Efficient GSVD Based Multi-user MIMO Linear

Efficient GSVD Based Multi-user MIMO Linear Precoding and Antenna Selection SchemeJaehyun Park∗,Joohwan Chun∗,and Haesun Park†∗Division of Electrical Engineering,School of Electrical Engineering and Computer ScienceKorea Advanced Institute of Science and Technology,Daejeon,KoreaEmail:{jhpark,chun}@sclab.kaist.ac.kr†Division of Computational Science and Engineering,College of ComputingGeorgia Institute of Technology,Atlanta,Georgia30332Email:hpark@Abstract—We present a multi-user multiple-input multiple-output(MU-MIMO)precoding scheme utilizing generalized sin-gular value decomposition(GSVD).Our work is motivated by the precoding scheme developed by Sadek,Tarighat,and Sayed that maximizes the signal to leakage and noise ratio(SLNR),in which the precoding weight is obtained by the generalized eigenvalue decomposition(GEVD).However,the covariance matrix utilized in GEVD becomes close to be singular as the signal to noise ratio goes high.To improve the numerical accuracy,a GSVD based algorithm is exploited in this paper and a novel method is derived to compute the precoding weights for multiple users by removing redundant computational loads rather than repeating the GSVD algorithm for each user.In addition,we propose a technique to speed up the GSVD based precoding algorithm by preprocessing using the QR decomposition.Finally,to improve the system performance,the antenna selection scheme using GSVD is also proposed which does not require the exhaustive search to choose the active antennas.I.I NTRODUCTIONRecently,multiple-input multiple-output(MIMO)broad-casting scenario with the multiple antennas at the transmit side and the multi-user(MU)at the receive side has been studied extensively since the higher performance can be achieved by simply increasing the number of antennas at the transmit side [1].Previously,due to the simplicity in implementation and good performance,several linear precoding schemes have been proposed to suppress the cochannel interference[2]–[4].In [4],an iterative method is developed to compute precoding weights.In[2,3],a closed-form solution is proposed by maximizing the signal to leakage and noise ratio(SLNR)[2,3] (or the signal to interference and noise ratio at the transmitter (Tx-SINR)in[5]).SLNR maximizing precoding weight can be computed by the generalized eigenvalue decomposition (GEVD)of two covariance matrices of each user’s channel and interferences’channel plus noise.However computation of GEVD may become numerically unstable because the covariance matrix of interference and noise becomes closer to be singular as the signal to noise ratio(SNR)goes high. In this paper,we propose a GSVD based precoding algo-rithm using generalized singular value decomposition(GSVD) for MU-MIMO system,which produces numerically better and computationally fast solution.The GSVD algorithm developed in[6,7]resolves the singularity problem in the linear discrim-inant analysis(LDA)for high dimensional(or undersampled) data successfully.By modifying the LDA/GSVD algorithm for MU-MIMO,we show how the singularity in covariance matrix can be handled properly.In addition,we show how the GSVD algorithm can be utilized to compute the precoding weights for all users rather than repeating the GSVD for each user.Finally,we also propose the method to speed up our precoding algorithm when the number of transmit antennas is greater than the sum of all users’receive antennas,in which the dimension of precoding weight is reduced from preprocessing by the QR decomposition(QRD)[8].In[3],when applying the SLNR criterion to design the linear MU-MIMO precoding, the probability density function(PDF)of SLNR and its mean are derived for the various numbers of receive antennas,which shows that the SLNR decreases when the number of receive antennas increases.Similarly,the performance of the proposed scheme may be degraded when the number of receive antennas increases since the increase in the number of the receive antennas causes the interferences to other users.Accordingly, in this paper,the antenna selection scheme using GSVD is developed to reduce the number of the active antennas for each user.The rest of this paper is organized as follows.In Section II the system model considered in this paper is described and in Section III SLNR maximizing precoding scheme is reviewed briefly.In Section IV we review the GSVD due to Paige and Saunders[9].We then propose MU-MIMO precoding scheme based on GSVD for single user and modify it for multiple users.Furthermore,the fast GSVD based precoding algorithms using QRD is developed.In Section V the antenna selection scheme using GSVD is proposed.Several simulation results, showing the bit error rate(BER)curves for various parameters, are given in Section VI.Concluding remarks are made in Section VII.The superscripts in A T,A H,and A†denote,respectively, the transposition,conjugate transposition,and pseudo-inverse of a matrix A. A F and tr(A)denote Frobenious norm and trace of A,respectively.I K and0K,M denote a K×K identity matrix and a zero K×M matrix,respectively. diag{a1,...,a N}denotes a diagonal matrix with diagonal978-1-4244-3435-0/09/$25.00 ©2009 IEEEelements{a1,...,a N}.Finally A(i:j,k:l)denotes a submatrix of A with elements from the i th row to the j th row and from the k th column to the l th column.II.S YSTEM MODELWe consider N transmit antennas at base station and K users with M receive antennas per each user.The channel is assumed to be frequencyflat fading and constant over a transmission frame duration.The received signal for the k th user can then be represented asy k=H k x+n k,H k∈C M ×N,x∈C N×1(1)where y k=y(k)1,...,y(k)MT.The precoded transmit signalfor all users is denoted as x and n k is the complex whiteGaussian noise vector having a varianceσ2.Here,entries of H k are assumed to be complex Gaussian variables with zero-mean and unit-variance.Concatenating the received signals ofall users,we can rewrite(1)as:y=Hx+n,(2)where y=⎡⎢⎣y1...y K⎤⎥⎦,n=⎡⎢⎣n1...n K⎤⎥⎦,H=⎡⎢⎣H1...H K⎤⎥⎦,(3)and M=KM .Since we only consider the linear precoding scheme,the precoded signal x can be written as:x=Kk=1W k s k,W k∈C N×m k,s k∈C m k×1(4)where s k and W k are,respectively,the data stream vector and the precoding matrix for the k th user with m k≤M .Here we assume that s k and W k are normalized as follows:E[s k s H k]=I m k,tr(W k W H k)=1,for k=1,...,K.(5) III.SLNR MAXIMIZING P RECODING SCHEMEIn this section,we briefly review the precoding scheme to maximize SLNR[2],which gives the closed form solution. The SLNR of each user is defined as the ratio between the desired signal power summation for each user and the interference introduced to the other users and noise power summation as follows[2]:J(W k)=H k W k 2FKi=1,i=kH i W k 2F+M σ2=trW H k H H k H k W ktrW H kM σ2I N+˜H H k˜H kW k,(6)with the constraint that tr(W H k W k)=1.Here,˜H k is defined as[H1;...;H k−1;H k+1;...;H K](∈C(M−M )×N),which is the sub-channel of all users except the k th user.Since M σ2I N+˜H H k˜H k is symmetric positive definite, there exists a nonsingular matrix X k∈C N×N such that[10] X H k H H k H k X k=Λk=Diag{λ1,k,λ2,k,...,λN,k}X H kM σ2I N+˜H H k˜H kX k=I N,(7) withλ1,k≥λ2,k≥...≥λN,k≥0.Therefore,the equation(6)can be rewritten asJ(W k)=trW H k X−H kΛk X−1k W ktrW H k X−H k X−1k W k.(8)We then set˜W k=X−1k W k and substitute its thin SVD(= U kΣk V H k,where U k∈C N×m k andΣk,V k∈C m k×m k) into(8)asJ(W k)=trV kΣH k U H kΛk U kΣk V H ktrV kΣH k U H k U kΣk V H k=trΣH k U H kΛk U kΣkmki=1d2i,k,(9)where d i,k is the i th diagonal element ofΣk.Note that J(W k)is independent of V k.By using the approach in[2], we can obtain U k maximizing J(W k)asU k=I mk0N−mk,m k,(10)and{d1,k,...,d mk,k}={1,...,1},so that W k is given by the first m k rows of X H k multiplied by the scaling factor to satisfy tr(W k W H k)=m k,i.e.,[2]W k=αk G k,G k∈C N×m k,(11) whereαk is proper scaling factor for the k th user and the columns of G k are the eigenvectors corresponding to the m k maximum generalized eigenvalues of the matrix pairH H k H k,M σ2I N+˜H H k˜H k.The solution of the above method tends to be numerically less accurate especially whenM σ2I N+˜H H k˜H kbecomes close to be singular at high SNR.In the next section we introduce the GSVD developed by Paige and Saunders[9],which is applied to circumvent the nonsingularity problem in LDA[6,7]and propose a precoding scheme based on GSVD that offers better numerical stability. The idea is based on the observation that(11)can be obtained by the GSVD of the matrix pair(H k,[˜H k;√M σI N]),sinceM σ2I N+˜H H k˜H k=˜H H k√M σI N˜H k√M σI N.(12) IV.G ENERALIZED S INGULAR V ALUE D ECOMPOSITION AND MU-MIMO PRECODING SCHEME BASED ON GSVD A.Generalized Singular Value DecompositionAlthough there are several kinds of GSVD formulations[10, 11],we review the GSVD algorithm due to Paige and Saunders [9],which has a more complicated but a less restrictive form. Theorem1:[Paige and Saunders’s GSVD]Suppose any two matrices K b∈C m×n and K w∈C p×n,with the same number of columns,are given.Then there exist unitarymatrices U ∈C m ×n ,V ∈C p ×p ,and a nonsingular matrix X ∈C n ×n such thatU H K b X =(Σb ,0)and V H K w X =(Σw ,0),(13)where t =rank K bK wandΣb m ×t =⎛⎝I r D b 0b ⎞⎠,Σw p ×t=⎛⎝0wD w I t −r −s ⎞⎠,where r =rank K bK w −rank (K w ),s =rank (K b )+rank (K w )−rankK bK w,and the matrices 0b ∈R (m −r −s )×(t −r −s )and 0w ∈R (p −t +r )×r are zero ma-trices with possibly no rows or no columns.The ma-trices,D b =diag (αr +1,...,αr +s )∈R s ×s and D w =diag (βr +1,...,βr +s )∈R s ×s satisfy 1>αr +1≥...≥αr +s >0,0<βr +1≤...≤βr +s <1,and α2i+β2i =1for i =r +1,...,r +s . According to the constructive proof of Theorem 1by Paige and Saunders,the computation of the GSVD starts with the complete orthogonal decomposition [10]of K ≡ K bK w,i.e.,P HK KQ K = R K 000,(14)where P K ∈C (m +p )×(m +p )and Q K ∈C n ×n are unitary and R K ∈C t ×t is nonsingular.It then proceeds with the SVDs of submatrices of P K ,partitioned asP K = P 11P 12P 21P 22 ,P 11∈C m ×t ,P 21∈C p ×t .(15)Note that P 11 2≤1and hence the singular values of P 11do not exceed one.We can then write the SVD of P 11as U H P 11W =Σb ,where U ∈C m ×m and W ∈C t ×t are unitary and Σb has the form shown in (13).Then,P 21W is decomposed as P 21W =VL ,where V ∈C p ×p is unitary and L ∈C p ×t is lower triangular which can be obtainedin the similar way as QRD.Since P 11P 21 has orthonormalcolumns,the columns ofΣbLshould be also orthonormal,which implies L =Σw .Using these results,we can get the following equation:K bK w Q K =P 11R K 0P 21R K 0 = UΣb W H R K 0VΣw W H R K 0.(16)Here,the first t columns of X are given by Q K R −1K W.Defining αi =1,βi =0for i =1,...,r and αi =0,βi =1for i =r +s +1,...,t ,we can easily obtainβ2iK H b K b x i =α2i K H w K w x i ,(17)Algorithm 1MU-MIMO precoding based on GSVD for single user Given a channel H =[H 1;...;H K ]for K users as in (3),this algorithmcomputes the precoding matrix W k ∈C N ×m k for the k th user.1)Set K k = H k ;˜Hk ;√M σI N ∈C (M +N )×N .2)Compute the reduced QRD of K k ,i.e.,P H k K k =R k ,where P k ∈C(M +N )×N has orthonormal columns and R k ∈C N ×N is upper triangular.3)Compute ˇVk from the SVD of P k (1:M ,1:N ),i.e.,ˇU H kP k (1:M ,1:N )ˇV k =ˇΣk .4)Solve the triangular system R k W k =ˇVk (:,1:m k )for W k ,which is the precoding matrix for the k th user.for 1≤i ≤t ,where x i represents the i th column of X .Here,the columns of X are the generalized right singular vectors for the matrix pair (K b ,K w ).B.MU-MIMO precoding scheme based on GSVDPrecoding scheme for single user:Since x i in (17)is a gen-eralized eigenvector for the matrix pair (K H b K b ,K Hw K w ),we can find the precoding matrix by applying the GSVD algorithm to the matrix pair (H k ,[˜H k ;√M σI N ]).The GSVD algorithmfor the precoding matrix is summarized in Algorithm 1.Since rank (K k )is always full due to the term √M σI N in K k ,the reduced QRD in Step 2of Algorithm 1is utilized as the rank revealing decomposition in (14),which is similar to the regularized LDA/GSVD [8]and makes the algorithm much simpler as in [8].Algorithm 1can be applied without any restriction on M or N .Algorithm 1requires computation of one QRD (Step 2),one SVD of M ×N matrix (Step 3),and m k backward substitutions (Step 4)and overall takes much less computa-tional complexities than the GEVD of an N ×N matrix pair.By utilizing the GSVD based precoding scheme described above,we can deal with better-conditioned matrices 1and avoid numerical errors during the matrix multiplications.Precoding scheme for multiple users:For total K users in the system we need to repeat Algorithm 1K times to compute the precoding matrices.To reduce its computational loads,the follwoing property is useful:Property 1:Let A ∈C M ×N ,M >N ,have a QRD as:A =Q 1R ,Q 1∈C M ×N ,R ∈C N ×N(18)where Q H 1Q 1=I N and R is upper triangular.The QRD ofUA ,where U ∈C M ×M is unitary,is then given byUA=Q R ,(19)where Q =UQ 1D and R =DR ,and D =diag {±1,...,±1}∈R N ×N .Even though the QRD of UA is not unique,by setting D =I N ,it can be acheived that R =R and Q =UQ 1,which is a key factor in the following derivation.Suppose that the precoding matrix W 1for the first user is computed by Algorithm 1.The matrix K 1in Step 2is then1Generally,the condition number of A H A is the square of that of A .given byK 1=⎡⎣H 1˜H 1√M σI N⎤⎦= H √M σI N=P 1R 1,(20)where P 1∈C (M +N )×N has orthonormal columns and R 1∈C N ×N is upper triangular.For the second user,K 2=⎡⎣H 2˜H 2√M σI N ⎤⎦=S 2,1K 1=P 2R 2,(21)where P 2∈C (M +N )×N has orthonormal columns and R 2∈C N ×N is upper triangular,andS 2,1=⎡⎣0M ×MI M I M0M ×M I M +N −2M ⎤⎦.(22)It can then be easily derived that:R 2=R 1,(23)P 2(1:M,1:N )=P 1(M+1:2M,1:N ).(24)Therefore,given the QRD of K 1,it is unnecessary to computethe QRD for the second user in Step 2of Algorithm 1.Similarly,for the k th user,R k=R 1,P k (1:M ,1:N )=P 1((k −1)M +1:k M ,1:N ).Based on these observations,an efficient precoding algorithm for multiple users can be developed,which is summarized in Algorithm 2.Note that Algorithm 2requires only one QRD in Step 2for K users,thus eliminating the redundant computational complexities when computing the precoding matrices of K users by employing Algorithm 1repeatedly.The interesting thing is that,if the channel plus noise matrix K in Step 2of Algorithm 2is not transformed into the unitary matrix P by R −1,Algorithm 2gives the equivalent precoding weight to the optimal solution for the single user MIMO precoding.C.MU-MIMO precoding scheme based on GSVD with QRD preprocessingThe computational complexity of the precoding scheme proposed in Section IV increases as the number of transmit antenna (N )increases,which is undesirable when M ≤N .2Accordingly we propose a fast MU-MIMO precoding scheme based on GSVD using QRD preprocessing,where QRD reduces the dimension of the precoding weight during the computing process.Since Q H 1HH=R by using reduced QRD,where Q 1∈CN ×M,M ≤N ,the N -dimensional channel vectors H H for users can be represented in an M -dimensional space as R by the dimension reducing transformation Q H 1.2Itcould often arise in general MU-MIMO problem.In addition,the case of M >N can be sometimes forced to M ≤N by scheduling and random beamforming schemes [12,13].Algorithm 2MU-MIMO precoding algorithm based on GSVD for multiple usersGiven a channel H =[H 1;...;H K ]for K users as in (3),this algorithmcomputes the precoding matrix W k ∈C N ×m k for all users.1)Set K = H ;√M σI N ∈C (M +N )×N .2)Compute the reduced QRD of K ,i.e.,P H K =R ,where P ∈C (M +N )×N has orthonormal columns and R ∈C N ×N is upper triangular.3)For k =1:K3.1)Compute ˇVk from the SVD of P ((k −1)M +1:kM ,1:N ),i.e.,ˇU H kP ((k −1)M +1:kM ,1:N )ˇV k =ˇΣk .4)Solve the triangular system R W k =ˇVk (:,1:m k )for W k ,k =1,...,K .Suppose we find the weight matrix ˆWk for the dimension reduced channel matrix R .The cost function can then be written as:J ˆWk =tr ˆW H k R k R H kˆW ktr ˆW H k M σ2I M +˜R k ˜R H k ˆW k,(25)where R k =Q H 1H H k and ˜R k =Q H 1˜H H k ,which are given bythe corresponding columns of R .SinceM σ2I M +˜R k ˜R H k =Q H 1 M σ2I N +˜H H k ˜H kQ 1,(25)can be rewritten asJ ˆW k =tr ˆW H k Q H 1H H k H k Q 1ˆW ktr ˆW H k Q H 1 M σ2I N +˜H H k ˜H k Q 1ˆW k ,(26)which means that W k =Q 1ˆWk provides the solution for (6).The above derivation shows that the reduced QRD preprocessing to H H enables us to manipulate the matrices of smaller size M ×M rather than M ×N in computing the GSVD based precoding weights.In addition,the above prerpocessing can work together with the efficient algorithm for multiple users discussed in Secction IV-B.Based on these results,the new algorithm is summarized in Algorithm 3.Note that the SVDs of M ×N matrices for k =1,...,K in Step 3of Algorithm 1or Algorithm 2,the dominant part of the computational complexities,are replaced with those of much smaller M ×M matrices.To detect symbol s k from (1),we use the simplematched filter W H kH H k as in [2]because W H k H H k H k W k =diag {λ1,k ,...,λm k ,k }.V.A NTENNA SELECTION SCHEME BASED ON GSVD Since the increase in the number of the receive antennas causes the interferences to other users,the performance of the proposed scheme could be affected by the number of receive antennas.In [3],the PDF of SLNR is characterized for the number of receive antennas which concludes that the increase in the number of the active antennas reduces the SLNR.Based on this observation,we propose the antenna selection algorithm to reduce the number of the active antennas.Since M ≥m k ,the minimum required number of the receive antenna for k th user is m k .Therefore,the proposed antennaAlgorithm 3MU-MIMO precoding algorithm based on GSVD withQRD preprocessing for multiple users Given a channel H =[H 1;...;H K ]for K users as in (3)and M ≤N ,this algorithm computes the precoding matrix W k ∈C N ×m k for all users.1)Compute the reduced QRD of H H ,i.e,H H =Q 1R ,where Q 1has orthonormal columns and R is upper triangular.2)Set ˆK= R H ;√M σI M ∈C 2M ×M .3)Compute the reduced QRD of ˆK,i.e.,ˆP T ˆK =ˆR ,where ˆP has orthonormal columns and ˆRis upper triangular.4)For k =1:K4.1)Compute ˇV k from the SVD of ˆP ((k −1)M +1:kM ,1:M ),i.e.,ˇU T k ˆP ((k −1)M +1:kM ,1:M )ˇV k =ˇΣ k.5)Solve the triangular system ˆR ˆW k =ˇV k(:,1:m k )for ˆW k .6)W k =Q 1ˆWk for k =1,...,K .selection algorithm chooses m k active antennas for each user.For simplicity,we assume that m k =m for all k .Let ˇΣk in Step 3of Algorithm 2as ˇΣk =diag {σ(1)k ,...,σ(M )k },where σ(1)k ≥...≥σ(M )k .From Theorem 1,the generalized eigenvalues of the matrix pair H H k H k , M σ2I N +˜H H k˜H k are then given as:λ(i )k=(σ(i )k )21−(σ(i )k )2,(27)which implies that,for a large σ(i )k ,λ(i )k also becomes large.Therefore,it is natural that the data streams of the k th user are precoded by the linear combination of the first m k columns of ˇVk in Step 4of Algorithm 2,since the SLNR of the i th stream of k th user is λ(i )k .However,the data streams leak throughthe transformed channel corresponding to σ(m +1)k ,...,σ(M )k causing the interferences,which can be represented as:E k =σ(m +1)kˇUk (:,m +1)ˇV k (:,m +1)H +...+σ(M )kˇU k (:,M )ˇVk (:,M )H .(28)Therefore,the antennas corresponding to the rows of P ((k −1)M +1:kM ,1:N )geometrically close to E k may yield more interferences and need to be deactivated.That is,let C (k )i = P E k P ((k −1)M +i,1:N )H 2for i =1,...,M ,where P E k is the projection matrix given by P E k =E †k E k ,and define C (k )(1),C (k )(2),...,C (k )(M )be the ascending ordering of C (k )1,C (k )2,...,C (k )M .The active antenna index set is then given as:I (k )Active ={j |C (k )j=C (k )(i ),i =1,...,m ,m ≤M }.(29)Note that the proposed selection method requires mainly K additional SVD computations.On the other hand,the exhaus-tive search needs (M C m )K GSVD precoding computations which results in a prohibitive complexity.VI.S IMULATION R ESULTSWe assess performance of the proposed MU-MIMO precod-ing scheme based on GSVD through Monte-Carlo simulationfor different conditions.Each entry of MIMO channel H is generated according to complex white Gaussian distribution with zero-mean and unit-variance.In all simulations,each user’s data are modulated by quadrature phase shift keying (QPSK)signaling and for the matrix decompositions such as QRD,GEVD,and SVD,MATLAB built-in functions are used.Fig.1shows the BER performance of the proposed scheme for N =10,K =4,M =2,and m i =2for i =1,2,3,4.For the reference of comparison we also evaluate the BER performances of the iterative nullspace-directed SVD (iterative null SVD)scheme in [4]and the SLNR maximizing precoding scheme in [2].The SLNR maximizing scheme outperforms the iterative null SVD scheme especially at low SNR since the SLNR maximizing scheme considers the noise variance.At low SNR,SLNR maximizing precoding scheme and GSVD based scheme have similar BERs,but,when SNR >0dB,the latter scheme has lower BERs and the difference becomes larger as SNR increases,i.e.,the GSVD based scheme has a steeper BER curve due to the fact that the GEVD in (11)could be computed inaccurately as the SNR goes high.Fig.2shows BER curves for N =12,K =3,M =3,and m i =3for i =1,2,3.Since N >M ,we additionally evaluate the BER for the GSVD based precoding scheme with QRD preprocessing.The figure exhibits that the preprocessing does not affect the BER performance of the GSVD based scheme,having lower BERs than the SLNR maximizing precoding scheme.It is interesting to note that the performance gap between the proposed schemes and the SLNR maximizing scheme is more apparent than the result shown in Fig.1.Therefore,as N ,M ,and m k increase,it tends to be more unstable to compute the precoding weights in (11)resulting the more performance gain of the proposed scheme compared with the SLNR maximizing precoding scheme.Fig.3shows BER curves for N =9,K =3,M =4,and m i =3for i =1,2,3.We can also find that the GSVD based precoding has lower BERs than the SLNR maximizing scheme.However,since M >N causing more interferences to other users,both schemes can not maintain their slopes of BER curves at high SNR.On the other hand,the iterative null SVD has the lowest BERs.Note that,as pointed in [2],the iterative null SVD scheme is orders of magnitude more complex than the SLNR based precoding or GSVD based precoding schemes which provide closed form solutions.From the figure,the proposed antenna selection scheme improves the BERs of the GSVD based precoding scheme close to those of the iterative null SVD scheme.VII.C ONCLUDING REMARKSIn this paper,we have proposed the MU-MIMO precoding scheme based on GSVD to detect multiple users’symbols which has the improved numerical properties and computation-ally efficiency.The precoding weight of our proposed scheme transforms the MU-MIMO channels to maximize the separa-bility of the effective channels among the users in sending the multiple data streams to the multiple users.Moreover,we have shown how the GSVD based precoding scheme forsingle user can be extended to compute the precoding weights for multiple users efficiently rather than repeating the GSVD algorithm for each user.The simulation results demonstrate that the MU-MIMO precoding scheme based on GSVD has good performance gain especially at high SNR where the covariance matrix of interference and noise gets closer to be singular,and that the QRD preprocessing scheme has no performance degradation compared with the GSVD based precoding scheme even while reducing the computational 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