
                 
               
               } 
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               C 
               
                 2 
                 , 
                 1 
                 , 
                 
                   5 
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   8 
                 
               
             
             = 
             
               
                 { 
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                       
                       
                         
                           
                              
                             
                               
                                 - 
                                 jπ 
                               
                               / 
                               32 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 - 
                                 j2π 
                               
                               / 
                               32 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 - 
                                 j3π 
                               
                               / 
                               32 
                             
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           1 
                         
                       
                       
                         
                           
                              
                             
                               jπ 
                               / 
                               32 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               j2π 
                               / 
                               32 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               j3π 
                               / 
                               32 
                             
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           1 
                         
                       
                       
                         
                           
                              
                             
                               
                                 - 
                                 j3π 
                               
                               / 
                               32 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 - 
                                 6 
                               
                                
                               
                                 π 
                                 / 
                                 32 
                               
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 - 
                                 j9π 
                               
                                
                               .32 
                             
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           1 
                         
                       
                       
                         
                           
                              
                             
                               j3π 
                               / 
                               32 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               j6π 
                               / 
                               32 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               j9π 
                               / 
                               32 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
               . 
             
           
         
       
     
     For rank 1, each of the second codewords included in the second codebook C 2  may include diagonal elements present in at least one matrix of 
     
       
         
           
             
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                        
                       jθ 
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                        
                       j2θ 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                        
                       j3θ 
                     
                   
                 
               
               ] 
             
             , 
             
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                        
                       jθ 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                        
                       jθ 
                     
                   
                 
               
               ] 
             
             , 
             
               
                 and 
                  
                 
                     
                 
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                          
                         jθ 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                          
                         jθ 
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     For rank 2, each of the second codewords included in the second codebook C 2  may include diagonal elements present in at least one matrix of 
     
       
         
           
             
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                        
                       jθ 
                     
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                        
                       j2θ 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                        
                       j3θ 
                     
                   
                 
               
               ] 
             
             , 
             
               [ 
               
                 
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     1 
                   
                   
                     0 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                        
                       jθ 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     0 
                   
                   
                     0 
                   
                   
                     
                        
                       jθ 
                     
                   
                 
               
               ] 
             
             , 
             
               
                 and 
                  
                 
                     
                 
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                          
                         jθ 
                       
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       
                          
                         jθ 
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     In the above example, the second codebook C 2  may be given as 
     
       
         
           
             
                 
             
              
             
               
                 C 
                 
                   2 
                   , 
                   2 
                   , 
                   
                     1 
                      
                     
                         
                     
                      
                     … 
                      
                     
                         
                     
                      
                     4 
                   
                 
               
               = 
               
                 { 
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                             - 
                             1 
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                             - 
                             1 
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                             - 
                             1 
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           j 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           j 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           j 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           j 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                             - 
                             j 
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                             - 
                             j 
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           1 
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                             - 
                             j 
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                             - 
                             j 
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
           
         
       
       
         
           
             
                 
             
              
             and 
           
         
       
       
         
           
             
               C 
               
                 2 
                 , 
                 2 
                 , 
                 
                   5 
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   8 
                 
               
             
             = 
             
               { 
               
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                            
                           
                             
                               - 
                               jπ 
                             
                             / 
                             32 
                           
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                            
                           
                             
                               - 
                               j2π 
                             
                             / 
                             32 
                           
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                            
                           
                             
                               - 
                               j3π 
                             
                             / 
                             32 
                           
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             
                               - 
                               jπ 
                             
                             / 
                             32 
                           
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             
                               - 
                               j2π 
                             
                             / 
                             32 
                           
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             
                               - 
                               j3π 
                             
                             / 
                             32 
                           
                         
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                            
                           
                             jπ 
                             / 
                             32 
                           
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                            
                           
                             j2π 
                             / 
                             32 
                           
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                            
                           
                             j3π 
                             / 
                             32 
                           
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             jπ 
                             / 
                             32 
                           
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             j2π 
                             / 
                             32 
                           
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             j3π 
                             / 
                             32 
                           
                         
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                            
                           
                             
                               - 
                               j3π 
                             
                             / 
                             32 
                           
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                            
                           
                             
                               - 
                               j6π 
                             
                             / 
                             32 
                           
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                            
                           
                             
                               - 
                               j9π 
                             
                             / 
                             32 
                           
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         1 
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             
                               - 
                               j3π 
                             
                             / 
                             32 
                           
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             
                               - 
                               j6π 
                             
                             / 
                             32 
                           
                         
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                            
                           
                             
                               - 
                               j9π 
                             
                             / 
                             32 
                           
                         
                       
                     
                   
                   ] 
                 
               
               } 
             
           
         
       
     
     In this example, the T matrix may correspond to a diagonal matrix for rank 1. The T matrix T 1  may have a form of the following non-diagonal matrix. 
     
       
         
           
             
               T 
               1 
             
             = 
             
               [ 
               
                 
                   
                     1 
                   
                   
                     
                       
                         ρ 
                         0 
                       
                        
                       
                          
                         jθ0 
                       
                     
                   
                   
                     
                       
                         ρ 
                         0 
                         2 
                       
                        
                       
                          
                         
                           j2θ 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         ρ 
                         0 
                         3 
                       
                        
                       
                          
                         
                           j3θ 
                           0 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ρ 
                         0 
                       
                        
                       
                          
                         
                           - 
                           
                             jθ 
                             0 
                           
                         
                       
                     
                   
                   
                     1 
                   
                   
                     
                       
                         ρ 
                         0 
                       
                        
                       
                          
                         
                           jθ 
                           0 
                         
                       
                     
                   
                   
                     
                       
                         ρ 
                         0 
                         2 
                       
                        
                       
                          
                         
                           j2θ 
                           0 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ρ 
                         0 
                         2 
                       
                        
                       
                          
                         
                           - 
                           
                             j2θ 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         ρ 
                         0 
                       
                        
                       
                          
                         
                           - 
                           
                             jθ 
                             0 
                           
                         
                       
                     
                   
                   
                     1 
                   
                   
                     
                       
                         ρ 
                         0 
                       
                        
                       
                          
                         
                           jθ 
                           0 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ρ 
                         0 
                         3 
                       
                        
                       
                          
                         
                           - 
                           
                             j3θ 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         ρ 
                         0 
                         2 
                       
                        
                       
                          
                         
                           - 
                           
                             j2θ 
                             0 
                           
                         
                       
                     
                   
                   
                     
                       
                         ρ 
                         0 
                       
                        
                       
                          
                         
                           - 
                           
                             jθ 
                             0 
                           
                         
                       
                     
                   
                   
                     1 
                   
                 
               
               ] 
             
           
         
       
     
     In the T 1  matrix, ρ 0  may correspond to a value associated with a channel correlation and be quantized to a plurality of candidate values. 
     Adaptive Codebook 
     Exemplary embodiments may provide an adaptive codebook used in an adaptive mode. According to exemplary embodiments, a precoding matrix W used by the transmitter may be expressed by W=W 1 W 2 . In this example, W 2  may be well defined based on various standards, for example, a 3GPP LTE Release 8 standard and the like. Matrices W 1  may be selected as non-diagonal matrices T1. The precoding matrix W may also be defined as W=W 1 W 2 /∥W 1 W 2 ∥. 
       FIG. 3  illustrates a conceptual relationship between two codebooks and a precoding matrix according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 3 , both a transmitter and a receiver may store a first codebook C 1    310  and a second codebook C 2    320 . The receiver may select a preferred first codeword from the first codebook C 1    310 , and may select a preferred second codeword from the second codebook C 2    320 . A first precoding matrix indicator may be fed back to the transmitter as an index of the preferred first codeword, and a second precoding matrix indicator may be fed back to the transmitter as an index of the preferred second codeword. 
     The transmitter may find the preferred first codeword from the first codebook C 1    310  based on the first precoding matrix indicator and the second precoding matrix indicator, and may find the preferred second codeword from the second codebook C 2    320 . The transmitter may determine a precoding matrix W=f(W 1 , W 2 ) based on the preferred first codeword and the preferred second codeword. 
       FIG. 4  illustrates a communication method of a receiver and a transmitter to share channel information using two codebooks according to an exemplary embodiment of the present invention. 
     A rotation-based differential codebook W 2 W 1  case and a transformation-based differential codebook will be described with reference to  FIG. 4 . 
     Rotation-Based Differential Codebook W 2 W 1  Case: 
     At step  410 , the transmitter and the receiver may maintain a memory storing a first codebook C 1  defined in known standards and a second codebook C 2  that may be defined based on Table 1 through Table 3. 
     At step  420 , when the receiver measures a channel formed between the transmitter and the receiver, the receiver may generate a first precoding matrix indicator corresponding to W 1  selected from the first codebook C 1 , and may generate a second precoding matrix indicator corresponding to W 2  selected from the second codebook C 2 . 
     At step  430 , the receiver may feed back the first precoding matrix indicator and the second precoding matrix indicator to the transmitter. The receiver may further feed back channel quality information and a rank indicator. 
     At step  440 , the transmitter may find W 1  and W 2  based on the first precoding matrix indicator and the second precoding matrix indicator and generate a precoding matrix W based on W=W 2 W 1 . 
     At step  450 , the transmitter may precode a data stream based on the precoding matrix W and may transmit data using a plurality of transmit antennas, for example, 2, 4, 8, and the like. 
     Transformation-Based Differential Codebook Case: 
     At step  410 , the transmitter and the receiver may maintain a memory storing a first codebook C 1  and a second codebook C 2  including a plurality of C 2,r,k . The first codebook C 1  may correspond to a codebook defined in known standards, and may also correspond to a codebook including a plurality of candidates of T matrix T 1 . For example, since T 1  is a function of W 1  in W=T 1 W 2 , the transmitter and the receiver may use a codebook including candidates with respect to W 1  and may also use a codebook including candidates with respect to T 1 . Hereinafter, the first precoding matrix indicator may be assumed to be generated from the codebook including candidates with respect to W 1 . However, it is only an example and thus, the exemplary embodiments are not limited thereto. 
     At step  420 , the receiver may generate the first precoding matrix indicator from the first codebook C 1  and generate the second precoding matrix indicator from the second codebook C 2 . 
     At step  430 , the receiver may feed back the first precoding matrix indicator and the second precoding matrix indicator to the transmitter. The receiver may further feed back channel quality information and a rank indicator. 
     Although not illustrated in  FIG. 4 , the transmitter may find W 1  based on the first precoding matrix indicator and may find the T matrix based on T 1 =diag(W 1 ) and T 1 =[diag(w 1 )diag(w 2 )]. 
     At step  440 , the transmitter may generate a precoding matrix W=T 1 W 2  based on T matrix and W 2 . 
     At step  450 , the transmitter may precode a data stream based on the precoding matrix W and may transmit data using a plurality of transmit antennas, for example, 2, 4, 8, and the like. 
     The processes, functions, methods and/or software described above may be recorded, stored, or fixed in one or more computer-readable storage media that includes program instructions to be implemented by a computer to cause a processor to execute or perform the program instructions. The media may also include, alone or in combination with the program instructions, data files, data structures, and the like. The media and program instructions may be those specially designed and constructed, or they may be of the kind well-known and available to those having skill in the computer software arts. Examples of computer-readable storage media include magnetic media such as hard disks, floppy disks, and magnetic tape, optical media such as CD ROM disks and DVDs, magneto-optical media such as optical disks, and hardware devices that are specially configured to store and perform program instructions, such as Read-Only Memory (ROM), Random Access Memory (RAM), flash memory, and the like. Examples of program instructions include both machine code, such as produced by a compiler, and files containing higher level code that may be executed by the computer using an interpreter. The described hardware devices may be configured to act as one or more software modules that are recorded, stored, or fixed in one or more computer-readable storage media, in order to perform the operations and methods described above, or vice versa. In addition, a computer-readable storage medium may be distributed among computer systems connected through a network and non-transitory computer-readable codes or program instructions may be stored and executed in a decentralized manner. 
       FIG. 5  illustrates a communication apparatus according to an exemplary embodiment of the present invention. 
     Referring to  FIG. 5 , the communication apparatus may include a memory  510 , a processor  520 , and a communication interface  530 . The communication apparatus may be installed in any of a receiver and a transmitter. 
     When the Communication Apparatus is Installed in the Receiver: 
     The memory  510  may store a first codebook including first codewords and a second codebook including second codewords. 
     Based on the first codewords and the second codewords stored in the memory  510 , the processor  520  may generate a first precoding matrix indicator corresponding to one of the first codewords and may generate a second precoding matrix indicator corresponding to one of the second codewords. 
     Assuming W=W 2 W 1  or W=T 1 W 2 , the processor  520  may generate the first precoding matrix indicator and the second precoding matrix indicator. The processor  520  may determine T 1 . 
     The communication interface  530  may transmit, to the transmitter, the first precoding matrix indicator corresponding to W 1  or T 1  and the second precoding matrix indicator corresponding to W 2 . 
     When the Communication Apparatus is Installed in the Transmitter: 
     The memory  510  may store the first codebook including first codewords and the second codebook including second codewords. 
     The communication interface  530  may receive the first precoding matrix indicator and the second precoding matrix indicator. 
     The processor  520  may extract, from the memory  510 , W 1  corresponding to the first precoding matrix indicator and W 2  corresponding to the second precoding matrix indicator. 
     The processor  530  may generate a precoding matrix W based on W 1  and W 2 . 
     Descriptions made above with reference to  FIG. 1  through  FIG. 4  are applicable to the communication apparatus of  FIG. 5  and thus, further descriptions will be omitted here. 
     While the invention has been shown and described with reference to certain exemplary embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims their equivalents.