Patent Publication Number: US-2023163818-A1

Title: 4tx codebook enhancement in lte

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of application Ser. No. 17/403,587 filed Aug. 16, 2021; which is a continuation of application Ser. No. 16/989,587 filed Aug. 10, 2020, now U.S. Pat. No. 11,095,345 granted Aug. 17, 2021; which is a continuation of application Ser. No. 16/518,597 filed Jul. 22, 2019, now U.S. Pat. No. 10,742,280 granted Aug. 11, 2020; which is a continuation of application Ser. No. 15/698,015, filed Sep. 7, 2017, now U.S. Pat. No. 10,361,760, granted Jul. 23, 2019; which is a continuation of application Ser. No. 15/132,024, filed Apr. 18, 2016, now U.S. Pat. No. 9,780,850, granted Oct. 3, 2017; which is a continuation of application Ser. No. 14/863,156, filed Sep. 23, 2015, now U.S. Pat. No. 9,319,121, granted Apr. 19, 2016; which is a continuation of application Ser. No. 14/177,547, filed Feb. 11, 2014, now U.S. Pat. No. 9,281,881, granted Mar. 8, 2016; which claims the benefit of U.S. Provisional Patent Application No. 61/763,804, filed on Feb. 12, 2013; U.S. Provisional Patent Application No. 61/768,851, filed on Feb. 25, 2013; U.S. Provisional Patent Application No. 61/769,463, filed on Feb. 26, 2013; U.S. Provisional Patent Application No. 61/770,705, filed on Feb. 28, 2013; U.S. Provisional Patent Application No. 61/777,664, filed on Mar. 12, 2013; U.S. Provisional Patent Application No. 61/807,647, filed on Apr. 2, 2013; U.S. Provisional Patent Application No. 61/812,459, filed on Apr. 16, 2013; and U.S. Provisional Patent Application No. 61/817,657, filed on Apr. 30, 2013, all titled “4Tx Codebook Enhancement in Release 12.” The disclosures of all of the above-mentioned applications are incorporated herein by reference in their entirety. 
    
    
     TECHNICAL FIELD 
     The technical field of this invention is wireless communication such as wireless 
     telephony. 
     BACKGROUND 
     The present embodiments relate to wireless communication systems and, more particularly, to the precoding of Physical Downlink Shared Channel (PDSCH) data and dedicated reference signals with codebook-based feedback for multi-input multi-output (MIMO) transmissions. 
     With Orthogonal Frequency Division Multiplexing (OFDM), multiple symbols are transmitted on multiple carriers that are spaced apart to provide orthogonality. An OFDM modulator typically takes data symbols into a serial-to-parallel converter, and the output of the serial-to-parallel converter is considered as frequency domain data symbols. The frequency domain tones at either edge of the band may be set to zero and are called guard tones. These guard tones allow the OFDM signal to fit into an appropriate spectral mask. Some of the frequency domain tones are set to values which will be known at the receiver. Among these are Channel State Information Reference Signals (CSI-RS) and Dedicated or Demodulating Reference Signals (DMRS). These reference signals are useful for channel estimation at the receiver. 
     In multi-input multi-output (MIMO) communication systems with multiple transmit/receive antennas, the data transmission is performed via precoding. Here, precoding refers to a linear (matrix) transformation of L-stream data into P-stream where L denotes the number of layers (also termed the transmission rank) and P denotes the number of transmit antennas. With the use of dedicated (i.e. user-specific) DMRS, a transmitter, such as a base station or eNodeB (eNodeB), can perform precoding operations that are transparent to user equipment (UE) acting as receivers. It is beneficial for the base station to obtain a precoding matrix recommendation from the user equipment. This is particularly the case for frequency-division duplexing (FDD) where the uplink and downlink channels occupy different parts of the frequency bands, i.e. the uplink and downlink are not reciprocal. Hence, a codebook-based feedback from the UE to the eNodeB is preferred. To enable a codebook-based feedback, a precoding codebook needs to be designed. 
     The Long-Term Evolution (LTE) specification includes codebooks for 2-antenna, 4-antenna, and 8-antenna transmissions. While those codebooks are designed efficiently, the present inventors recognize that still further improvements in downlink (DL) spectral efficiency are possible. Accordingly, the preferred embodiments described below are directed toward these problems as well as improving upon the prior art. 
     SUMMARY 
     Systems and methods for channel state information (CSI) and precoding matrix indicator (PMI) feedback in a wireless communication system are disclosed. A precoding matrix is generated for multi-antenna transmission based on a precoding matrix indicator (PMI) feedback from at least one remote receiver wherein the PMI indicates a choice of precoding matrix derived from a matrix multiplication of two matrices from a first codebook and a second codebook. One or more layers of a data stream are precoded with the precoding matrix and transmitted to the remote receiver. 
     In one embodiment, a method of CSI feedback and transmitting data in a wireless communication system comprises receiving one or more precoding matrix indicator (PMI) signals from a remote transceiver. The PMI signals indicating a choice of a precoding matrix W. The system generates the precoding matrix W from a matrix multiplication of two matrices W 1  and W 2 . Matrix W is termed the composite precoder. Matrix W 1  targets wideband/long-term channel properties, and matrix W 2  targets frequency-selective/short-term channel properties. Each of the components W 1 , W 2  is assigned a codebook. Hence, two distinct codebooks are needed: C 1  and C 2 . Matrix W 1  is selected from a first codebook C 1  based on a first group of bits in the PMI signals, and matrix W 2  is selected from a second codebook C 2  based on a second group of bits in the PMI signals. 
     Proposed first and second codebooks C 1  and C 2  are defined below for different ranks and different PMI bit lengths. 
     In one embodiment, one or more layers of a data stream are precoded by multiplication with the precoding matrix W. The precoded layers in the data stream are then transmitted to the remote receiver. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       These and other aspects of this invention are illustrated in the drawings, in which: 
         FIG.  1    illustrates an exemplary wireless telecommunications network. 
         FIG.  2    illustrates a uniform linear array (ULA) or four pairs of ULA elements. 
         FIG.  3    illustrates four pairs of cross-polarized arrays. 
         FIG.  4    illustrates an example in which a grid of beams is shifted by m=2 beams. 
         FIG.  5    illustrates an example of using a row selection vector to prune antenna pairs in an 8Tx array. 
         FIG.  6    illustrates the technique used in downlink LTE-Advanced (LTE-A). 
         FIG.  7    is a block diagram illustrating internal details of a mobile UE and an eNodeB in an exemplary network system. 
     
    
    
     DETAILED DESCRIPTION 
     The invention(s) now will be described more fully hereinafter with reference to the accompanying drawings. The invention(s) may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention(s) to a person of ordinary skill in the art. A person of ordinary skill in the art may be able to use the various embodiments of the invention(s). 
       FIG.  1    illustrates an exemplary wireless telecommunications network  100 . Network  100  includes a plurality of base stations  101 ,  102  and  103 . In operation, a telecommunications network necessarily includes many more base stations. Each base station  101 ,  102  and  103  (eNodeB) is operable over corresponding coverage areas  104 ,  105  and  106 . Each base station&#39;s coverage area is further divided into cells. In the illustrated network, each base station&#39;s coverage area is divided into three cells  104   a - c ,  105   a - c ,  106   a - c . User equipment (UE)  107 , such as telephone handset, is shown in Cell A  104   a . Cell A  104   a  is within coverage area  104  of base station  101 . Base station  101  transmits to and receives transmissions from UE  107 . As UE  107  moves out of Cell A  104   a  and into Cell B  105   b , UE  107  may be handed over to base station  102 . Because UE  107  is synchronized with base station  101 , UE  107  can employ non-synchronized random access to initiate handover to base station  102 . 
     Non-synchronized UE  107  also employs non-synchronous random access to request allocation of uplink  108  time or frequency or code resources. If UE  107  has data ready for transmission, which may be traffic data, measurements report, tracking area update, UE  107  can transmit a random access signal on uplink  108 . The random access signal notifies base station  101  that UE  107  requires uplink resources to transmit the UEs data. Base station  101  responds by transmitting to UE  107  via downlink  109 , a message containing the parameters of the resources allocated for UE  107  uplink transmission along with a possible timing error correction. After receiving the resource allocation and a possible timing advance message transmitted on downlink  109  by base station  101 , UE  107  optionally adjusts its transmit timing and transmits the data on uplink  108  employing the allotted resources during the prescribed time interval. 
     Base station  101  configures UE  107  for periodic uplink sounding reference signal (SRS) transmission. Base station  101  estimates uplink channel quality information (CSI) from the SRS transmission. The preferred embodiments of the present invention provide improved communication through precoded multi-antenna transmission with codebook-based feedback. In a cellular communication system, a UE is uniquely connected to and served by a single cellular base station or eNodeB at a given time. An example of such a system is the 3GPP LTE system, which includes the LTE-Advanced (LTE-A) system. With an increasing number of transmit antennas at the eNodeB, the task of designing an efficient codebook with desirable properties is challenging. 
     CSI consists of Channel Quality Indicator (CQI), precoding matrix indicator (PMI), precoding type indicator (PTI), and/or rank indication (RI). The time and frequency resources that can be used by the UE to report CSI are controlled by the eNodeB. 
     In one embodiment, a dual-stage codebook for CSI feedback is based on the product structure proposed in: 
         W=W   1   W   2   (1)
 
     where W 1  targets wideband/long-term channel properties and W 2  targets frequency-selective/short-term channel properties. Each of the components W 1 , W 2  is assigned a codebook. Hence, two distinct codebooks are needed: CB 1  and CB 2 . W is termed the composite precoder. The choice of W 1  and W 2  are indicated via PMI 1  and PMI 2 . 
     The following principles are enforced for codebook designs: 
     (1) Finite alphabet for W: each matrix element belongs to a finite set of values or constellation (e.g., M-PSK alphabet). 
     (2) Constant modulus for W: all elements in a precoding matrix have the same magnitude. This is important to facilitate power amplifier (PA) balance property in all scenarios. Note that constant modulus is a sufficient condition for PA balance, but not a necessary condition. However, enforcing constant modulus property tends to result in a simpler codebook design. Note also that while the precoding codebook (for feedback) conforms to the constant modulus property, this does not restrict the eNodeB from using non-constant modulus precoder. This is possible due to the use of UE-specific RS for demodulation. 
     (3) Nested property for W: every matrix/vector of rank-n is a sub-matrix of a rank-(n+1) precoding matrix, n=1, 2, . . . , N−1 where N is the maximum number of layers. While this property is desirable as it allows to reduce the complexity of PMI selection, it is not necessary to facilitate rank override if UE-specific RS is used. 
     (4) The associated feedback signaling overhead should be minimized. This is achieved by a balance between the overhead associated with W 1  (wideband, long-term) and W 2  (sub-band, short-term). Here, both the time (feedback rate) and frequency (feedback granularity) dimensions are important.
         (a) Blindly increasing the size of CB 1  (while reducing the size of CB 2 ) does not guarantee reducing the overall feedback overhead if a certain level of performance is expected. If the codebook CB 1  is meant to cover a certain precoder sub-space with a given spatial resolution, increasing the size of CB 1  demands an increase in feedbacks signaling associated with W 1 , both in time and frequency. This is because CB 1  starts to capture shorter-term channel properties, which are meant to be parts of CB 2 .   (b) To ensure that CB 1  does not need to be updated too frequently (in time and frequency), CB 1  should capture long-term channel properties such as the antenna setup and a range of values of angle of departure (AoD) which are associated with spatial correlation.   (c) The design should strive to keep the maximum overhead associated with W 2 /CB 2  the same as the Release 8 PMI overhead (i.e., ≤4 bits).       

     (5) Unitary precoder for W: (the column vectors of a precoder matrix must be pair-wise orthogonal to one another), while not necessary, is a sufficient condition for maintaining constant average transmitted power. This constraint is also used in designing the codebook at least for some relevant ranks. 
     The design of a 4Tx codebook for LTE disclosed herein targets an enhancement for multi-user (MU) MIMO over the 4Tx codebook in LTE Release 8. Rather than redesigning the 4Tx codebook, the enhancements disclosed herein focus on enhancing the MU-MIMO performance because the LTE Release 8 4Tx codebook was already designed to offer competitive performance for single-user (SU) MIMO (while keeping MU-MIMO in mind with the support of 8 discrete Fourier transform (DFT) vectors in the rank-1 codebook). Based on this consideration, the 4Tx enhancement focuses on rank-1 and at most rank-2 where MU-MIMO becomes relevant. 
     In terms of antenna setups, three setups may be considered:
         two dual-polarized elements with λ/2 (half wavelength) spacing between two elements,   two dual-polarized elements with 4λ. (larger) spacing between two elements, and   uniform linear array (ULA) with λ/2 (half wavelength) spacing.       

     The first and the second setups have the highest priority. Good performance should be ensured for dual (i.e., cross) polarized antenna arrays with both small and large spacing. 
     The antenna element indexing shown in  FIGS.  2  and  3    is use enumerate the spatial channel coefficients H n,m , where n and m are the receiver and transmitter antenna indices, respectively.  FIG.  2    illustrates a ULA or four pairs of ULA elements indexed 1-8.  FIG.  3    illustrates four pairs of cross-polarized arrays. The indexing for the four pairs of cross-polarized antennas represents grouping two antennas with the same polarization, which tend to be more correlated. This is analogous to the indexing of 4 pairs of ULA in  FIG.  2   . 
     Proposed Codebook Structure 
     The following notation is used to define the codebooks below:
         W: 4Tx feedback precoding matrix.   W 1 : first feedback precoding matrix   W 2 : second feedback precoding matrix   i 1 : PMI index of W 1      i 2 : PMI index of W 2      N: maximum number of layers   N TXA : number of transmit antennas   I k : (k×k)-dimensional identity matrix       

     Following the guideline of using the same principle for 4Tx enhancement and 8Tx, the block diagonal grid-of-beam (GoB) structure is used. This structure is common between 4Tx and 8Tx. 
     W 1  and the associated codebook can be written as follows: 
     
       
         
           
             
               
                 
                   	 
                   
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     Here, different W 1  matrices represent a partitioning (without overlap) in terms of beam angles:
         W 1  is a block diagonal matrix of size X where X is a (N TXA /2)×Nb matrix. Nb denotes the number of adjacent (N TXA /2)−Tx DFT beams contained in X. Such a design is able to synthesize N (N TXA /2)−Tx DFT beams within each polarization group. For a given N, the spatial oversampling factor is essentially (N/2). The overall (N TXA /2)−Tx DFT beam collections are captured in the (N TXA /2)×N matrix B.   By using co-phasing in W 2  (described below), the composite precoder W can synthesize up to N DFT beams.   For 4Tx, it should be noted that the Release 8 rank-1 codebook already contains eight 4Tx DFT beams.   The set of W 1  matrices represents (N/Nb)-level partitioning (hence non-overlapping) of the N beam angles (in X, i.e. each polarization group).   This design results in a codebook size of (N/Nb) for W 1 .       

     If some overlap in the set of beam angles is desired between two different W 1  matrices, the above formulation can be slightly modified such that two consecutive X matrices consists of some overlapping beam angles. Overlapping in beam angles may be beneficial to reduce “edge effects”, i.e. to ensure that a common W 1  matrix can be better chosen for different resource blocks (RBs) within the same precoding sub-band when sub-band precoding or CSI feedback is used. Herein a sub-band refers to a set of continuous physical resource blocks (PRB). With overlapping, 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
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     This enhancement targets MU-MIMO improvement and, therefore, is designed for rank-1 (and at most rank-2). At the same time, the Release 8 4Tx codebook should still be used at least for SU-MIMO. Keeping in mind that dynamic switching (switching without RRC configuration) between SU-MIMO and MU-MIMO is the baseline assumption for Release 12, an eNodeB should be able to use the Release 8 4Tx and the enhanced components interchangeably (i.e. the switching between the two components should be dynamic). Thanks to the dual-stage feedback structure and in particular the W=W 1 *W 2  structure, this can be realized in a simple and natural manner. The enhanced component can be augmented or combined with the Release 8 codebook as follows:
         the Release 8 4Tx codebook is used as the codebook for W 2  and associated with W 1 =identity matrix;   when PMI 1  indicates that W 1 =identity matrix is chosen, CB 2  is chosen as the original Rel-8 codebook;   else, when PMI 1  indicates some other W 1 , W 1  and CB 2  are chosen as the enhanced component.       

     The above switching/augmentation mechanism features the following:
         best possible 4Tx MU-MIMO codebook enhancement opportunity. This is because the optimization effort for the new components can be focused on improving MU-MIMO without the need for considering the SU-MIMO performance (which is covered by the Release 8 4Tx codebook). Furthermore, the new components can be designed “from scratch” as the above augmentation mechanism with Release 8 4Tx codebook can be done without constraining any of the structures of the new components.   maintain best possible performance for 4Tx SU-MIMO without additional standardization effort. This comes from the use of Release 8 4Tx codebook. It is noted that the Release 8 4Tx codebook offers competitive performance in various antenna and channel setups including dual-polarized arrays as pointed out partly due to the inherent block diagonal structure in a number of the precoder matrices/vectors.   achieve flexible frequency-selective precoding with physical uplink shared channel (PUSCH) mode 3-2. If Release 8 codebook is not re-used and the new codebook is entirely based on the W=W 1 W 2  structure, all subband PMI must fall in the same Grid-of-Beam (GoB) due to the wideband W 1  constraint. This inevitably limits the precoding gains of mode 3-2 and system performance. On the contrary, by augmenting the Release 8 codebook, all Release 8 PMI vectors can be used on each subband independently without any constraint. This is critical to ensure flexible subband precoding.       

     The design for W 2  (based on co-phasing and selection) follows the structure used for 8Tx codebook design. Co-phasing allows some phase adjustment between the two polarization groups and generation of 4Tx DFTTx DFT vectors from two block diagonal 2Tx DFTTx DFT matrices. The (group) selection operation allows refinement/adjustment of beam angles across RBs within the same sub-band thereby maximizing frequency-selective precoding gain. 
     The combination of beam selection and co-phasing in W 2 —combined with W 1 —should result in a unitary precoder W=W 1 W 2 . 
     An example of a complete design with non-overlapping block diagonal GoB augmentation of (N,Nb) codebook is given in the subsequent sections. It is straightforward to extend the proposed designs to include adjacent W 1  matrix overlapping, but is omitted here due to simplicity. 
     
       
         
           
             
               
                 
                   
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                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     X 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   ∈ 
                   
                     { 
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           
                                             N 
                                             b 
                                           
                                           ⁢ 
                                           k 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       mod 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                                 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               N 
                                               b 
                                             
                                             ⁢ 
                                             k 
                                           
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       mod 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               N 
                                               b 
                                             
                                             ⁢ 
                                             k 
                                           
                                           + 
                                           
                                             N 
                                             b 
                                           
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       mod 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           : 
                           k 
                         
                         = 
                         0 
                       
                       , 
                       
                         
                           … 
                           ⁢ 
                                
                           N 
                           / 
                           
                             N 
                             b 
                           
                         
                         - 
                         1 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Rank 1 
     Assuming (N, Nb)=(8,4) as an example. 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         I 
                         4 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     →size-3 (Rel-8 codebook augmented with block diagonal GoB). 
     When W 1 =I 4 , then W 2 ∈C 2,R8Tx4r1 , 
     where C 2,R8Tx4r1  denotes the Release 8 4Tx rank-1 codebook used for W 2 . 
     When 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       k 
                       = 
                       0 
                     
                     , 
                     1 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             
               
                 
                   W 
                   2 
                 
                 ∈ 
                 
                   CB 
                   2 
                 
               
               = 
               
                 { 
                 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                       
                       
                         
                           Y 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                       
                       
                         
                           jY 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                       
                       
                         
                           
                             - 
                             Y 
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                       
                       
                         
                           
                             - 
                             jY 
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
             , 
             
               Y 
               ∈ 
               
                 { 
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
           
         
       
     
     The rank-1 PMI overhead for this example is shown in Table 1. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 W1 overhead (wideband) 
                 W2 overhead (subband) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Rank-1 
                 log2(3)-bits 
                 4-bit 
               
               
                   
                   
               
            
           
         
       
     
     Other values of (N, Nb) are not precluded if justified by sufficient system performance gain and reasonable feedback overhead. 
     Note that the block-diagonal enhancement components are sub-matrices of a subset of the Release 10 8Tx codebook. As such, the 4Tx GoB components can be obtained by pruning the 8Tx codebook for 4Tx MIMO feedback. This is discussed in more detail below. 
     N, Nb can also take other values. For example if (N, Nb)=(16, 4). 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         I 
                         4 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     →size-5 (Rel-8 codebook augmented with block diagonal GoB). 
     When W 1 =I 4 , then W 2 ∈C 2,R8Tx4r1 . 
     where C 2,R8Tx4r1  denotes the Release 8 4Tx rank-1 codebook used for W 2 . 
     When 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       k 
                       = 
                       0 
                     
                     , 
                     1 
                     , 
                     2 
                     , 
                     3 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             
               
                 
                   W 
                   2 
                 
                 ∈ 
                 
                   CB 
                   2 
                 
               
               = 
               
                 { 
                 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                       
                       
                         
                           Y 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                       
                       
                         
                           jY 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                       
                       
                         
                           
                             - 
                             Y 
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                       
                       
                         
                           
                             - 
                             jY 
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
             , 
             
               Y 
               ∈ 
               
                 { 
                 
                   
                     [ 
                     
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   , 
                   
                     [ 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
           
         
       
     
     The rank-1 PMI overhead for this example is shown in Table 2. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
               
                   
                 W1 overhead (wideband) 
                 W2 overhead (subband) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Rank-1 
                 log2(5)-bits 
                 4-bit 
               
               
                   
                   
               
            
           
         
       
     
     It is possible in some embodiments that the Release 12 4Tx codebook does not comprise the Release 8 codebook. In that case, the identity matrix I 4  is removed from the W 1  codebook C 1 . 
     It is also possible to increase the value of N (e.g. to 32 or 64). This however increases the C 1  codebook size and feedback overhead, and reduces the span of angular spread that Nb adjacent beams can cover. 
     Rank 2: 
     Assuming (N,Nb)=(8,4) as an example: 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         I 
                         4 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     →size-3 (Rel-8 codebook augmented with block diagonal GoB). 
     When W 1 =I 4 , then W 2 ∈C 2,R8Tx4r2 , 
     where C 2,R8Tx4r2  denotes the Release 8 4Tx rank-2 codebook used for W 2 . 
     When 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                     
                 
                   ( 
                   
                     
                       k 
                       = 
                       0 
                     
                     , 
                     1 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             
               
                 
                   W 
                   2 
                 
                 ∈ 
                 
                   C 
                   ⁢ 
                   
                     B 
                     2 
                   
                 
               
               = 
               
                 { 
                 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                         
                           Y 
                         
                       
                       
                         
                           Y 
                         
                         
                           
                             - 
                             Y 
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                         
                           Y 
                         
                       
                       
                         
                           jY 
                         
                         
                           
                             - 
                             jY 
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
             , 
           
         
       
       
         
           
             Y 
             ∈ 
             
               { 
               
                 
                   [ 
                   
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                   
                   ] 
                 
               
               } 
             
           
         
       
     
     One minor shortcoming of this design is that the W 2  overhead is not constant, but varying dependent on the W 1  matrix. In particular,
         if W 1 =I, W 2  overhead is 4-bits per subband; and   if W 1  corresponds to the block-diagonal component, W 2  overhead is 3-bits per subband.       

     Because W 1  and W 2  are jointly encoded in the PUSCH feedback mode, joint blind decoding of W 1 /W 2  is required at the eNodeB, which may increase the eNodeB implementation complexity. 
     As one solution, the two columns in a rank-2 precoding matrix W may be chosen from different beams in one grid. 
     As an example, the following design is possible. 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         I 
                         4 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     →size-3 (Rel-8 codebook augmented with block diagonal GoB). 
     When W 1 =I 4 : then W 2 ∈C 2,R8Tx4r2 , 
     where C 2,R8Tx4r2  denotes the Release 8 4Tx rank-2 codebook used for W 2 . 
     When 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                     
                 
                   ( 
                   
                     
                       k 
                       = 
                       0 
                     
                     , 
                     1 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             	 
             
               
                 
                   
                     W 
                     2 
                   
                   ∈ 
                   
                     C 
                     ⁢ 
                     
                       B 
                       2 
                     
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         1 
                         
                           2 
                         
                       
                       [ 
                       
                         
                           
                             
                               Y 
                               1 
                             
                           
                           
                             
                               Y 
                               2 
                             
                           
                         
                         
                           
                             
                               Y 
                               1 
                             
                           
                           
                             
                               - 
                               
                                 Y 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                     , 
                     
                       
                         1 
                         
                           2 
                         
                       
                       [ 
                       
                         
                           
                             
                               Y 
                               1 
                             
                           
                           
                             
                               Y 
                               2 
                             
                           
                         
                         
                           
                             
                               jY 
                               1 
                             
                           
                           
                             
                               - 
                               
                                 jY 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   } 
                 
               
               , 
             
           
         
       
       
         
           
             
               ( 
               
                 
                   Y 
                   1 
                 
                 , 
                 
                   Y 
                   2 
                 
               
               ) 
             
             ∈ 
             
               { 
               
                 
                   ( 
                   
                     
                       e 
                       1 
                       4 
                     
                     , 
                     
                       e 
                       1 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       2 
                       4 
                     
                     , 
                     
                       e 
                       2 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       3 
                       4 
                     
                     , 
                     
                       e 
                       3 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       4 
                       4 
                     
                     , 
                     
                       e 
                       4 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       1 
                       4 
                     
                     , 
                     
                       e 
                       2 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       2 
                       4 
                     
                     , 
                     
                       e 
                       3 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       1 
                       4 
                     
                     , 
                     
                       e 
                       4 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       2 
                       4 
                     
                     , 
                     
                       e 
                       4 
                       4 
                     
                   
                   ) 
                 
               
               } 
             
           
         
       
     
     where e q   p  is a p×1 column vector with all elements equivalent to zero, except the q-th element which is 1. The W 2  overhead is 4-bits per subband, consistent with the Release 8 codebook overhead. Note that any other (Y 1 , Y 2 ) pair, denoted by (e m   4 , e n   4 ), are equally applicable, where 1≤m≤4, 1≤n≤4, m≠n. 
     Note that in any of the (Y 1 , Y 2 ) pair above, the two selection vectors inside the brackets [ ] can be permuted. For example, (e 1   4 , e 2   4 ) can be replaced by (e 2   4 , e 1   4 ), and the resulting codebook is equivalently applicable. 
     It should be noted that that the block-diagonal enhancement components are sub-matrices of a subset of the Release 10 8Tx codebook. 
     As another solution, it can be resolved by adopting a (N, Nb)=(16, 8) codebook enhancement. 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         I 
                         4 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     →size-3 (Rel-8 codebook augmented with block diagonal GoB). 
     When W 1 =I 4 , then W 2 ∈C 2,R8Tx4r2 , 
     where C 2,R8Tx4r2  denotes the Release 8 4Tx rank-2 codebook used for W 2 . 
     When 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                     
                 
                   ( 
                   
                     
                       k 
                       = 
                       0 
                     
                     , 
                     1 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             
               
                 
                   W 
                   2 
                 
                 ∈ 
                 
                   C 
                   ⁢ 
                   
                     B 
                     2 
                   
                 
               
               = 
               
                 { 
                 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                         
                           Y 
                         
                       
                       
                         
                           Y 
                         
                         
                           
                             - 
                             Y 
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           Y 
                         
                         
                           Y 
                         
                       
                       
                         
                           jY 
                         
                         
                           
                             - 
                             jY 
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
             , 
           
         
       
       
         
           
             Y 
             ∈ 
             
               { 
               
                 
                   [ 
                   
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                     
                       
                         0 
                       
                     
                   
                   ] 
                 
                 , 
                 
                   [ 
                   
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                     
                     
                       
                         1 
                       
                     
                   
                   ] 
                 
                 , 
               
               } 
             
           
         
       
     
     The rank-2 PMI overhead for this example is shown in Table 3. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 3 
               
               
                   
                   
               
               
                   
                 W1 overhead (wideband) 
                 W2 overhead (subband) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Rank-2 
                 log2(3)-bit 
                 4-bit 
               
               
                   
                   
               
            
           
         
       
     
     Alternatively, it is possible that the Release 12 4Tx codebook does not comprise of the Release 8 codebook. In that case, the identity matrix is removed from the W 1  codebook C 1 . 
     Another possible design is to use (N,Nb)=(16,4) as 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         I 
                         4 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   1 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     →size-5 (Rel-8 codebook augmented with block diagonal GoB). 
     When W 1 =I 4 , then W 2 ∈C 2,R8Tx4r2 , 
     where C 2,R8Tx4r2  denotes the Release 8 4Tx rank-2 codebook used for W 2 . When 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                     
                 
                   ( 
                   
                     
                       k 
                       = 
                       0 
                     
                     , 
                     … 
                         
                     , 
                     3 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             	 
             
               
                 
                   
                     W 
                     2 
                   
                   ∈ 
                   
                     C 
                     ⁢ 
                     
                       B 
                       2 
                     
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         1 
                         
                           2 
                         
                       
                       [ 
                       
                         
                           
                             
                               Y 
                               1 
                             
                           
                           
                             
                               Y 
                               2 
                             
                           
                         
                         
                           
                             
                               Y 
                               1 
                             
                           
                           
                             
                               - 
                               
                                 Y 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                     , 
                     
                       
                         1 
                         
                           2 
                         
                       
                       [ 
                       
                         
                           
                             
                               Y 
                               1 
                             
                           
                           
                             
                               Y 
                               2 
                             
                           
                         
                         
                           
                             
                               jY 
                               1 
                             
                           
                           
                             
                               - 
                               
                                 jY 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   } 
                 
               
               , 
             
           
         
       
       
         
           
             
               ( 
               
                 
                   Y 
                   1 
                 
                 , 
                 
                   Y 
                   2 
                 
               
               ) 
             
             ∈ 
             
               { 
               
                 
                   ( 
                   
                     
                       e 
                       1 
                       4 
                     
                     , 
                     
                       e 
                       1 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       2 
                       4 
                     
                     , 
                     
                       e 
                       2 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       3 
                       4 
                     
                     , 
                     
                       e 
                       3 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       4 
                       4 
                     
                     , 
                     
                       e 
                       4 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       1 
                       4 
                     
                     , 
                     
                       e 
                       2 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       2 
                       4 
                     
                     , 
                     
                       e 
                       3 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       1 
                       4 
                     
                     , 
                     
                       e 
                       4 
                       4 
                     
                   
                   ) 
                 
                 , 
                 
                   ( 
                   
                     
                       e 
                       2 
                       4 
                     
                     , 
                     
                       e 
                       4 
                       4 
                     
                   
                   ) 
                 
               
               } 
             
           
         
       
     
     It is also possible to increase the value of N (e.g. to 32 or 64). This however increases the C 1  codebook size and feedback overhead, and reduces the span of angular spread that Nb adjacent beams can cover. 
     Rank 3: 
     The simplest solution is to reuse the Release 8 codebook as-is for Release 12. 
         W   1   =I   4   (16)
 
     →size-1 (Rel-8 codebook only). 
         W   2    ∈C   2,R8Tx4r3 , 
     where C 2,R8Tx4r3  denotes the Release 8 4Tx rank-3 codebook used for W 2 . 
     If enhancement based on (N,Nb) structure is warranted by sufficient performance gain, it can be done in a similar fashion as for rank-1 and rank-2. For instance, based on a (N,Nb)=(4,4) design: 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         I 
                         4 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     →size-2 (Rel-8 codebook augmented with block diagonal GoB). 
     When W 1 =I 4 , then W 2 ∈C 2,R8Tx4r3 , 
     where C 2,R8Tx4r3  denotes the Release 8 4Tx rank-3 codebook used for W 2 . 
     When 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                     
                 
                   ( 
                   
                     k 
                     = 
                     0 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             	 
             
               
                 
                   W 
                   2 
                 
                 ∈ 
                 
                   C 
                   ⁢ 
                   
                     B 
                     2 
                   
                 
               
               = 
               
                 { 
                 
                   
                     1 
                     
                       2 
                     
                   
                   [ 
                   
                     
                       
                         
                           Y 
                           1 
                         
                       
                       
                         
                           Y 
                           2 
                         
                       
                     
                     
                       
                         
                           Y 
                           1 
                         
                       
                       
                         
                           - 
                           
                             Y 
                             2 
                           
                         
                       
                     
                   
                   ] 
                 
                 } 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   
                     Y 
                     1 
                   
                   , 
                   
                     Y 
                     2 
                   
                 
                 ) 
               
               ∈ 
               
                 { 
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               e 
                               1 
                               4 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               e 
                               2 
                               4 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               e 
                               3 
                               4 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               e 
                               4 
                               4 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                       
                     
                   
                   
                     
                       
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               e 
                               1 
                               4 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               e 
                               2 
                               4 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               e 
                               3 
                               4 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               e 
                               4 
                               4 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 } 
               
             
             , 
           
         
       
     
     corresponding to i 2 =0, . . . , 7. 
     To make the W 2  overhead consistent for different W 1  matrices, i 2 =8, . . . , 15 shall be reserved for W 2  corresponding to W 1  matrices that are enhancement components (e.g. block diagonal). 
     Alternatively, when 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                     
                 
                   ( 
                   
                     k 
                     = 
                     0 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             
               
                 
                   W 
                   2 
                 
                 ∈ 
                 
                   C 
                   ⁢ 
                   
                     B 
                     2 
                   
                 
               
               = 
               
                 { 
                 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           
                             Y 
                             1 
                           
                         
                         
                           
                             Y 
                             2 
                           
                         
                       
                       
                         
                           
                             Y 
                             1 
                           
                         
                         
                           
                             - 
                             
                               Y 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           
                             Y 
                             1 
                           
                         
                         
                           
                             Y 
                             2 
                           
                         
                       
                       
                         
                           
                             j 
                             ⁢ 
                             
                               Y 
                               1 
                             
                           
                         
                         
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             
                               Y 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
             , 
           
         
       
       
         
           
             
               ( 
               
                 
                   Y 
                   1 
                 
                 , 
                 
                   Y 
                   2 
                 
               
               ) 
             
             ∈ 
             
               
                 { 
                 
                   
                     
                       
                         
                           ( 
                           
                             
                               e 
                               1 
                               4 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               e 
                               2 
                               4 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               e 
                               3 
                               4 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               e 
                               4 
                               4 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               e 
                               1 
                               4 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               e 
                               2 
                               4 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               e 
                               3 
                               4 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               e 
                               4 
                               4 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 } 
               
               . 
             
           
         
       
     
     In this case W2 overhead is 4-bits per sub-band 
     The rank-3 PMI overhead for this example is shown in Table 4. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 4 
               
               
                   
                   
               
               
                   
                 W1 overhead (wideband) 
                 W2 overhead (subband) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Rank-3 
                 0-bit (w/o augmentation) 
                 4-bit 
               
               
                   
                   
                 1-bit (w/augmentation) 
               
               
                   
                   
               
            
           
         
       
     
     It is possible that the Release 12 4Tx codebook does not comprise of the Release 8 codebook. In that case, the identity matrix is removed from the W 1  codebook C 1 . 
     Note that in any of the (Y 1 , Y 2 ) pairs above, the two selection vectors inside the brackets can be permuted. For example, (e 1   4 , [e 1   4 , e 3   4 ]) can be replaced by (e 1   4 , [e 3   4 , e 1   4 ]), and the resulting codebook is equivalently applicable. 
     It is not precluded that the Release 12 4Tx rank-3 codebook is redesigned using the GoB component as proposed above, and not include any Release 8 4Tx rank-3 precoding matrices. 
     Another alternative GoB design is (N,Nb)=(8,8), where 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   ( 
                   
                     k 
                     = 
                     0 
                   
                   ) 
                 
               
             
             , 
               
             and 
           
         
       
       
         
           
             
               
                 
                   W 
                   2 
                 
                 ∈ 
                 
                   C 
                   ⁢ 
                   
                     B 
                     2 
                   
                 
               
               = 
               
                 { 
                 
                   
                     1 
                     
                       2 
                     
                   
                   [ 
                   
                     
                       
                         
                           Y 
                           1 
                         
                       
                       
                         
                           Y 
                           2 
                         
                       
                     
                     
                       
                         
                           Y 
                           1 
                         
                       
                       
                         
                           - 
                           
                             Y 
                             2 
                           
                         
                       
                     
                   
                   ] 
                 
                 } 
               
             
             , 
               
             or 
           
         
       
       
         
           
             
               
                 W 
                 2 
               
               ∈ 
               
                 C 
                 ⁢ 
                 
                   B 
                   2 
                 
               
             
             = 
             
               { 
               
                 
                   
                     1 
                     
                       2 
                     
                   
                   [ 
                   
                     
                       
                         
                           Y 
                           1 
                         
                       
                       
                         
                           Y 
                           2 
                         
                       
                     
                     
                       
                         
                           Y 
                           1 
                         
                       
                       
                         
                           - 
                           
                             Y 
                             2 
                           
                         
                       
                     
                   
                   ] 
                 
                 , 
                 
                   
                     1 
                     
                       2 
                     
                   
                   [ 
                   
                     
                       
                         
                           Y 
                           1 
                         
                       
                       
                         
                           Y 
                           2 
                         
                       
                     
                     
                       
                         
                           j 
                           ⁢ 
                           
                             Y 
                             1 
                           
                         
                       
                       
                         
                           
                             - 
                             j 
                           
                           ⁢ 
                           
                             Y 
                             2 
                           
                         
                       
                     
                   
                   ] 
                 
               
               } 
             
           
         
       
       
         
           
             
               ( 
               
                 
                   Y 
                   1 
                 
                 , 
                 
                   Y 
                   2 
                 
               
               ) 
             
             ∈ 
             
               { 
               
                 
                   
                     
                       
                         ( 
                         
                           
                             e 
                             1 
                             4 
                           
                           , 
                           
                             [ 
                             
                               
                                 e 
                                 1 
                                 4 
                               
                               , 
                               
                                 e 
                                 5 
                                 4 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             2 
                             4 
                           
                           , 
                           
                             [ 
                             
                               
                                 e 
                                 2 
                                 4 
                               
                               , 
                               
                                 e 
                                 6 
                                 4 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             3 
                             4 
                           
                           , 
                           
                             [ 
                             
                               
                                 e 
                                 3 
                                 4 
                               
                               , 
                               
                                 e 
                                 7 
                                 4 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             4 
                             4 
                           
                           , 
                           
                             [ 
                             
                               
                                 e 
                                 4 
                                 4 
                               
                               , 
                               
                                 e 
                                 8 
                                 4 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                       , 
                     
                   
                 
                 
                   
                     
                       
                         ( 
                         
                           
                             e 
                             5 
                             4 
                           
                           , 
                           
                             [ 
                             
                               
                                 e 
                                 1 
                                 4 
                               
                               , 
                               
                                 e 
                                 5 
                                 4 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             6 
                             4 
                           
                           , 
                           
                             [ 
                             
                               
                                 e 
                                 2 
                                 4 
                               
                               , 
                               
                                 e 
                                 6 
                                 4 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             7 
                             4 
                           
                           , 
                           
                             [ 
                             
                               
                                 e 
                                 3 
                                 4 
                               
                               , 
                               
                                 e 
                                 7 
                                 4 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             3 
                             4 
                           
                           , 
                           
                             [ 
                             
                               
                                 e 
                                 4 
                                 4 
                               
                               , 
                               
                                 e 
                                 8 
                                 4 
                               
                             
                             ] 
                           
                         
                         ) 
                       
                       , 
                     
                   
                 
                 
                   
                     
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 e 
                                 1 
                                 4 
                               
                               , 
                               
                                 e 
                                 5 
                                 4 
                               
                             
                             ] 
                           
                           , 
                           
                             e 
                             1 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 e 
                                 2 
                                 4 
                               
                               , 
                               
                                 e 
                                 6 
                                 4 
                               
                             
                             ] 
                           
                           , 
                           
                             e 
                             2 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 e 
                                 3 
                                 4 
                               
                               , 
                               
                                 e 
                                 7 
                                 4 
                               
                             
                             ] 
                           
                           , 
                           
                             e 
                             3 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 e 
                                 4 
                                 4 
                               
                               , 
                               
                                 e 
                                 8 
                                 4 
                               
                             
                             ] 
                           
                           , 
                           
                             e 
                             4 
                             4 
                           
                         
                         ) 
                       
                       , 
                     
                   
                 
                 
                   
                     
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 e 
                                 1 
                                 4 
                               
                               , 
                               
                                 e 
                                 5 
                                 4 
                               
                             
                             ] 
                           
                           , 
                           
                             e 
                             5 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 e 
                                 2 
                                 4 
                               
                               , 
                               
                                 e 
                                 6 
                                 4 
                               
                             
                             ] 
                           
                           , 
                           
                             e 
                             6 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 e 
                                 3 
                                 4 
                               
                               , 
                               
                                 e 
                                 7 
                                 4 
                               
                             
                             ] 
                           
                           , 
                           
                             e 
                             7 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             [ 
                             
                               
                                 e 
                                 4 
                                 4 
                               
                               , 
                               
                                 e 
                                 8 
                                 4 
                               
                             
                             ] 
                           
                           , 
                           
                             e 
                             8 
                             4 
                           
                         
                         ) 
                       
                       , 
                     
                   
                 
               
               } 
             
           
         
       
     
     Rank 4: 
     The simplest solution is to reuse Release 8 codebook as-is for Release 12. 
         W   1   =I   4   (18)
 
     →Size-1 (Rel-8 codebook only). 
         W   2   ∈C   2,R8Tx4r4 , 
     where C 2,R8Tx4r4  denotes the Release 8 4Tx rank-4 codebook used for W 2 . 
     If enhancement based on (N,Nb) structure is warranted by sufficient performance gain, it can be done in a similar fashion as for rank-1 and rank-2. For instance, based on a (N,Nb)=(4,4) design, 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         I 
                         4 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     →Size-2 (Rel-8 codebook augmented with block diagonal GoB) 
     When W 1 =I 4 , then W 2 ∈C 2,R8Tx4r4 , 
     where C 2,R8Tx4r4  denotes the Release 8 4Tx rank-4 codebook used for W 2 . 
     When 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                    
                 
                   ( 
                   
                     k 
                     = 
                     0 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             
               
                 
                   W 
                   2 
                 
                 ∈ 
                 
                   C 
                   ⁢ 
                   
                     B 
                     2 
                   
                 
               
               = 
               
                 { 
                 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           
                             Y 
                             1 
                           
                         
                         
                           
                             Y 
                             2 
                           
                         
                       
                       
                         
                           
                             Y 
                             1 
                           
                         
                         
                           
                             - 
                             
                               Y 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           
                             Y 
                             1 
                           
                         
                         
                           
                             Y 
                             2 
                           
                         
                       
                       
                         
                           
                             j 
                             ⁢ 
                             
                               Y 
                               1 
                             
                           
                         
                         
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             
                               Y 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
             , 
           
         
       
       
         
           where 
         
       
       
         
           
             
               
                 ( 
                 
                   
                     Y 
                     1 
                   
                   , 
                   
                     Y 
                     2 
                   
                 
                 ) 
               
               ∈ 
               
                 { 
                 
                   
                     
                       
                         ( 
                         
                           [ 
                           
                             e 
                             1 
                             4 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 e 
                                 3 
                                 4 
                               
                               ] 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                       
                     
                   
                   
                     
                       
                         ( 
                         
                           [ 
                           
                             e 
                             1 
                             4 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 e 
                                 3 
                                 4 
                               
                               ] 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   e 
                                   2 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   4 
                                   4 
                                 
                               
                               ] 
                             
                             , 
                             
                               [ 
                               
                                 
                                   e 
                                   1 
                                   4 
                                 
                                 , 
                                 
                                   e 
                                   3 
                                   4 
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 } 
               
             
             , 
           
         
       
     
     corresponding to i 1 =0, . . . , 7. 
     Note there are a total of 8 W 2  matrices (i 2 =0, . . . , 7) for 
     
       
         
           
             
               W 
               1 
             
             = 
             
               [ 
               
                 
                   
                     
                       X 
                       
                         
                           〈 
                           k 
                         
                         ) 
                       
                     
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       X 
                       
                         
                           〈 
                           k 
                         
                         ) 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     To make the W 2  overhead consistent across all W 1 , W 2  can be reserved for i 2 =8, . . . , 15. 
     Alternatively, one may adopt a (N, Nb)=(8,8) codebook as below: 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         I 
                         4 
                       
                       , 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     →Size-2 (Rel-8 codebook augmented with block diagonal GoB) 
     When W 1 =I 4 , then W 2 ∈C 2,R8Tx4r4 , 
     where C 2,R8Tx4r4  denotes the Release 8 4Tx rank-4 codebook used for W 2 . 
     When 
     
       
         
           
             
               
                 W 
                 1 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           X 
                           
                             
                               〈 
                               k 
                             
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   ( 
                   
                     k 
                     = 
                     0 
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     then 
     
       
         
           
             
               
                 
                   W 
                   2 
                 
                 ∈ 
                 
                   C 
                   ⁢ 
                   
                     B 
                     2 
                   
                 
               
               = 
               
                 { 
                 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           
                             Y 
                             1 
                           
                         
                         
                           
                             Y 
                             2 
                           
                         
                       
                       
                         
                           
                             Y 
                             1 
                           
                         
                         
                           
                             - 
                             
                               Y 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                   , 
                   
                     
                       1 
                       
                         2 
                       
                     
                     [ 
                     
                       
                         
                           
                             Y 
                             1 
                           
                         
                         
                           
                             Y 
                             2 
                           
                         
                       
                       
                         
                           
                             j 
                             ⁢ 
                             
                               Y 
                               1 
                             
                           
                         
                         
                           
                             
                               - 
                               j 
                             
                             ⁢ 
                             
                               Y 
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                 
                 } 
               
             
             , 
               
             and 
           
         
       
       
         
           
             
               ( 
               
                 
                   Y 
                   1 
                 
                 , 
                 
                   Y 
                   2 
                 
               
               ) 
             
             ∈ 
             
               
                 { 
                 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 [ 
                                 
                                   e 
                                   1 
                                   8 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       e 
                                       5 
                                       8 
                                     
                                     ] 
                                   
                                   , 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         1 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         5 
                                         8 
                                       
                                     
                                     ] 
                                   
                                 
                                 ) 
                               
                               , 
                               
                                 ( 
                                 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         2 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         6 
                                         8 
                                       
                                     
                                     ] 
                                   
                                   , 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         2 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         6 
                                         8 
                                       
                                     
                                     ] 
                                   
                                 
                                 ) 
                               
                               , 
                               
                                 ( 
                                 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         3 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         7 
                                         8 
                                       
                                     
                                     ] 
                                   
                                   , 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         3 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         7 
                                         8 
                                       
                                     
                                     ] 
                                   
                                   , 
                                   
                                     ( 
                                     
                                       
                                         [ 
                                         
                                           
                                             e 
                                             4 
                                             8 
                                           
                                           , 
                                           
                                             e 
                                             8 
                                             8 
                                           
                                         
                                         ] 
                                       
                                       , 
                                       
                                         [ 
                                         
                                           
                                             e 
                                             4 
                                             8 
                                           
                                           , 
                                           
                                             e 
                                             8 
                                             8 
                                           
                                         
                                         ] 
                                       
                                     
                                     ) 
                                   
                                   , 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 [ 
                                 
                                   e 
                                   1 
                                   8 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       e 
                                       5 
                                       8 
                                     
                                     ] 
                                   
                                   , 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         2 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         6 
                                         8 
                                       
                                     
                                     ] 
                                   
                                 
                                 ) 
                               
                               , 
                               
                                 ( 
                                 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         1 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         5 
                                         8 
                                       
                                     
                                     ] 
                                   
                                   , 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         3 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         7 
                                         8 
                                       
                                     
                                     ] 
                                   
                                 
                                 ) 
                               
                               , 
                               
                                 ( 
                                 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         2 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         6 
                                         8 
                                       
                                     
                                     ] 
                                   
                                   , 
                                   
                                     [ 
                                     
                                       
                                         e 
                                         4 
                                         8 
                                       
                                       , 
                                       
                                         e 
                                         8 
                                         8 
                                       
                                     
                                     ] 
                                   
                                   , 
                                   
                                     ( 
                                     
                                       
                                         [ 
                                         
                                           
                                             e 
                                             3 
                                             8 
                                           
                                           , 
                                           
                                             e 
                                             7 
                                             8 
                                           
                                         
                                         ] 
                                       
                                       , 
                                       
                                         
                                           [ 
                                           
                                             
                                               e 
                                               4 
                                               8 
                                             
                                             , 
                                             
                                               e 
                                               8 
                                               8 
                                             
                                           
                                           ] 
                                         
                                         ⁢ 
                                         
                                            
                                           
                                              
                                              
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
                 } 
               
               . 
             
           
         
       
     
     Note that in any of the (Y 1 , Y 2 ) pair above, the two selection vectors inside the brackets [ ] can be permuted. 
     Also note that any (Y 1 , Y 2 ) pair above can be replaced by a different pair denoted as ([e m   N     b   , e m+N/2   N     b   ], [e n   N     b   , e n+N/2   N     b   ]), where 1≤m≤N/2, 1≤n≤N/2. 
     The rank-4 PMI overhead for this example is shown in Table 5. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 5 
               
               
                   
                   
               
               
                   
                 W1 overhead (wideband) 
                 W2 overhead (sub-band) 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Rank-4 
                 0-bit (w/o augmentation) 
                 4-bit 
               
               
                   
                   
                 1-bit (w/augmentation) 
               
               
                   
                   
               
            
           
         
       
     
     It is possible that the Release 12 4Tx codebook does not comprise of the Release 8 codebook. In that case, the identity matrix is removed from the W 1  codebook C 1 . 
     It is not precluded that the Release 12 4Tx rank-4 codebook is redesigned using the GoB component as proposed above, and not includes any Release 8 4Tx rank-4 precoding matrices. 
     The final 4Tx codebook comprises of rank-r codebook, r=1, 2, 3, 4. For each rank r, the corresponding rank-r codebook can be constructed by methods discussed above regarding rank-1 to rank-4 codebooks. It is not precluded that codebook is enhanced for certain rank(s), while for the other ranks, Release 8 codebook is reused. 
     Reformulation of Proposed Rank-1 to Rank-4 Codebooks 
     Using (N, Nb)=(16,4) as an example, the proposed codebook in the sections above may be re-formulated by the equations shown in the tables below. It should be noted that these tables can be easily extended to other (N, Nb) values. 
     Rank 1-2 
     If Release 12 4Tx codebook for rank-1 and rank-2 is redesigned by the GoB framework with adjacent beams overlapping, and does not include the Release 8 codebook, the 4Tx codebooks can be expressed by equations in Tables 6-1 and 6-2. 
     A first PMI value of n 1 ∈{0, 1, . . . , ƒ(υ)−1) and a second PMI value of n 2 ∈{0, 1, . . . , g(υ)−1} correspond to the codebook indices n 1  and n 2  given in Table. 6-j, where υ is equal to the associated rank value and where j=υ, ƒ(υ)={8,8} and g(υ)={16,16}. Interchangeably, the first and second precoding matrix indicators are expressed by i 1  and i 2 . 
     The quantities φ n  and v m  are expressed by 
       φ n   =e   jπn/2   (21)
 
         v   m =[1;  e   j2πn/16 ]  (22)
 
     Table 6-1 illustrates a codebook for 1-layer CSI reporting according to one embodiment. 
     
       
         
           
               
               
             
               
                 TABLE 6-1 
               
               
                   
               
             
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 i 1   
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
               
                 0-7 
                 W 2i     1   , 0   (1)   
                 W 2i     1   , 1   (1)   
                 W 2i     1   , 2   (1)   
                 W 2i     1   , 3   (1)   
                 W 2i     1     +1,0   (1)   
                 W 2i     1     +1,1   (1)   
                 W 2i     1     +1,2   (1)   
                 W 2i     1     +1,3   (1)   
               
            
           
           
               
               
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 i 1   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
               
                 0-7 
                 W 2i     1     +2,0   (1)   
                 W 2i     1     +2,1   (1)   
                 W 2i     1     +2,2   (1)   
                 W 2i     1     +2,3   (1)   
                 W 2i     1     +3,0   (1)   
                 W 2i     1     +3,1   (1)   
                 W 2i     1     +3,2   (1)   
                 W 2i     1     +3,3   (1)   
               
               
                   
               
            
           
           
               
            
               
                 where 
               W     m   ,   n       (   1   )       =       1     4       [           v   m                 φ   n     ⁢     v   m             ]                       
 
               
               
                   
               
            
           
         
       
     
     Table 6-2 illustrates a codebook for 2-layer CSI reporting according to one embodiment. 
     
       
         
           
               
               
             
               
                 TABLE 6-2 
               
               
                   
               
             
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 0 
                 1 
                 2 
                 3 
               
               
                   
               
               
                 0-7 
                 W 2i     1     ,2i     1     ,0   (2)   
                 W 2i     1     ,2i     1     ,1   (2)   
                 W 2i     1     ,1,2i     1     +1,0   (2)   
                 W 2i     1     +1,2i     1     +1,1   (2)   
               
            
           
           
               
               
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
               
                 0-7 
                 W 2i     1     +2,2i     1     +2,0   (2)   
                 W 2i     1     +2,2i     1     +2,1   (2)   
                 W 2i     1     +3,2i     1     +3,0   (2)   
                 W 2i     1     +3,2i     1     +3,1   (2)   
               
            
           
           
               
               
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                 0-7 
                 W 2i     1     ,2i     1     +1,0   (2)   
                 W 2i     1     ,2i     1     +1,1   (2)   
                 W 2i     1     +1,2i     1     +2,0   (2)   
                 W 2i     1     +1,2i     1     +2,1   (2)   
               
            
           
           
               
               
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
               
                 0-7 
                 W 2i     1     ,2i     1     +3,0   (2)   
                 W 2i     1     ,2i     1     +3,1   (2)   
                 W 2i     1     +1,2i     1     +3,0   (2)   
                 W 2i     1     +1,2i     1     +3,1   (2)   
               
               
                   
               
            
           
           
               
            
               
                 where 
               W     m   ,     m   ′     ,   n       (   2   )       =       1     8       [           v   m           v     m   ′                   φ   n     ⁢     v   m               -     φ   n       ⁢     v     m   ′               ]                       
 
               
               
                   
               
            
           
         
       
     
     If the Release 12 4Tx codebook for rank-1 and rank-2 is designed by augmenting the existing Release 8 codebook with GoB components, then the Release 12 4Tx codebook can be expressed as in Table 6-3 and Table 6-4. 
     Table 6-3 illustrates a codebook for 1-layer CSI reporting according to one embodiment. 
     
       
         
           
               
               
             
               
                 TABLE 6-3 
               
               
                   
               
             
            
               
                 i 1   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 0 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
            
               
                   
                 Use precoding matrix of index i 2  of Release 8 4Tx rank-1 codebook 
               
               
                 i 1   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 1-8 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
            
               
                   
                 Corresponding to i 1  = 0-7 in table 6-1 
               
               
                   
                   
               
            
           
         
       
     
     Table 6-4 illustrates a codebook for 2-layer CSI reporting according to one embodiment. 
     
       
         
           
               
               
             
               
                 TABLE 6-4 
               
               
                   
               
             
            
               
                 i 1   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 0 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
            
               
                   
                 Use precoding matrix of index i 2  of Release 8 4Tx rank-1 codebook 
               
               
                 i 1   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 1-8 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
            
               
                   
                 Corresponding to i 1  = 0-7 in table 6-2 
               
               
                   
                   
               
            
           
         
       
     
     Rank 3-4 
     If the Release 12 4Tx codebook for rank-3 and rank-3 is redesigned by the GoB framework with adjacent beams overlapping, and not includes the Release 8 codebook, the 4Tx codebooks can be expressed by equations in Table 6-5 and 6-6. 
     Table 6-5 illustrates a codebook for 3-layer CSI reporting according to one embodiment. 
     
       
         
           
               
               
             
               
                 TABLE 6-5 
               
               
                   
               
             
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 0 
                 1 
                 2 
                 3 
               
               
                   
               
               
                   
                 W 16i     1     ,16i     1     ,16i     1     +8,0   (3)   
                 W 16i     1     ,16i     1     ,16i     1     +8,1   (3)   
                 W 16i     1     +4,16i     1     +4,16i     1     +12,0   (3)   
                 W 16i     1     +4,16i     1     +4,16i     1     +12,1   (3)   
               
            
           
           
               
               
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
               
                   
                 W 16i     1     +8,16i     1     ,16i     1     +8,0   (3)   
                 W 16i     1     +8,16i     1     ,16i     1     +8,1   (3)   
                 W 16i+12     1     ,16i     1     +4,16i     1     +12,0   (3)   
                 W 16i     1     +12,16i     1     +4,16i     1     +12,1   (3)   
               
            
           
           
               
               
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 8 
                 9 
                 10 
                 11 
               
               
                   
               
               
                   
                 {tilde over (W)} 16i     1     ,16i     1     +8,16i     1     ,0   (3)   
                 {tilde over (W)} 16i     1     ,16i     1     +8,16i     1     ,1   (3)   
                 {tilde over (W)} 16i     1     +4,16i     1     +12,16i     1     +4,0   (3)   
                 {tilde over (W)} 16i     1     +4,16i     1     +12,16i     1     +4,1   (3)   
               
            
           
           
               
               
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
               
                   
                 {tilde over (W)} 16i     1     ,16i     1     +8,16i     1     +8,0   (3)   
                 {tilde over (W)} 16i     1     ,16i     1     +8,16i     1     +8,1   (3)   
                 {tilde over (W)} 16i     1     +4,16i     1     +12,16i     1     +12,0   (3)   
                 {tilde over (W)} 16i     1     +4,16i     1     +12,16i     1     +12,1   (3)   
               
               
                   
               
            
           
           
               
            
               
                 where 
                 W     m   ,     m   ′     ,     m   ″     ,   n       (   3   )       =       1       1   ⁢   2         [           v   m           v     m   ′             v     m   ″                   φ   n     ⁢     v   m               -     φ   n       ⁢     v     m   ′                 -     φ   n       ⁢     v     m   ″               ]       ,                     
 
               
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   W 
                                   ~ 
                                 
                                 
                                   m 
                                   , 
                                   
                                     
                                       m 
                                       ′ 
                                     
                                     ⁢ 
                                     
                                       m 
                                       ″ 
                                     
                                   
                                   , 
                                   n 
                                 
                                 
                                   ( 
                                   3 
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   1 
                                   
                                     
                                       1 
                                       ⁢ 
                                       2 
                                     
                                   
                                 
                                 [ 
                                 
                                   
                                     
                                       
                                         v 
                                         m 
                                       
                                     
                                     
                                       
                                         v 
                                         
                                           m 
                                           ′ 
                                         
                                       
                                     
                                     
                                       
                                         v 
                                         
                                           m 
                                           ″ 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           φ 
                                           n 
                                         
                                         ⁢ 
                                         
                                           v 
                                           m 
                                         
                                       
                                     
                                     
                                       
                                         
                                           φ 
                                           n 
                                         
                                         ⁢ 
                                         
                                           v 
                                           
                                             m 
                                             ′ 
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             φ 
                                             n 
                                           
                                         
                                         ⁢ 
                                         
                                           v 
                                           
                                             m 
                                             ″ 
                                           
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                   
               
            
           
         
       
     
     Table 6-6 illustrates a codebook for 4-layer CSI reporting according to one embodiment. 
     
       
         
           
               
               
             
               
                 TABLE 6-6 
               
               
                   
               
             
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 0 
                 1 
                 2 
                 3 
               
               
                   
               
               
                 0 
                 W 16i     1     ,16i     1     +8,16i     1     ,16i     1     +8,0   (4)   
                 W 16i     1     ,16i     1     +8,16i     1     ,16i     1     +8,1   (4)   
                 W 16i     1     +4,16i     1     +12,16i     1     +4,16i     1     +12,0   (4)   
                 W 16i     1     +4,16i     1     +12,16i     1     +4,16i     1     +12,1   (4)   
               
            
           
           
               
               
            
               
                   
                 i 2   
               
            
           
           
               
               
               
               
               
            
               
                 i 1   
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
               
                 0 
                 W 16i     1     ,16i     1     +8,16i     1     +4,16i     1     +12,0   (4)   
                 W 16i     1     ,16i     1     +8,16i     1     +4,16i     1     +12,1   (4)   
                 W 16i     1     +4,16i     1     +12,16i     1     ,16i     1     +8,0   (4)   
                 W 16i     1     +4,16i     1     +12,16i     1     ,16i     1     +8,0   (4)   
               
               
                   
               
            
           
           
               
            
               
                 where 
               W     m   ,     m   ′     ,     m   ″     ,     m   ′′′     ,   n       (   4   )       =       1       1   ⁢   6         [           v   m           v     m   ′             v     m   ″             v     m   ′′′                   φ   n     ⁢     v   m               φ   n     ⁢     v     m   ′                 -     φ   n       ⁢     v     m   ″                 -     φ   n       ⁢     v     m   ′′′               ]                       
 
               
               
                   
               
            
           
         
       
     
     If the Release 12 4Tx codebook for rank-3 and rank-3 is designed by augmenting the existing Release 8 codebook with GoB components, the Release 12 4Tx codebook can be expressed as in Table 6-7 and Table 6-8. 
     Table 6-7 illustrates a codebook for 3-layer CSI reporting according to one embodiment. 
     
       
         
           
               
               
             
               
                 TABLE 6-7 
               
               
                   
               
             
            
               
                 i 1   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 0 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
            
               
                   
                 Use precoding matrix of index i 2  of Release 8 4Tx rank-3 codebook 
               
               
                 i 1   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 1-8 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
            
               
                   
                 Corresponding to i 1  = 0-7 in Table 6-5. 
               
               
                   
                   
               
            
           
         
       
     
     Table 6-8 illustrates a codebook for 4-layer CSI reporting according to one embodiment. 
     
       
         
           
               
               
             
               
                 TABLE 6-8 
               
               
                   
               
             
            
               
                 i 1   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 0 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
            
               
                   
                 Use precoding matrix of index i 2  of Release 8 4Tx rank-4 codebook 
               
               
                 i 1   
                 i 2   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 1-8 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
            
               
                   
                 For i 2  = 0-7: corresponding to i 1  = 0-7 in Table 6-6. 
               
               
                   
                 For i 2  = 8-15: reserved. 
               
               
                   
                   
               
            
           
         
       
     
     Alternative Codebook Designs 
     For the W 1  codebook C 1  proposed above, the GoB component is expressed in the form of a block diagonal matrix as 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       [ 
                       
                         
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     where without overlapping: 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       
                         
                           〈 
                           k 
                         
                         ) 
                       
                     
                     ∈ 
                     
                       { 
                       
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       b 
                                       
                                         
                                           ( 
                                           
                                             
                                               N 
                                               b 
                                             
                                             ⁢ 
                                             k 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         m 
                                         ⁢ 
                                         o 
                                         ⁢ 
                                         d 
                                         ⁢ 
                                           
                                         N 
                                       
                                     
                                   
                                   
                                     
                                       b 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 N 
                                                 b 
                                               
                                               ⁢ 
                                               k 
                                             
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         m 
                                         ⁢ 
                                         o 
                                         ⁢ 
                                         d 
                                         ⁢ 
                                           
                                         N 
                                       
                                     
                                   
                                   
                                     … 
                                   
                                   
                                     
                                       b 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 N 
                                                 b 
                                               
                                               ⁢ 
                                               k 
                                             
                                             + 
                                             
                                               N 
                                               b 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         m 
                                         ⁢ 
                                         o 
                                         ⁢ 
                                         d 
                                         ⁢ 
                                           
                                         N 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             : 
                             k 
                           
                           = 
                           0 
                         
                         , 
                         1 
                         , 
                           
                         … 
                            
                         , 
                         
                           
                             N 
                             
                               N 
                               b 
                             
                           
                           - 
                           1 
                         
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     and with overlapping: 
     
       
         
           
             
               
                 
                   
                     X 
                     
                       
                         〈 
                         k 
                       
                       ) 
                     
                   
                   ∈ 
                   
                     
                       { 
                       
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       b 
                                       
                                         
                                           ( 
                                           
                                             
                                               N 
                                               b 
                                             
                                             ⁢ 
                                             k 
                                             ⁢ 
                                             l 
                                             ⁢ 
                                             2 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         m 
                                         ⁢ 
                                         o 
                                         ⁢ 
                                         d 
                                         ⁢ 
                                           
                                         N 
                                       
                                     
                                   
                                   
                                     
                                       b 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 N 
                                                 b 
                                               
                                               ⁢ 
                                               k 
                                               ⁢ 
                                               l 
                                               ⁢ 
                                               2 
                                             
                                             + 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         m 
                                         ⁢ 
                                         o 
                                         ⁢ 
                                         d 
                                         ⁢ 
                                           
                                         N 
                                       
                                     
                                   
                                   
                                     … 
                                   
                                   
                                     
                                       b 
                                       
                                         
                                           ( 
                                           
                                             
                                               
                                                 N 
                                                 b 
                                               
                                               ⁢ 
                                               k 
                                               ⁢ 
                                               l 
                                               ⁢ 
                                               2 
                                             
                                             + 
                                             
                                               N 
                                               b 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         m 
                                         ⁢ 
                                         o 
                                         ⁢ 
                                         d 
                                         ⁢ 
                                           
                                         N 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             : 
                             k 
                           
                           = 
                           0 
                         
                         , 
                         1 
                         , 
                         … 
                         , 
                         
                           
                             
                               2 
                               ⁢ 
                               N 
                             
                             
                               N 
                               b 
                             
                           
                           - 
                           1 
                         
                       
                       } 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     Each X (k)  represents a group of Nb adjacent beams that model a certain angle of arrival and angular spread. 
     Alternative designs are possible where the block diagonal sub-matrix of W 1  (i.e. x (k) ) is replaced by linear or non-linear transformations of x (k) , for example: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           W 
                           1 
                         
                         ( 
                         
                           n 
                           , 
                           m 
                           , 
                           k 
                         
                         ) 
                       
                       ∈ 
                       
                         C 
                         1 
                       
                     
                     = 
                     
                       { 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     f 
                                     n 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       X 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   
                                     g 
                                     m 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       X 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           k 
                           = 
                           0 
                         
                         , 
                         … 
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     where ƒ n ( ), g m ( ), n=0, . . . , m=0, . . . are linear/non-linear transformation functions. 
     In the following section several such possible designs are described assuming (N,Nb)=(16,4) as an example, but extension to other (N, Nb) values are straightforward. 
     It is also assumed that the Release 12 4Tx codebook is redesigned using the GoB structure and does not include the Release 8 4Tx codebook; however it is straightforward to augment the Release 8 4Tx codebook by the designs proposed below. 
     Example 1: Beam Shifting 
     In one embodiment, 
     
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         1 
                       
                       ( 
                       
                         n 
                         , 
                         m 
                         , 
                         k 
                       
                       ) 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     f 
                                     n 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       X 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   
                                     g 
                                     m 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       X 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           k 
                           = 
                           0 
                         
                         , 
                         … 
                       
                       } 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 
                                   D 
                                   ⁡ 
                                   ( 
                                   m 
                                   ) 
                                 
                                 ⁢ 
                                 
                                   X 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Alternatively, 
     
       
         
           
             
               
                 
                   
                     C 
                     1 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 m 
                                 ) 
                               
                               ⁢ 
                               
                                 X 
                                 
                                   
                                     〈 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               X 
                               
                                 
                                   〈 
                                   k 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     D(m) (m=0, 1 . . . N −1) is a Nt/2×Nt/2 diagonal matrix, in the case of Nt=4 denoted as 
     
       
         
           
             
               
                 
                   D 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     m 
                                   
                                   N 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     Herein D(m) performs beam shifting. For instance, as the first sub-matrix x (k)  comprises of Nb adjacent beams b (N     b     k)mod N  b (N     b     k+1)mod N  . . . b (N     b     k+N     b     −1)mod N , the second sub-matrix D(m)X (k)  comprises a different grid of Nb beams as 
         b   (N     b     k+m)mod N    b   (N     b     k+l+m)mod N    . . . b   (N     b     k+N     b     −1+m)mod N . 
     In other words, the second grid of beam is shifted by m beams, where m can take values from 0 to N−1. If m=0, . . . N−1, the W 1  codebook size is increased to 
     
       
         
           
             N 
             ⁢ 
             
               N 
               
                 N 
                 b 
               
             
           
         
       
     
     without W 1  overlapping, and 
     
       
         
           
             N 
             ⁢ 
             
               
                 2 
                 ⁢ 
                 N 
               
               
                 N 
                 b 
               
             
           
         
       
     
     with W 1  overlapping. Note that the codebook proposed in the initial sections above is a special case where m=0, in which case there is no beam shifting. 
       FIG.  4    illustrates an example in which a grid of beams is shifted by m=2 beams. A first grid of Nb beams  401  is selected from N beams  402 . The second grid of beams  403  is selected from the N beams  402 , but is shifted by two beams  404 ,  405  relative to the first grid of beams  401 . 
     It is not precluded that a subset of D(m) matrices are used in constructing codebook C 1 , where m∈Π, Π⊆{0, . . . N−1}. For instance, when Π={1}, the Nb beams in the second grid of beams (e.g., for the vertical polarization array  402 ) are all shifted by  1  beam, which is half of the number of overlapping beams (Nb/2=2) between two consecutive W 1  matrices. As another example, when Π={0,1}, the second grid of Nb beams may be unshifted, or shifted by one beam. 
     In another embodiment, both the first and second grid of beams may be shifted. This is expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           W 
                           1 
                         
                         ( 
                         
                           n 
                           , 
                           m 
                           , 
                           k 
                         
                         ) 
                       
                       ∈ 
                       
                         C 
                         1 
                       
                     
                     = 
                     
                       
                         { 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       f 
                                       n 
                                     
                                     ( 
                                     
                                       X 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       g 
                                       
                                         m 
                                            
                                       
                                     
                                     ( 
                                     
                                       X 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           , 
                           
                             k 
                             = 
                             0 
                           
                           , 
                           … 
                         
                            
                         } 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   G 
                                   ⁡ 
                                   ( 
                                   n 
                                   ) 
                                 
                                 ⁢ 
                                 
                                   X 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 
                                   D 
                                   ⁡ 
                                   ( 
                                   m 
                                   ) 
                                 
                                 ⁢ 
                                 
                                   X 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 where 
                 ⁢ 
                     
                 
                   G 
                   ⁡ 
                   ( 
                   n 
                   ) 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         e 
                         
                           j 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               n 
                             
                             N 
                           
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               ( 
               
                 
                   n 
                   = 
                   0 
                 
                 , 
                 
                   
                     1 
                     ⁢ 
                         
                     … 
                     ⁢ 
                         
                     N 
                   
                   - 
                   1 
                 
               
               ) 
             
             , 
             
               
                 D 
                 ⁡ 
                 ( 
                 m 
                 ) 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       
                         e 
                         
                           j 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               m 
                             
                             N 
                           
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               
                 ( 
                 
                   
                     m 
                     = 
                     0 
                   
                   , 
                   
                     
                       1 
                       ⁢ 
                           
                       … 
                       ⁢ 
                           
                       N 
                     
                     - 
                     1 
                   
                 
                 ) 
               
               . 
             
           
         
       
     
     Similarly, it is possible to use a subset of G(n) and D(m) matrices in generating the codebook C 1 . 
     Example 2: Beam Permutation 
     It is further possible to permute the selected beams in one or both sub-matrices of W 1 . As one example, the W 1  codebook is expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           W 
                           1 
                         
                         ( 
                         
                           l 
                           , 
                           m 
                           , 
                           k 
                         
                         ) 
                       
                       ∈ 
                       
                         C 
                         1 
                       
                     
                     = 
                     
                       { 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 
                                   D 
                                   ⁡ 
                                   ( 
                                   m 
                                   ) 
                                 
                                 ⁢ 
                                 
                                   X 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   P 
                                   ⁡ 
                                   ( 
                                   l 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     3 
                     ⁢ 
                     l 
                   
                   ) 
                 
               
             
           
         
       
     
     where P(l) is a 4×4 column permutation of an N b  ×N b  identity matrix I N     b   . 
     One example of P(l) is 
     
       
         
           
             
               P 
               ⁡ 
               ( 
               l 
               ) 
             
             = 
             
               
                 [ 
                 
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       0 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     By multiplying P(l) to D(m)X (k) , the Nb beams in the second sub-matrix D(m)X (k)  are permuted before being co-phased with the first grid of beam X (k) , providing further diversity gain in the W 1  codebook. 
     Permutation can be performed for both sub-matrices of W 1 . In another example, the W 1  codebook is expressed as 
     
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         1 
                       
                       ( 
                       
                         j 
                         , 
                         l 
                         , 
                         n 
                         , 
                         m 
                         , 
                         k 
                       
                       ) 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       f 
                                       n 
                                     
                                     ( 
                                     
                                       X 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       P 
                                       1 
                                     
                                     ( 
                                     j 
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   
                                     
                                       g 
                                       
                                         m 
                                           
                                       
                                     
                                     ( 
                                     
                                       X 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     
                                       P 
                                       2 
                                     
                                     ( 
                                     l 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           k 
                           = 
                           0 
                         
                         , 
                         … 
                       
                           
                       } 
                     
                     = 
                     
                       [ 
                       ⁠ 
                       
                         
                           
                             
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 n 
                                 ) 
                               
                               ⁢ 
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   P 
                                   1 
                                 
                                 ( 
                                 j 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 m 
                                 ) 
                               
                               ⁢ 
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   P 
                                   2 
                                 
                                 ( 
                                 l 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     where P 1 (j) and P 2 (l) perform column permutations for the first and the second grid of beams, respectively. 
     Note that the permutation operation P(l) can be applied without beam shifting (e.g. D(m)). 
     Example 3: Phase Rotation 
     In another embodiment, 
     
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         1 
                       
                       ( 
                       
                         n 
                         , 
                         m 
                         , 
                         k 
                       
                       ) 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     f 
                                     n 
                                   
                                   ( 
                                   
                                     X 
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   
                                     g 
                                     m 
                                   
                                   ( 
                                   
                                     X 
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           k 
                           = 
                           0 
                         
                         , 
                         … 
                       
                          
                       } 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 m 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     Alternatively, 
     
       
         
           
             
               
                 
                   
                     C 
                     1 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               X 
                               
                                 
                                   〈 
                                   k 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               D 
                               ⁡ 
                               ( 
                               m 
                               ) 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             X 
                             
                               
                                 〈 
                                 k 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     D(m) (m=0, 1 . . . N−1) is a Nb×Nb diagonal matrix denoted as 
     
       
         
           
             
               
                 
                   D 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     m 
                                   
                                   N 
                                 
                               
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             … 
                           
                           
                               
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       π 
                                       ⁡ 
                                       ( 
                                       
                                         
                                           N 
                                           
                                             b 
                                               
                                           
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                   
                                   N 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     Herein D(m) performs phase correction to the Nb beams. For instance, as the first sub-matrix X (k)  comprises Nb adjacent beams defined as: 
         b   (N     b     k)mod N    b   (N     b     k+1)mod N    . . . b   (N     b     k+N     b     −1)mod N , 
     the second sub-matrix D(m)X (k)  comprises a different grid of Nb beams defined as 
     
       
         
           
             
               
                 
                   
                     b 
                     
                       
                         ( 
                         
                           
                             
                               N 
                               b 
                             
                             ⁢ 
                             k 
                           
                           + 
                           m 
                         
                         ) 
                       
                       ⁢ 
                       mod 
                       ⁢ 
                       N 
                     
                   
                 
                 
                   
                     
                       e 
                       
                         j 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             m 
                           
                           N 
                         
                       
                     
                     ⁢ 
                     
                       b 
                       
                         
                           ( 
                           
                             
                               
                                 N 
                                 b 
                               
                               ⁢ 
                               k 
                             
                             + 
                             m 
                           
                           ) 
                         
                         ⁢ 
                         mod 
                         ⁢ 
                         N 
                       
                     
                   
                 
                 
                   ⋯ 
                 
                 
                   
                     
                       e 
                       
                         j 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             
                               π 
                               ⁡ 
                               ( 
                               
                                 
                                   N 
                                   b 
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                             ⁢ 
                             m 
                           
                           N 
                         
                       
                     
                     ⁢ 
                     
                       b 
                       
                         
                           ( 
                           
                             
                               
                                 N 
                                 b 
                               
                               ⁢ 
                               k 
                             
                             + 
                             
                               N 
                               b 
                             
                             - 
                             1 
                           
                           ) 
                         
                         ⁢ 
                         mod 
                         ⁢ 
                         N 
                       
                     
                   
                 
               
             
             . 
           
         
       
     
     In other words, the m-th beam in the second grid (m=0, 1, . . . , Nb- 1 ) is phase rotated by 
     
       
         
           
             
               2 
               ⁢ 
               
                 π 
                 ⁡ 
                 ( 
                 
                   
                     N 
                     b 
                   
                   - 
                   1 
                 
                 ) 
               
               ⁢ 
               m 
             
             N 
           
         
       
     
     degrees. 
     It is also possible to apply the phase rotation to both the first and the second grid of beams. 
     An alternative formulation of the phase rotation matrix D(m) is given as 
     
       
         
           
             
               
                 
                   D 
                   = 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             e 
                             
                               j 
                               ⁢ 
                               
                                 
                                   π 
                                     
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                     N 
                                     b 
                                   
                                 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           … 
                         
                         
                             
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             e 
                             
                               j 
                               ⁢ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         N 
                                         
                                           b 
                                             
                                         
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   π 
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                     N 
                                     b 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     where Nb phase correction components 
     
       
         
           
             ( 
             
               
                 e 
                 . 
                 g 
                 . 
               
               , 
               1 
               , 
               
                 e 
                 
                   j 
                   ⁢ 
                   
                     π 
                     
                       2 
                       ⁢ 
                       
                         N 
                         b 
                       
                     
                   
                 
               
               , 
               
                 … 
                 ⁢ 
                     
                 
                   e 
                   
                     j 
                     ⁢ 
                     
                       
                         
                           ( 
                           
                             
                               N 
                               b 
                             
                             - 
                             1 
                           
                           ) 
                         
                         ⁢ 
                         π 
                       
                       
                         2 
                         ⁢ 
                         
                           N 
                           b 
                         
                       
                     
                   
                 
               
             
             ) 
           
         
       
     
     uniformly sample a 90-degree sector. As such, co-phasing between two grids of beams is no longer limited to the QPSK alphabet as in the W2 codebook, but can take values of 
     
       
         
           
             
               { 
               
                 0 
                 , 
                 
                   π 
                   
                     2 
                     ⁢ 
                     
                       N 
                       b 
                     
                   
                 
                 , 
                 
                   … 
                   ⁢ 
                       
                   
                     
                       
                         ( 
                         
                           
                             N 
                             b 
                           
                           - 
                           1 
                         
                         ) 
                       
                       ⁢ 
                       π 
                     
                     
                       2 
                       ⁢ 
                       
                         N 
                         b 
                       
                     
                   
                 
               
               } 
             
             + 
             
               
                 { 
                 
                   0 
                   , 
                   
                     π 
                     / 
                     2 
                   
                   , 
                   π 
                   , 
                   , 
                   
                     3 
                     ⁢ 
                     π 
                     / 
                     2 
                   
                   , 
                 
                 } 
               
               . 
             
           
         
       
     
     It will be further understood that combination of beam shifting, beam permutation, and/or phase rotation schemes can be used in constructing the W 1  codebook. 
     Pruning the 8Tx Codebook for 4Tx MIMO 
     The LTE Release 10 8Tx codebook is designed using the GoB structure. Specifically:
         each 4Tx polarization array is over-sampled by N DFT beams&#39;   each wideband W 1  matrix comprises Nb adjacent DFT beams to cover a certain AoD and angular spread, and   narrowband W 2  performs beam selection and co-phasing.       

     As such, the 4Tx GoB codebook components can be selected as sub-matrices of a subset of the 8Tx codebook in Release 12. In other words, each 4Tx GoB precoder may correspond to four selected rows of a Release 10 8Tx precoder (i.e., pruning the 8Tx codebook down to four rows). 
     To illustrate this pruning, the Release 10 8Tx codebook is denoted as 
         C   (8)   =C   1   (8)   ×C   2   (8) , 
     where C (8) ={W (8) }, C 1   (8) ={W 1   (8) } and C 2   (8) ={W 2   (8) } are the first and second codebook. 
     Subsequently, the 4Tx GoB codebook can be written as 
         C   (4)   ={W   (4)   }=C   1   (4)   ×C   2   (4) ,  (37)
 
     where C (4) ={W (4) }⊆{W (8)   [(n1,n2,ne,n4),:] }, and 
     where, for an 8×R matrix H, H (n     i     ,:)  denotes the ni-th row of H, 
     
       
         
           
             
               
                 H 
                 
                   [ 
                   
                     
                       ( 
                       
                         
                           n 
                           ⁢ 
                           1 
                         
                         , 
                         
                           n 
                           ⁢ 
                           2 
                         
                         , 
                         
                           n 
                           ⁢ 
                           3 
                         
                         , 
                         
                           n 
                           ⁢ 
                           4 
                         
                       
                       ) 
                     
                     , 
                     : 
                   
                   ] 
                 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         H 
                         
                           ( 
                           
                             
                               n 
                               ⁢ 
                               1 
                             
                             , 
                             : 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         H 
                         
                           ( 
                           
                             
                               n 
                               ⁢ 
                               2 
                             
                             , 
                             : 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         H 
                         
                           ( 
                           
                             
                               n 
                               ⁢ 
                               3 
                             
                             , 
                             : 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         H 
                         
                           ( 
                           
                             
                               n 
                               ⁢ 
                               4 
                             
                             , 
                             : 
                           
                           ) 
                         
                       
                     
                   
                 
                 ) 
               
             
             , 
           
         
       
     
     and ( n1,n2,n3,n4 ) is the row selection vector for codebook pruning. 
     The 4Tx GoB codebooks proposed above may be pruned from the Release 10 8Tx codebook. The following notations are used herein: 
     i 1   4 : index of first precoder W 1  for 4Tx 
     i 2   4  index of second precoder W 2  for 4Tx 
     i 1   8  index of first precoder W 1  for 8Tx 
     i 2   8  index of second precoder W 1  for 8Tx 
     As such, each 8Tx precoding matrix W (8) =W 1   (8) W 2   (8)  is denoted by a pair of 8Tx codebook indices (i 1   8 , i 2   8 ), and each 4Tx precoding matrix W (4) =W 1   (4) W 2   (4)  is denoted by a pair of 4Tx codebook indices (i 1   4 , i 2   4 ), 
     Rank-1 
     For a (N,Nb)=(N,Nb) 4Tx GoB codebook with overlapping beams, where N&lt;=32, Nb=4, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-1 precoding matrix denoted by (i 1   8 , i 2   8 ), where 
     
       
         
           
             
               
                 i 
                 1 
                 4 
               
               = 
               
                 
                   
                     3 
                     ⁢ 
                     2 
                   
                   N 
                 
                 ⁢ 
                 
                   i 
                   1 
                   8 
                 
               
             
             , 
           
         
       
       
         
           
             
               
                 i 
                 2 
                 4 
               
               = 
               
                 i 
                 2 
                 8 
               
             
             , 
             
               
                 i 
                 2 
                 4 
               
               = 
               0 
             
             , 
             … 
                 
             , 
             15 
           
         
       
     
     More specifically: 
     For (N,Nb)=(8,4) 4Tx codebook with overlapping, each 4Tx precoding matrix denoted by a pair of 4Tx codebook indices (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-1 precoding matrix denoted by a pair of 8Tx codebook indices (i 1   8 , i 2   8 ), as given in Table 7. 
     
       
         
           
               
               
               
             
               
                 TABLE 7 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-3 
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 4i 1   4   
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-3 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 4i 1   4   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(16,4) 4Tx codebook with overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-1 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 8. 
     
       
         
           
               
               
               
             
               
                 TABLE 8 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-7 
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 2i 1   4   
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 2i 1   4   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
            
           
         
       
     
     For (N,Nb)=(32,4) 4Tx codebook with overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-1 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 9. 
     
       
         
           
               
               
               
             
               
                 TABLE 9 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-15 
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 i 1   4   
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-15 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 i 1   4   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(N,Nb) 4Tx codebook without overlapping beams, where N&lt;=32, and Nb=4, each 4Tx precoding matrix denoted by a pair of 4Tx codebook indices (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-1 precoding matrix denoted by a pair of 8Tx codebook indices (i 1   8 , i 2   8 ), where 
     
       
         
           
             
               
                 i 
                 1 
                 4 
               
               = 
               
                 
                   
                     6 
                     ⁢ 
                     4 
                   
                   N 
                 
                 ⁢ 
                 
                   i 
                   1 
                   8 
                 
               
             
             , 
           
         
       
       
         
           
             
               
                 i 
                 2 
                 4 
               
               = 
               
                 i 
                 2 
                 8 
               
             
             , 
             
               
                 i 
                 2 
                 4 
               
               = 
               0 
             
             , 
             … 
                 
             , 
             15. 
           
         
       
     
     More specifically: 
     For a (N,Nb)=(8,4) 4Tx codebook without overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-1 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 10. 
     
       
         
           
               
               
               
             
               
                 TABLE 10 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-1 
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 8i 1   4   
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-1 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 8i 1   4   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(16,4) 4Tx codebook without overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-1 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 11. 
     
       
         
           
               
               
               
             
               
                 TABLE 11 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-3 
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                 7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 4i 1   4   
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                 7 
               
            
           
           
               
               
               
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-3 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15  
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 4i 1   4   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15.  
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(32,4) 4Tx codebook without overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-1 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 12. 
     
       
         
           
               
               
               
             
               
                 TABLE 12 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-7 
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 2i 1   4   
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 2i 1   4   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
            
           
         
       
     
     Rank-2 
     Exemplary Codebook 1 
     In this section, we assume the following 4Tx rank-2 codebook, where the W 2  codebook is size-8. 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     where (k=0, . . . N/N b −1) without overlapping beams, and (k=0, . . . 2N/N b −1) with overlapping beams. 
     
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         2 
                       
                       ∈ 
                       
                         C 
                         ⁢ 
                         
                           B 
                           2 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             1 
                             
                               2 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     Y 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           
                             1 
                             
                               2 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jY 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jY 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       } 
                     
                   
                   , 
                     
                   
                     
                       where 
                       ⁢ 
                       
                             
                       
                       
                         ( 
                         
                           
                             Y 
                             1 
                           
                           , 
                           
                             Y 
                             2 
                           
                         
                         ) 
                       
                     
                     ∈ 
                     
                       
                         { 
                         
                           
                             ( 
                             
                               
                                 e 
                                 1 
                                 4 
                               
                               , 
                               
                                 e 
                                 1 
                                 4 
                               
                             
                             ) 
                           
                           , 
                           
                             ( 
                             
                               
                                 e 
                                 2 
                                 4 
                               
                               , 
                               
                                 e 
                                 2 
                                 4 
                               
                             
                             ) 
                           
                           , 
                           
                             ( 
                             
                               
                                 e 
                                 3 
                                 4 
                               
                                 
                               , 
                               
                                 e 
                                 3 
                                 4 
                               
                             
                             ) 
                           
                           , 
                           
                             ( 
                             
                               
                                 e 
                                 4 
                                 4 
                               
                                 
                               , 
                               
                                 e 
                                 4 
                                 4 
                               
                             
                             ) 
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     For a (N,Nb)=(N,Nb) 4Tx GoB codebook with overlapping beams, where N&lt;=32, Nb=4, each 4Tx precoding matrix denoted by a pair of 4Tx codebook indices (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by a pair of 8Tx codebook indices (i 1   8 , i 2   8 ), where 
     
       
         
           
             
               
                 i 
                 1 
                 4 
               
               = 
               
                 
                   
                     3 
                     ⁢ 
                     2 
                   
                   N 
                 
                 ⁢ 
                 
                   i 
                   1 
                   8 
                 
               
             
             , 
           
         
       
       
         
           
             
               
                 i 
                 2 
                 4 
               
               = 
               
                 i 
                 2 
                 8 
               
             
             , 
             
               
                 i 
                 2 
                 4 
               
               = 
               0 
             
             , 
             … 
                 
             , 
             7 
           
         
       
     
     More specifically: 
     For a (N,Nb)=(8,4) 4Tx codebook with overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 13. 
     
       
         
           
               
               
               
             
               
                 TABLE 13 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-3 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 4i 1   4   
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(16,4) 4Tx codebook with overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 14. 
     
       
         
           
               
               
               
             
               
                 TABLE 14 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-7 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 2i 1   4   
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(32,4) 4Tx codebook with overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 15. 
     
       
         
           
               
               
               
             
               
                 TABLE 15 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-15 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 i 1   4   
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(N,Nb) 4Tx codebook without overlapping beams, where N&lt;=32, Nb=4, each 4Tx precoding matrix denoted by a pair of 4Tx codebook indices (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by a pair of 8Tx codebook indices (i 1   8 , i 2   8 ), where 
     
       
         
           
             
               
                 i 
                 1 
                 4 
               
               = 
               
                 
                   
                     6 
                     ⁢ 
                     4 
                   
                   N 
                 
                 ⁢ 
                 
                   i 
                   1 
                   8 
                 
               
             
             , 
           
         
       
       
         
           
             
               
                 i 
                 2 
                 4 
               
               = 
               
                 i 
                 2 
                 8 
               
             
             , 
             
               
                 i 
                 2 
                 4 
               
               = 
               0 
             
             , 
             … 
                 
             , 
             7 
           
         
       
     
     More specifically: 
     For a (N,Nb)=(8,4) 4Tx codebook without overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 16. 
     
       
         
           
               
               
               
             
               
                 TABLE 16 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-1 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 8i 1   4   
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(16,4) 4Tx codebook without overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 17. 
     
       
         
           
               
               
               
             
               
                 TABLE 17 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-3 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 4i 1   4   
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(32,4) 4Tx codebook without overlapping beams, each 4Tx precoding matrix denoted by (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by (i 1   8 , i 2   8 ), as given in Table 18. 
     
       
         
           
               
               
               
             
               
                 TABLE 18 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-7 
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 2i 1   4   
                 0 
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
               
               
                   
               
            
           
         
       
     
     Exemplary Codebook 2 
     In this section, we assume the following 4Tx GoB codebook where the W 2  codebook is size-16. 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     where (k=0, . . . N/N b −1) without overlapping beams, and (k=0, . . . 2N/N b −1) with overlapping beams. 
     
       
         
           
             
               
                 
                                    
                   
                     
                       
                         
                           W 
                           2 
                         
                         ∈ 
                         
                           C 
                           ⁢ 
                           
                             B 
                             2 
                           
                         
                       
                       = 
                       
                         { 
                         
                           
                             
                               1 
                               
                                 2 
                               
                             
                             [ 
                             
                               
                                 
                                   
                                     Y 
                                     1 
                                   
                                 
                                 
                                   
                                     Y 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     Y 
                                     1 
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           , 
                           
                             
                               1 
                               
                                 2 
                               
                             
                             [ 
                             
                               
                                 
                                   
                                     Y 
                                     1 
                                   
                                 
                                 
                                   
                                     Y 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     j 
                                     ⁢ 
                                     
                                       Y 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       jY 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         } 
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               ( 
               
                 
                   Y 
                   1 
                 
                 , 
                 
                   Y 
                   2 
                 
               
               ) 
             
             ∈ 
             
               
                 { 
                 
                   
                     ( 
                     
                       
                         e 
                         1 
                         4 
                       
                       , 
                       
                         e 
                         1 
                         4 
                       
                     
                     ) 
                   
                   , 
                   
                     ( 
                     
                       
                         e 
                         2 
                         4 
                       
                       , 
                       
                         e 
                         2 
                         4 
                       
                     
                     ) 
                   
                   , 
                   
                     ( 
                     
                       
                         e 
                         3 
                         4 
                       
                       , 
                       
                         e 
                         3 
                         4 
                       
                     
                     ) 
                   
                   , 
                   
                     ( 
                     
                       
                         e 
                         4 
                         4 
                       
                         
                       , 
                       
                         e 
                         4 
                         4 
                       
                     
                     ) 
                   
                   , 
                   
                     ( 
                     
                       
                         e 
                         1 
                         4 
                       
                         
                       , 
                       
                         e 
                         2 
                         4 
                       
                     
                     ) 
                   
                   , 
                   
                     ( 
                     
                       
                         e 
                         2 
                         4 
                       
                         
                       , 
                       
                         e 
                         3 
                         4 
                       
                     
                     ) 
                   
                   , 
                   
                     ( 
                     
                       
                         e 
                         1 
                         4 
                       
                         
                       , 
                       
                         e 
                         4 
                         4 
                       
                     
                     ) 
                   
                   , 
                   
                     ( 
                     
                       
                         e 
                         2 
                         4 
                       
                         
                       , 
                       
                         e 
                         4 
                         4 
                       
                     
                     ) 
                   
                 
                 } 
               
               . 
             
           
         
       
     
     Such a 4Tx codebook cannot be completely pruned out of the 8Tx codebook unless the oversampling rate is N=32, which is equivalent to the 8Tx oversampling rate. In this case, when there is W 1  overlapping for 4Tx, each 4Tx precoding matrix denoted by a pair of 4Tx codebook indices (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by a pair of 8Tx codebook indices (i 1   8 , i 2   8 ), as given in Table 19, where 
     i 1   4 =i 1   8 , i 1   4 =0, . . . 15, 
     i 2   4 =i 2   8 , i 2   3 =0, . . . , 15 
     
       
         
           
               
               
               
             
               
                 TABLE 19 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-15 
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 i 1   4   
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-15 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 i 1   4   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
            
           
         
       
     
     For a (N,Nb)=(32,4) 4Tx GoB codebook without overlapping beams, each 4Tx precoding matrix denoted by a pair of 4Tx codebook indices (i 1   4 , i 2   4 ) can be pruned from a corresponding 8Tx rank-2 precoding matrix denoted by a pair of 8Tx codebook indices (i 1   8 , i 2   8 ), as given in Table 20, where 
     i 1   4 =2i 1   8 , i 1   4 =0, . . . , 7, 
     i 2   4 =i 2   8 , i 2   4 =0, . . . , 15 
     
       
         
           
               
               
               
             
               
                 TABLE 20 
               
               
                   
               
             
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-7 
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 2i 1   4   
                 0 
                 1 
                  2 
                  3 
                  4 
                  5 
                  6 
                  7 
               
            
           
           
               
               
               
            
               
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                   
                 0-7 
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
            
           
           
               
               
               
            
               
                 Corresponding 
                 i 1   8   
                 i 2   8   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 8Tx indices 
                 2i 1   4   
                 8 
                 9 
                 10 
                 11 
                 12 
                 13 
                 14 
                 15 
               
               
                   
               
            
           
         
       
     
     Rank-3 
     A rank-3 4Tx GoB codebook cannot be pruned from the rank-3 8Tx codebook. This can be seen from the 8Tx design, which followed the (N,Nb)=(16,8) structure, where 
     
       
         
           
             
               B 
               = 
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               b 
                               0 
                             
                           
                           
                             
                               b 
                               1 
                             
                           
                           
                             … 
                           
                           
                             
                               b 
                               15 
                             
                           
                         
                       
                       ] 
                     
                         
                     [ 
                     B 
                     ] 
                   
                   
                     
                       1 
                       + 
                       m 
                     
                     , 
                     
                       1 
                       + 
                       n 
                     
                   
                 
                 = 
                 
                   e 
                   
                     j 
                     ⁢ 
                     
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         mn 
                       
                       16 
                     
                   
                 
               
             
             , 
             
               m 
               = 
               0 
             
             , 
             1 
             , 
             2 
             , 
             3 
             , 
             
               n 
               = 
               0 
             
             , 
             1 
             , 
             … 
                 
             , 
             15 
           
         
       
       
         
           
             
               X 
               
                 ( 
                 k 
                 ) 
               
             
             ∈ 
             
               { 
               
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 b 
                                 
                                   4 
                                   ⁢ 
                                   kmod 
                                   ⁢ 
                                   32 
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         4 
                                         ⁢ 
                                         k 
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   mod 
                                   ⁢ 
                                   32 
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         4 
                                         ⁢ 
                                         k 
                                       
                                       + 
                                       2 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   mod 
                                   ⁢ 
                                   32 
                                 
                               
                             
                             
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         
                                           4 
                                           ⁢ 
                                           k 
                                         
                                         + 
                                         3 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     mod 
                                     ⁢ 
                                     32 
                                   
                                 
                                 ⁢ 
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         
                                           4 
                                           ⁢ 
                                           k 
                                         
                                         + 
                                         4 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     mod 
                                     ⁢ 
                                     32 
                                   
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         4 
                                         ⁢ 
                                         k 
                                       
                                       + 
                                       5 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   mod 
                                   ⁢ 
                                   32 
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         4 
                                         ⁢ 
                                         k 
                                       
                                       + 
                                       6 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   mod 
                                   ⁢ 
                                   32 
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         4 
                                         ⁢ 
                                         k 
                                       
                                       + 
                                       7 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   mod 
                                   ⁢ 
                                   32 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                 
                 
                   
                     
                       
                         k 
                         = 
                         0 
                       
                       , 
                       
                         … 
                         ⁢ 
                             
                         3 
                       
                     
                   
                 
               
               } 
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         W 
                         1 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               
                                   
                                 0 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       Codebook 
                       ⁢ 
                       1 
                       : 
                       
                         C 
                         1 
                       
                       ⁢ 
                       
                         { 
                         
                           
                             W 
                             1 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           , 
                           
                             W 
                             1 
                             
                               ( 
                               1 
                               ) 
                             
                           
                           , 
                           
                             W 
                             1 
                             
                               ( 
                               2 
                               ) 
                             
                           
                           , 
                           
                             W 
                             1 
                             
                               ( 
                               3 
                               ) 
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         
                           W 
                           2 
                         
                         ∈ 
                         
                           CB 
                           2 
                         
                       
                       = 
                       
                         { 
                         
                           
                             1 
                             
                               2 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     Y 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         } 
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       ( 
                       
                         
                           Y 
                           1 
                         
                         , 
                         
                           Y 
                           2 
                         
                       
                       ) 
                     
                     ∈ 
                     
                       
                         { 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       e 
                                       1 
                                       4 
                                     
                                     , 
                                     
                                       [ 
                                       
                                         
                                           e 
                                           1 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           5 
                                           4 
                                         
                                       
                                       ] 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       2 
                                       4 
                                     
                                     , 
                                     
                                       [ 
                                       
                                         
                                           e 
                                           2 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           6 
                                           4 
                                         
                                       
                                       ] 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       3 
                                       4 
                                     
                                     , 
                                     
                                       [ 
                                       
                                         
                                           e 
                                           3 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           7 
                                           4 
                                         
                                       
                                       ] 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       4 
                                       4 
                                     
                                     , 
                                     
                                       [ 
                                       
                                         
                                           e 
                                           4 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           8 
                                           4 
                                         
                                       
                                       ] 
                                     
                                   
                                   ) 
                                 
                                 , 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       e 
                                       5 
                                       4 
                                     
                                     , 
                                     
                                       [ 
                                       
                                         
                                           e 
                                           1 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           5 
                                           4 
                                         
                                       
                                       ] 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       6 
                                       4 
                                     
                                     , 
                                     
                                       [ 
                                       
                                         
                                           e 
                                           2 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           6 
                                           4 
                                         
                                       
                                       ] 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       7 
                                       4 
                                     
                                     , 
                                     
                                       [ 
                                       
                                         
                                           e 
                                           3 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           7 
                                           4 
                                         
                                       
                                       ] 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       e 
                                       8 
                                       4 
                                     
                                     , 
                                     
                                       [ 
                                       
                                         
                                           e 
                                           4 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           8 
                                           4 
                                         
                                       
                                       ] 
                                     
                                   
                                   ) 
                                 
                                 , 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       [ 
                                       
                                         
                                           e 
                                           1 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           5 
                                           4 
                                         
                                       
                                       ] 
                                     
                                     , 
                                     
                                       e 
                                       1 
                                       4 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       [ 
                                       
                                         
                                           e 
                                           2 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           6 
                                           4 
                                         
                                       
                                       ] 
                                     
                                     , 
                                     
                                       e 
                                       2 
                                       4 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       [ 
                                       
                                         
                                           e 
                                           3 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           7 
                                           4 
                                         
                                       
                                       ] 
                                     
                                     , 
                                     
                                       e 
                                       3 
                                       4 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       [ 
                                       
                                         
                                           e 
                                           4 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           8 
                                           4 
                                         
                                       
                                       ] 
                                     
                                     , 
                                     
                                       e 
                                       4 
                                       4 
                                     
                                   
                                   ) 
                                 
                                 , 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       [ 
                                       
                                         
                                           e 
                                           1 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           5 
                                           4 
                                         
                                       
                                       ] 
                                     
                                     , 
                                     
                                       e 
                                       5 
                                       4 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       [ 
                                       
                                         
                                           e 
                                           2 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           6 
                                           4 
                                         
                                       
                                       ] 
                                     
                                     , 
                                     
                                       e 
                                       6 
                                       4 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       [ 
                                       
                                         
                                           e 
                                           3 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           7 
                                           4 
                                         
                                       
                                       ] 
                                     
                                     , 
                                     
                                       e 
                                       7 
                                       4 
                                     
                                   
                                   ) 
                                 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       [ 
                                       
                                         
                                           e 
                                           4 
                                           4 
                                         
                                         , 
                                         
                                           e 
                                           8 
                                           4 
                                         
                                       
                                       ] 
                                     
                                     , 
                                     
                                       e 
                                       8 
                                       4 
                                     
                                   
                                   ) 
                                 
                                 , 
                               
                             
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     The column vectors of each rank-3 8Tx precoding matrix comprise critically sampled (e.g. sampling rate 4) 4Tx DFT vectors in order to ensure that a rank-3 8Tx precoding matrix satisfies the unitary constraint. For 4Tx, in order to achieve the unitary constraint, column vectors of each rank-3 4Tx precoding matrix must comprise critically sampled (e.g. sampling rate 2) 2Tx DFT vectors. As such, the column vectors of any 4Tx GoB matrix will always span across different W 1  matrices of the 8Tx codebook, leaving it impossible to construct the rank-3 4-Tx codebook by pruning the 8Tx codebook. 
     However the rank-3 4Tx GoB component can be pruned from the rank-6 8Tx codebook, where 
         C   (4)   ={W   (4) } rank3   ⊆{W   (8)   [(n1,n2,n3,n4),(m1,m2,m3)] } rank6   (45)
 
     e.g., the ( n1,n2,n3,n4 )-th row and ( m1, m2, m3 )-th column of a rank-6 8Tx precoding matrix corresponding to a pair of codebook index (i 1   8 , i 2   8 ) are used to construct a 4Tx precoding matrix corresponding to a pair of codebook indices (i 1   4 , i 2   4 ). 
     For instance, for the (N,Nb)=(4,4) GoB codebook, the pruning shown in Table 21 is possible: 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 21 
               
               
                   
                   
               
             
            
               
                   
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
               
                   
                   
                 0 
                 arbitrary 
               
               
                   
                 Corresponding 8Tx 
                 i 1   8   
                 i 2   8   
               
               
                   
                 indices 
                 0 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     Note that the column-selection method ( m1, m2, m3 ) may depend on the 4Tx GoB precoding matrix. For instance: 
       ( Y   1   ,Y   2 )∈{( e   1   4 [ e   1   4   , e   3   4 ])}⇒( m 1,  m 2,  m 3)=(1,2,6)  (46)
 
       ( Y   1   ,Y   2 )∈{( e   3   4 [ e   1   4   , e   3   4 ])}⇒( m 1,  m 2,  m 3)=(5,2,6)  (47)
 
       ( Y   1   ,Y   2 )∈{([ e   1   4   ,e   3   4 ],  e   1   4 )}⇒( m 1,  m 2,  m 3)=(1,5,6)  (48)
 
       ( Y   1   ,Y   2 )∈{([ e   1   4   ,e   3   4 ], e   3   4 )}⇒( m 1,  m 2,  m 3)=(1,5,6)  (49)
 
     Rank-4 
     Similarly, the rank-4 4Tx GoB component cannot be pruned from the rank-4 8Tx codebook. However, a rank-4 4Tx GoB component can be pruned from the rank-8 8Tx codebook, where 
         C   (4)   ={W   (4) } rank-4   ⊆{W   (8)   [(n1,n2,n3,n4),(m1,m2,m3,m4)] } rank-8   (50)
 
     e.g., the ( n1,n2,n3,n4 )-th row and ( m1, m2, m3, m4) -th column of a rank-8 8Tx precoding matrix corresponding to a pair of codebook index (i 1   8 , i 2   8 ) is used to construct a 4Tx precoding matrix corresponding to a pair of codebook indices (i 1   4 , i 2   4 ). For instance for the (N,Nb)=(4,4) GoB codebook, the pruning shown in Table 22 is possible. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 22 
               
               
                   
                   
               
             
            
               
                   
                 4Tx indices 
                 i 1   4   
                 i 2   4   
               
               
                   
                   
                 0 
                 arbitrary 
               
               
                   
                 Corresponding 8Tx 
                 i 1   8   
                 i 2   8   
               
               
                   
                 indices 
                 0 
                 0 
               
               
                   
                   
               
            
           
         
       
     
     Note that the column-selection method  (m1, m2, m3, m4)  may depend on the 4Tx GoB precoding matrix. For instance: 
       ( Y   1   , Y   2 )∈{([ e   1   4   ,e   3   4 ],[ e   1   4   ,e   3   4 ])}⇒( m 1, m 2, m 3, m 4)=(1,5,2,6)  (51)
 
       ( Y   1   , Y   2 )∈{([ e   2   4   ,e   4   4 ],[ e   2   4   ,e   4   4 ])}⇒( m 1, m 2, m 3, m 4)=(3,7,4,8)  (52)
 
       ( Y   1   , Y   2 )∈{([ e   1   4   ,e   3   4 ],[ e   2   4   ,e   4   4 ])}⇒( m 1, m 2, m 3, m 4)=(1,5,4,8)  (53)
 
       ( Y   1   , Y   2 )∈{([ e   2   4   ,e   4   4 ],[ e   1   4   ,e   3   4 ],)}⇒( m 1, m 2, m 3, m 4)=(3,7,2,6)  (54)
 
     Row Selection Vector (n1,n2,n3,n4) for Pruning 
     In one embodiment, the row selection vector ( n1,n2,n3,n4 ) is hard-coded/fixed in the specification, and is not signalled to the UE.  FIG.  5    illustrates an example of using a row selection vector to prune antenna pairs in an 8Tx array. Eight antennas 1-8 are arranged in four cross-polarized pairs  501 - 504 . Using the row selection vector ( n1,n2,n3,n4 ) =[1,2,5,6]  prunes the first two cross-polarized antenna pairs  501 ,  502  from the 8Tx array  505  for a 4Tx deployment  506 . 
     In another embodiment, the row selection vector ( n1,n2,n3,n4 ) may be semi-statically radio resource control (RRC)-configured for a UE, and can be different for different UEs. There are multiple possible methods for RRC signalling ( n1,n2,n3,n4 ) 
     Example 1 
     For the k-th antenna port of a 4-Tx MIMO system (k=1, 2, 3, 4), the eNodeB signals the corresponding virtual antenna port index n k  in an 8-Tx MIMO system. This requires a signaling overhead of log 2(8)×4=12 bits. 
     Example 2 
     As there are a total of C 4   8 =70 ways of choosing 4-antenna ports out of an 8-antenna port system, log 2(70)=7-bits are used to signal the combinatorial index of the antenna selection vector ( n1,n2,n3,n4 ). This achieves 5-bits of overhead savings. 
     It is further possible to down-size the set of candidate antenna selection vectors { (n1,n2,n3,n4) }, which may further reduce the RRC signalling overhead. For instance, the eNodeB may use 1-bit to configure the UE to assume ( n1,n2,n3,n4 ) =[1,2,5,6]  or ( n1,n2,n3,n4 ) =[1,4,5,8] . With ( n1,n2,n3,n4 ) =[1,4,5,6] , two adjacent cross-polarized antenna pairs are configured, which models 4Tx cross-polarized antennas with small antenna spacing. With ( n1,n2,n3,n4 ) =[1,4,5,8] , two antenna pairs with large antenna spacing are modelled. This may be important for some wireless operators that must deploy wide antenna spacing due to their GSM/HSPA/LTE spectrum farming limitation, where two separate antenna radomes with large spacing are used for 4Tx MIMO. 
     For a UE configured in transmission mode  10 , where the UE is configured with multiple CSI-RS resources, the row selection vector ( n1,n2,n3,n4 ) can be independently configured for each 4-Tx CSI-RS resource. 
     Example of 4Tx Codebook from 8Tx Pruning 
     Using the methods proposed above for rank-1,2,3,4Tx GoB codebooks, one example of a 4Tx codebook is summarized below. 
     Release 12 enhancement is achieved by augmenting the Release 8 codebook with double-codebook (DCB) components.
         component 1 (Release 8): W1=identity matrix, W2 selected from Release 8 codebook,   component 2 (DCB): . . . pruned Release 10 8Tx codebook,
           select four rows out of eight rows from the 8Tx precoding matrices (i.e., a subset of the 8Tx matrices).   
               

     Rank-1: 
     Component 1: Release 8 
         W   1   =I   4 ,  (55)
 
         W   2   ∈C   2,R8−4Tx−R1   (56)
 
     Component 2: DCB: 
     W 1 : i 1 =0, . . . 15, each W 1  is the (1,2,5,6) th  row of the (i 1 ) th  8Tx W 1  codebook. 
     W 2 : i 2 =0, . . . 15, same as 8Tx W 2  codebook
         Note: the 4Tx DCB components are the (1,2,5,6) th  row of the 8Tx DCB codebook. This is equivalent to a (N,Nb)=(32,4) codebook with adjacent W 1  overlapping.       

     Rank-2: 
     Component 1: Release 8 
         W   1   =I   4 ,  (57)
 
         W   2   ∈C   2,R8−4Tx−R2   (58)
 
     Component 2: DCB: 
     W 1 : i 1 =0, . . . 15, each W 1  is the (1,2,5,6) th  row of the (i 1 ) th  8Tx W 1  codebook. 
     W 2 : i 2 =0, . . . 15, same as 8Tx W 2  codebook.
         Note: the 4Tx DCB components are the (1,2,5,6) th  row of the 8Tx DCB codebook. This is equivalent to a (N,Nb)=(32,4) codebook with adjacent W 1  overlapping.       

     Rank-3/4: 
     Re-use the Release 8 codebook. 
     Further Enhancement of 4Tx Codebook 
     Any of the codebooks proposed above can be further enhanced by adding more 4Tx precoding matrices. Note that in the sections above, the Release 8 4Tx codebook is inherited in Release 12 4Tx codebook in the form of W=W 1 W 2 , where W 1  is a 4×4 identity matrix and W 2  are taken from Release 8 codebook. An extension of this design is possible, where the W 1  codebook not only includes the 4×4 identity matrix  14 , but also includes other size 4×4 matrices. One exemplary extension is where the W 1  codebook comprises of a set of diagonal matrices, where each diagonal element performs phase rotation for each row of W 2  matrix. 
     Following this design principle, an example of a further-enhanced 4Tx codebook is given below. 
     Rank-1/2: 
         C   1   ={W   1   (0)   ,W   1   (1)   ,W   1   (2)   , . . . ,W   1   N−1   (59)
 
     where: 
     For i 1 =0, . . . , N/2−1, W 1   (0) , W 1   (1) , W 1   (2)  . . . W 1   (N/2−1)  are diagonal matrices (e.g. performing phase rotation), and W 2  inherits the Release 8 4Tx codebook. In particular, for i 1  =0, W 1   (0) =I 4 . This allows the Release 8 codebook to be re-used without change in Release 12. 
     When i 1 =(N/2), . . . , N−1, then W 1   (N/2)  . . . W 1   (N-1)  are block diagonal matrices designed with double-codebook structure. In this case, W 1  and W 2  can use any of the DCB codebook components proposed in the sections above. As one example, W 1 /W 2  may be pruned 8Tx codebooks, where N=32, and 
     W 1 : i 1 =16, . . . 31, each W 1  is the (1,2,5,6) th  row of the (i 1 -15) th  8Tx W 1  matrix, 
     W 2 : i 2 =0, . . . 15, same as 8Tx W 2  codebook. 
     Herein, the W 1  overhead is 5-bit, and W 2  overhead is 4-bits per subband. 
     Rank-3 and rank-4 may follow the same enhancement design if necessary. 
     Alternative Design for Rank-2 
     The GoB designs proposed above assumed that each W 1  matrix comprises a group of adjacent DFT beams. It is noted that the beams in each W 1  grid are not necessarily adjacent, which makes other designs possible. In this section we propose an example rank-2 design with non-adjacent beams in W 1 . 
     To recapture the notations, note that 
     
       
         
           
             
               
                 
                   
                     W 
                     1 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             X 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             X 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   60 
                   ) 
                 
               
             
           
         
       
     
     where X (k)  comprises of multiple 2×1 DFT beams, k=i 1 . In the following section, we assume an over-sampling ratio N=16, but the proposed design can be easily generalized to other N values. 
     Alternative Design 1 
     It is noted that an N-times oversampled 2×1 DFT beam expressed as 
     
       
         
           
             
               
                 b 
                 lmodN 
               
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                   
                   
                     
                       
                         e 
                         
                           j 
                           ⁢ 
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             l 
                             / 
                             N 
                           
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               l 
               = 
               0 
             
             , 
             
               
                 … 
                 ⁢ 
                     
                 N 
               
               - 
               1 
             
           
         
       
     
     is orthogonal to 
         b   (1+N/2)mod N . 
     Therefore each W 1  grid may comprise two orthogonal DFT beams expressed as 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   61 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       X 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           b 
                           lmodN 
                         
                         , 
                         
                           b 
                           
                             
                               ( 
                               
                                 l 
                                 + 
                                 
                                   N 
                                   / 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             modN 
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     k 
                     = 
                     1. 
                   
                 
               
               
                 
                   ( 
                   62 
                   ) 
                 
               
             
           
         
       
     
     The W 2  codebook may comprise beam selection and co-phasing matrixes, for example, denoted as: 
     
       
         
           
             
               
                 
                   
                     W 
                     2 
                   
                   = 
                   
                     
                       C 
                       2 
                     
                     = 
                     
                       Ω 
                       ∘ 
                       
                         { 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     e 
                                     1 
                                   
                                 
                                 
                                   
                                     e 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     e 
                                     1 
                                   
                                 
                                 
                                   
                                     e 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                           , 
                           
                             ( 
                             
                               
                                 
                                   
                                     e 
                                     1 
                                   
                                 
                                 
                                   
                                     e 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     e 
                                     1 
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       e 
                                       2 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           , 
                           
                             ( 
                             
                               
                                 
                                   
                                     e 
                                     1 
                                   
                                 
                                 
                                   
                                     e 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       e 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     e 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                           , 
                           
                             ( 
                             
                               
                                 
                                   
                                     e 
                                     1 
                                   
                                 
                                 
                                   
                                     e 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       e 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       e 
                                       2 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
           
         
       
     
     where ∘ denotes the Schur product, e i   2  is a 2×1 column vector with all zero entries except the i-th element being 1, and 
     
       
         
           
             
               
                 
                   Ω 
                   = 
                   
                     
                       { 
                       
                         ( 
                         
                           
                             
                               
                                 I 
                                 2 
                               
                             
                             
                                 
                             
                           
                           
                             
                                 
                             
                             
                               
                                 
                                   e 
                                   
                                     j 
                                     ⁢ 
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                       k 
                                       / 
                                       N 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   I 
                                   2 
                                 
                               
                             
                           
                         
                         ) 
                       
                       } 
                     
                     
                       k 
                       = 
                       Ξ 
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
           
         
       
     
     are diagonal matrices that perform co-phasing. 
     As an example, 
     
       
         
           
             
               
                 
                   
                     Ω 
                     = 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   I 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   - 
                                   
                                     I 
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   jI 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   - 
                                   
                                     jI 
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
           
         
       
     
     which results in a 4-bits W 2  codebook. Correspondingly, for each X (k) =[b l mod N , b (l+N/2)mod N ], k=l=i 1 , the corresponding sixteen composite matrices W=W i     1     ,i     2   , i 2 =0, . . . 15 are expressed as shown in Table 23. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 23 
               
               
                   
                   
               
             
            
               
                   
                 i 2  = 0, . . . , 3 
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             0 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       
                                         〈 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       
                                         〈 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             1 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       
                                         〈 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     modN 
                                   
                                 
                               
                             
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             2 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     b 
                                     lmodN 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     modN 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             3 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 i 2  = 4, . . . , 7 
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             4 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     modN 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             5 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     mod 
                                     ⁢ 
                                       
                                     N 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             6 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             7 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 i 2  = 8, . . . , 11 
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             8 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             9 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             10 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             11 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 i 2  = 12, . . . , 15 
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             12 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             13 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             14 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             15 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
            
           
         
       
     
     It is noted that W i     1     ,i     2   , i 2 =4, . . . ,7 are identical to W i     1     ,i     2   , i 2 =0, . . . , 3, and W i     1     ,i     2   , i 2 =12, . . . , 15 are identical to W i     1     ,i     2   , i 2 =8, . . . , 11. Therefore it&#39;s possible to reduce the W 2  size to 3-bits (i 2 =0, . . . , 7), where 
     
       
         
           
             
               
                 
                   
                     Ω 
                     = 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   I 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   jI 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   66 
                   ) 
                 
               
             
           
         
       
     
     and the composite precoder W=W i     1     ,i     2    is given by the values shown in Table 24. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 24 
               
               
                   
                   
               
             
            
               
                   
                 i 2  = 0, . . . , 3 
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             0 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             1 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     modN 
                                   
                                 
                               
                             
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             2 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     modN 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             3 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                 i 2  = 4, . . . , 7 
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             4 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             5 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           
                                             + 
                                             N 
                                           
                                           / 
                                           2 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             6 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   j 
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           W 
                           
                             k 
                             , 
                             7 
                           
                         
                         = 
                         
                           ( 
                           
                             
                               
                                 
                                   b 
                                   
                                     l 
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                               
                                 
                                   b 
                                   
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           N 
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     m 
                                     ⁢ 
                                     o 
                                     ⁢ 
                                     d 
                                     ⁢ 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       l 
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     j 
                                   
                                   ⁢ 
                                   
                                     b 
                                     
                                       
                                         ( 
                                         
                                           l 
                                           + 
                                           
                                             N 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       m 
                                       ⁢ 
                                       o 
                                       ⁢ 
                                       d 
                                       ⁢ 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                   
                   
               
            
           
         
       
     
     Note that other co-phasing methods Ω are possible, for example 
     
       
         
           
             
               
                 
                   Ω 
                   = 
                   
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   I 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   jI 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   
                                     e 
                                     
                                       j 
                                       ⁢ 
                                       
                                         π 
                                         4 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     I 
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 
                                   I 
                                   2 
                                 
                               
                               
                                   
                               
                             
                             
                               
                                   
                               
                               
                                 
                                   
                                     e 
                                     
                                       j 
                                       ⁢ 
                                       
                                         
                                           3 
                                           ⁢ 
                                           π 
                                         
                                         4 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     I 
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                         , 
                       
                       } 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   67 
                   ) 
                 
               
             
           
         
       
     
     In this case, the W 2  codebook size is 4-bits. 
     Generalization 
     The precoders proposed above as alternative design 1 may be combined with the precoders in the previous sections to construct the final rank-2 codebook. For example, assuming over-sampling ratio N (e.g. N=16), N b =4, the rank-2 codebook may be expressed as 
     
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         1 
                       
                       ∈ 
                       
                         C 
                         1 
                       
                     
                     = 
                     
                       { 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   X 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   X 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         . 
                       
                       } 
                     
                   
                   , 
                   
                     
                       i 
                       1 
                     
                     = 
                     
                       k 
                       = 
                       0 
                     
                   
                   , 
                   
                     
                       … 
                       ⁢ 
                           
                       2 
                       ⁢ 
                       N 
                       / 
                       
                         N 
                         b 
                       
                     
                     - 
                     1 
                   
                 
               
               
                 
                   ( 
                   68 
                   ) 
                 
               
             
           
         
       
     
     -&gt;size N/2 
     When i 2 =0, . . . , 7: 
     
       
         
           
             
               
                 
                   
                     X 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             b 
                             
                               
                                 ( 
                                 
                                   
                                     N 
                                     b 
                                   
                                   ⁢ 
                                   
                                     k 
                                     / 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               modN 
                             
                           
                         
                         
                           
                             b 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       N 
                                       b 
                                     
                                     ⁢ 
                                     
                                       k 
                                       / 
                                       2 
                                     
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               modN 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             b 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       N 
                                       b 
                                     
                                     ⁢ 
                                     
                                       k 
                                       / 
                                       2 
                                     
                                   
                                   + 
                                   
                                     N 
                                     b 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               modN 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   69 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         2 
                       
                       ∈ 
                       
                         C 
                         ⁢ 
                         
                           B 
                           2 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             1 
                             
                               2 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     Y 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           
                             1 
                             
                               2 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jY 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jY 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   70 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         Y 
                         1 
                       
                       , 
                       
                         Y 
                         2 
                       
                     
                     ) 
                   
                   ∈ 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             e 
                             1 
                             4 
                           
                           , 
                           
                             e 
                             1 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             2 
                             4 
                           
                           , 
                           
                             e 
                             2 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             3 
                             4 
                           
                             
                           , 
                           
                             e 
                             3 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             4 
                             4 
                           
                             
                           , 
                           
                             e 
                             4 
                             4 
                           
                         
                         ) 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   71 
                   ) 
                 
               
             
           
         
       
     
     When i 2 =8, . . . , 15: 
     
       
         
           
             
               
                 
                                    
                   
                     
                       X 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           b 
                           
                             
                               ( 
                               
                                 k 
                                 - 
                                 
                                   N 
                                   / 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             modN 
                           
                         
                         , 
                         
                           b 
                           kmodN 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   72 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         2 
                       
                       ∈ 
                       
                         C 
                         ⁢ 
                         
                           B 
                           2 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           2 
                         
                       
                       × 
                       
                         { 
                         
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           - 
                                           
                                             Y 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             Y 
                                             1 
                                           
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             Y 
                                             1 
                                           
                                         
                                       
                                       
                                         
                                           - 
                                           
                                             Y 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                               
                             
                           
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           jY 
                                           1 
                                         
                                       
                                       
                                         
                                           jY 
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           jY 
                                           1 
                                         
                                       
                                       
                                         
                                           - 
                                           
                                             jY 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             jY 
                                             1 
                                           
                                         
                                       
                                       
                                         
                                           jY 
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             jY 
                                             1 
                                           
                                         
                                       
                                       
                                         
                                           - 
                                           
                                             jY 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                               
                             
                           
                         
                         } 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   73 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       ( 
                       
                         
                           Y 
                           1 
                         
                         , 
                         
                           Y 
                           2 
                         
                       
                       ) 
                     
                     ∈ 
                     
                       { 
                       
                         ( 
                         
                           
                             e 
                             1 
                             2 
                           
                           , 
                           
                             e 
                             2 
                             2 
                           
                         
                         ) 
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   74 
                   ) 
                 
               
             
           
         
       
     
     The W 1  overhead is log 2(N/2)=3-bits, and the W 2  overhead is 4-bits. 
     Note that the combined codebook can be re-expressed below. 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         … 
                             
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   75 
                   ) 
                 
               
             
           
         
       
     
     →size N 
     When 
     
       
         
           
             
               
                 
                   
                     W 
                     1 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             
                               X 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           k 
                           = 
                           
                             
                               i 
                               1 
                             
                             = 
                             0 
                           
                         
                         , 
                         
                           
                             … 
                             ⁢ 
                                 
                             N 
                             / 
                             2 
                           
                           - 
                           1 
                         
                       
                       ) 
                     
                     : 
                   
                 
               
               
                 
                   ( 
                   76 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     X 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             b 
                             
                               
                                 ( 
                                 
                                   
                                     N 
                                     b 
                                   
                                   ⁢ 
                                   
                                     k 
                                     / 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               modN 
                             
                           
                         
                         
                           
                             b 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       N 
                                       b 
                                     
                                     ⁢ 
                                     
                                       k 
                                       / 
                                       2 
                                     
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               modN 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             b 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       N 
                                       b 
                                     
                                     ⁢ 
                                     
                                       k 
                                       / 
                                       2 
                                     
                                   
                                   + 
                                   
                                     N 
                                     b 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               modN 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   77 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       W 
                       2 
                     
                     ∈ 
                     
                       C 
                       ⁢ 
                       
                         B 
                         2 
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               
                                 Y 
                                 1 
                               
                             
                             
                               
                                 Y 
                                 2 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 1 
                               
                             
                             
                               
                                 - 
                                 
                                   Y 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               
                                 Y 
                                 1 
                               
                             
                             
                               
                                 Y 
                                 2 
                               
                             
                           
                           
                             
                               
                                 jY 
                                 1 
                               
                             
                             
                               
                                 - 
                                 
                                   jY 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   78 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         Y 
                         1 
                       
                       , 
                       
                         Y 
                         2 
                       
                     
                     ) 
                   
                   ∈ 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             e 
                             1 
                             4 
                           
                           , 
                           
                             e 
                             1 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             2 
                             4 
                           
                           , 
                           
                             e 
                             2 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             3 
                             4 
                           
                           , 
                           
                             e 
                             3 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             4 
                             4 
                           
                             
                           , 
                           
                             e 
                             4 
                             4 
                           
                         
                         ) 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   79 
                   ) 
                 
               
             
           
         
       
     
     When 
     
       
         
           
             
               
                 
                                    
                   
                     
                       
                         W 
                       
                       1 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             k 
                             = 
                             
                               
                                 i 
                                 1 
                               
                               = 
                               
                                 N 
                                 2 
                               
                             
                           
                           , 
                           … 
                               
                           , 
                           
                             N 
                             - 
                             1 
                           
                         
                         ) 
                       
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       X 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           b 
                           
                             
                               ( 
                               
                                 k 
                                 - 
                                 
                                   N 
                                   / 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             modN 
                           
                         
                         , 
                         
                           b 
                           kmodN 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         2 
                       
                       ∈ 
                       
                         C 
                         ⁢ 
                         
                           B 
                           2 
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           2 
                         
                       
                       × 
                       
                         { 
                         
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           - 
                                           
                                             Y 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             Y 
                                             1 
                                           
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             Y 
                                             1 
                                           
                                         
                                       
                                       
                                         
                                           - 
                                           
                                             Y 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                               
                             
                           
                           
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           jY 
                                           1 
                                         
                                       
                                       
                                         
                                           jY 
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           jY 
                                           1 
                                         
                                       
                                       
                                         
                                           - 
                                           
                                             jY 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             jY 
                                             1 
                                           
                                         
                                       
                                       
                                         
                                           jY 
                                           2 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           Y 
                                           1 
                                         
                                       
                                       
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             jY 
                                             1 
                                           
                                         
                                       
                                       
                                         
                                           - 
                                           
                                             jY 
                                             2 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 , 
                               
                             
                           
                         
                         } 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   82 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       ( 
                       
                         
                           Y 
                           1 
                         
                         , 
                         
                           Y 
                           2 
                         
                       
                       ) 
                     
                     ∈ 
                     
                       { 
                       
                         ( 
                         
                           
                             e 
                             1 
                             2 
                           
                           , 
                           
                             e 
                             2 
                             2 
                           
                         
                         ) 
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
           
         
       
     
     In this case, the W 1  overhead is log 2(N)=4-bits, and the W 2  overhead is 3-bits. 
     Another possible combinatorial design is 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     ∈ 
                     
                       C 
                       1 
                     
                   
                   = 
                   
                     { 
                     
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         … 
                             
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   84 
                   ) 
                 
               
             
           
         
       
         
         
           
             →size N 
           
         
       
    
     When 
     
       
         
           
             
               
                 
                                    
                   
                     
                       
                         W 
                       
                       1 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             k 
                             = 
                             
                               
                                 i 
                                 1 
                               
                               = 
                               0 
                             
                           
                           , 
                           … 
                               
                           , 
                           
                             
                               N 
                               / 
                               2 
                             
                             - 
                             1 
                           
                         
                         ) 
                       
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   85 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       X 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               b 
                               
                                 
                                   ( 
                                   
                                     
                                       N 
                                       b 
                                     
                                     ⁢ 
                                     
                                       k 
                                       / 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 modN 
                               
                             
                           
                           
                             
                               b 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         N 
                                         b 
                                       
                                       ⁢ 
                                       
                                         k 
                                         / 
                                         2 
                                       
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 modN 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               b 
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         N 
                                         b 
                                       
                                       ⁢ 
                                       
                                         k 
                                         / 
                                         2 
                                       
                                     
                                     + 
                                     
                                       N 
                                       b 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 modN 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   86 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       
                         
                           W 
                           2 
                         
                         ∈ 
                         
                           C 
                           ⁢ 
                           
                             B 
                             2 
                           
                         
                       
                       = 
                       
                         { 
                         
                           
                             
                               1 
                               
                                 2 
                               
                             
                             [ 
                             
                               
                                 
                                   
                                     Y 
                                     1 
                                   
                                 
                                 
                                   
                                     Y 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     Y 
                                     1 
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           , 
                           
                             
                               1 
                               
                                 2 
                               
                             
                             [ 
                             
                               
                                 
                                   
                                     Y 
                                     1 
                                   
                                 
                                 
                                   
                                     Y 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     jY 
                                     1 
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       jY 
                                       2 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                         } 
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   87 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         Y 
                         1 
                       
                       , 
                       
                         Y 
                         2 
                       
                     
                     ) 
                   
                   ∈ 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             e 
                             1 
                             4 
                           
                           , 
                           
                             e 
                             1 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             2 
                             4 
                           
                           , 
                           
                             e 
                             2 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             3 
                             4 
                           
                             
                           , 
                           
                             e 
                             3 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             4 
                             4 
                           
                             
                           , 
                           
                             e 
                             4 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             1 
                             4 
                           
                             
                           , 
                           
                             e 
                             2 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             2 
                             4 
                           
                             
                           , 
                           
                             e 
                             3 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             1 
                             4 
                           
                             
                           , 
                           
                             e 
                             4 
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             e 
                             2 
                             4 
                           
                             
                           , 
                           
                             e 
                             4 
                             4 
                           
                         
                         ) 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   88 
                   ) 
                 
               
             
           
         
       
     
     When 
     
       
         
           
             
               
                 
                                   
                   
                     
                       
                         W 
                       
                       1 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 X 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             k 
                             = 
                             
                               
                                 i 
                                 1 
                               
                               = 
                               
                                 N 
                                 2 
                               
                             
                           
                           , 
                           … 
                               
                           , 
                           
                             N 
                             - 
                             1 
                           
                         
                         ) 
                       
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   89 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       X 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           b 
                           
                             
                               ( 
                               
                                 k 
                                 - 
                                 
                                   N 
                                   / 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             modN 
                           
                         
                         , 
                         
                           b 
                           kmodN 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   90 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       W 
                       2 
                     
                     ∈ 
                     
                       C 
                       ⁢ 
                       
                         B 
                         2 
                       
                     
                   
                   = 
                   
                     
                       1 
                       
                         2 
                       
                     
                     × 
                     
                       { 
                       
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         - 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         - 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         jY 
                                         1 
                                       
                                     
                                     
                                       
                                         jY 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         jY 
                                         1 
                                       
                                     
                                     
                                       
                                         - 
                                         
                                           jY 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           jY 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         jY 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           jY 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         - 
                                         
                                           jY 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                               π 
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                               π 
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                               π 
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             
                                               j 
                                               ⁢ 
                                               
                                                 π 
                                                 4 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             
                                               j 
                                               ⁢ 
                                               
                                                 π 
                                                 4 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                               π 
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             
                                               j 
                                               ⁢ 
                                               
                                                 π 
                                                 4 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             
                                               j 
                                               ⁢ 
                                               
                                                 π 
                                                 4 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                               
                                                 3 
                                                 ⁢ 
                                                 π 
                                               
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                               
                                                 3 
                                                 ⁢ 
                                                 π 
                                               
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                               
                                                 3 
                                                 ⁢ 
                                                 π 
                                               
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             
                                               j 
                                               ⁢ 
                                               
                                                 
                                                   3 
                                                   ⁢ 
                                                   π 
                                                 
                                                 4 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             
                                               j 
                                               ⁢ 
                                               
                                                 
                                                   3 
                                                   ⁢ 
                                                   π 
                                                 
                                                 4 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           e 
                                           
                                             j 
                                             ⁢ 
                                             
                                               
                                                 3 
                                                 ⁢ 
                                                 π 
                                               
                                               4 
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             
                                               j 
                                               ⁢ 
                                               
                                                 
                                                   3 
                                                   ⁢ 
                                                   π 
                                                 
                                                 4 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           - 
                                           
                                             e 
                                             
                                               j 
                                               ⁢ 
                                               
                                                 
                                                   3 
                                                   ⁢ 
                                                   π 
                                                 
                                                 4 
                                               
                                             
                                           
                                         
                                         ⁢ 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   91 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       ( 
                       
                         
                           Y 
                           1 
                         
                         , 
                         
                           Y 
                           2 
                         
                       
                       ) 
                     
                     ∈ 
                     
                       { 
                       
                         ( 
                         
                           
                             e 
                             1 
                             2 
                           
                           , 
                           
                             e 
                             2 
                             2 
                           
                         
                         ) 
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   92 
                   ) 
                 
               
             
           
         
       
     
     The W 1  overhead is log 2(N)=4-bits, and the W 2  overhead is 4-bits. 
     Inheritance of the Release 8 Codebook in Release 12 
     Using Release 8 Precoders for W 2    
     As proposed in the previous sections, it is possible for the Release 12 codebook to inherit the Release 8 codebook as a subset. This can be achieved by using the Release 8 4Tx precoder as the codebook for W 2  which is associated with W1=4×4 identity matrix  14 . This is applicable for rank-i to rank-4, where the subband W 2  overhead is 4-bits per subband. 
     It is also possible to divide the Release 8 precoders into N groups, where each group has 16/N Release 8 precoders that form a different W 2  codebook. For each W 2  codebook, the W 1  matrix is equivalent to a 4×4 identity matrix. The subband W 2  overhead will therefore be reduced to log 2(16/N) bits. 
     For example
         The Release 8 codebook is inherited in Release 12 codebook, corresponding to two W 1  matrices, e.g., W 1   k =I 4 , k=0, 1, For W 1   0 , the W 2  codebook comprises the first eight Release 8 precoders. For W 1   1 , the W 2  codebook comprises the last eight Release 8 precoders.   Other values of N are possible to adapt the subband PMI bit-width. For example, N=2 corresponds to a subband size of log 2(8)=3-bits, N=4 corresponds to a subband size of log 2(4)=2 bits.   The above designs can be applied for rank-1, rank-2, rank-3, and rank-4.       

     Constructing W 1  with Release 8 Precoders and W 2  with Column Selection 
     It is further possible to inherit the Release 8 codebook in Release 12 by constructing the W 1  matrix with Release 8 precoders, and constructing the W 2  codebook with column selection matrices. 
     For example:
         For rank-1, W 1  includes all or a subset of Release 8 rank-1 vectors. The size of W 1  is given by 4×L, where 1&lt;=L&lt;=16 is the number of rank-1 Release 8 codebook vectors that are included in W 1 . The W 2  codebook comprises L column-selection vectors [e 1 , e 2  . . . e L ], where e i  is an L×1 vector of all zero entries, except for the i-th entry, which is equivalent to 1.   For rank-r (r=2, 3, 4), the columns of W 1  comprise all or a subset of Release 8 rank-r vectors, e.g. this may be denoted as       

         W   1 =[ W   s(0)   Rel.8-rank-r   W   s(1)   Rel.8-rank-r    . . . W   s(L−1)   Rel.8-rank-r ]  (93)
 
     where W s(1)   Rel.8-rank-r  is the s(1)-th Release 8 precoder. The size of W 1  is 4×rL, where 1&lt;=L&lt;=16 is the number of rank-r Release 8 codebook matrices in W 1 . The W 2  codebook comprises L column-selection matrices, where the 1-th W 2  matrix (1&lt;=1&lt;=L) is 
     
       
         
           
             
               
                 
                   
                     W 
                     ⁢ 
                     2 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             0 
                             
                               
                                 r 
                                 ⁡ 
                                 ( 
                                 
                                   l 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               × 
                               r 
                             
                           
                         
                       
                       
                         
                           
                             I 
                             
                               r 
                               × 
                               r 
                             
                           
                         
                       
                       
                         
                           
                             0 
                             
                               
                                 r 
                                 ⁡ 
                                 ( 
                                 
                                   L 
                                   - 
                                   l 
                                 
                                 ) 
                               
                               × 
                               r 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   94 
                   ) 
                 
               
             
           
         
       
     
     Alternatively, the W 1  matrix can be constructed in a block diagonal manner. 
     For example, for rank-1: 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             A 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             B 
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   95 
                   ) 
                 
               
             
           
         
       
     
     where A and B are of sizes 2×16, where the 1-th column of A is the first two rows of the 1-th Release 8 precoder, and the 1-th column of B is the last two rows of the 1-th Release 8 precoder, 1 =1, . . . , 16. 
     The W 2  matrix can be written in the form of 
     
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         2 
                         k 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 e 
                                 k 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 e 
                                 k 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       k 
                       = 
                       1 
                     
                   
                   , 
                   … 
                       
                   , 
                   16 
                   , 
                 
               
               
                 
                   ( 
                   96 
                   ) 
                 
               
             
           
         
       
     
     where e k  is the k-th column of a 16×16 identity matrix. 
     For rank-r, r=2, . . . , 4: 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       1 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             A 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             B 
                           
                         
                       
                       ] 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   97 
                   ) 
                 
               
             
           
         
       
     
     where A and B are of sizes 2×16r, and where 
     A [W 1   Rel.8-rank-r  W 2   Rel.8-rank-r  . . . W 16   Rel.8-rank-r ] (1:2) , P (1:2)  is the first and second row of matrix P, and 
     B [W 1   Rel.8-rank-r  W 2   Rel.8-rank-r  . . . W 16   Rel.8-rank-r ] (3:4) , P (3:4)  is the third and fourth row of matrix P. 
     For the W 2  codebook, 
     
       
         
           
             
               
                 
                   
                     
                       
                         W 
                         2 
                         k 
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 e 
                                 k 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 e 
                                 k 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
 
                     
                       k 
                       = 
                       1 
                     
                     , 
                     
                       … 
                       ⁢ 
                           
                       16 
                     
                     , 
                     
 
                     where 
                   
                   ⁢ 
                   
 
                   
                     
                       e 
                       k 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 0 
                                 
                                   
                                     r 
                                     ⁡ 
                                     ( 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   × 
                                   r 
                                 
                               
                             
                           
                           
                             
                               
                                 I 
                                 
                                   r 
                                   × 
                                   r 
                                 
                               
                             
                           
                           
                             
                               
                                 0 
                                 
                                   
                                     r 
                                     ⁡ 
                                     ( 
                                     
                                       L 
                                       - 
                                       k 
                                     
                                     ) 
                                   
                                   × 
                                   r 
                                 
                               
                             
                           
                         
                         ] 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   98 
                   ) 
                 
               
             
           
         
       
     
     4Tx Codebook Enhancements for LTE 
     In the following section, possible 4Tx codebook enhancement alternatives for LTE Release 12 are disclosed. In these examples, {tilde over (e)} i , is a 4×1 vector with all zero entries except for the i-th element which has value 1. 
     Rank-1/2 
     For a rank-1/2 codebook, the following two alternatives are possible. 
     Alt-1: 
     Reusing the 8Tx GoB design with N=32 over-sampled beams and 4 adjacent beams per grid, the following 4Tx codebook can be considered. 
                                            B   =     [           b   0           b   1         ⋯         b   31           ]       ,   
   	         [   B   ]         1   +   m     ,     1   +   n         =     e     j   ⁢       2   ⁢     π   ⁢   mn       32           ,   
   	     m   =   0     ,       …   ⁢             N   i     2       -   1     ,   
   	     n   =   0     ,   1   ,   ⋯   ,   31             (   99   )                             X     (   k   )       ∈     {           [           b     2   ⁢     kmod   ⁢   32               b       (       2   ⁢   k     +   1     )     ⁢     mod   ⁢   32               b       (       2   ⁢   k     +   2     )     ⁢     mod   ⁢   32               b       (       2   ⁢   k     +   3     )     ⁢     mod   ⁢   32               ]     :   k     =   0     ,   1   ,   ⋯   ,   15     }             (   100   )                                                  W   1     (   k   )       =     [           X     (   k   )           0           0         X     (   k   )             ]               (   101   )               Codebook 1:  C   1   ={W   1   (0)   , W   1   (1)   , W   1   (2) ), . . . ,  W   1   (15) }  (102)
 
     Rank-1: 4-bit) 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       2 
                     
                     ∈ 
                     
                       C 
                       2 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               Y 
                             
                           
                           
                             
                               Y 
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               Y 
                             
                           
                           
                             
                               jY 
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               Y 
                             
                           
                           
                             
                               
                                 - 
                                 Y 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               Y 
                             
                           
                           
                             
                               
                                 - 
                                 jY 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   103 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Y 
                   ∈ 
                   
                     { 
                     
                       
                         
                           e 
                           ~ 
                         
                         1 
                       
                       , 
                       
                         
                           e 
                           ~ 
                         
                         2 
                       
                       , 
                       
                         
                           e 
                           ~ 
                         
                         3 
                       
                       , 
                       
                         
                           e 
                           ~ 
                         
                         4 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   104 
                   ) 
                 
               
             
           
         
       
     
     Rank-2: 4-bit) 
     
       
         
           
             
               
                 
                                    
                   
                     
                       
                         W 
                         2 
                       
                       ∈ 
                       
                         C 
                         2 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             1 
                             
                               2 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     Y 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           
                             1 
                             
                               2 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jY 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jY 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   105 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           Y 
                           1 
                         
                         , 
                         
                           Y 
                           2 
                         
                       
                       ) 
                     
                     ∈ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               1 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               1 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               2 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               2 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               3 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               3 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               4 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               4 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               1 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               2 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               2 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               3 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               1 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               4 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               2 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               4 
                             
                           
                           ) 
                         
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   106 
                   ) 
                 
               
             
           
         
       
     
     Rank-2: 3-bit) 
     For rank-2, if a (3-bit) W 2  is preferred, (Y 1 , Y 2 ) can be changed to: 
       ( Y   1   ,Y   2 )∈{( {tilde over (e)}   1   ,{tilde over (e)}   1 ),( {tilde over (e)}   2   ,{tilde over (e)}   2 ),( {tilde over (e)}   3   ,{tilde over (e)}   3 ),( {tilde over (e)}   4   ,{tilde over (e)}   4 )}.  (107)
 
     Each W 1  matrix is constructed by a set of adjacent DFT beams and covers a narrow range of angle-of-departures/arrivals. The fundamental consideration is based the feedback of wideband/long-term channel based on W 1  matrix, and the fed back W 1  matrix in a properly designed C 1  codebook should be able to reflect the wideband AoA/AoDs with sufficient accuracy. For instance, most macro base stations in cellular communication systems are elevated on a cell tower and likely have a direct line-of-sign to the UE, where the angle of incoming radio signals to a UE is within in a small range. Hence, the wideband W 1  comprising a set of adjacent beams can be used to cover the range of incoming radio signals on the wideband, whereas the narrowband W 2  codebook can be used to select a specific beam on each subband. This W 1  design is particular suitable to macro base station with narrowly spaced antennas, propagation channel with sufficient line-of-sights, and perfectly calibrated base station antennas. 
     Alt-2: 
     A rank-1/2 codebook comprises two components, where W 1  structures are different in each component. For the first 8 W 1  matrices, Xn comprise four adjacent DFT beams with over-sampling rate of N=16. For the last 8 W 1  matrices, Xn comprise four distributed non-adjacent DFT beams uniformly sampling the [0, 360] angle of arrival sub-space. This provides wider angular spread coverage and may be beneficial to large timing misalignment error. 
     The W 1  codebook therefore can be given by: 
     i 1 =0, 1, . . . , 7: 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       
                         ( 
                         
                           i 
                           1 
                         
                         ) 
                       
                     
                     ∈ 
                     
                       { 
                       
                         [ 
                         
                           
                             
                               
                                 b 
                                 
                                   2 
                                   ⁢ 
                                   
                                     i 
                                     1 
                                   
                                   ⁢ 
                                   
                                     mod 
                                     ⁢ 
                                     16 
                                   
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           i 
                                           1 
                                         
                                       
                                       + 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     mod 
                                     ⁢ 
                                     16 
                                   
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           i 
                                           1 
                                         
                                       
                                       + 
                                       2 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     mod 
                                     ⁢ 
                                     16 
                                   
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           i 
                                           1 
                                         
                                       
                                       + 
                                       3 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     mod 
                                     ⁢ 
                                     16 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   108 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         b 
                         n 
                       
                       ( 
                       
                         m 
                         + 
                         1 
                       
                       ) 
                     
                     = 
                     
                       e 
                       
                         j 
                         ⁢ 
                         
                           
                             2 
                             ⁢ 
                             
                               π 
                               ⁢ 
                               mn 
                             
                           
                           16 
                         
                       
                     
                   
                   , 
                   
 
                   
                     n 
                     = 
                     0 
                   
                   , 
                   1 
                   , 
                   ⋯ 
                   , 
                   15 
                   , 
                   
 
                   
                     m 
                     = 
                     0 
                   
                   , 
                   .1 
                 
               
               
                 
                   ( 
                   109 
                   ) 
                 
               
             
           
         
       
     
     i 1 =8, 9, . . . , 15: 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       
                         ( 
                         
                           i 
                           1 
                         
                         ) 
                       
                     
                     ∈ 
                     
                       { 
                       
                         [ 
                         
                           
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         i 
                                         1 
                                       
                                       - 
                                       8 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     mod 
                                     ⁢ 
                                     32 
                                   
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         i 
                                         1 
                                       
                                       - 
                                       8 
                                     
                                     ) 
                                   
                                   + 
                                   
                                     8 
                                     ⁢ 
                                     
                                       mod 
                                       ⁢ 
                                       32 
                                     
                                   
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         i 
                                         1 
                                       
                                       - 
                                       8 
                                     
                                     ) 
                                   
                                   + 
                                   
                                     16 
                                     ⁢ 
                                     
                                       mod 
                                       ⁢ 
                                       32 
                                     
                                   
                                 
                               
                             
                             
                               
                                 b 
                                 
                                   
                                     ( 
                                     
                                       
                                         i 
                                         1 
                                       
                                       - 
                                       8 
                                     
                                     ) 
                                   
                                   + 
                                   
                                     24 
                                     ⁢ 
                                     
                                       mod 
                                       ⁢ 
                                       32 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   110 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                     
                   
                     
                       
                         
                           b 
                           n 
                         
                         ( 
                         
                           m 
                           + 
                           1 
                         
                         ) 
                       
                       = 
                       
                         e 
                         
                           j 
                           ⁢ 
                           
                             
                               2 
                               ⁢ 
                               
                                 π 
                                 ⁢ 
                                 mn 
                               
                             
                             32 
                           
                         
                       
                     
                     , 
                     
 
                     	 
                     
                       n 
                       = 
                       0 
                     
                     , 
                     1 
                     , 
                     ⋯ 
                     , 
                     31 
                     , 
                     
 
                     	 
                     
                       m 
                       = 
                       0 
                     
                     , 
                     .1 
                   
                 
               
               
                 
                   ( 
                   111 
                   ) 
                 
               
             
           
         
       
     
     As can be seen, for the last eight W 1  matrices, each W 1  matrix in codebook C 1  is comprised of four non-adjacent DFT beams. The four DFT beams in each W 1  matrix are widely spaced and uniformly distributed in the [0,360] DFT subspace to cover a wide range of angle-of-arrivals/departures. The DFT beams in a first W 1  matrix are rotated by a small angle against four DFT beams in a second W 1  matrix. More specifically, the W 1  matrices comprising widely spaced DFT beams are summarized as 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       1 
                       
                         ( 
                         8 
                         ) 
                       
                     
                     ∈ 
                     
                       { 
                       
                         [ 
                         
                           
                             
                               
                                 b 
                                 0 
                               
                             
                             
                               
                                 b 
                                 8 
                               
                             
                             
                               
                                 b 
                                 16 
                               
                             
                             
                               
                                 b 
                                 24 
                               
                             
                           
                         
                         ] 
                       
                       } 
                     
                   
                   ⁢ 
                   
 
                   
                     
                       X 
                       1 
                       
                         ( 
                         9 
                         ) 
                       
                     
                     ∈ 
                     
                       { 
                       
                         [ 
                         
                           
                             
                               
                                 b 
                                 1 
                               
                             
                             
                               
                                 b 
                                 9 
                               
                             
                             
                               
                                 b 
                                 17 
                               
                             
                             
                               
                                 b 
                                 25 
                               
                             
                           
                         
                         ] 
                       
                       } 
                     
                   
                   ⁢ 
                   
 
                   … 
                   ⁢ 
                   
 
                   
                     
                       X 
                       1 
                       
                         ( 
                         15 
                         ) 
                       
                     
                     ∈ 
                     
                       { 
                       
                         [ 
                         
                           
                             
                               
                                 b 
                                 7 
                               
                             
                             
                               
                                 b 
                                 15 
                               
                             
                             
                               
                                 b 
                                 23 
                               
                             
                             
                               
                                 b 
                                 31 
                               
                             
                           
                         
                         ] 
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   112 
                   ) 
                 
               
             
           
         
       
     
     Such a design framework is particularly beneficial in uses cases (e.g., widely spaced antenna components, un-calibrated antenna array, rich multipath scattering environment) where the angle-of-arrival/departures of the incoming wireless signals are distributed in a wide range. For instance, in a heterogeneous deployment scenario where dense small cells are overlaid on top of macro base station on the same frequency, the multipath radio signals received by a UE are reflected by numerous scattering objects surrounding the UE (e.g., building, cars). Have widely spaced and non-adjacent DFT beams in W 1  ensures that all incoming signals from different angles can be properly captured. As another use case, it is noted that the transmitter timing at different antennas on a base station shall be synchronized by careful timing alignment calibration. In practice, perfect timing alignment cannot always be guaranteed at a base station, especially for low-cost low-power small base stations (e.g. pico-cells, femto-cells) with cheaper RF components. In 3GPP LTE, a maximum 65 nano-second downlink transmission timing misalignment requirement is set forth for all base stations. A consequence of misaligned antenna timing in the time domain is the increased channel variation on different OFDM subcarriers in the frequency domain, and the main DFT beam angle on one frequency subband can be significantly different than the main DFT beam angle on another subband. In this case, having non-adjacent widely spaced DFT beams in the W 1  matrix ensures that the wideband angle-of-arrival/departures can be more reliably covered, resulting in higher feedback accuracy. 
     The W 2  codebook for rank-1 (4-bit) may be given by: 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       2 
                     
                     ∈ 
                     
                       C 
                       2 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               Y 
                             
                           
                           
                             
                               Y 
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               Y 
                             
                           
                           
                             
                               jY 
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               Y 
                             
                           
                           
                             
                               
                                 - 
                                 Y 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               Y 
                             
                           
                           
                             
                               
                                 - 
                                 jY 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   113 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   Y 
                   ∈ 
                   
                     { 
                     
                       
                         
                           e 
                           ~ 
                         
                         1 
                       
                       , 
                       
                         
                           e 
                           ~ 
                         
                         2 
                       
                       , 
                       
                         
                           e 
                           ~ 
                         
                         3 
                       
                       , 
                       
                         
                           e 
                           ~ 
                         
                         4 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   114 
                   ) 
                 
               
             
           
         
       
     
     The W 2  codebook for rank-2 (4-bit) may be given by: 
     For W 2  corresponding to i 1 =0,1, . . . , 7 
     
       
         
           
             
               
                 
                                    
                   
                     
                       
                         W 
                         2 
                       
                       ∈ 
                       
                         C 
                         2 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             1 
                             
                               2 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     Y 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         , 
                         
                           
                             1 
                             
                               2 
                             
                           
                           [ 
                           
                             
                               
                                 
                                   Y 
                                   1 
                                 
                               
                               
                                 
                                   Y 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   jY 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   
                                     jY 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   115 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         Y 
                         1 
                       
                       , 
                       
                         Y 
                         2 
                       
                     
                     ) 
                   
                   ∈ 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             1 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             1 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             2 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             2 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             3 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             3 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             4 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             1 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             2 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             2 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             3 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             1 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             4 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             2 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             4 
                           
                         
                         ) 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   116 
                   ) 
                 
               
             
           
         
       
     
     If a (3-bit) W 2  is preferred, (Y 1 , Y 2 ) may be changed to: 
       ( Y   1   ,Y   2 )∈{( {tilde over (e)}   1   ,{tilde over (e)}   1 ),( {tilde over (e)}   2   ,{tilde over (e)}   2 ),( {tilde over (e)}   3   ,{tilde over (e)}   3 ),( {tilde over (e)}   4   ,{tilde over (e)}   4 )}(117)
 
     For W 2  corresponding to i 1 =8, 9, . . . , 15 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       2 
                     
                     ∈ 
                     
                       C 
                       2 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 1 
                                 
                                   2 
                                 
                               
                               [ 
                               
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             , 
                             
                               
                                 1 
                                 
                                   2 
                                 
                               
                               [ 
                               
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               
                                 1 
                                 
                                   2 
                                 
                               
                               [ 
                               
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       
                                         Y 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             , 
                             
                               
                                 1 
                                 
                                   2 
                                 
                               
                               [ 
                               
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       
                                         Y 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               
                                 1 
                                 
                                   2 
                                 
                               
                               [ 
                               
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       jY 
                                       1 
                                     
                                   
                                   
                                     
                                       jY 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             , 
                             
                               
                                 1 
                                 
                                   2 
                                 
                               
                               [ 
                               
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       jY 
                                       1 
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         jY 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               
                                 1 
                                 
                                   2 
                                 
                               
                               [ 
                               
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       
                                         jY 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       jY 
                                       2 
                                     
                                   
                                 
                               
                               ] 
                             
                             , 
                             
                               
                                 1 
                                 
                                   2 
                                 
                               
                               [ 
                               
                                 
                                   
                                     
                                       Y 
                                       1 
                                     
                                   
                                   
                                     
                                       Y 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       
                                         jY 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         jY 
                                         2 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   118 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         Y 
                         1 
                       
                       , 
                       
                         Y 
                         2 
                       
                     
                     ) 
                   
                   ∈ 
                   
                     { 
                     
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             1 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             3 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             2 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             4 
                           
                         
                         ) 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   119 
                   ) 
                 
               
             
           
         
       
     
     If a (3-bit) W 2  is preferred, the W 2  codebook may be changed to 
     
       
         
           
             
               
                 
                   
                     
                       W 
                       2 
                     
                     ∈ 
                     
                       C 
                       2 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               
                                 Y 
                                 1 
                               
                             
                             
                               
                                 Y 
                                 2 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 1 
                               
                             
                             
                               
                                 Y 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               
                                 Y 
                                 1 
                               
                             
                             
                               
                                 Y 
                                 2 
                               
                             
                           
                           
                             
                               
                                 Y 
                                 1 
                               
                             
                             
                               
                                 - 
                                 
                                   Y 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               
                                 Y 
                                 1 
                               
                             
                             
                               
                                 Y 
                                 2 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   Y 
                                   1 
                                 
                               
                             
                             
                               
                                 Y 
                                 2 
                               
                             
                           
                         
                         ] 
                       
                       , 
                       
                         
                           1 
                           
                             2 
                           
                         
                         [ 
                         
                           
                             
                               
                                 Y 
                                 1 
                               
                             
                             
                               
                                 Y 
                                 2 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   Y 
                                   1 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   Y 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   120 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       ( 
                       
                         
                           Y 
                           1 
                         
                         , 
                         
                           Y 
                           2 
                         
                       
                       ) 
                     
                     ∈ 
                     
                       { 
                       
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               1 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               3 
                             
                           
                           ) 
                         
                         , 
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               2 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               4 
                             
                           
                           ) 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   121 
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             or 
           
         
       
       
         
           
             
               
                 
                                     
                   
                     
                       
                         W 
                         2 
                       
                       ∈ 
                       
                         C 
                         2 
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         - 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           Y 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         - 
                                         
                                           Y 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         jY 
                                         1 
                                       
                                     
                                     
                                       
                                         jY 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         jY 
                                         1 
                                       
                                     
                                     
                                       
                                         - 
                                         
                                           jY 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           jY 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         jY 
                                         2 
                                       
                                     
                                   
                                 
                                 ] 
                               
                               , 
                               
                                 
                                   1 
                                   
                                     2 
                                   
                                 
                                 [ 
                                 
                                   
                                     
                                       
                                         Y 
                                         1 
                                       
                                     
                                     
                                       
                                         Y 
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         - 
                                         
                                           jY 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         - 
                                         
                                           jY 
                                           2 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   122 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                     
                   
                     
                       
                         ( 
                         
                           
                             Y 
                             1 
                           
                           , 
                           
                             Y 
                             2 
                           
                         
                         ) 
                       
                       ∈ 
                       
                         { 
                         
                           ( 
                           
                             
                               
                                 e 
                                 ~ 
                               
                               1 
                             
                             , 
                             
                               
                                 e 
                                 ~ 
                               
                               3 
                             
                           
                           ) 
                         
                         } 
                       
                     
                     ⁢ 
                     
 
                     	 
                     or 
                     ⁢ 
                     
 
                     	 
                     
                       { 
                       
                         ( 
                         
                           
                             
                               e 
                               ~ 
                             
                             2 
                           
                           , 
                           
                             
                               e 
                               ~ 
                             
                             4 
                           
                         
                         ) 
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   123 
                   ) 
                 
               
             
           
         
       
     
     In one embodiment, the new rank-1/2 codebook designs set forth above may be incorporated into an LTE system complying with LTE Release 12. A subset of precoders in the proposed codebooks above may be used to construct a new Release 12 4Tx codebook, for either rank-1 or rank-2. For example, the W 1  codebook may be constructed by W 1  matrices only comprising widely spaced DFT beams, e.g., i 2 =8, 9, . . . 15 in Equation (110)-(111). 
       FIG.  6    illustrates the precoding matrix/vector selection process according to one embodiment. The final precoding matrix/vector is a function of two PMIs: 
         W =ƒ( PMI   1   ,PMI   2 )  (124)
 
     where PMI 1  is updated at a significantly less frequent rate than PMI 2 . PMI 1  is intended for the entire system bandwidth while PMI 2  can be frequency-selective. 
       FIG.  6    illustrates the technique used in downlink LTE-Advanced (LTE-A). The UE selects PMI 1  and PMI 2  and hence W 1  and W 2  in a manner similar to the LTE feedback paradigm. 
     The UE first selects the first precoder codebook W 1  in block  601  based on the long-term channel properties such as spatial covariance matrix such as in a spatial correlation domain from an input of PMI 1 . This is done in a long-term basis consistent with the fact that spatial covariance matrix needs to be estimated over a long period of time and in a wideband manner. 
     Conditioned upon W 1 , the UE selects W 2  based on the short-term (instantaneous) channel. This is a two stage process. In block  602 , a set of codebooks CB 2   (0)  to CB 2   (N-1)  is selected based upon the PMI 1  input. Block  603  selects one precoder corresponding to the selected codebook CB 2   (PM1   1   )  and PMI 2 . This selection may be conditioned upon the selected rank indicator (RI). Alternatively, RI can be selected jointly with W 2 . Block  604  takes the selected W 1  and W 2  and forms the function ƒ(W 1 , W 2 ). 
     PMI 1  and PMI 2  are reported to the base station (eNodeB) at different rates and/or different frequency resolutions. 
     Based on this design framework, several types of codebook design are described herein. While each type can stand alone, it is also possible to use different types in a single codebook design especially if the design is intended for different scenarios. A simple yet versatile design can be devised as follows: 
     PMI 1  selects one of the N codebooks W 1  as indicated above. 
     PMI 2  selects at least one of the column vectors of W, wherein the number of selected column vectors is essentially the recommended transmission rank (RI). 
     This design allows construction of N different scenarios where the codebook W 1  for each scenario is chosen to contain a set of basis vectors for a particular spatial channel characteristic W 2 . While any two-dimensional function can be used in equation (124), this disclosure assumes a product (matrix multiplication) function f(x,y)=xy. Thus the final short-term precoding matrix/vector is computed as a matrix product of W 1  and W 2 : W=W 1 W 2 . 
       FIG.  7    is a block diagram illustrating internal details of a mobile UE  701  and an eNodeB  702  in the network system of  FIG.  1   . Mobile UE  701  may represent any of a variety of devices such as a server, a desktop computer, a laptop computer, a cellular phone, a Personal Digital Assistant (PDA), a smart phone or other electronic devices. In some embodiments, the electronic mobile UE  701  communicates with eNodeB  702  based on a LTE or Evolved Universal Terrestrial Radio Access Network (E-UTRAN) protocol. Alternatively, another communication protocol now known or later developed can be used. 
     Mobile UE  701  comprises a processor  703  coupled to a memory  704  and a transceiver  705 . The memory  704  stores (software) applications  706  for execution by the processor  703 . The applications could comprise any known or future application useful for individuals or organizations. These applications could be categorized as operating systems (OS), device drivers, databases, multimedia tools, presentation tools, Internet browsers, emailers, Voice-Over-Internet Protocol (VOIP) tools, file browsers, firewalls, instant messaging, finance tools, games, word processors or other categories. Regardless of the exact nature of the applications, at least some of the applications may direct the mobile UE  701  to transmit UL signals to eNodeB (base-station)  702  periodically or continuously via the transceiver  705 . In at least some embodiments, the mobile UE  701  identifies a Quality of Service (QoS) requirement when requesting an uplink resource from eNodeB  702 . In some cases, the QoS requirement may be implicitly derived by eNodeB  702  from the type of traffic supported by the mobile UE  701 . As an example, VOIP and gaming applications often involve low-latency uplink (UL) transmissions while High Throughput (HTP)/Hypertext Transmission Protocol (HTTP) traffic can involve high-latency uplink transmissions. 
     Transceiver  705  includes uplink logic which may be implemented by execution of instructions that control the operation of the transceiver. Some of these instructions may be stored in memory  704  and executed when needed by processor  703 . As would be understood by one of skill in the art, the components of the uplink logic may involve the physical (PHY) layer and/or the Media Access Control (MAC) layer of the transceiver  705 . Transceiver  705  includes one or more receivers  707  and one or more transmitters  708 . 
     Processor  703  may send or receive data to various input/output devices  709 . A subscriber identity module (SIM) card stores and retrieves information used for making calls via the cellular system. A Bluetooth baseband unit may be provided for wireless connection to a microphone and headset for sending and receiving voice data. Processor  703  may send information to a display unit for interaction with a user of mobile UE  701  during a call process. The display may also display pictures received from the network, from a local camera, or from other sources such as a Universal Serial Bus (USB) connector. Processor  703  may also send a video stream to the display that is received from various sources such as the cellular network via RF transceiver  705  or the camera. 
     During transmission and reception of voice data or other application data, transmitter  707  may be or become non-synchronized with its serving eNodeB. In this case, it sends a random access signal. As part of this procedure, it determines a preferred size for the next data transmission, referred to as a message, by using a power threshold value provided by the serving eNodeB, as described in more detail above. In this embodiment, the message preferred size determination is embodied by executing instructions stored in memory  704  by processor  703 . In other embodiments, the message size determination may be embodied by a separate processor/memory unit, by a hardwired state machine, or by other types of control logic, for example. 
     eNodeB  702  comprises a processor  710  coupled to a memory  711 , symbol processing circuitry  712 , and a transceiver  713  via backplane bus  714 . The memory stores applications  715  for execution by processor  710 . The applications could comprise any known or future application useful for managing wireless communications. At least some of the applications  715  may direct eNodeB  702  to manage transmissions to or from mobile UE  701 . 
     Transceiver  713  comprises an uplink Resource Manager, which enables eNodeB  702  to selectively allocate uplink Physical Uplink Shared CHannel (PUSCH) resources to mobile UE  701 . As would be understood by one of skill in the art, the components of the uplink resource manager may involve the physical (PHY) layer and/or the Media Access Control (MAC) layer of the transceiver  713 . Transceiver  713  includes at least one receiver  715  for receiving transmissions from various UEs within range of eNodeB  702  and at least one transmitter  716  for transmitting data and control information to the various UEs within range of eNodeB  702 . 
     The uplink resource manager executes instructions that control the operation of transceiver  713 . Some of these instructions may be located in memory  711  and executed when needed on processor  710 . The resource manager controls the transmission resources allocated to each UE  701  served by eNodeB  702  and broadcasts control information via the PDCCH. 
     Symbol processing circuitry  712  performs demodulation using known techniques. Random access signals are demodulated in symbol processing circuitry  712 . 
     During transmission and reception of voice data or other application data, receiver  715  may receive a random access signal from a UE  701 . The random access signal is encoded to request a message size that is preferred by UE  701 . UE  701  determines the preferred message size by using a message threshold provided by eNodeB  702 . In this embodiment, the message threshold calculation is embodied by executing instructions stored in memory  711  by processor  710 . In other embodiments, the threshold calculation may be embodied by a separate processor/memory unit, by a hardwired state machine, or by other types of control logic, for example. Alternatively, in some networks the message threshold is a fixed value that may be stored in memory  711 , for example. In response to receiving the message size request, eNodeB  702  schedules an appropriate set of resources and notifies UE  701  with a resource grant. 
     Many modifications and other embodiments of the invention(s) will come to mind to one skilled in the art to which the invention(s) pertain having the benefit of the teachings presented in the foregoing descriptions, and the associated drawings. Therefore, it is to be understood that the invention(s) are not to be limited to the specific embodiments disclosed. Although specific terms are employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation.