A New SLNR-based Linear Precoding for Downlink Multi-User Multi-Stream MIMO Systems
aa r X i v : . [ c s . I T ] A ug A New SLNR-based Linear Precoding forDownlink Multi-User Multi-Stream MIMOSystems
Peng Cheng, Meixia Tao and Wenjun Zhang
Abstract
Signal-to-leakage-and-noise ratio (SLNR) is a promising criterion for linear precoder design inmulti-user (MU) multiple-input multiple-output (MIMO) systems. It decouples the precoder designproblem and makes closed-form solution available. In this letter, we present a new linear precodingscheme by slightly relaxing the SLNR maximization for MU-MIMO systems with multiple data streamsper user. The precoding matrices are obtained by a general form of simultaneous diagonalization of twoHermitian matrices. The new scheme reduces the gap between the per-stream effective channel gains,an inherent limitation in the original SLNR precoding scheme. Simulation results demonstrate that theproposed precoding achieves considerable gains in error performance over the original one for multi-stream transmission while maintaining almost the same achievable sum-rate.
Index Terms
Signal-to-leakage-and-noise ratio (SLNR), linear precoding, multi-user MIMO.
I. I
NTRODUCTION
The significance of a downlink multi-user multiple-input-multiple-output (MU-MIMO) systemis to allow a base station (BS) to communicate with several co-channel mobile stations (MS)simultaneously and thereby considerably increase the system throughput. To utilize the benefit, itis essential to suppress co-channel interference (CCI). Among many CCI suppression schemes,
The authors are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, P. R. China.Emails: { cp2001cp, mxtao, zhangwenjun } @sjtu.edu.cn linear precoding gains the popularity because of its simplicity for implementation and goodperformance. To design the optimal linear MU-MIMO precoding scheme, it is often desirable tomaximize the output signal-to-interference-plus-noise ratio (SINR) for each user. However, thisproblem is known to be challenging due to its coupled nature and no closed-form solution isavailable yet. A more tractable but suboptimal design is to enforce a zero-CCI requirement foreach user, such as block diagonalization (BD) [1] and coordinated beamforming (CB) [2].In [3], the authors propose a so-called signal-to-leakage-and-noise ratio (SLNR) as the op-timization metric for linear precoder design. This metric transforms a coupled optimizationproblem into a completely decoupled one, for which a closed-form solution is available. Unlikethe BD approach, it does not impose a restriction on the number of transmit antennas at the BS.Moreover, it is applicable for any number of users and data streams in contrast to CB scheme.Specifically, the SLNR based linear precoding weights in [3] are obtained by the generalizedeigenvalue decomposition (GED) of the channel covariance matrix and the leakage channel-plus-noise covariance matrix of each user. However, a drawback of such GED based precodingscheme is that, when each user has multiple data streams, the effective channel gain for eachstream can be severely unbalanced. If power control or adaptive modulation and coding cannotbe applied, the overall error performance of each user will suffer significant loss.In this letter, we present a new linear precoding scheme based on the SLNR criterion for adownlink MU-MIMO system with multiple data streams per user. The design goal is to reducethe margin between the effective SINRs of multiple data streams. To do this, we introduce a slightrelaxation for pursuing SLNR maximization (Note that maximizing SLNR at the transmitter sidedoes not necessarily lead to output SINR maximization at each receiver). Thereby, we obtaina general form of simultaneous diagonalization of two covariance matrices linked to the user’schannel and leakage-plus-noise. Based on that, the new precoding matrices are then obtained.We also present a simple and low-complexity algorithm to compute the precoding matrix foreach user. Simulation results confirm that, compared with the original scheme, our schemedemonstrates sizable performance gains in error rate performance for multi-stream transmissionwhile maintaining almost the same sum-rate performance. Notations : E ( · ) , Tr ( · ) , ( · ) − , and ( · ) H denote expectation, trace, inverse, and conjugatetranspose, respectively. k·k F represents the Frobenius norm. I N is the N × N identity matrix. diag ( a , · · · , a N ) is the diagonal matrix with element a n on the n -th diagonal. Besides, C M × N represents the set of M × N matrices in complex field.II. S YSTEM M ODEL
We consider a downlink MU-MIMO system with N transmit antennas and M receive antennasat each of the K active users. Let H k ∈ C M × N denotes the channel from the BS to the MS k and ¯H k = (cid:2) H H , · · · , H Hk − , H Hk +1 , · · · , H HK (cid:3) H ∈ C ( K − M × N represent the correspondingconcatenated leakage channel. A spatially uncorrelated flat Rayleigh fading channel is assumed.The elements of H k are modeled as independent and identically distributed complex Gaussianvariables with zero-mean and unit-variance. In addition, we assume H k , and also ¯H k , have fullrank with probability one. For a specific vector time, the transmitted vector symbol of user k isdenoted as s k ∈ C L × , where L ( ≤ M ) is the number of data streams supported for user k andis assumed equal for all the users for simplicity. The vector symbol satisfies the power constraint E (cid:0) s k s Hk (cid:1) = I L . Before entering into the MIMO channel, the vector s k is pre-multiplied by aprecoding matrix F k ∈ C N × L . Here, power allocation and rate adaptation among data streamscan be applied. However, the signal design or feedback support may be relatively complex andthus we resort to precoding design only in this work. Then, for a given user k , the receivedsignal vector can be written as r k = H k F k s k + H k X Ki =1 ,i = k F i s i + n k (1)in which the second term represents CCI and the third term is the additive white Gaussian noisewith E (cid:0) n k n Hk (cid:1) = σ I M .We review the original SLNR based precoding scheme in [3]. Recall that the SLNR is definedas the ratio of received signal power at the desired MS to received signal power at the otherterminals (the leakage) plus noise power without considering receive matrices, given bySLNR k = Tr (cid:0) F Hk H Hk H k F k (cid:1) Tr (cid:0) F Hk (cid:0) M/Lσ I + ¯H Hk ¯H k (cid:1) F k (cid:1) , (2)for k = 1 , · · · , K. According to the SLNR criterion, the precoding matrix F k is designed basedon the following metric F opt k = arg max F k ∈ C N × L SLNR k (3)with Tr (cid:0) F k F Hk (cid:1) = L for power limitation. Since H Hk H k is Hermitian and positive semidefi-nite (HPSD) and M/Lσ I + ¯H Hk ¯H k is Hermitian and positive definite (HPD), by generalized eigenvalue decomposition, there exists an invertible matrix T k ∈ C N × N such that T Hk H Hk H k T k = Λ k = diag ( λ , · · · , λ N ) (4) T Hk (cid:0) M/Lσ I + ¯H Hk ¯H k (cid:1) T k = I N (5)with λ ≥ λ ≥ · · · ≥ λ N ≥ . Here, the columns of T k and the diagonal entries of Λ k are thegeneralized eigenvectors and eigenvalues of the pair (cid:8) H Hk H k , M/Lσ I + ¯H Hk ¯H k (cid:9) , respectively.It is then shown in [3] that the optimal precoder which is able to maximize the objective function(3) can be obtained by extracting the leading L columns of T k as F opt k = ρ T k [ I L ; ] , (6)where ρ is a scaling factor so that Tr (cid:0) F k F Hk (cid:1) = L . The resulting maximum SLNR value isgiven by SLNR max k = P Li =1 λ i /L . Along with the realization of the precoder, the matched-filtertype receive matrix, denoted as G k = ( H k F k ) H , is applied at each user receiver, resulting ininter-stream interference free. Note that better performance could be achieved if a multi-userMMSE type receiver is adopted. In this letter, we still adopt MF-type detector at the receiver asin [3] for implementation simplicity and analytical convenience.A drawback of such GED based precoding scheme is that, when L ≥ , the effective channelgain for each stream can be severely unbalanced as shall be illustrated in Section III-C. It isknown that the overall performance of a user with multiple streams is dominated by the streamwith the worst channel condition. Hence, such channel imbalance would lead to poor overallerror performance for a user. In the next section, we allow a slight relaxation on the SLNRmaximization, which provides additional degrees of freedom to design a new precoding schemeso as to overcome this drawback.III. P ROPOSED P RECODING S CHEME
A. Design Principle by Matrix Theory
The expressions in (4) and (5) by the GED approach motivate us to find a more general formof simultaneous diagonalization of two matrices. Before introducing our results in Proposition1, we review the following Lemma [4, Ch. 4, 4.5.8]:
Lemma 1 : Let A , B ∈ C n × n be Hermitian. There is a non-singular matrix S ∈ C n × n suchthat S H AS = B if and only if A and B have the same inertia, that is, have the same numberof positive, negative, and zero eigenvalues. Proposition 1 : For the pair of matrices (cid:8) H Hk H k , M/Lσ I + ¯H Hk ¯H k (cid:9) , there is a non-singularmatrix P k ∈ C N × N such that P Hk H Hk H k P k = Θ k (7) P Hk (cid:0) M/Lσ I + ¯H Hk ¯H k (cid:1) P k = Ω k (8)in which Θ k = diag ( θ , θ , · · · θ N ) and Ω k = diag ( ω , ω , · · · ω N ) with the entries satisfying > θ ≥ · · · ≥ θ M > , θ M +1 = · · · = θ N = 0 and < ω ≤ · · · ≤ ω M < , ω M +1 = · · · = ω N = 1 as well as θ i + ω i = 1 for i = 1 , , · · · , N . Proof:
Denote A k = H Hk H k , B k = M/Lσ I + ¯H Hk ¯H k and C k = A k + B k . Let theeigenvalues λ i ( A k ) , λ i ( B k ) and λ i ( C k ) , i = 1 , , · · · , N , be arranged in increasing order.Since A k is HPSD and B k is HPD, namely, λ i ( A k ) ≥ and λ i ( B k ) > , then by [4, 4.3.1], wehave λ i ( C k ) ≥ λ i ( A k ) + λ ( B k ) > , ∀ i . This implies that C k is HPD. Then, by the matrixtheory in [4, 4.5.8, Exercise], there must be a non-singular matrix Q k ∈ C N × N such that Q Hk C k Q k = Q Hk ( A k + B k ) Q k = I N . (9)Further, denote A ′ k = Q Hk A k Q k and B ′ k = Q Hk B k Q k . By Lemma 1, it can be shown that A ′ k and B ′ k have the same inertia with A k and B k , respectively. Thus, A ′ k is HPSD and B ′ k is HPD.Now, by using [4, 4.3.1] again, it is easy to show that > λ i ( A ′ k ) ≥ and ≥ λ i ( B ′ k ) > .Next, according to the eigen-decomposition (ED) of a Hermitian matrix [4], there must be aunitary matrix U k ∈ C N × N such that U Hk A ′ k U k = diag ( λ ( A ′ k ) , · · · , λ N ( A ′ k )) . (10)Applying U k in both sides of (9), we obtain U Hk ( A ′ k + B ′ k ) U k = I N . (11)Hence, observing (10) and (11), we find that it is necessary for U Hk B ′ k U k to satisfy U Hk B ′ k U k =diag ((1 − λ ( A ′ k )) , · · · , (1 − λ N ( A ′ k ))) . Clearly, as U k is unitary, then { − λ i ( A ′ k ) } Ni =1 mustbe the eigenvalues of B ′ k . To this end, we define P k = Q k U k . Since rank (cid:0) H Hk H k (cid:1) = rank ( H k ) = M and the rank is unchanged upon left or right multiplication by a nonsingular matrix, then wearrive at the results in (7) and (8). Algorithm 1
The specific design of precoder F ′ k for user k Input: A k = H Hk H k , and C k = (cid:0) H Hk H k + M/Lσ I + ¯H Hk ¯H k (cid:1)
1) Compute Cholesky decomposition on C k , as C k = G k G Hk , where G k ∈ C N × N is a lower triangular matrix with positivediagonal entries. Then, G − k can be easily obtained and we have (cid:0) G − k (cid:1) H = Q k in (9).2) Compute A ′ k = Q Hk A k Q k , then compute ED on A ′ k as A ′ k U k = U k Λ k . Note U k must be unitary and it can be alsoobtained by computing the left singular matrix of A ′ k in terms of SVD.3) Compute P k = Q k U k .Output: F ′ k = γ P k ( I L ; ) . B. Precoder Design
The simultaneous diagonalization in general form stated in Proposition 1 draws a significantdistinction from the original GED based deduction in (4) and (5). This allows us to design anew precoding scheme. In specific, the proposed precoder F ′ k and matched decoder G ′ k can bedesigned as F ′ k = γ P k [ I L ; ] , G ′ k = ( H k F ′ k ) H (12)in which γ is a normalization factor so that Tr (cid:0) F ′ k F ′ Hk (cid:1) = L . It is clear that G ′ k H k F k amountsto a certain diagonal matrix, also resulting in inter-stream-interference free.The remaining problem is how to compute a specific precoder F ′ k for each user. Based on ourproof of Proposition 1, we present a closed-form expression using a simple and low-complexalgorithm, outlined in Algorithm 1. In the next subsection, we reveal the superiority of theproposed precoding scheme through per-stream SINR discussion. C. Performance Discussion
Firstly, continuing to use the same symbols A k and B k as in the proof of Proposition 1, wecan show that A k t k i = λ i B k t k i and A k p k i = ( θ i /ω i ) B k p k i from (5) and (8), in which t k i and p k i correspond to the i -th column of T k and P k , respectively. Here, both λ i and θ i /ω i must bethe generalized eigenvalues of the pair { A k , B k } . It is then easy to see that λ j = θ j /ω j , j = 1 , , · · · , N (13)with { λ j } Nj =1 and { θ j } Nj =1 being sorted in descending order while { ω j } Nj =1 sorted in ascendingorder. Now we have SLNR k = (cid:16)P Ll =1 θ l (cid:17) / (cid:16)P Ll =1 (1 − θ l ) (cid:17) , which is slightly smaller than SLNR max k given in Section II.On the other hand, the ultimate performance is decided by post-SINR. Clearly, the decodedsignal should take the form ˆs k = G ′ k H k F k s k + G ′ k (cid:16) H k X Ki =1 ,i = k F i s i + n k (cid:17) . (14)Thanks to diagonal form in (7) and (8), the covariance matrix of noise vector is given by E (cid:0) G ′ k n k n Hk G ′ Hk (cid:1) = σ I L E (cid:0) G ′ k G ′ Hk (cid:1) = γ σ I L diag ( θ , · · · , θ L ) . Furthermore, it can be verifiedthrough numerical results (difficult via theoretical analysis though) that the residual CCI is muchsmaller than the noise power at high SNR. As such, the SINR on the l -th stream, η ′ l can beapproximately calculated as η ′ l = ( γ θ l ) / ( γ σ θ l ) = γ θ l /σ . Then, for any two streams l and m with l > m , the margin of ∆ ′ l,m between η ′ l and η ′ m in terms of decibel (dB) can be expressedas ∆ ′ l,m = 10log ( η ′ l /η ′ m ) = 10log ( θ l /θ m ) . (15)Following the same analysis, the margin of ∆ l,m for the original scheme can be analogouslycalculated as ∆ l,m = 10log ( η l /η m ) = 10log ( λ l /λ m ) . (16)According to (13), we have λ l /λ m = ( θ l ω m ) / ( θ m ω l ) . Further, we have that ω m > ω l for l > m by definition. It then ensures that the following inequality holds: ∆ ′ l,m < ∆ l,m . (17)This explicitly shows that the SINR margin between any two streams decreases by applying theproposed scheme. In other words, the effective channel gains between the multiple streams arenow less unbalanced. Its effectiveness will be further examined by simulation in the next section.IV. S IMULATION R ESULTS
Fig. 1 compares the simulated bit error rate (BER) per user in a MU-MIMO system withdifferent system configurations. Here, P denotes the proposed precoding scheme and O denotesthe original scheme in [3]. QPSK modulation with Gray mapping is employed and the BERcurves are plotted versus the transmit SNR (
L/σ ) . It is seen that the proposed scheme and theoriginal scheme for single-stream case ( L = 1 ) achieve the same BER performance. For multiplestreams ( L = 2 and 3), the former outperforms the latter with sizeable gains. In specific, a gain −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 22 2410 −5 −4 −3 −2 −1 SNR (dB) BE R O. L=1P. L=1O. L=2P. L=2O. L=3P. L=3
Fig. 1. Uncoded BER of a MU-MIMO system with N = 8 transmit antennas at the BS and K = 2 users each with M = 3 receive antennas. of around dB and dB can be achieved at BER= − for streams of L = 2 and L = 3 ,respectively. We also carried out the achievable sum-rate comparison. It is found that our schemeis almost the same as the original one. The results are omitted due to page limit.The above simulation results verify the effectiveness of the proposed precoding scheme overthe original SLNR based scheme when there are multiple data streams for each user.R EFERENCES [1] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multi-userMIMO channels,”
IEEE Trans. Signal Process. , vol. 52, no. 2, pp. 461–471, Feb. 2004.[2] C. B. Chae, D. Mazzarese, N. Jindal and R. W. Heath, Jr., “Coordinated Beamforming with Limited Feedback in the MIMOBroadcast Channel,”
IEEE J. Sel. Areas in Commun. , vol. 26, no. 8, pp. 1505-1515, Oct. 2008.[3] M. Sadek, A. Tarighat and A. H. Sayed, “A leakage-based precoding scheme for downlink multi-user MIMO Channels,”
IEEE Trans. Wireless Commun. , vol. 6, no. 5, pp. 1711–1721, May 2007.[4] R. A. Horn and C. R. Johnson,
Matrix analysis . New York: Cambridge University Press, 1985. r X i v : . [ c s . I T ] A ug A New SLNR-based Linear Precoding forDownlink Multi-User Multi-Stream MIMO Systems
Peng Cheng, Meixia Tao and Wenjun Zhang
Abstract —Signal-to-leakage-and-noise ratio (SLNR) is apromising criterion for linear precoder design in multi-user (MU)multiple-input multiple-output (MIMO) systems. It decouples theprecoder design problem and makes closed-form solution avail-able. In this letter, we present a new linear precoding scheme byslightly relaxing the SLNR maximization for MU-MIMO systemswith multiple data streams per user. The precoding matrices areobtained by a general form of simultaneous diagonalization of twoHermitian matrices. The new scheme reduces the gap between theper-stream effective channel gains, an inherent limitation in theoriginal SLNR precoding scheme. Simulation results demonstratethat the proposed precoding achieves considerable gains in errorperformance over the original one for multi-stream transmissionwhile maintaining almost the same achievable sum-rate.
Index Terms —Signal-to-leakage-and-noise ratio (SLNR), linearprecoding, multi-user MIMO.
I. I
NTRODUCTION T HE significance of a downlink multi-user multiple-input-multiple-output (MU-MIMO) system is to allow a basestation (BS) to communicate with several co-channel mo-bile stations (MS) simultaneously and thereby considerablyincrease the system throughput. To utilize the benefit, it isessential to suppress co-channel interference (CCI). Amongmany CCI suppression schemes, linear precoding gains thepopularity because of its simplicity for implementation andgood performance. To design the optimal linear MU-MIMOprecoding scheme, it is often desirable to maximize the outputsignal-to-interference-plus-noise ratio (SINR) for each user.However, this problem is known to be challenging due to itscoupled nature and no closed-form solution is available yet. Amore tractable but suboptimal design is to enforce a zero-CCIrequirement for each user, such as block diagonalization (BD)[1] and coordinated beamforming (CB) [2].In [3], the authors propose a so-called signal-to-leakage-and-noise ratio (SLNR) as the optimization metric for linearprecoder design. This metric transforms a coupled optimiza-tion problem into a completely decoupled one, for which aclosed-form solution is available. Unlike the BD approach,it does not impose a restriction on the number of transmitantennas at the BS. Moreover, it is applicable for any numberof users and data streams in contrast to CB scheme. Specif-ically, the SLNR based linear precoding weights in [3] areobtained by the generalized eigenvalue decomposition (GED)
The authors are with the Department of Electronic Engineering, ShanghaiJiao Tong University, Shanghai 200240, P. R. China. Emails: { cp2001cp,mxtao, zhangwenjun } @sjtu.edu.cnThis work is supported by the National Science Key Special Project ofChina under grants 2008ZX03003-004 and 2008BAH30B09. of the channel covariance matrix and the leakage channel-plus-noise covariance matrix of each user. However, a drawbackof such GED based precoding scheme is that, when eachuser has multiple data streams, the effective channel gain foreach stream can be severely unbalanced. If power control oradaptive modulation and coding cannot be applied, the overallerror performance of each user will suffer significant loss.In this letter, we present a new linear precoding schemebased on the SLNR criterion for a downlink MU-MIMOsystem with multiple data streams per user. The design goal isto reduce the margin between the effective SINRs of multipledata streams. To do this, we introduce a slight relaxation forpursuing SLNR maximization (Note that maximizing SLNR atthe transmitter side does not necessarily lead to output SINRmaximization at each receiver). Thereby, we obtain a generalform of simultaneous diagonalization of two covariance matri-ces linked to the user’s channel and leakage-plus-noise. Basedon that, the new precoding matrices are then obtained. We alsopresent a simple and low-complexity algorithm to compute theprecoding matrix for each user. Simulation results confirm that,compared with the original scheme, our scheme demonstratessizable performance gains in error rate performance for multi-stream transmission while maintaining almost the same sum-rate performance. Notations : E ( · ) , Tr ( · ) , ( · ) − , and ( · ) H denote expectation,trace, inverse, and conjugate transpose, respectively. k·k F represents the Frobenius norm. I N is the N × N identitymatrix. diag ( a , · · · , a N ) is the diagonal matrix with element a n on the n -th diagonal. Besides, C M × N represents the set of M × N matrices in complex field.II. S YSTEM M ODEL
We consider a downlink MU-MIMO system with N trans-mit antennas and M receive antennas at each of the K activeusers. Let H k ∈ C M × N denotes the channel from the BS tothe MS k and ¯H k = (cid:2) H H , · · · , H Hk − , H Hk +1 , · · · , H HK (cid:3) H ∈ C ( K − M × N represent the corresponding concatenated leak-age channel. A spatially uncorrelated flat Rayleigh fadingchannel is assumed. The elements of H k are modeled as inde-pendent and identically distributed complex Gaussian variableswith zero-mean and unit-variance. In addition, we assume H k ,and also ¯H k , have full rank with probability one. For a specificvector time, the transmitted vector symbol of user k is denotedas s k ∈ C L × , where L ( ≤ M ) is the number of data streamssupported for user k and is assumed equal for all the users forsimplicity. The vector symbol satisfies the power constraint E (cid:0) s k s Hk (cid:1) = I L . Before entering into the MIMO channel, the vector s k is pre-multiplied by a precoding matrix F k ∈ C N × L .Here, power allocation and rate adaptation among data streamscan be applied. However, the signal design or feedback supportmay be relatively complex and thus we resort to precodingdesign only in this work. Then, for a given user k , the receivedsignal vector can be written as r k = H k F k s k + H k X Ki =1 ,i = k F i s i + n k (1)in which the second term represents CCI and the third term isthe additive white Gaussian noise with E (cid:0) n k n Hk (cid:1) = σ I M .We review the original SLNR based precoding scheme in[3]. Recall that the SLNR is defined as the ratio of receivedsignal power at the desired MS to received signal power atthe other terminals (the leakage) plus noise power withoutconsidering receive matrices, given bySLNR k = Tr (cid:0) F Hk H Hk H k F k (cid:1) Tr (cid:0) F Hk (cid:0) M/Lσ I + ¯H Hk ¯H k (cid:1) F k (cid:1) , (2)for k = 1 , · · · , K. According to the SLNR criterion, theprecoding matrix F k is designed based on the following metric F opt k = arg max F k ∈ C N × L SLNR k (3)with Tr (cid:0) F k F Hk (cid:1) = L for power limitation. Since H Hk H k isHermitian and positive semidefinite (HPSD) and M/Lσ I + ¯H Hk ¯H k is Hermitian and positive definite (HPD), by gen-eralized eigenvalue decomposition, there exists an invertiblematrix T k ∈ C N × N such that T Hk H Hk H k T k = Λ k = diag ( λ , · · · , λ N ) (4) T Hk (cid:0) M/Lσ I + ¯H Hk ¯H k (cid:1) T k = I N (5)with λ ≥ λ ≥ · · · ≥ λ N ≥ . Here, the columns of T k andthe diagonal entries of Λ k are the generalized eigenvectorsand eigenvalues of the pair (cid:8) H Hk H k , M/Lσ I + ¯H Hk ¯H k (cid:9) ,respectively. It is then shown in [3] that the optimal precoderwhich is able to maximize the objective function (3) can beobtained by extracting the leading L columns of T k as F opt k = ρ T k [ I L ; ] , (6)where ρ is a scaling factor so that Tr (cid:0) F k F Hk (cid:1) = L . Theresulting maximum SLNR value is given by SLNR max k = P Li =1 λ i /L . Along with the realization of the precoder,the matched-filter type receive matrix, denoted as G k =( H k F k ) H , is applied at each user receiver, resulting in inter-stream interference free. Note that better performance couldbe achieved if a multi-user MMSE type receiver is adopted. Inthis letter, we still adopt MF-type detector at the receiver as in[3] for implementation simplicity and analytical convenience.A drawback of such GED based precoding scheme is that,when L ≥ , the effective channel gain for each stream canbe severely unbalanced as shall be illustrated in Section III-C. It is known that the overall performance of a user withmultiple streams is dominated by the stream with the worstchannel condition. Hence, such channel imbalance would leadto poor overall error performance for a user. In the next section,we allow a slight relaxation on the SLNR maximization,which provides additional degrees of freedom to design a newprecoding scheme so as to overcome this drawback. III. P ROPOSED P RECODING S CHEME
A. Design Principle by Matrix Theory
The expressions in (4) and (5) by the GED approachmotivate us to find a more general form of simultaneousdiagonalization of two matrices. Before introducing our resultsin Proposition 1, we review the following Lemma [4, Ch. 4,4.5.8]:
Lemma 1 : Let A , B ∈ C n × n be Hermitian. There is anon-singular matrix S ∈ C n × n such that S H AS = B if andonly if A and B have the same inertia, that is, have the samenumber of positive, negative, and zero eigenvalues. Proposition 1 : For the pair of matrices (cid:8) H Hk H k , M/Lσ I + ¯H Hk ¯H k (cid:9) , there is a non-singularmatrix P k ∈ C N × N such that P Hk H Hk H k P k = Θ k (7) P Hk (cid:0) M/Lσ I + ¯H Hk ¯H k (cid:1) P k = Ω k (8)in which Θ k = diag ( θ , θ , · · · θ N ) and Ω k = diag ( ω , ω , · · · ω N ) with the entries satisfying > θ ≥· · · ≥ θ M > , θ M +1 = · · · = θ N = 0 and < ω ≤ · · · ≤ ω M < , ω M +1 = · · · = ω N = 1 as well as θ i + ω i = 1 for i = 1 , , · · · , N . Proof:
Denote A k = H Hk H k , B k = M/Lσ I + ¯H Hk ¯H k and C k = A k + B k . Let the eigenvalues λ i ( A k ) , λ i ( B k ) and λ i ( C k ) , i = 1 , , · · · , N , be arranged in increasing order.Since A k is HPSD and B k is HPD, namely, λ i ( A k ) ≥ and λ i ( B k ) > , then by [4, 4.3.1], we have λ i ( C k ) ≥ λ i ( A k )+ λ ( B k ) > , ∀ i . This implies that C k is HPD. Then, by thematrix theory in [4, 4.5.8, Exercise], there must be a non-singular matrix Q k ∈ C N × N such that Q Hk C k Q k = Q Hk ( A k + B k ) Q k = I N . (9)Further, denote A ′ k = Q Hk A k Q k and B ′ k = Q Hk B k Q k . ByLemma 1, it can be shown that A ′ k and B ′ k have the sameinertia with A k and B k , respectively. Thus, A ′ k is HPSD and B ′ k is HPD. Now, by using [4, 4.3.1] again, it is easy to showthat > λ i ( A ′ k ) ≥ and ≥ λ i ( B ′ k ) > . Next, accordingto the eigen-decomposition (ED) of a Hermitian matrix [4],there must be a unitary matrix U k ∈ C N × N such that U Hk A ′ k U k = diag ( λ ( A ′ k ) , · · · , λ N ( A ′ k )) . (10)Applying U k in both sides of (9), we obtain U Hk ( A ′ k + B ′ k ) U k = I N . (11)Hence, observing (10) and (11), we find that it isnecessary for U Hk B ′ k U k to satisfy U Hk B ′ k U k =diag ((1 − λ ( A ′ k )) , · · · , (1 − λ N ( A ′ k ))) . Clearly, as U k is unitary, then { − λ i ( A ′ k ) } Ni =1 must be the eigenvaluesof B ′ k . To this end, we define P k = Q k U k . Since rank (cid:0) H Hk H k (cid:1) = rank ( H k ) = M and the rank is unchangedupon left or right multiplication by a nonsingular matrix, thenwe arrive at the results in (7) and (8). B. Precoder Design
The simultaneous diagonalization in general form stated inProposition 1 draws a significant distinction from the originalGED based deduction in (4) and (5). This allows us to design
Algorithm 1
The specific design of precoder F ′ k for user k Input: A k = H Hk H k , and C k = (cid:0) H Hk H k + M/Lσ I + ¯H Hk ¯H k (cid:1)
1) Compute Cholesky decomposition on C k , as C k = G k G Hk , where G k ∈ C N × N is a lower triangular matrix with positive diagonal entries. Then, G − k can be easily obtained and we have (cid:16) G − k (cid:17) H = Q k in (9).2) Compute A ′ k = Q Hk A k Q k , then compute ED on A ′ k as A ′ k U k = U k Λ k . Note U k must be unitary and it can be also obtained by computingthe left singular matrix of A ′ k in terms of SVD.3) Compute P k = Q k U k .Output: F ′ k = γ P k ( I L ; ) . a new precoding scheme. In specific, the proposed precoder F ′ k and matched decoder G ′ k can be designed as F ′ k = γ P k [ I L ; ] , G ′ k = ( H k F ′ k ) H (12)in which γ is a normalization factor so that Tr (cid:0) F ′ k F ′ Hk (cid:1) = L .It is clear that G ′ k H k F k amounts to a certain diagonal matrix,also resulting in inter-stream-interference free.The remaining problem is how to compute a specific pre-coder F ′ k for each user. Based on our proof of Proposition1, we present a closed-form expression using a simple andlow-complex algorithm, outlined in Algorithm 1. In the nextsubsection, we reveal the superiority of the proposed precodingscheme through per-stream SINR discussion. C. Performance Discussion
Firstly, continuing to use the same symbols A k and B k as inthe proof of Proposition 1, we can show that A k t k i = λ i B k t k i and A k p k i = ( θ i /ω i ) B k p k i from (5) and (8), in which t k i and p k i correspond to the i -th column of T k and P k ,respectively. Here, both λ i and θ i /ω i must be the generalizedeigenvalues of the pair { A k , B k } . It is then easy to see that λ j = θ j /ω j , j = 1 , , · · · , N (13)with { λ j } Nj =1 and { θ j } Nj =1 being sorted in descending orderwhile { ω j } Nj =1 sorted in ascending order. Now we haveSLNR k = (cid:16)P Ll =1 θ l (cid:17) / (cid:16)P Ll =1 (1 − θ l ) (cid:17) , which is slightlysmaller than SLNR max k given in Section II.On the other hand, the ultimate performance is decided bypost-SINR. Clearly, the decoded signal should take the form ˆs k = G ′ k H k F k s k + G ′ k (cid:18) H k X Ki =1 ,i = k F i s i + n k (cid:19) . (14)Thanks to diagonal form in (7) and (8), the covariancematrix of noise vector is given by E (cid:0) G ′ k n k n Hk G ′ Hk (cid:1) = σ I L E (cid:0) G ′ k G ′ Hk (cid:1) = γ σ I L diag ( θ , · · · , θ L ) . Furthermore,it can be verified through numerical results (difficult viatheoretical analysis though) that the residual CCI is muchsmaller than the noise power at high SNR. As such, theSINR on the l -th stream, η ′ l can be approximately calculatedas η ′ l = (cid:0) γ θ l (cid:1) / (cid:0) γ σ θ l (cid:1) = γ θ l /σ . Then, for any twostreams l and m with l > m , the margin of ∆ ′ l,m between η ′ l and η ′ m in terms of decibel (dB) can be expressed as ∆ ′ l,m = 10log ( η ′ l /η ′ m ) = 10log ( θ l /θ m ) . (15)Following the same analysis, the margin of ∆ l,m for the −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20 22 2410 −5 −4 −3 −2 −1 SNR (dB) BE R O. L=1P. L=1O. L=2P. L=2O. L=3P. L=3
Fig. 1. Uncoded BER of a MU-MIMO system with N = 8 transmit antennasat the BS and K = 2 users each with M = 3 receive antennas. original scheme can be analogously calculated as ∆ l,m = 10log ( η l /η m ) = 10log ( λ l /λ m ) . (16)According to (13), we have λ l /λ m = ( θ l ω m ) / ( θ m ω l ) .Further, we have that ω m > ω l for l > m by definition. Itthen ensures that the following inequality holds: ∆ ′ l,m < ∆ l,m . (17)This explicitly shows that the SINR margin between anytwo streams decreases by applying the proposed scheme. Inother words, the effective channel gains between the multiplestreams are now less unbalanced. Its effectiveness will befurther examined by simulation in the next section.IV. S IMULATION R ESULTS
Fig. 1 compares the simulated bit error rate (BER) per userin a MU-MIMO system with different system configurations.Here, P denotes the proposed precoding scheme and O denotesthe original scheme in [3]. QPSK modulation with Graymapping is employed and the BER curves are plotted versusthe transmit SNR (
L/σ ) . It is seen that the proposed schemeand the original scheme for single-stream case ( L = 1 ) achievethe same BER performance. For multiple streams ( L = 2 and3), the former outperforms the latter with sizeable gains. Inspecific, a gain of around dB and dB can be achieved atBER= − for streams of L = 2 and L = 3 , respectively.We also carried out the achievable sum-rate comparison. It isfound that our scheme is almost the same as the original one.The results are omitted due to page limit.The above simulation results verify the effectiveness of theproposed precoding scheme over the original SLNR basedscheme when there are multiple data streams for each user.R EFERENCES[1] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methodsfor downlink spatial multiplexing in multi-user MIMO channels,”
IEEETrans. Signal Process. , vol. 52, no. 2, pp. 461–471, Feb. 2004.[2] C. B. Chae, D. Mazzarese, N. Jindal and R. W. Heath, Jr., “CoordinatedBeamforming with Limited Feedback in the MIMO Broadcast Channel,”
IEEE J. Sel. Areas in Commun. , vol. 26, no. 8, pp. 1505-1515, Oct. 2008.[3] M. Sadek, A. Tarighat and A. H. Sayed, “A leakage-based precodingscheme for downlink multi-user MIMO Channels,”
IEEE Trans. WirelessCommun. , vol. 6, no. 5, pp. 1711–1721, May 2007.[4] R. A. Horn and C. R. Johnson,