Abstract:
The present invention relates to methods and apparatus for establishing a precoding codebook for a Multiple Input Multiple Output (MIMO) wireless communication system. The precoding codebook includes a plurality of codebook entries. Each codebook entry includes four sets of vectors for four respective corresponding transmission ranks. The vectors may be predetermined, or generated from source unitary matrices. In addition, the codebook is fully nested.

Description:
CLAIM OF PRIORITY 
     This application makes reference to, incorporates the same herein, and claims all benefits accruing under 35 U.S.C. §119 from a provisional application earlier filed in the U.S. Patent &amp; Trademark Office on 26 Jun. 2007 and there duly assigned Ser. No. 60/929,396. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to methods and apparatus for establishing a precoding codebook for a Multiple Input Multiple Output (MIMO) wireless communication system. 
     2. Description of the Related Art 
     Orthogonal Frequency Division Multiplexing (OFDM) is a popular wireless communication technology to multiplex data in frequency domain. 
     A multiple antenna communication system, which is often referred to as multiple input multiple output (MIMO) system, is widely used in combination with OFDM technology, in a wireless communication system to improve system performance. 
     In a MIMO system, both transmitter and receiver are equipped with multiple antennas. Therefore, the transmitter is capable of transmitting independent data streams simultaneously in the same frequency band. Unlike traditional means of increasing throughput (i.e., the amount of data transmitted per time unit) by increasing bandwidth or increasing overall transmit power, MIMO technology increases the spectral efficiency of a wireless communication system by exploiting the additional dimension of freedom in the space domain due to multiple antennas. Therefore MIMO technology can significantly increase the throughput and range of the system. 
     When the transmission channels between the transmitters and the receivers are relatively constant, it is possible to use a closed-loop MIMO scheme to further improve system performance. In a closed-loop MIMO system, the receiver informs the transmitter of feedback information regarding the channel condition. The transmitter utilizes this feedback information, together with other considerations such as scheduling priority, data and resource availability, to optimize the transmission scheme. 
     A popular closed-loop MIMO scheme is MIMO preceding. With precoding, the data streams to be transmitted are precoded, i.e., pre-multiplied by a precoding matrix, before being passed on to the multiple transmit antennas in a transmitter. 
     In a contemporary closed-loop MIMO precoding scheme, when a transmitter precodes data before transmitting the data to a receiver, the transmitter informs the receiver of the precoding information such as an identification of the precoding matrix by transmitting dedicated pilots (also referred to as reference signals) or explicit control information that carries the precoding information. 
     A significant problem with the contemporary single user precoding codebooks is that the codebooks require either too much memory for storage or too complicated math to generate in real time. 
     SUMMARY OF THE INVENTION 
     It is therefore an object of the present invention to provide an improved system and an improved method for transmitting data in a multiple input multiple output (MIMO) system. 
     It is another object to provide an improved system and an improved method to establish a simple precoding codebook. 
     According to one aspect of the present invention, a codebook is established to include a plurality of codebook entries. Each codebook entry includes four sets of vectors for four respective corresponding transmission ranks. The codebook is established by: 
                                                                                               Rank 1   Rank 2   Rank 3   Rank 4                                1   V 1     V 1     V 4     V 1     V 2     V 4     V 1     V 2     V 3     V 4         2   V 5     V 5     V 6     V 5     V 6     V 7     V 5     V 6     V 7     V 8         3   V 3     V 3     V 4     V 3     V 4     V 1     V 1     V 4     V 3     V 2         4   V 7     V 7     V 8     V 7     V 8     V 5     V 5     V 8     V 7     V 6         5   V 9     V 9     V 12     V 9     V 10     V 12     V 9     V 10     V 11     V 12         6   V 13     V 13     V 16     V 13     V 14     V 16     V 13     V 14     V 15     V 16         7   V 11     V 11     V 9     V 11     V 9     V 10     V 11     V 9     V 12     V 10         8   V 15     V 15     V 13     V 15     V 13     V 14     V 15     V 13     V 16     V 14         9   V 2     V 2     V 1     V 2     V 1     V 3     V 2     V 1     V 4     V 3         10   V 6     V 6     V 7     V 6     V 8     V 7     V 6     V 5     V 8     V 7         11   V 4     V 4     V 2     V 4     V 3     V 2     V 4     V 2     V 3     V 1         12   V 8     V 8     V 6     V 8     V 6     V 5     V 8     V 6     V 7     V 5         13   V 17     V 17     V 18     V 17     V 18     V 19     V 17     V 18     V 19     V 20         14   V 18     V 18     V 20     V 18     V 17     V 20     V 18     V 20     V 17     V 19         15   V 19     V 19     V 17     V 19     V 20     V 17     V 17     V 20     V 19     V 18         16   V 20     V 20     V 19     V 20     V 19     V 18     V 20     V 19     V 18     V 17                      
where V 1  through V 20  are certain vectors.
 
     The vectors V 1  through V 20  may be respectively established by:
 
 V   1   T =[1 1 1 1],
 
 V   2   T =[1 1 −1 1],
 
 V   3   T =[1 −1 1 −1],
 
 V   4   T =[1 −1 −1 1],
 
 V   5   T =[1  j  −1 − j],  
 
 V   6   T   =[−j  1 − j  1],
 
 V   7   T =[−1  j  1 − j],  
 
 V   8   T   =[j  1  j  1],
 
 V   9   T =└1(1+ j )/√{square root over (2)} j (−1 +j )/√{square root over (2)}┘,
 
 V   10   T =└1 −j )/√{square root over (2)}1(−1− j )/√{square root over (2)}− j┘,  
 
 V   11   T   =└−j (−1 +j )/√{square root over (2)}1(−1 −j )/√{square root over (2)}┘
 
 V   12   T =└(−1− j )/√{square root over (2)} j (−1 +j )/√{square root over (2)}1┘,
 
 V   13   T =└1(−1 +j )/√{square root over (2)}− j (1 +j )/√{square root over (2)}┘,
 
 V   14   T =└(−1− j )/√{square root over (2)}1(1− j )/√{square root over (2)} j┘,  
 
 V   15   T   =└j (1+ j )/√{square root over (2)}1(1− j )/√{square root over (2)}┘,
 
 V   16   T   =└( 1 −j )/√{square root over (2)}− j (1+ j )/√{square root over (2)}1┘,
 
 V   17   T =[1 1 1 −1],
 
 V   18   T =[1 1 −1 1],
 
 V   19   T =[1 −1 1 1], and
 
 V   20   T =[−1 1 1 1].
 
     Alternatively, the certain vectors V 1  through V 20  may be established from five matrices M 1  to M 5  generated from a Householder equation and five input vectors u 1  to u 5 . The five matrices M 1  to M 5  are given as: 
                       M   1     =       [           v   1           v   2           v   3           v   4           ]     =         I   4     -     2   ⁢           ⁢     u   1     ⁢       u   1   H     /            u   1          2           =     0.5   *     [         1       1       1       1           1       1         -   1           -   1             1         -   1         1         -   1             1         -   1           -   1         1         ]             ,                   M   2     =       [           v   5           v   6           v   7           v   8           ]     =         I   4     -     2   ⁢           ⁢     u   2     ⁢       u   2   H     /            u   2          2           =     0.5   *     [         1       j         -   1           -   j               -   j         1         -   j         1             -   1         j       1         -   j             j       1       j       1         ]             ,                   M   3     =       [           v   9           v   10           v   11           v   12           ]     =         I   4     -     2   ⁢           ⁢     u   3     ⁢       u   3   H     /            u   3          2           =     0.5   *     [         1           1   +   j       2           j             -   1     +   j       2                   1   -   j       2           1           1   -   j       2             -   j               -   j               -   1     +   j       2           1             -   1     -   j       2                     -   1     -   j       2           j             -   1     +   j       2           1         ]             ,                   M   4     =       [           v   13           v   14           v   15           v   16           ]     =         I   4     -     2   ⁢           ⁢     u   4     ⁢       u   4   H     /            u   4          2           =     0.5   *     [         1             -   1     +   j       2             -   j             1   +   j       2                     -   1     -   j       2           1           1   -   j       2           1           j           1   +   j       2           1           1   -   j       2                   1   -   j       2             -   j             1   +   j       2           1         ]             ,                   M   5     =       [           v   17           v   18           v   19           v   20           ]     =         I   4     -     2   ⁢           ⁢     u   5     ⁢       u   5   H     /            u   5          2           =     0.5   *     [         1       1       1         -   1             1       1         -   1         1           1         -   1         1       1             -   1         1       1       1         ]             ,               
where the five input vectors u 1  to u 5  are:
 
 u   1   T =[1 −1 −1 −1],
 
 u   2   T =[1  j  1  −j],  
 
 u   3   T =└1(−1 +j )/√{square root over (2)} j (1 +j )/√{square root over (2)}┘,
 
 u   4   T =└1(1 +j )/√{square root over (2)}− j (− 1 + j )/√{square root over (2)}┘, and
 
 u   5   T =[1 −1 −1 1].
 
     Still alternatively, the certain vectors V 1  through V 20  may be established from five matrices M 1  to M 5  generated from rotating certain unitary matrices. The rotating operation is established by a rotation matrix U rot  given as: 
               U   rot     =         1     2       ⁡     [         1       0         -   1         0           0       1       0         -   1             1       0       1       0           0       1       0       1         ]       .           
In addition, the five matrices M 1  to M 5  are established by
 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       1 
                     
                     = 
                       
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 1 
                               
                             
                             
                               
                                 v 
                                 2 
                               
                             
                             
                               
                                 v 
                                 3 
                               
                             
                             
                               
                                 v 
                                 4 
                               
                             
                           
                         
                         ] 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             U 
                             rot 
                           
                           * 
                           
                             
                               1 
                               
                                 2 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       - 
                                       1 
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           0.5 
                           * 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     
                       M 
                       2 
                     
                     = 
                       
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 5 
                               
                             
                             
                               
                                 v 
                                 6 
                               
                             
                             
                               
                                 v 
                                 7 
                               
                             
                             
                               
                                 v 
                                 8 
                               
                             
                           
                         
                         ] 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           
                             U 
                             rot 
                           
                           * 
                           
                             
                               1 
                               
                                 2 
                               
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     0 
                                   
                                   
                                     j 
                                   
                                   
                                     0 
                                   
                                   
                                     
                                       - 
                                       j 
                                     
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     j 
                                   
                                   
                                     0 
                                   
                                   
                                     j 
                                   
                                   
                                     0 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           0.5 
                           * 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   j 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   
                                     - 
                                     j 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                 
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     j 
                                   
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   j 
                                 
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     j 
                                   
                                 
                               
                               
                                 
                                   j 
                                 
                                 
                                   1 
                                 
                                 
                                   j 
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     
                       M 
                       3 
                     
                     = 
                       
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 9 
                               
                             
                             
                               
                                 v 
                                 10 
                               
                             
                             
                               
                                 v 
                                 11 
                               
                             
                             
                               
                                 v 
                                 12 
                               
                             
                           
                         
                         ] 
                       
                       = 
                         
                       ⁢ 
                       
                         0.5 
                         * 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       j 
                                     
                                     ) 
                                   
                                   / 
                                   
                                     2 
                                   
                                 
                               
                               
                                 j 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         1 
                                       
                                       + 
                                       j 
                                     
                                     ) 
                                   
                                   / 
                                   
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       j 
                                     
                                     ) 
                                   
                                   / 
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       j 
                                     
                                     ) 
                                   
                                   / 
                                   
                                     2 
                                   
                                 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   j 
                                 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         1 
                                       
                                       + 
                                       j 
                                     
                                     ) 
                                   
                                   / 
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         1 
                                       
                                       - 
                                       j 
                                     
                                     ) 
                                   
                                   / 
                                   
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         1 
                                       
                                       - 
                                       j 
                                     
                                     ) 
                                   
                                   / 
                                   
                                     2 
                                   
                                 
                               
                               
                                 j 
                               
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         1 
                                       
                                       + 
                                       j 
                                     
                                     ) 
                                   
                                   / 
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     
                       M 
                       4 
                     
                     = 
                       
                     ⁢ 
                     
                       
                         [ 
                         
                           
                             
                               
                                 v 
                                 13 
                               
                             
                             
                               
                                 v 
                                 14 
                               
                             
                             
                               
                                 v 
                                 15 
                               
                             
                             
                               
                                 v 
                                 16 
                               
                             
                           
                         
                         ] 
                       
                       = 
                         
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               
                                 
                                   ( 
                                   
                                     
                                       - 
                                       1 
                                     
                                     + 
                                     j 
                                   
                                   ) 
                                 
                                 / 
                                 
                                   2 
                                 
                               
                             
                             
                               
                                 - 
                                 j 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     j 
                                   
                                   ) 
                                 
                                 / 
                                 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       - 
                                       1 
                                     
                                     - 
                                     j 
                                   
                                   ) 
                                 
                                 / 
                                 
                                   2 
                                 
                               
                             
                             
                               1 
                             
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     j 
                                   
                                   ) 
                                 
                                 / 
                                 
                                   2 
                                 
                               
                             
                             
                               1 
                             
                           
                           
                             
                               j 
                             
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     j 
                                   
                                   ) 
                                 
                                 / 
                                 
                                   2 
                                 
                               
                             
                             
                               1 
                             
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     j 
                                   
                                   ) 
                                 
                                 / 
                                 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     j 
                                   
                                   ) 
                                 
                                 / 
                                 
                                   2 
                                 
                               
                             
                             
                               
                                 - 
                                 j 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     j 
                                   
                                   ) 
                                 
                                 / 
                                 
                                   2 
                                 
                               
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     M 
                     5 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             
                               v 
                               17 
                             
                           
                           
                             
                               v 
                               18 
                             
                           
                           
                             
                               v 
                               19 
                             
                           
                           
                             
                               v 
                               20 
                             
                           
                         
                       
                       ] 
                     
                     = 
                       
                     ⁢ 
                     
                       
                         
                           U 
                           rot 
                         
                         * 
                         
                           
                             1 
                             
                               2 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                       
                       = 
                         
                       ⁢ 
                       
                         0.5 
                         * 
                         
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                           . 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Still alternatively, the certain vectors V 1  through V 20  may be established from five matrices M 1  to M 5 . The five matrices M 1  to M 5  are established by: 
     
       
         
           
             
               
                 M 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           v 
                           1 
                         
                       
                       
                         
                           v 
                           2 
                         
                       
                       
                         
                           v 
                           3 
                         
                       
                       
                         
                           v 
                           4 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         1 
                       
                       
                         1 
                       
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         
                           - 
                           1 
                         
                       
                     
                     
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                     
                     
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 M 
                 2 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           v 
                           5 
                         
                       
                       
                         
                           v 
                           6 
                         
                       
                       
                         
                           v 
                           7 
                         
                       
                       
                         
                           v 
                           8 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         1 
                       
                       
                         1 
                       
                       
                         1 
                       
                     
                     
                       
                         1 
                       
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         
                           - 
                           1 
                         
                       
                     
                     
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                     
                     
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 M 
                 3 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           v 
                           9 
                         
                       
                       
                         
                           v 
                           10 
                         
                       
                       
                         
                           v 
                           11 
                         
                       
                       
                         
                           v 
                           12 
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                           - 
                           j 
                         
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         j 
                       
                     
                     
                       
                         1 
                       
                       
                         1 
                       
                       
                         j 
                       
                       
                         1 
                       
                     
                     
                       
                         
                           - 
                           1 
                         
                       
                       
                         
                           - 
                           j 
                         
                       
                       
                         1 
                       
                       
                         j 
                       
                     
                     
                       
                         
                           - 
                           j 
                         
                       
                       
                         1 
                       
                       
                         
                           - 
                           j 
                         
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 M 
                 4 
               
               = 
                 
               ⁢ 
               
                 
                   [ 
                   
                     
                       
                         
                           v 
                           13 
                         
                       
                       
                         
                           v 
                           14 
                         
                       
                       
                         
                           v 
                           15 
                         
                       
                       
                         
                           v 
                           16 
                         
                       
                     
                   
                   ] 
                 
                 = 
                   
                 ⁢ 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                           
                             ( 
                             
                               
                                 - 
                                 1 
                               
                               - 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         j 
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 - 
                                 1 
                               
                               + 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         1 
                       
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         
                           - 
                           j 
                         
                       
                     
                     
                       
                         
                           - 
                           j 
                         
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         1 
                       
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         j 
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
               
             
             , 
             and 
           
         
       
       
         
           
             
               M 
               5 
             
             = 
               
             ⁢ 
             
               
                 [ 
                 
                   
                     
                       
                         v 
                         17 
                       
                     
                     
                       
                         v 
                         18 
                       
                     
                     
                       
                         v 
                         19 
                       
                     
                     
                       
                         v 
                         20 
                       
                     
                   
                 
                 ] 
               
               = 
                 
               ⁢ 
               
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         
                           - 
                           j 
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 - 
                                 1 
                               
                               - 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         1 
                       
                       
                         
                           
                             ( 
                             
                               
                                 - 
                                 1 
                               
                               + 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         j 
                       
                     
                     
                       
                         
                           - 
                           j 
                         
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         1 
                       
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 - 
                                 1 
                               
                               + 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         
                           - 
                           j 
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 - 
                                 1 
                               
                               - 
                               j 
                             
                             ) 
                           
                           / 
                           
                             2 
                           
                         
                       
                       
                         1 
                       
                     
                   
                   ] 
                 
                 . 
               
             
           
         
       
     
     A subset of the vectors in the codebook may be phase shifted to generate a new codebook: 
     
       
         
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                   
               
               
                   
                 Rank 1 
                 Rank 2 
                 Rank 3 
                 Rank 4 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 1 
                 V 1   
                 V 1   
                 V 4   
                 V 1   
                 V 2   
                 V 4   
                 V 1   
                 V 2   
                 V 3   
                 V 4   
               
               
                 2 
                 V 5   
                 V 5   
                 V 6   
                 V 5   
                 V 6   
                 V 7   
                 V 5   
                 V 6   
                 V 7   
                 V 8   
               
               
                 3 
                 V 3   
                 V 3   
                 −V 4   
                 V 3   
                 −V 4   
                 V 1   
                 V 1   
                 −V 4   
                 V 3   
                 −V 2   
               
               
                 4 
                 V 7   
                 −V 7   
                 V 8   
                 −V 7   
                 V 8   
                 −V 5   
                 −V 5   
                 V 8   
                 −V 7   
                 V 6   
               
               
                 5 
                 V 9   
                 V 9   
                 V 12   
                 V 9   
                 V 10   
                 V 12   
                 V 9   
                 V 10   
                 V 11   
                 V 12   
               
               
                 6 
                 V 13   
                 V 13   
                 V 16   
                 V 13   
                 V 14   
                 V 16   
                 V 13   
                 V 14   
                 V 15   
                 V 16   
               
               
                 7 
                 V 11   
                 −jV 11   
                 jV 9   
                 −jV 11   
                 jV 9   
                 −jV 10   
                 −jV 11   
                 jV 9   
                 jV 12   
                 −jV 10   
               
               
                 8 
                 V 15   
                 jV 15   
                 −jV 13   
                 jV 15   
                 −jV 13   
                 jV 14   
                 jV 15   
                 −jV 13   
                 −jV 16   
                 jV 14   
               
               
                 9 
                 V 2   
                 V 2   
                 V 1   
                 V 2   
                 V 1   
                 −V 3   
                 V 2   
                 V 1   
                 −V 4   
                 −V 3   
               
               
                 10 
                 V 6   
                 −jV 6   
                 −jV 7   
                 −jV 6   
                 jV 8   
                 −jV 7   
                 −jV 6   
                 jV 5   
                 jV 8   
                 −jV 7   
               
               
                 11 
                 V 4   
                 V 4   
                 −V 2   
                 V 4   
                 −V 3   
                 −V 2   
                 V 4   
                 −V 2   
                 −V 3   
                 V 1   
               
               
                 12 
                 V 8   
                 jV 8   
                 −jV 6   
                 jV 8   
                 −jV 6   
                 −jV 5   
                 jV 8   
                 −jV 6   
                 jV 7   
                 −jV 5   
               
               
                 13 
                 V 17   
                 V 17   
                 V 18   
                 V 17   
                 V 18   
                 V 19   
                 V 17   
                 V 18   
                 V 19   
                 V 20   
               
               
                 14 
                 V 18   
                 V 18   
                 V 20   
                 V 18   
                 V 17   
                 V 20   
                 V 18   
                 V 20   
                 V 17   
                 V 19   
               
               
                 15 
                 V 19   
                 V 19   
                 V 17   
                 V 19   
                 V 20   
                 V 17   
                 V 17   
                 V 20   
                 V 19   
                 V 18   
               
               
                 16 
                 V 20   
                 −V 20   
                 −V 19   
                 −V 20   
                 −V 19   
                 −V 18   
                 −V 20   
                 −V 19   
                 −V 18   
                 −V 17   
               
               
                   
               
             
          
         
       
     
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete appreciation of the invention, and many of the attendant advantages thereof, will be readily apparent as the same becomes better understood by reference to the following detailed description when considered in conjunction with the accompanying drawings in which like reference symbols indicate the same or similar components, wherein: 
         FIG. 1  schematically illustrates an Orthogonal Frequency Division Multiplexing (OFDM) transceiver chain; 
         FIG. 2  schematically illustrates a Multiple Input Multiple Output (MIMO) transceiver chain; 
         FIG. 3  schematically illustrates a single codeword MIMO transmission scheme; 
         FIG. 4  schematically illustrates a multiple codeword MIMO transmission scheme; 
         FIG. 5  schematically illustrates a multiple codeword MIMO transmission scheme for two transmission layers; 
         FIG. 6  schematically illustrates a multiple codeword MIMO transmission scheme for three transmission layers; 
         FIG. 7  schematically illustrates a multiple codeword MIMO transmission scheme for four transmission layers; 
         FIG. 8  schematically illustrates an example of a single user MIMO system; 
         FIG. 9  schematically illustrates an example of a multiple-user MIMO system; and 
         FIG. 10  schematically illustrates a feedback-based MIMO precoding and decoding system. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 1  illustrates an Orthogonal Frequency Division Multiplexing (OFDM) transceiver chain. In a communication system using OFDM technology, at transmitter chain  110 , control signals or data  111  is modulated by modulator  112  and is serial-to-parallel converted by Serial/Parallel (S/P) converter  113 . Inverse Fast Fourier Transform (IFFT) unit  114  is used to transfer the signal from frequency domain to time domain. Cyclic prefix (CP) or zero prefix (ZP) is added to each OFDM symbol by CP insertion unit  116  to avoid or mitigate the impact due to multipath fading. Consequently, the signal is transmitted by transmitter (Tx) front end processing unit  117 , such as an antenna (not shown), or alternatively, by fixed wire or cable. At receiver chain  120 , assuming perfect time and frequency synchronization are achieved, the signal received by receiver (Rx) front end processing unit  121  is processed by CP removal unit  122 . Fast Fourier Transform (FFT) unit  124  transfers the received signal from time domain to frequency domain for further processing. 
     The total bandwidth in an OFDM system is divided into narrowband frequency units called subcarriers. The number of subcarriers is equal to the FFT/IFFT size N used in the system. In general, the number of subcarriers used for data is less than N because some subcarriers at the edge of the frequency spectrum are reserved as guard subcarriers. In general, no information is transmitted on guard subcarriers. 
     Multiple Input Multiple Output (MIMO) schemes use multiple transmission antennas and multiple receive antennas to improve the capacity and reliability of a wireless communication channel. A MIMO system promises linear increase in capacity with K where K is the minimum of number of transmit (M) and receive antennas (N), i.e. K=min(M,N). A simplified example of a 4×4 MIMO system is shown in  FIG. 2 . In this example, four different data streams are transmitted separately from four transmission antennas. The transmitted signals are received at four receive antennas. Some form of spatial signal processing is performed on the received signals in order to recover the four data streams. An example of spatial signal processing is vertical Bell Laboratories Layered Space-Time (V-BLAST) which uses the successive interference cancellation principle to recover the transmitted data streams. Other variants of MIMO schemes include schemes that perform some kind of space-time coding across the transmission antennas (e.g., diagonal Bell Laboratories Layered Space-Time (D-BLAST)) and also beamforming schemes such as Spatial Division multiple Access (SDMA). 
     The MIMO channel estimation consists of estimating the channel gain and phase information for links from each of the transmission antennas to each of the receive antennas. Therefore, the channel for M×N MIMO system consists of an N×M matrix: 
                   H   =     [           h   11           h   12         …         h     1   ⁢           ⁢   M                 h   21           h   22         …         h     2   ⁢           ⁢   M               ⋮       ⋮       …       ⋮             h     N   ⁢           ⁢   1             h     M   ⁢           ⁢   2           …         h   NM           ]             (   1   )               
where h ij  represents the channel gain from transmission antenna j to receive antenna i. In order to enable the estimations of the elements of the MIMO channel matrix, separate pilots are transmitted from each of the transmission antennas.
 
     An example of single-code word MIMO scheme is given in  FIG. 3 . In case of single-code word MIMO transmission, a cyclic redundancy check (CRC) is added to a single information block and then coding, for example, using turbo codes and low-density parity check (LDPC) code, and modulation, for example, by quadrature phase-shift keying (QPSK) modulation scheme, are performed. The coded and modulated symbols are then demultiplexed for transmission over multiple antennas. 
     In case of multiple codeword MIMO transmission, shown in  FIG. 4 , the information block is de-multiplexed into smaller information blocks. Individual CRCs are attached to these smaller information blocks and then separate coding and modulation is performed on these smaller blocks. After modulation, these smaller blocks are respectively demultiplexed into even smaller blocks and then transmitted through corresponding antennas. It should be noted that in case of multi-code word MIMO transmissions, different modulation and coding can be used on each of the individual streams, and thus resulting in a so-called Per Antenna Rate Control (PARC) scheme. Also, multi-code word transmission allows for more efficient post-decoding interference cancellation because a CRC check can be performed on each of the code words before the code word is cancelled from the overall signal. In this way, only correctly received code words are cancelled, and thus avoiding any interference propagation in the cancellation process. 
     In the third Generation Partnership Project (3GPP) Long Term Evolution (LTE) systems, a maximum of two codewords are used for transmission of two, three or four MIMO layers. As shown in  FIG. 5 , for rank-2 or two layers transmission, codeword- 1  (CW 1 ) is transmitted from Layer- 0  while CW 2  is transmitted from Layer- 1 . For rank-3 or three layers transmission, codeword- 1  (CW 1 ) is transmitted from Layer- 0  while CW 2  is transmitted from Layer- 1  and Layer- 2 . For rank-4 or four layers transmission, codeword- 1  (CW 1 ) is transmitted from Layer- 0  and Layer- 1  while CW 2  is transmitted from Layer- 2  and Layer- 3 . 
     An example of single-user (SU) MIMO system is shown in  FIG. 8 . In case of single-user MIMO, all the MIMO layers are transmitted to a single user. In case of multi-user (MU) MIMO system as shown in  FIG. 9 , however, the total MIMO layers are shared among multiple units of user equipment (UEs). 
     In a closed-loop MIMO precoding system, for each transmission antenna size we construct a set of preceding matrices (i.e., codewords) and let this set be known at both the Node-B (i.e., the base station) and the user equipment (UE). We call this set of matrices as the “codebook” and denote it P={(P 1 , . . . , P L }. Here L=2 q  denotes the size of the codebook and q is the number of (feedback) bits needed to index the codebook. In a limited feedback precoding MIMO system illustrated in  FIG. 10 , once the codebook is specified for a MIMO system, the receiver observes a channel realization, selects the best precoding matrix (i.e., codeword) to be used at the moment, and feeds back the index of the codeword to the transmitter. 
     An example of precoding is discrete Fourier transform (DFT)-based or Fourier precoding. A Fourier matrix is a N×N square matrix with entries given by:
 
 P   N   =e   j2πmn/N    m,n= 0, 1 . . . ( N− 1)  (2)
 
     For example, a 2×2 Fourier matrix can be expressed as: 
     
       
         
           
             
               
                 
                   
                     P 
                     2 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               1 
                             
                           
                           
                             
                               1 
                             
                             
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Similarly, a 4×4 Fourier matrix can be expressed as: 
     
       
         
           
             
               
                 
                   
                     P 
                     4 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             1 
                           
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   / 
                                   2 
                                 
                               
                             
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                             
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 3 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   / 
                                   2 
                                 
                               
                             
                           
                         
                         
                           
                             1 
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                             
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                             
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 3 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                             
                           
                         
                         
                           
                             1 
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 3 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   / 
                                   2 
                                 
                               
                             
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 3 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 π 
                               
                             
                           
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 9 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   / 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               1 
                             
                             
                               1 
                             
                             
                               1 
                             
                           
                           
                             
                               1 
                             
                             
                               j 
                             
                             
                               
                                 - 
                                 1 
                               
                             
                             
                               
                                 - 
                                 j 
                               
                             
                           
                           
                             
                               1 
                             
                             
                               
                                 - 
                                 1 
                               
                             
                             
                               1 
                             
                             
                               
                                 - 
                                 1 
                               
                             
                           
                           
                             
                               1 
                             
                             
                               
                                 - 
                                 j 
                               
                             
                             
                               
                                 - 
                                 1 
                               
                             
                             
                               j 
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Multiple Fourier matrices can be defined by introducing a shift parameter (g/G) in the Fourier matrix. The entry of the multiple Fourier matrices is given by: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       mn 
                     
                     = 
                     
                       
                         ⅇ 
                         
                           j 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                           ⁢ 
                           
                             
                               m 
                               N 
                             
                             · 
                             
                               ( 
                               
                                 n 
                                 + 
                                 
                                   g 
                                   G 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                     
                   
                   , 
                   
                     n 
                     = 
                     0 
                   
                   , 
                   1 
                   , 
                   
                     … 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           N 
                           - 
                           1 
                         
                         ) 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     A set of four 2×2 Fourier matrices can be defined by taking G=4, and g=0, 1, 2 and 3, and are written as: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       2 
                       0 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             1 
                           
                           
                             
                               - 
                               1 
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       P 
                       2 
                       1 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   / 
                                   4 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 ⅇ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     π 
                                     / 
                                     4 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       P 
                       2 
                       2 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             1 
                           
                         
                         
                           
                             
                               ⅇ 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   / 
                                   2 
                                 
                               
                             
                           
                           
                             
                               - 
                               
                                 ⅇ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     π 
                                     / 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       P 
                       2 
                       3 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                             
                               1 
                             
                           
                           
                             
                               
                                 ⅇ 
                                 
                                   j 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   3 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     π 
                                     / 
                                     4 
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   ⅇ 
                                   
                                     j 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     3 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       π 
                                       / 
                                       4 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     In a first embodiment according to the principles of the present invention, we design a SU-MIMO codebook as a function of a set of input vectors. Given a set of input vectors as shown Table 1, we can generate the codebooks per rank as shown in Table 2. 
     
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Basic Vectors to Generate the Codebook according to the principles 
               
               
                 of the current invention 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 v 1   
                 1 
                 1 
                 1 
                 1 
               
               
                 v 2   
                 1 
                 1 
                 −1  
                 −1  
               
               
                 v 3   
                 1 
                 −1  
                 1 
                 −1  
               
               
                 v 4   
                 1 
                 −1  
                 −1  
                 1 
               
               
                 v 5   
                 1 
                 j 
                 −1  
                 −j 
               
               
                 v 6   
                 −j 
                 1 
                 −j 
                 1 
               
               
                 v 7   
                 −1  
                 j 
                 1 
                 −j 
               
               
                 v 8   
                 j 
                 1 
                 j 
                 1 
               
               
                 v 9   
                 1 
                 (1 + j)/√2 
                 j 
                 (−1 + j))/√2 
               
               
                 v 10   
                 (1 − j)/√2 
                 1 
                 (−1 − j)/√2 
                 −j 
               
               
                 v 11   
                 −j 
                 (−1 + j)/√2 
                 1 
                 (−1 − j)/√2 
               
               
                 v 12   
                 (−1 − j)/√2 
                 j 
                 (−1 + j)/√2 
                 1 
               
               
                 v 13   
                 1 
                 (−1 + j)/√2 
                 −j 
                 (1 + j)/√2 
               
               
                 v 14   
                 (−1 − j)/√2 
                 1 
                 (1 − j)/√2 
                 j 
               
               
                 v 15   
                 j 
                 (1 + j)/√2 
                 1 
                 (1 − j)/√2 
               
               
                 v 16   
                 (1 − j)/√2 
                 −j 
                 (1 + j)√2 
                 1 
               
               
                 v 17   
                 1 
                 1 
                 1 
                 −1  
               
               
                 v 18   
                 1 
                 1 
                 −1  
                 1 
               
               
                 v 19   
                 1 
                 −1  
                 1 
                 1 
               
               
                 v 20   
                 −1  
                 1 
                 1 
                 1 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Required Vectors for each Rank Codebook 
               
             
          
           
               
                   
                 Rank 1 
                 Rank 2 
                 Rank 3 
                 Rank 4 
               
               
                   
               
             
          
           
               
                 1 
                 v 1   
                 v 1   
                 v 4   
                 v 1   
                 v 2   
                 v 4   
                 v 1   
                 v 2   
                 v 3   
                 v 4   
               
               
                 2 
                 v 5   
                 v 5   
                 v 6   
                 v 5   
                 v 6   
                 v 7   
                 v 5   
                 v 6   
                 v 7   
                 v 8   
               
               
                 3 
                 v 3   
                 v 3   
                 v 4   
                 v 3   
                 v 4   
                 v 1   
                 v 1   
                 v 4   
                 v 3   
                 v 2   
               
               
                 4 
                 v 7   
                 v 7   
                 v 8   
                 v 7   
                 v 8   
                 v 5   
                 v 5   
                 v 8   
                 v 7   
                 v 6   
               
               
                 5 
                 v 9   
                 v 9   
                 v 12   
                 v 9   
                 v 10   
                 v 12   
                 v 9   
                 v 10   
                 v 11   
                 v 12   
               
               
                 6 
                 v 13   
                 v 13   
                 v 16   
                 v 13   
                 v 14   
                 v 16   
                 v 13   
                 v 14   
                 v 15   
                 v 16   
               
               
                 7 
                 v 11   
                 v 11   
                 v 9   
                 v 11   
                 v 9   
                 v 10   
                 v 11   
                 v 9   
                 v 12   
                 v 10   
               
               
                 8 
                 v 15   
                 v 15   
                 v 13   
                 v 15   
                 v 13   
                 v 14   
                 v 15   
                 v 13   
                 v 16   
                 v 14   
               
               
                 9 
                 v 2   
                 v 2   
                 v 1   
                 v 2   
                 v 1   
                 v 3   
                 v 2   
                 v 1   
                 v 4   
                 v 3   
               
               
                 10 
                 v 6   
                 v 6   
                 v 7   
                 v 6   
                 v 8   
                 v 7   
                 v 6   
                 v 5   
                 v 8   
                 v 7   
               
               
                 11 
                 v 4   
                 v 4   
                 v 2   
                 v 4   
                 v 3   
                 v 2   
                 v 4   
                 v 2   
                 v 3   
                 v 1   
               
               
                 12 
                 v 8   
                 v 8   
                 v 6   
                 v 8   
                 v 6   
                 v 5   
                 v 8   
                 v 6   
                 v 7   
                 v 5   
               
               
                 13 
                 v 17   
                 v 17   
                 v 18   
                 v 17   
                 v 18   
                 v 19   
                 v 17   
                 v 18   
                 v 19   
                 v 20   
               
               
                 14 
                 v 18   
                 v 18   
                 v 20   
                 v 18   
                 v 17   
                 v 20   
                 v 18   
                 v 20   
                 v 17   
                 v 19   
               
               
                 15 
                 v 19   
                 v 19   
                 v 17   
                 v 19   
                 v 20   
                 v 17   
                 v 17   
                 v 20   
                 v 19   
                 v 18   
               
               
                 16 
                 v 20   
                 v 20   
                 v 19   
                 v 20   
                 v 19   
                 v 18   
                 v 20   
                 v 19   
                 v 18   
                 v 17   
               
               
                   
               
             
          
         
       
     
     In a second embodiment according to the principles of the present invention, we can apply certain phase shifts to the vectors in order to maximize the chordal distance between each pair of matrices in each rank. Chordal distance is a distance between a pair of matrices. Exemplary phase shifts are given in Table 3. 
     
       
         
               
             
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Required Phase shifts (degrees) to Vectors for each Rank Codebook 
               
             
          
           
               
                   
                 Rank 2 
                 Rank 3 
                 Rank 4 
               
               
                   
               
             
          
           
               
                 1 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 2 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 3 
                 0 
                 180 
                 0 
                 180 
                 0 
                 0 
                 180 
                 0 
                 180 
               
               
                 4 
                 180 
                 0 
                 180 
                 0 
                 180 
                 180 
                 0 
                 180 
                 0 
               
               
                 5 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 6 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 7 
                 −90 
                 90 
                 −90 
                 90 
                 −90 
                 −90 
                 90 
                 90 
                 −90 
               
               
                 8 
                 90 
                 −90 
                 90 
                 −90 
                 90 
                 90 
                 −90 
                 −90 
                 90 
               
               
                 9 
                 0 
                 0 
                 0 
                 0 
                 180 
                 0 
                 0 
                 180 
                 180 
               
               
                 10 
                 −90 
                 −90 
                 −90 
                 90 
                 −90 
                 −90 
                 90 
                 90 
                 −90 
               
               
                 11 
                 0 
                 180 
                 0 
                 180 
                 180 
                 0 
                 180 
                 180 
                 0 
               
               
                 12 
                 90 
                 −90 
                 90 
                 −90 
                 −90 
                 90 
                 −90 
                 90 
                 −90 
               
               
                 13 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 14 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 15 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 16 
                 180 
                 180 
                 180 
                 180 
                 180 
                 180 
                 180 
                 180 
                 180 
               
               
                   
               
             
          
         
       
     
     These two tables can be compressed into a more compact form as shown in the table below. 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 CodeBook Matrices as a function of input vectors 
               
             
          
           
               
                   
                 Rank 1 
                 Rank 2 
                 Rank 3 
                 Rank 4 
               
               
                   
                 Matrices 
                 Matrices 
                 Matrices 
                 Matrices 
               
               
                   
               
             
          
           
               
                 1 
                 v 1   
                 v 1   
                 v 4   
                 v 1   
                 v 2   
                 v 4   
                 v 1   
                 v 2   
                 v 3   
                 v 4   
               
               
                 2 
                 v 5   
                 v 5   
                 v 6   
                 v 5   
                 v 6   
                 v 7   
                 v 5   
                 v 6   
                 v 7   
                 v 8   
               
               
                 3 
                 v 3   
                 v 3   
                 −v 4   
                 v 3   
                 −v 4   
                 v 1   
                 v 1   
                 −v 4   
                 v 3   
                 −v 2   
               
               
                 4 
                 v 7   
                 −v 7   
                 v 8   
                 −v 7   
                 v 8   
                 −v 5   
                 −v 5   
                 v 8   
                 −v 7   
                 v 6   
               
               
                 5 
                 v 9   
                 v 9   
                 v 12   
                 v 9   
                 v 10   
                 v 12   
                 v 9   
                 v 10   
                 v 11   
                 v 12   
               
               
                 6 
                 v 13   
                 v 13   
                 v 16   
                 v 13   
                 v 14   
                 v 16   
                 v 13   
                 v 14   
                 v 15   
                 v 16   
               
               
                 7 
                 v 11   
                 −jv 11   
                 jv 9   
                 −jv 11   
                 jv 9   
                 −jv 10   
                 −jv 11   
                 jv 9   
                 jv 12   
                 −jv 10   
               
               
                 8 
                 v 15   
                 jv 15   
                 −jv 13   
                 jv 15   
                 −jv 13   
                 jv 14   
                 jv 15   
                 −jv 13   
                 −jv 16   
                 jv 14   
               
               
                 9 
                 v 2   
                 v 2   
                 v 1   
                 v 2   
                 v 1   
                 −v 3   
                 v 2   
                 v 1   
                 −v 4   
                 −v 3   
               
               
                 10 
                 v 6   
                 −jv 6   
                 −jv 7   
                 −jv 6   
                 jv 8   
                 −jv 7   
                 −jv 6   
                 jv 5   
                 jv 8   
                 −jv 7   
               
               
                 11 
                 v 4   
                 v 4   
                 −v 2   
                 v 4   
                 −v 3   
                 −v 2   
                 v 4   
                 −v 2   
                 −v 3   
                 v 1   
               
               
                 12 
                 v 8   
                 jv 8   
                 −jv 6   
                 jv 8   
                 −jv 6   
                 −jv 5   
                 jv 8   
                 −jv 6   
                 jv 7   
                 −jv 5   
               
               
                 13 
                 v 17   
                 v 17   
                 v 18   
                 v 17   
                 v 18   
                 v 19   
                 v 17   
                 v 18   
                 v 19   
                 v 20   
               
               
                 14 
                 v 18   
                 v 18   
                 v 20   
                 v 18   
                 v 17   
                 v 20   
                 v 18   
                 v 20   
                 v 17   
                 v 19   
               
               
                 15 
                 v 19   
                 v 19   
                 v 17   
                 v 19   
                 v 20   
                 v 17   
                 v 17   
                 v 20   
                 v 19   
                 v 18   
               
               
                 16 
                 v 20   
                 −v 20   
                 −v 19   
                 −v 20   
                 −v 19   
                 −v 18   
                 −v 20   
                 −v 19   
                 −v 18   
                 −v 17   
               
               
                   
               
             
          
         
       
     
     In a third embodiment according to the principles of the present invention, the basis vectors v 1  to v 20  in Table 2 and Table 4 may be generated from source unitary matrices M 1  to M 5  generated from a Householder equation and five input vectors u 1  to u 5 , or any unitary rotation thereof. The input vectors u 1  to u 5  are given as:
 
 u   1   T =[1 −1 −1 −1]  (7)
 
 u   2   T =[1  j  1  −j]   (8)
 
 u   3   T =└1(−1 +j )/√{square root over (2)} j (1+ j )/√{square root over (2)}┘  (9)
 
 u   4   T =└1(1 +j )/√{square root over (2)}− j (−1+ j )/√{square root over (2)}┘  (10)
 
 u   5   T =[1 −1 −1 1]  (11)
 
Accordingly, the matrices M 1  to M 5  generated from the Householder equation and the input vectors u 1  to u 5  are established as:
 
     
       
         
           
             
               
                 
                   
                     M 
                     1 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               v 
                               1 
                             
                           
                           
                             
                               v 
                               2 
                             
                           
                           
                             
                               v 
                               3 
                             
                           
                           
                             
                               v 
                               4 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           I 
                           4 
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             1 
                           
                           ⁢ 
                           
                             
                               u 
                               1 
                               H 
                             
                             / 
                             
                               
                                  
                                 
                                   u 
                                   1 
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                       = 
                       
                         0.5 
                         * 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     2 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               v 
                               5 
                             
                           
                           
                             
                               v 
                               6 
                             
                           
                           
                             
                               v 
                               7 
                             
                           
                           
                             
                               v 
                               8 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           I 
                           4 
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             2 
                           
                           ⁢ 
                           
                             
                               u 
                               2 
                               H 
                             
                             / 
                             
                               
                                  
                                 
                                   u 
                                   2 
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                       = 
                       
                         0.5 
                         * 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 j 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   j 
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                               
                                 1 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 j 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                             
                             
                               
                                 j 
                               
                               
                                 1 
                               
                               
                                 j 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     3 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               v 
                               9 
                             
                           
                           
                             
                               v 
                               10 
                             
                           
                           
                             
                               v 
                               11 
                             
                           
                           
                             
                               v 
                               12 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           I 
                           4 
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             3 
                           
                           ⁢ 
                           
                             
                               u 
                               3 
                               H 
                             
                             / 
                             
                               
                                  
                                 
                                   u 
                                   3 
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                       = 
                       
                         0.5 
                         * 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   
                                     1 
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 j 
                               
                               
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     1 
                                     - 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   
                                     1 
                                     - 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   j 
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                     - 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                     - 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 j 
                               
                               
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     4 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               v 
                               13 
                             
                           
                           
                             
                               v 
                               14 
                             
                           
                           
                             
                               v 
                               15 
                             
                           
                           
                             
                               v 
                               16 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           I 
                           4 
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             4 
                           
                           ⁢ 
                           
                             
                               u 
                               4 
                               H 
                             
                             / 
                             
                               
                                  
                                 
                                   u 
                                   4 
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                       = 
                       
                         0.5 
                         * 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                               
                                 
                                   
                                     1 
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                     - 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   
                                     1 
                                     - 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                             
                             
                               
                                 j 
                               
                               
                                 
                                   
                                     1 
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   
                                     1 
                                     - 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     1 
                                     - 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                               
                                 
                                   
                                     1 
                                     + 
                                     j 
                                   
                                   
                                     2 
                                   
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     5 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               v 
                               17 
                             
                           
                           
                             
                               v 
                               18 
                             
                           
                           
                             
                               v 
                               19 
                             
                           
                           
                             
                               v 
                               20 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         
                           I 
                           4 
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             u 
                             5 
                           
                           ⁢ 
                           
                             
                               u 
                               5 
                               H 
                             
                             / 
                             
                               
                                  
                                 
                                   u 
                                   5 
                                 
                                  
                               
                               2 
                             
                           
                         
                       
                       = 
                       
                         0.5 
                         * 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     In addition, matrices M 1 , M 2  and M 5  may be rotated to generated new matrices M 1 ′, M 2 ′ and M 5 ′, respectively, to be used as source matrices. The rotation operation may be established by a rotation matrix U rot  given as: 
                     U   rot     =         1     2       ⁡     [         1       0         -   1         0           0       1       0         -   1             1       0       1       0           0       1       0       1         ]       .             (   17   )               
Accordingly, the rotated matrices M 1 ′, M 2 ′ and M 5 ′, are established as:
 
     
       
         
           
             
               
                 
                   
                     M 
                     1 
                     ′ 
                   
                   = 
                   
                     
                       
                         U 
                         rot 
                       
                       * 
                       
                         M 
                         1 
                       
                     
                     = 
                     
                       
                         
                           U 
                           rot 
                         
                         * 
                         
                           
                             1 
                             
                               2 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                       
                       = 
                       
                         0.5 
                         * 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     2 
                     ′ 
                   
                   = 
                   
                     
                       
                         U 
                         rot 
                       
                       * 
                       
                         M 
                         2 
                       
                     
                     = 
                     
                       
                         
                           U 
                           rot 
                         
                         * 
                         
                           
                             1 
                             
                               2 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   0 
                                 
                                 
                                   j 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     j 
                                   
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   j 
                                 
                                 
                                   0 
                                 
                                 
                                   j 
                                 
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                       
                       = 
                       
                         0.5 
                         * 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 j 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   j 
                                 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 j 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   j 
                                 
                               
                             
                             
                               
                                 j 
                               
                               
                                 1 
                               
                               
                                 j 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     5 
                     ′ 
                   
                   = 
                   
                     
                       
                         U 
                         rot 
                       
                       * 
                       
                         M 
                         5 
                       
                     
                     = 
                     
                       
                         
                           U 
                           rot 
                         
                         * 
                         
                           
                             1 
                             
                               2 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                               
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                       
                       = 
                       
                         0.5 
                         * 
                         
                           [ 
                           
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                             
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     We could also use as source a combination of a DFT matrix M2 as given by Equation (13), a Walsh matrix M1 as given by Equation (12), block diagonal matrix and any unitary rotation thereof. Block diagonal matrix M B  refers to a class of matrices that are given by the following 4 by 4 matrix: 
                       M   B     =     [         A                                   B         ]       ,           (   21   )               
where A and B are both 2 by 2 matrices.
 
     Given these source matrices M 1  to M 5 , and M 1 ′, M 2 ′ and M 5 ′, we can generate the codebook as rotated column subset selections of these matrices as listed in Table 4. 
     One of the properties of the codebooks shown in Table 2 and 4 is that these codebook are fully nested. In other words, the rank-1 vectors are column subsets of the rank-2 matrices, the rank-2 matrices are a subset of the rank-3 matrices, and the rank-3 matrices are a subset of the rank-4 precoding matrices. The advantage of this is that when the precoded channel is calculated for a given rank, no further complex multiplications and additions are required to calculate the precoded channel of the lower ranks. In fact the lower rank precoded channels are column subsets of the higher rank precoded channels. 
     While the present invention has been shown and described in connection with the preferred embodiments, it will be apparent to those skilled in the art that modifications and variations can be made without departing from the spirit and scope of the invention as defined by the appended claims.