Patent Publication Number: US-10763933-B1

Title: Precoding method, base station and computing circuit

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. provisional application No. 62/866,642, filed on Jun. 26, 2019, which is incorporated herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present application relates to a precoding method, a base station and a computing circuit, and more particularly, to a precoding method, a base station and a computing circuit capable of enhancing a sum rate with low computation complexity. 
     2. Description of the Prior Art 
     As the fifth generation (5G) wireless networks grow, the millimeter wave (mmWave) communication system becomes a promising solution to increase network capacity. An advantage of the mmWave system is its short wavelength enables plenty of antennas to be packed in a certain physical dimension, allowing for large-scale spatial multiplexing and highly directional beamforming. A design challenge of the mmWave systems is that millimeter wave would experience severe path loss, penetration loss and rain fading as compared to the current cellular bands. 
     Since a number of antennas is large, it is luxurious to allocate one radio frequency (RF) chain dedicated for one antenna. Practically, a number of RF chains is less than a number of (transmit) antennas. Hybrid beamforming, partitioning the signal processing operation into digital precoding and analog precoding, is a new signal processing concept for the mmWave communication system, which can reduce the number of RF chains. To achieve good performance, designing digital precoder and analog precoder in an iterative fashion has been proposed. However, designing digital and analog precoders iteratively is too complicated. 
     Therefore, it is necessary to improve the prior art. 
     SUMMARY OF THE INVENTION 
     It is therefore a primary objective of the present application to provide a precoding method, a base station and a computing circuit capable of enhancing a sum rate with low computation complexity. 
     An embodiment of the present application provides a precoding method, applied in a base station. The base station comprises a computing circuit, an analog precoder, multiple antennas and multiple radio frequency (RF) chains. The analog precoder is coupled between the plurality of antennas and the plurality of RF chains. The base station transmits multiple data streams toward the plurality of users. The precoding method comprises computing a relaxed beamforming matrix according to multiple desired channel correlation matrices and multiple interfering channel correlation matrices, wherein the relaxed beamforming matrix comprises a plurality of relaxed beamforming sub-matrices corresponding to the plurality of users, one relaxed beamforming sub-matrix is generated according to a desired channel correlation matrix and an interfering channel correlation matrix corresponding to one user; computing an approximated beamforming matrix according to the relaxed beamforming matrix, wherein entries within the approximated beamforming matrix have a constant magnitude; computing multiple degradations corresponding to the plurality of data streams according to multiple relaxed beamforming vectors within the relaxed beamforming matrix and multiple approximated beamforming vectors within the approximated beamforming matrix; selecting a selected data stream index according to the plurality of degradations; decomposing a selected relaxed beamforming vector corresponding to the selected data stream index into a first vector and a second vector, wherein entries within the first vector and the second vector have a constant magnitude; and updating the approximated beamforming matrix according to the first vector and augmenting the approximated beamforming matrix according to the second vector, to obtain an updated-and-augmented beamforming matrix; wherein the analog precoder performs a first precoding operation according to the updated-and-augmented beamforming matrix. 
     An embodiment of the present application provides a base station, comprising multiple antennas; multiple radio frequency (RF) chains; an analog precoder, coupled between the antennas and the RF chains, configured to perform a first precoding operation; a computing circuit, configured to perform the following steps: computing a relaxed beamforming matrix according to multiple desired channel correlation matrices and multiple interfering channel correlation matrices, wherein the relaxed beamforming matrix comprises multiple relaxed beamforming sub-matrices corresponding to the users, one relaxed beamforming sub-matrix is generated according to a desired channel correlation matrix and an interfering channel correlation matrix corresponding to one user; computing an approximated beamforming matrix according to the relaxed beamforming matrix, wherein entries within the approximated beamforming matrix have a constant magnitude; computing multiple degradations corresponding to the data streams according to multiple relaxed beamforming vectors within the relaxed beamforming matrix and multiple approximated beamforming vectors within the approximated beamforming matrix; selecting a selected data stream index according to the degradations; decomposing a selected relaxed beamforming vector corresponding to the selected data stream index into a first vector and a second vector, wherein entries within the first vector and the second vector have a constant magnitude; and updating the approximated beamforming matrix according to the first vector and augmenting the approximated beamforming matrix according to the second vector, to obtain an updated-and-augmented beamforming matrix; wherein the analog precoder performs the first precoding operation according to the updated-and-augmented beamforming matrix. 
     An embodiment of the present application provides a computing circuit, disposed within a base station. The base station comprises a computing circuit, an analog precoder, multiple antennas and multiple radio frequency (RF) chains. The analog precoder is coupled between the antennas and the RF chains. The computing circuit comprises a processing unit; and a memory, configured to store a program code, wherein the computing circuit is further configured to perform the following steps: computing a relaxed beamforming matrix according to multiple desired channel correlation matrices and multiple interfering channel correlation matrices, wherein the relaxed beamforming matrix comprises a plurality of relaxed beamforming sub-matrices corresponding to the users, one relaxed beamforming sub-matrix is generated according to a desired channel correlation matrix and an interfering channel correlation matrix corresponding to one user; computing an approximated beamforming matrix according to the relaxed beamforming matrix, wherein entries within the approximated beamforming matrix have a constant magnitude; computing multiple degradations corresponding to the data streams according to multiple relaxed beamforming vectors within the relaxed beamforming matrix and multiple approximated beamforming vectors within the approximated beamforming matrix; selecting a selected data stream index according to the degradations; decomposing a selected relaxed beamforming vector corresponding to the selected data stream index into a first vector and a second vector, wherein entries within the first vector and the second vector have a constant magnitude; and updating the approximated beamforming matrix according to the first vector and augmenting the approximated beamforming matrix according to the second vector, to obtain an updated-and-augmented beamforming matrix; wherein the analog precoder performs a first precoding operation according to the updated-and-augmented beamforming matrix. 
     These and other objectives of the present disclosure will no doubt become obvious to those of ordinary skill in the art after reading the following detailed description of the preferred embodiment that is illustrated in the various figures and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic diagram of a base station according to an embodiment of the present application. 
         FIG. 2  is a schematic diagram of a computing device according to an embodiment of the present application. 
         FIG. 3  is a schematic diagram of a process according to an embodiment of the present application. 
         FIG. 4  is a schematic diagram of a system model of a downlink system. 
         FIG. 5  is a schematic diagram of a system model of a virtual uplink system equivalent to the downlink system in  FIG. 4 . 
     
    
    
     DETAILED DESCRIPTION 
     In this disclosure, the wordings of “beamforming” and “precoding” are used interchangeably. A linear operation (e.g., precoding operation) is represented by its corresponding (precoding) matrix. User k indicates the k-th user. Data stream d indicates the d-th data stream. 
       FIG. 1  is a schematic diagram of a base station  10  according to an embodiment of the present application. The base station  10  may be applied in a millimeter wave (mmWave) communication system, which is configured to transmit a plurality of data streams toward the plurality of users. The term “user” in the present application is referred to “user equipment”, electronic device receiving data transmitted from the base station  10 , which may be mobile phone, tablet computer, wearable device, for example. The base station  10  comprises a computing circuit  12 , an analog precoder  14 , a digital precoder  16 , a plurality of radio frequency (RF) chains  11  and a plurality of antennas  13 . The digital precoder  16  receives the plurality of data streams and is configured to perform a digital precoding operation on a (baseband) data s of the plurality of data streams. The RF chains  11  up-converts the baseband data s toward a radio frequency. The analog precoder  14 , coupled between the RF chains  11  and the antennas  13 , is configured to perform an analog precoding operation on outputs of the RF chains. The antennas  13  transmit outputs of the analog precoder  14  toward the users. The computing circuit  12  is configured to compute an analog beamforming matrix V RF  for the analog precoder  14  to perform the analog precoding operation and compute a digital beamforming matrix V BB  for the digital precoder  16  to perform the digital precoding operation. 
       FIG. 2  is a schematic diagram of the computing circuit  12  according to an embodiment of the present application. The computing circuit  12  may comprise a processing unit  120  and a memory  122 . The memory  122  is configured to store a program code  124  to instruct the processing unit  120  to compute V RF  and V BB . The memory  122  may be a non-volatile memory (NVM), e.g., an electrically erasable programmable read only memory (EEPROM) or a flash memory, and not limited thereto. The processing unit  120  may be a process, e.g., a digital signal processor (DSP) or a central processing unit (CPU), and not limited thereto. 
     In an embodiment, the digital precoder  16  may be realized by ASIC (Application-Specific Integrated Circuit) or DSP. 
     In an embodiment, the analog precoder  14  may be realized by a plurality of phase shifters. In this regard, entries within/of the beamforming matrix V RF  may all have a constant magnitude. In an embodiment, every entry within the beamforming matrix V RF  may have a magnitude equal to 1, i.e., |V RF (p, q)|=1 for all p, q, where V RF (p, q) denotes the (p, q) th  entry of the beamforming matrix V RF . 
     The computing circuit  12  computes V RF  and V BB  to maximize an achievable sum rate, given that the analog precoder  14  is realized by the phase shifters. Mathematically, the computing circuit  12  intends to solve the problem shown in equations (1a)-(1c) below. In this application, solving an optimization problem may be not only referred to “finding a global optimal solution”, but also referred to “approaching an optimum by a sub-optimal solution”. 
     
       
         
           
             
               
                 
                   
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     In equations (1a)-(1c), R k  denotes an achievable rate of a user k (where k is a user index), and x denotes a transmitted signal transmitted by the antennas  13 . Equation (1b) represents a transmission power constraint, and equation (1c) represents a constant magnitude constraint. 
     Specifically, the transmitted signal x can be expressed as 
               x   =         V   RF     ⁢     V   BB     ⁢   s     =       V   RF     ⁢       ∑     k   =   1     K     ⁢       V     BB   ,   k       ⁢     s   k               ,         
where s=[s l   T , . . . , s K   T ] T  and V BB =└V BB,1 , . . . , V BB,K ┘. Signal vectors is a (n S ×1) vector and comprises transmit data of the plurality of data streams corresponding to a specific time, where n s  denotes a number of the plurality of data streams intended for the plurality of users, and K denotes a number of users. Signal vector s k  represents a data vector intended for the user k. V BB  is a (n RF ×n S ) matrix, where n RF  represents a number of the RF chains. V BB,k  represents a beamforming sub-matrix corresponding to the user k. In an embodiment, D k  data stream(s) is(are) provided for the user k and Σ k  D k =n S . In an embodiment, the n S  data streams may be equally provided to the K users, and each user is served by D data streams, i.e., n S =KD. In an embodiment, n S ≤n RF &lt;n T , wherein n T  denotes a number of transmit antennas, i.e., the antennas  13 . It means that, the base station  10  may be equipped with more RF chains than the data streams, and equipped with more antennas than the RF chains.
 
     A received signal y k  at the user k can be expressed as equation (2). H k  denotes channel matrix from the base station  10  to the user k, and n k  denotes noise received by the user k, where n k  may obey complex Gaussian distribution with variance σ n   2 , e.g., n k ˜CN (0, σ n   2 I) . The achievable rate R k  in (1a) can be expressed as equation (3), where C k  in (3) can be expressed as equation (4). 
     
       
         
           
             
               
                 
                   
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     A strategy to solve problem (1) includes, a) assuming V BB  as an initial constant V BB   (0) , substituting V BB   (0)  into equations (1a)-(1c) and solving problem (1) by treating V RF  as variable, in order to obtain a solution of V RF , denoted as V RF   (S) ; b) substituting V RF   (S)  into equations (1a)-(1c) as constant and solving problem (1) by treating V BB  as variable, in order to obtain a solution of V BB , denoted as V BB   (S) . 
     Note that, in paragraphs regarding solving the problem of the present application, the symbols V RF  and V BB  represent matrix variables; in  FIG. 1  and related paragraphs, the symbol V RF /V BB  defines the precoding operation the precoder  14 / 16  performs. In practical precoding operation, the solutions V RF   (S)  and V BB   (S)  are utilized to perform the precoding operations, which means V RF =V RF   (S)  and V BB =V BB   (S)  when the precoding operations are performed. 
     To solve problem (1), it is assumed that V BB   (0) =[γI 0] T , where I within V BB   (0)  is an identity matrix with size n S , 0 is a zero matrix with size (n RF −n S )×n S , which can be viewed that the n S  data streams are bypassed to the first n S  RF chains, and the rest (n RF −n S ) RF chains currently have zero inputs. 
     In addition, by treating V RF  as variable, the problem (1) is still a non-convex optimization problem, since the constant magnitude constraint (1c) is non-convex. One way to deal with such non-convexity is to relax the non-convex constant magnitude constraint and to solve the relaxed problem (1), and therefore a relaxed solution (beamforming matrix) {tilde over (V)} RF  is obtained. Then, the computing circuit  12  may approximate the relax solution {tilde over (V)} RF  by an approximated solution (beamforming matrix) {circumflex over (V)} RF  which satisfies the constant magnitude constraint. 
       FIG. 3  is a schematic diagram of a process  30  according to an embodiment of the present application. The process  30  may be compiled as the program code  124  and executed by the processing unit  120 , to generate V RF  and V BB . 
     Step  302 : Compute a relaxed beamforming matrix according to a plurality of desired channel correlation matrices and a plurality of interfering channel correlation matrices. 
     Step  304 : Compute an approximated beamforming matrix according to the relaxed beamforming matrix. 
     Step  306 : Compute a plurality of degradations corresponding to the plurality of data streams. 
     Step  308 : Select selected data stream indices according to the plurality of degradations 
     Step  310 : Decompose selected relaxed beamforming vectors corresponding to the selected data stream indices into first vectors and second vectors. 
     Step  312 : Update the approximated beamforming matrix according to the first vectors and augment the approximated beamforming matrix according to the second vectors. 
     Step  314 : Compute a digital beamforming matrix according to the updated-and-augmented beamforming matrix. 
     In Step  302 , the computing circuit  12  computes the relaxed beamforming matrix {tilde over (V)} RF  according to a plurality of desired channel correlation matrices G 1 , . . . , G K  and a plurality of interfering channel correlation matrices Q 1 , . . . , Q K . 
     Note that, another issue of solving the relaxed problem (1) is that the precoding designs of maximizing sum rate for all users are inter-related, and therefore the computation complexity is barely affordable. Fortunately, the well-known uplink-downlink duality can be exploited to decouple the precoding designs for different users. That is, the relaxed problem (1) can be transformed into K independent problems. Each individual problem is corresponding to one user to maximize a virtual/equivalent uplink signal-to-interference-plus-noise ratio (SINR). 
     The uplink-downlink duality is known in the art.  FIG. 4  depicts a system model of a downlink system, and  FIG. 5  depicts a virtual uplink system which is equivalent to the downlink system depicted in  FIG. 4 . 
     In the uplink direction, a virtual uplink received signal corresponding to the user k can be expressed as equation (5). In (5), γ is referred to power allocated to each data stream and γ may be assumed to be γ=P max /n RF  for simplicity. V RF,k  can be viewed as a beamforming sub-matrix corresponding to the user k, where V RF =└ 8  V RF, 1 , . . . , V RF,K ┘. Given (5), a virtual uplink SINR can be expressed as equation (6), where s k   (UL) ˜CN(0, I) is assumed. In (6), G k =γ 2 H k   H H k  is proportional to a correlation matrix of the desired channel H k  (in the user k&#39;s perspective), and named as desired channel correlation matrix (corresponding to the user k). Q k  (corresponding to the user k) is named as interfering channel correlation matrix, which is expressed as equation (7). Mathematically, the relaxed SINR maximization problem for the user k can be formulated as equation (8). 
     
       
         
           
             
               
                 
                   
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     In Step  302 , the computing circuit  12  may solve the relaxed/unconstrained problem (8) for all users, and therefore obtains relaxed solutions (beamforming sub-matrices) {tilde over (V)} RF,1 , . . . , {tilde over (V)} RF,K . Hence, the computing circuit  12  would obtain the relaxed beamforming matrix {tilde over (V)} RF =[{tilde over (V)} RF,1 , . . . , {tilde over (V)} RF,K ]. 
     In an embodiment, the problem (8) can be reformulated into a QCQP (quadratically constrained quadratic program), which can be expressed as equations (9a)-(9b), where {tilde over (V)} RF,k   [d     k]    denotes a beamforming vector corresponding to the d k -th data stream of the user k within the beamforming sub-matrix {tilde over (V)} RF,k ,as equation (10) shows. 
     
       
         
           
             
               
                 
                   
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                   10 
                   ) 
                 
               
             
           
         
       
     
     Solving the problem (9) is to equate a first order derivative of a Lagrangian function of the problem (9) to 0, which is expressed in (11). The problem (9) would turn into a generalized eigenvalue problem expressed in (12). In other words, if {tilde over (V)} RF,k   [d     k]    satisfies (12), then {tilde over (V)} RF,k   [d     k]    is an optimal solution of the problem (9). In addition, if {tilde over (V)} RF,k   [d     k ]    satisfies (12), then {tilde over (V)} RF,k   [d     k]    is a generalized eigenvector of a matrix pair (C k , Q k ) corresponding eigenvalue λ d , where the matrix pair (C k , Q k ) comprises C k  and Q k . Note that, λ d , which plays roles of both Lagrangian multiplier and eigenvalue, can be viewed as an objective virtual uplink SINR corresponding to the d-th data stream of user k, which can be understood by multiplying both sides of equation (12) by ({tilde over (V)} RF,k   [d     k ]   ) H  and dividing the multiplied both sides by 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           V 
                           ~ 
                         
                         
                           RF 
                           , 
                           k 
                         
                         
                           [ 
                           
                             d 
                             k 
                           
                           ] 
                         
                       
                       ) 
                     
                     H 
                   
                   ⁢ 
                   
                     Q 
                     k 
                   
                   ⁢ 
                   
                     
                       
                         V 
                         ~ 
                       
                       
                         RF 
                         , 
                         k 
                       
                       
                         [ 
                         
                           d 
                           k 
                         
                         ] 
                       
                     
                     . 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         ∂ 
                         
                           
                             V 
                             ~ 
                           
                           
                             RF 
                             , 
                             k 
                           
                           
                             [ 
                             
                               d 
                               k 
                             
                             ] 
                           
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   V 
                                   ~ 
                                 
                                 
                                   RF 
                                   , 
                                   k 
                                 
                                 
                                   [ 
                                   
                                     d 
                                     k 
                                   
                                   ] 
                                 
                               
                               ) 
                             
                             H 
                           
                           ⁢ 
                           
                             G 
                             k 
                           
                           ⁢ 
                           
                             
                               V 
                               ~ 
                             
                             
                               RF 
                               , 
                               k 
                             
                             
                               [ 
                               
                                 d 
                                 k 
                               
                               ] 
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 λ 
                                 d 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     V 
                                     ~ 
                                   
                                   
                                     RF 
                                     , 
                                     k 
                                   
                                   
                                     [ 
                                     
                                       d 
                                       k 
                                     
                                     ] 
                                   
                                 
                                 ) 
                               
                             
                             H 
                           
                           ⁢ 
                           
                             Q 
                             k 
                           
                           ⁢ 
                           
                             
                               V 
                               ~ 
                             
                             
                               RF 
                               , 
                               k 
                             
                             
                               [ 
                               
                                 d 
                                 k 
                               
                               ] 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       G 
                       k 
                     
                     ⁢ 
                     
                       
                         V 
                         ~ 
                       
                       
                         RF 
                         , 
                         k 
                       
                       
                         [ 
                         
                           d 
                           k 
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       λ 
                       d 
                     
                     ⁢ 
                     
                       Q 
                       k 
                     
                     ⁢ 
                     
                       
                         V 
                         ~ 
                       
                       
                         RF 
                         , 
                         k 
                       
                       
                         [ 
                         
                           d 
                           k 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     In an embodiment, Step  302  comprises finding the generalized eigenvector of matrices C k  and Q k , which is expressed in (13), where eig (d) (A,B) returns a generalized eigenvector corresponding to the d-th large generalized eigenvalue of the matrices A and B.
 
 {tilde over (V)}   RF,k   [d     k]   =eig (d     k     ) ( G   k   ,Q   k )   (13)
 
     Further, in Step  302 , the computing circuit  12  computes equation (13) for all d k =1, . . . , D k , to obtain {tilde over (V)} RF,k  according to (10), and computes equation (13) for all k, to obtain {tilde over (V)} RF  as {tilde over (V)} RF =[V RF,1 , . . . , {tilde over (V)} RF,K ]. Thus, the computing circuit  12  may obtain the relaxed beamforming matrix {tilde over (V)} RF . 
     Note that, the beamforming matrix {tilde over (V)} RF  is a solution comprising the beamforming sub-matrices {tilde over (V)} RF,k  which maximize the objectives SINR k   (UL) . However, the solution {tilde over (V)} RF  is usually not feasible for implementing the analog precoder  14  by phase shifters, since {tilde over (V)} RF  usually does not satisfy the constant magnitude constraint. 
     In this regard, the computing circuit  12  may further approach {tilde over (V)} RF  by an approximated beamforming matrix {circumflex over (V)} RF  to satisfy the constant magnitude constraint. The computing circuit  12  may find {circumflex over (V)} RF  which is closest to {tilde over (V)} RF  and satisfies the constant magnitude constraint, e.g., equation (1c). 
     In an embodiment, the computing circuit  12  may turn to solve problem (14), and the problem (14) is equivalent to problem (15) because of equation (14b). In addition, a global optimal solution of the problem (15) can be expressed as equation (16a), where arg(X) is an element-wise argument/phase operator. 
     
       
         
           
             
               
                 
                   
                     min 
                     
                       
                         V 
                         ^ 
                       
                       RF 
                     
                   
                   ⁢ 
                   
                      
                     
                       
                         
                           V 
                           ~ 
                         
                         RF 
                       
                       - 
                       
                         
                           V 
                           ^ 
                         
                         RF 
                       
                     
                      
                   
                 
               
               
                 
                   ( 
                   
                     14 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       s 
                       . 
                       t 
                       . 
                       
                           
                       
                       ⁢ 
                       
                          
                         
                           
                             
                               V 
                               ^ 
                             
                             RF 
                           
                           ⁡ 
                           
                             ( 
                             
                               p 
                               , 
                               q 
                             
                             ) 
                           
                         
                          
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     ∀ 
                     p 
                   
                   , 
                   q 
                 
               
               
                 
                   ( 
                   
                     14 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     min 
                     
                       
                         V 
                         ^ 
                       
                       RF 
                     
                   
                   ⁢ 
                   
                     tr 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             V 
                             ^ 
                           
                           RF 
                           H 
                         
                         ⁢ 
                         
                           
                             V 
                             ~ 
                           
                           RF 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     15 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       s 
                       . 
                       t 
                       . 
                       
                           
                       
                       ⁢ 
                       
                          
                         
                           
                             
                               V 
                               ^ 
                             
                             RF 
                           
                           ⁡ 
                           
                             ( 
                             
                               p 
                               , 
                               q 
                             
                             ) 
                           
                         
                          
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     ∀ 
                     p 
                   
                   , 
                   q 
                 
               
               
                 
                   ( 
                   
                     15 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       V 
                       ^ 
                     
                     RF 
                   
                   = 
                   
                     exp 
                     ⁢ 
                     
                       { 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           arg 
                           ⁡ 
                           
                             ( 
                             
                               
                                 V 
                                 ~ 
                               
                               RF 
                             
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   
                     16 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     In another embodiment, for finite resolution phase shifters within the analog precoder  14 , the computing circuit  12  may compute equation (16b) to obtain a quantized value b and use the quantized value b and equation (16c) to obtain {circumflex over (V)} RF , where B represents a number of bits corresponding to the finite resolution phase shifter and BS={2 u |u=0, . . . , B}. 
     
       
         
           
             
               
                 
                   
                     b 
                     ⁡ 
                     
                       ( 
                       
                         p 
                         , 
                         q 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         arg 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         min 
                       
                       
                         b 
                         ∈ 
                         BS 
                       
                     
                     ⁢ 
                     
                        
                       
                         
                           arg 
                           ⁡ 
                           
                             ( 
                             
                               
                                 
                                   V 
                                   ~ 
                                 
                                 RF 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   p 
                                   , 
                                   
                                       
                                   
                                   ⁢ 
                                   q 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             b 
                           
                           
                             2 
                             B 
                           
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   
                     16 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         V 
                         ^ 
                       
                       RF 
                     
                     ⁡ 
                     
                       ( 
                       
                         p 
                         , 
                         q 
                       
                       ) 
                     
                   
                   = 
                   
                     exp 
                     ⁢ 
                     
                       { 
                       
                         j 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               π 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               b 
                             
                             
                               2 
                               B 
                             
                           
                           ) 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   
                     16 
                     ⁢ 
                     c 
                   
                   ) 
                 
               
             
           
         
       
     
     Note that, performing (16a) can be viewed as one kind of element-wise approximation operation on the relaxed beamforming matrix {tilde over (V)} RF , while performing (16b) can be viewed as another. Performing (16a) and/or (16b) can guarantee that |arg({tilde over (V)} RF (p, q))−arg({circumflex over (V)} RF (p, q))|&lt;ε, where ε stands for a certain threshold. 
     In short, in Step  304 , the computing circuit  12  computes the approximated beamforming matrix {circumflex over (V)} RF  by equation (16). Note that, after the approximation in Step  304  is performed, {circumflex over (V)} RF  is no longer optimal. In other words, SINR k   (UL) ({circumflex over (V)} RF )&lt;SINR k   (UL) ({tilde over (V)} RF ). A difference between SINR k   (UL) ({tilde over (V)} RF ) and SINR k   (UL) ({circumflex over (V)} RF ) e.g., SINR k   (UL) ({tilde over (V)} RF )−SINR k   (UL) ({circumflex over (V)} RF ) may be viewed as a kind of SINR degradation caused by the approximation in Step  304 . 
     Note that, in Steps  302  and  304 , {tilde over (V)} RF  and {circumflex over (V)} RF  are assumed to be (n T ×n S ) matrices, which means that outputs of the first n S  RF chains are (about to be) precoded (via {circumflex over (V)} RF ), given n RF &gt;n S . Furthermore, outputs of the rest (n RF −n S ) RF chains may be further precoded to compensate the SINR loss/degradation of the approximation made by Step  304 . 
     In this regard, the computing circuit  12  may select the data streams whose SINR degrades worst due to the approximation stated in Step  302 , and incorporate the outputs of the rest (n RF −n S ) RF chains into the precoding operation of the analog precoder  14  to compensate SINR degradation for those data streams. 
     In Step  306 , the computing circuit  12  may compute a degradation Δ d  corresponding to the data stream d, where d is data stream index. The degradation Δ d  may be computed as equation (17). In (17), the data stream d represents the d k -th data stream of user k, and d herein may be represented as 
             d   =           ∑       k   ′     =   1       k   -   1       ⁢     D     k   ′         +       d   k     ⁢           ⁢   or   ⁢           ⁢   d       =           (     k   -   1     )     ⁢   D     +       d   k     ⁢           ⁢   when   ⁢           ⁢     D   k         =     D   ⁢     ∀     k   .                   
The computing circuit  12  may compute equation (17) for all d=1, . . . , n S , and therefore obtain degradations Δ 1 , . . . , Δ nS  for all d=1, . . . , n S .
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           Δ 
                           d 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               SINR 
                               k 
                               
                                 ( 
                                 UL 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   V 
                                   ~ 
                                 
                                 
                                   RF 
                                   , 
                                   K 
                                 
                                 
                                   [ 
                                   
                                     d 
                                     k 
                                   
                                   ] 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               SINR 
                               k 
                               
                                 ( 
                                 UL 
                                 ) 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   V 
                                   ^ 
                                 
                                 
                                   RF 
                                   , 
                                   k 
                                 
                                 
                                   [ 
                                   
                                     d 
                                     k 
                                   
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               λ 
                               d 
                             
                             - 
                             
                               
                                 tr 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             V 
                                             ~ 
                                           
                                           
                                             RF 
                                             , 
                                             k 
                                           
                                           
                                             [ 
                                             
                                               d 
                                               k 
                                             
                                             ] 
                                           
                                         
                                         ) 
                                       
                                       H 
                                     
                                     ⁢ 
                                     
                                       G 
                                       k 
                                     
                                     ⁢ 
                                     
                                       V 
                                       
                                         RF 
                                         , 
                                         k 
                                       
                                       
                                         [ 
                                         
                                           d 
                                           k 
                                         
                                         ] 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               
                                 tr 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             V 
                                             ~ 
                                           
                                           
                                             RF 
                                             , 
                                             k 
                                           
                                           
                                             [ 
                                             
                                               d 
                                               k 
                                             
                                             ] 
                                           
                                         
                                         ) 
                                       
                                       H 
                                     
                                     ⁢ 
                                     
                                       Q 
                                       k 
                                     
                                     ⁢ 
                                     
                                       V 
                                       
                                         RF 
                                         , 
                                         k 
                                       
                                       
                                         [ 
                                         
                                           d 
                                           k 
                                         
                                         ] 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         , 
                         where 
                       
                     
                   
                   
                     
                       
                         
                           λ 
                           d 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     V 
                                     ~ 
                                   
                                   
                                     RF 
                                     , 
                                     k 
                                   
                                   
                                     [ 
                                     d 
                                     ] 
                                   
                                 
                                 ) 
                               
                               H 
                             
                             ⁢ 
                             
                               G 
                               k 
                             
                             ⁢ 
                             
                               
                                 V 
                                 ~ 
                               
                               
                                 RF 
                                 , 
                                 k 
                               
                               
                                 [ 
                                 d 
                                 ] 
                               
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     V 
                                     ~ 
                                   
                                   
                                     RF 
                                     , 
                                     k 
                                   
                                   
                                     [ 
                                     d 
                                     ] 
                                   
                                 
                                 ) 
                               
                               H 
                             
                             ⁢ 
                             
                               Q 
                               k 
                             
                             ⁢ 
                             
                               
                                 V 
                                 ~ 
                               
                               
                                 RF 
                                 , 
                                 k 
                               
                               
                                 [ 
                                 d 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Note that, SINR k   (UL) ({tilde over (V)} RF,k   [d     k     ] ) and λ d  represent an optimal objective value according to the objective and the relaxed beamforming vector {tilde over (V)} RF,k   [d     k     ] ) corresponding to the data stream d, and SINR k   (UL) ({circumflex over (V)} RF,k   [d     k     ] ) represents an approximated objective value according to the objective and the approximated beamforming vector {circumflex over (V)} RF,k   [d     k     ] corresponding to the data stream d. 
     Among the degradations Δ d  for d=1, . . . , n S , in Step  308 , the computing circuit  12  may select (n RF −n S ) data stream indices corresponding to the (n RF −n S ) largest degradations. Suppose that entries within a set {Δ d |d=1, . . . , n S } is sorted and ordered such that a series of degradations Δ &lt;1&gt;  . . . ≥Δ &lt;nS&gt;  comply with Δ &lt;1&gt; ≥Δ &lt;2&gt; ≥ . . . ≥Δ &lt;nS&gt; . In such case, the computing circuit  12  may select data stream indices &lt;1&gt;, . . . , &lt;(n RF −n S )&gt; in Step  308 , and form an index set S={&lt;1&gt;, . . . , &lt;(n RF −n S )&gt;}. 
     It is known that a non-magnitude-1 complex number c=re jα  with 0≤r≤2 can be decomposed into two magnitude-1 complex number e jθ  and e jϕ . That is, c=re jα  (with 0≤r≤2) can be represented/decomposed as c=e jθ +e jϕ . Given that, in Step  310 , the computing circuit  12  decomposes a selected relaxed beamforming vector {tilde over (V)} RF,k   [d]  for d ∈S (i.e., corresponding a selected data stream index) into a first vector a d  and a second vector b d , such that V RF,k   [d] =a d +b d , where a d  and b d  comprises entries with magnitude-1. Supposed that V RF,k   [d] =[r 1 e jα     1   , . . . ,r n e jα     nT   ] T  (where {tilde over (V)} RF,k    [d]  may be normalized to satisfy 0≤r≤2), a d  and b d  can be expressed as a d =[e jθ     1   , . . . , e jθ     nT   ] T  and b d =[e jϕ     1   , . . . , e jϕ     nT   ] T  (wherein nT within the above equations of a d  and b d  means n T ), the entries within a d  and b d  can be obtained by θ m =α m +cos −1 (r m /2) and ϕ m =α m −cos −1 (r m /2). In addition, the decomposition of {tilde over (V)} RF,k   [d] =a d +b d  may be performed for all d ∈S (or equivalently, {tilde over (V)} RF,k   [&lt;t&gt;] =a &lt;t&gt; +b &lt;t&gt; , &lt;t&gt;∈S, where t denotes an index of the ordered degradation set S). 
     In Step  312 , the computing circuit  12  updates the approximated beamforming matrix according to the first vectors a d  and augments the approximated beamforming matrix according to the second vectors b d . For example, supposed that the approximated beamforming matrix {circumflex over (V)} RF  may be expressed as {circumflex over (V)} RF =└{circumflex over (V)} RF   [1] , . . . , {circumflex over (V)} RF   [d] , . . . , {circumflex over (V)} RF   [nS] ┘ (wherein nS within the equation of {circumflex over (V)} RF  above means n S ) for some d ∈S, the computing circuit  12  may replace the approximated beamforming vector {circumflex over (V)} RF   [d]  with the first vector a d  and augments {circumflex over (V)} RF  by the second b d . Taking d=&lt;1&gt; (the first element within the set S) as an example, the computing circuit  12  may obtain a temporary beamforming matrix as └{circumflex over (V)} RF   [1] , . . . , a d , . . . , {circumflex over (V)} RF   [nS] , b d ┘. 
     The updating and augmenting operation may be performed for all d ∈S. Finally, the computing circuit  12  may obtain an updated-and-augmented beamforming matrix V RF,UnA , where the d-th column of V RF,UnA , denoted as V RF,UnA   [d]  may be expressed as equation (18). In (18) 
     
       
         
           
             
               
                 
                   
                     V 
                     
                       RF 
                       , 
                       UnA 
                     
                     
                       [ 
                       d 
                       ] 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 V 
                                 ^ 
                               
                               RF 
                               
                                 [ 
                                 d 
                                 ] 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               d 
                               ≤ 
                               
                                 n 
                                 s 
                               
                             
                             , 
                             
                               d 
                               ∉ 
                               S 
                             
                             , 
                           
                         
                       
                       
                         
                           
                             
                               
                                 a 
                                 d 
                               
                               = 
                               
                                 a 
                                 
                                   〈 
                                   t 
                                   〉 
                                 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               d 
                               ≤ 
                               
                                 n 
                                 s 
                               
                             
                             , 
                             
                               d 
                               = 
                               
                                 
                                   〈 
                                   t 
                                   〉 
                                 
                                 ∈ 
                                 S 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               b 
                               
                                 〈 
                                 t 
                                 〉 
                               
                             
                             , 
                           
                         
                         
                           
                             d 
                             = 
                             
                               
                                 n 
                                 s 
                               
                               + 
                               t 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     The updated-and-augmented beamforming matrix V RF,UnA  may be a solution V RF   (S)  obtained by the computing circuit  12 . In Step  314 , the computing circuit  12  may substitute V RF =V RF   (S) =V RF,unA  into equations (1a)-(1c) as constant and solve problem (1) by treating V BB  as variable, in order to obtain a solution of V BB   (S) , a digital precoding matrix, such that the digital precoder  16  may perform the precoding operation according to the digital precoding matrix V BB   (s) . 
     Specifically, in Step  314 , the computing circuit  12  may compute effect channels  H   k =H k V RF  from the base station  10  to the user k for all k and solve the problem expressed in (19). Solving the problem (19) involves zero-forcing beamforming/precoding and water-filling power allocation techniques, which are known in the art and not narrated herein for brevity. 
     
       
         
           
             
               
                 
                   
                     max 
                     
                       V 
                       BB 
                     
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       K 
                     
                     ⁢ 
                     
                        
                       
                         I 
                         + 
                         
                           
                             
                               
                                 H 
                                 _ 
                               
                               k 
                             
                             ⁢ 
                             
                               V 
                               
                                 BB 
                                 , 
                                 k 
                               
                             
                             ⁢ 
                             
                               V 
                               
                                 BB 
                                 , 
                                 k 
                               
                               H 
                             
                             ⁢ 
                             
                               
                                 H 
                                 _ 
                               
                               k 
                               H 
                             
                           
                           
                             
                               ∑ 
                               
                                 
                                   
                                     k 
                                     ′ 
                                   
                                   = 
                                   1 
                                 
                                 , 
                                 
                                   
                                     k 
                                     ′ 
                                   
                                   ≠ 
                                   k 
                                 
                               
                               K 
                             
                             ⁢ 
                             
                               
                                 
                                   H 
                                   _ 
                                 
                                 k 
                               
                               ⁢ 
                               
                                 V 
                                 
                                   BB 
                                   , 
                                   
                                     k 
                                     ′ 
                                   
                                 
                               
                               ⁢ 
                               
                                 V 
                                 
                                   BB 
                                   , 
                                   
                                     k 
                                     ′ 
                                   
                                 
                                 H 
                               
                               ⁢ 
                               
                                 
                                   H 
                                   _ 
                                 
                                 k 
                                 H 
                               
                             
                           
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   
                     19 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     s 
                     . 
                     t 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       tr 
                       ⁡ 
                       
                         ( 
                         
                           xx 
                           H 
                         
                         ) 
                       
                     
                   
                   ≤ 
                   
                     P 
                     max 
                   
                 
               
               
                 
                   ( 
                   
                     19 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Performing the process  30  may achieve better performance in terms of sum rate (which outperforms over an existing low-complexity hybrid precoding in massive multiuser MIMO solution, published by Liang et al. in  IEEE Wireless Communication Letter , pp. 653-656, 2014 and an existing iterative hybrid precoding solution, published by Sohrabi et al. in  IEEE Journal Selected Topics in Signal Processing , vol. 10(3), pp. 501-516, 2016, denoted as Iterative solution) while consume lower computation complexity, compared to the Iterative solution). 
     In summary, the present application relaxes the constant magnitude constraint and utilizes the uplink-downlink duality to solve the unconstrained maximum sum rate precoding problem to obtain the relaxed beamforming matrix, approximates the relaxed beamforming matrix to obtain the approximated beamforming matrix, chooses the data stream with objective SINR performance degrades worst, and exploits extra RF chain to compensate the degradation. 
     Those skilled in the art will readily observe that numerous modifications and alterations of the device and method may be made while retaining the teachings of the invention. Accordingly, the above disclosure should be construed as limited only by the metes and bounds of the appended claims.