Simplified Algorithms for Optimizing Multiuser Multi-Hop MIMO Relay Systems

Transcription

1 2896 IEEE TRANSACTIONS ON COMMUNICATIONS, VO. 59, NO. 10, OCTOBER 2011 Simpified Agorithms for Optimizing Mutiuser Muti-op MIMO Reay Systems Yue Rong, Senior Member, IEEE Abstract In this paper, we address the issue of mutiaccess communication through muti-hop inear non-regenerative reays, where a users, a reay nodes, and the destination node may have mutipe antennas. Using a inear ima mean-squared error MMSE receiver at the destination node, we demonstrate that the optima ampifying matrix at each reay node can be viewed as a inear MMSE fiter concatenated with another inear fiter. As a consequence, the MSE matrix of the signa waveform estimation at the destination node is decomposed into the sum of the MSE matrices at a reay nodes. We show that at a high signa-to-noise ratio SNR environment, this MSE matrix decomposition significanty simpifies the soution to the probem of optimizing the source precoding matrices and reay ampifying matrices. Simuation resuts show that even at the ow to medium SNR range, the simpified optimization agorithms have ony a margina performance degradation but a greaty reduced computationa compexity and signaing overhead compared with the existing optima iterative agorithm, and thus are of great interest for practica reay systems. Index Terms MIMO reay, muti-hop reay, MMSE, mutiuser. I. INTRODUCTION NON-REGENERATIVE mutipe-input mutipe-output MIMO reay communication systems recenty have attracted much research interest [1]-[7]. For a singe-user muti-hop MIMO reay system with any number of hops, the optimaity of channe diagonaization has been proved in [3]. Based on this diagonaization property, an iterative agorithm was deveoped in [3] to optimize the power aocation at a data streams and a nodes. For a mutiuser reay system, the achievabe sum rate has been derived in [4], assug that each user is equipped with a singe antenna. In [5], the optima reay ampifying matrix and source precoding matrices were deveoped to maximize the sum source-destination mutua information of a two-hop mutiuser reay system, where the users and the reay node are equipped with mutipe antennas. Recenty, a ima mean-squared error MMSE-based optima mutiuser MIMO reay system has been proposed [6]. The quaity-of-service constraints in a muti-antenna reay broadcast channe were investigated in [7]. In this paper, we focus on mutiaccess communication through muti-hop inear non-regenerative reays. In contrast Paper approved by O. Oyman, the Editor for Cooperative and eterogeneous Networks of the IEEE Communications Society. Manuscript received December 7, 2010; revised March 15, 2011 and May 9, Y. Rong is with the Department of Eectrica and Computer Engineering, Curtin University, Bentey, WA 6102, Austraia e-mai: y.rong@curtin.edu.au. This research was supported under the Austraian Research Counci s Discovery Projects funding scheme project numbers DP , DP Digita Object Identifier /TCOMM /11$25.00 c 2011 IEEE to [4], we consider a reay system where a users, a reay nodes, and the destination node may have mutipe antennas. Using a inear MMSE receiver at the destination node, we show that the optima ampifying matrix at each reay node can be viewed as a inear MMSE fiter concatenated with another inear fiter. As a consequence, the MSE matrix of the signa waveform estimation at the destination node can be decomposed into the sum of the MSE matrices at a reay nodes. A very usefu appication of such decomposition is that it greaty simpifies the source precoding matrices and reay ampifying matrices optimization probem at a moderatey high signa-to-noise ratio SNR environment. In particuar, it enabes the power aocation optimization to be performed at each reay node in a distributed manner, which requires ony oca channe state information CSI knowedge. Thus the simpified reay agorithms proposed in this paper have a significant reduction in both the computationa compexity and the signaing overhead compared with the iterative agorithm deveoped in [6]. Simuation resuts show that even at the ow to medium SNR range, the simpified optimization agorithms ony sighty increase the MSE of the signa waveform estimation and the system bit-error-rate BER, but greaty reduce the computationa compexity ess than the compexity of carrying out one iteration of the agorithm in [6]. Thus, the simpified optimization agorithms are of great interest for practica reay systems. The mutiuser muti-hop reay agorithms deveoped in this paper can be appied in mutihop wireess backhau networks where high data rate wireess inks need to be estabished over a ong distance with the aid of fixed reay nodes. Note that muti-hop wireess backhau networks are being considered by severa industry standards such as IEEE802.16j [8], [9]. We woud ike to mention that the decomposition of the MSE matrix was first discovered in [10] for a singe-user two-hop MIMO reay system. Our paper generaizes [10] from singe-user two-hop MIMO reay system to mutiuser muti-hop MIMO reay systems with any number of hops and any number of users. Note that due to the introduction of mutiusers and mutipe reay nodes, a rigorous proof of the MSE matrix decomposition for muti-hop MIMO reay system is much more chaenging than that for the two-hop MIMO channe, and is one contribution of this paper. The generaization from a singe-user two-hop MIMO system to mutiuser muti-hop MIMO reay systems is significant. Note that athough in this paper we focus on upink mutiaccess systems, the downink broadcast system can be designed by expoiting the upink-downink duaity for muti-hop inear non-regenerative MIMO reay systems estabished in [11].

2 RONG: SIMPIFIED AGORITMS FOR OPTIMIZING MUTIUSER MUTI-OP MIMO REAY SYSTEMS 2897 s 1 si s Nu B 1 B i B Nu G 1 Gi G Nu v 1 y 1 x2 F2 2 v 2 y 2 x F 3 3 v y Fig. 1. Bock diagram of an N u-user -hop inear non-regenerative MIMO reay communication system. In this paper, for notationa convenience, we consider a narrow band singe-carrier system. owever, our resuts can be straightforwardy generaized to broadband muti-carrier muti-hop MIMO reay systems as in the case of two-hop MIMO reay system shown in [2]. The rest of this paper is organized as foows. In Section II, we introduce the mode of a muti-hop inear non-regenerative mutiaccess MIMO reay communication system. The proposed source and reay design agorithms are presented in Section III. In Section IV, we show some numerica exampes. Concusions are drawn in Section V. II. SYSTEM MODE We consider a mutiaccess upink system where N u users simutaneousy transmit information to a common destination node equipped with a inear receiver as shown in Fig. 1. Due to the ong source-destination distance, 1 reay nodes are appied in seria to reay signas from a users to the destination node, where the th reay node is equipped with N antennas, = 1,, 1, and the destination node has N antennas. The ith user transmits M i independent data streams using M i antennas, i =1,,N u. We denote N 0 = N u M i as the tota number of independent data streams from a users. For a inear non-regenerative MIMO reay system, there shoud be N 0 N 1,,N,since otherwise the system can not support N 0 active symbos in each transmission. Such condition is imposed by the inherent physica property of the MIMO channe which is true aso for cassica singe-hop MIMO communication systems [12], but not from our agorithms deveoped ater. In systems with more than 1 potentia reays, reay nodes having enough number of antennas shoud be chosen to estabish an -hop communication system. In this paper, we focus on the non-regenerative reay strategy as in [1]-[7] and [10]-[11] due to the foowing two reasons. First, in the non-regenerative strategy, the reay node ony ampifies and retransmits its received signa. Thus, the compexity of the non-regenerative strategy is much ower than that of the regenerative strategy. This advantage is particuary important when a nodes are equipped with mutipe antennas, since decoding mutipe data streams invoves much more computationa efforts than decoding a singe data stream. Second, for muti-hop reay networks, the deay introduced by the regenerative strategy due to decoding is much arger than that of the non-regenerative strategy. At the ith user, the M i 1 moduated signa vector s i is ineary precoded by the M i M i source precoding matrix B i, and the precoded signa vector u i = B i s i is transmitted W ŝ to the first reay node. The received signa vector at the first reay node is given by N u y 1 = G i u i + v 1 1 x 1 + v 1 1 where G i, i =1,,N u,isthen 1 M i MIMO channe matrix between the first reay node and the ith user, v 1 is the N 1 1 independent and identicay distributed i.i.d. additive white Gaussian noise AWGN vector at the first reay node, x 1 = F 1 s, s [ s T 1,, st N u ] T,and 1 [G 1,, G Nu ], F 1 bdb 1,, B Nu. 2 ere 1 is the equivaent N 1 N 0 first-hop MIMO channe, F 1 is the equivaent N 0 N 0 bock diagona source precoding matrix, s is an N 0 1 vector containing source symbos from a users, bd stands for a bock diagona matrix, and T denotes matrix vector transpose. We assume that E[ss ]=I N0,whereE[ ] stands for the statistica expectation, denotes the ermitian transpose, and I n is an n n identity matrix. Due to its simpicity, a inear nonregenerative reay matrix is used at each reay as in [1]-[7]. The input-output reationship at the th reay nodes is x +1 = F +1 y, =1,, 1 3 where F +1, =1,, 1, isthen N ampifying matrix at the th reay node, and y, =1,, 1, isthe N 1 signa vector received at the th reay node written as y = x + v, =1,, 1 4 where, =1,, 1, isthen N MIMO channe matrix of the th hop, and v is the i.i.d. AWGN vector at the th reay node. Finay, at the ast hop, the signa vector received at the destination node is given by 4 with =. We assume that a noises are compex circuary symmetric with zero mean and unit variance. From 1-4, we have [3] y = A s + v, =1,, where A is the equivaent MIMO channe matrix from the source to the th hop, and v is the equivaent noise vector given by [3] 1 A = i F i, =1,, 5 i= v 1 = v 1, v = j i F i v j +v, =2,,.6 j=2 i= ere for matrices X i, k i= X i X X k. From 6, the covariance matrix of v, C =E[ v v ], = 1,,,isgivenby C 1 =I N1 j C = i F i j=2 i= F i i +I N,=2,,. i=j We woud ike to mention that the system mode in this paper extends the mode of mutiuser MIMO reay systems in [4],

3 2898 IEEE TRANSACTIONS ON COMMUNICATIONS, VO. 59, NO. 10, OCTOBER 2011 [5], [7] from two hops to mutipe hops. Such extension is important in the case of ong source-destination distance where a two-hop reay is not sufficient and a muti-hop reay is necessary to estabish a reiabe source-destination ink. III. PROPOSED SOURCE AND REAY DESIGN AGORITMS With a inear receiver at the destination node, the estimated signa vector is given by ŝ = W y,wherew is the N N 0 weight matrix. The weight matrix of the inear MMSE receiver [13] is W =A A +C A [3], where stands for the matrix inversion. Using this MMSE receiver, the MSE matrix E at the destination node is given by [3] E = I N0 + A C A [ = I N0 + F i i i F i =2 i= 1 F i i + I N i F i. 7 i= i= et us introduce matrices j D A A + C = i F i j=1 i= F i i + I N =1,, 1. 8 It can be shown from [3] that the transmission power consumed by the th reay node is tr E [ x +1 x +1] =tr F+1 D F +1,=1,, 9 where tr denotes matrix trace. Using 7 and 9, the probem of imizing the MSE of the signa waveform estimation at the destination node can be written as IN0 tr + A {F },{B C A 10 i} s.t. tr F D F p, =2,, 11 trb i B i q i, i =1,,N u 12 where 11 and 12 are the transmission power constraint at each reay node and each user, respectivey, p and q i are the corresponding power budget, {F } [F 2,, F ], and {B i } [B 1,, B Nu ]. The probem is nonconvex with matrix variabes, and a gobay optima soution is very difficut to obtain with a reasonabe computationa compexity non-exhaustive searching. In [6], an iterative procedure was deveoped to obtain at east a ocay optima soution of the probem 10-12, where in each iteration, the reay ampifying matrices are optimized with fixed source precoding matrices, and then the source precoding matrices are updated with the given reay ampifying matrices. owever, the computationa compexity and the signaing overhead of the iterative agorithm is quite high for practica reay systems. In the foowing, we propose simpified agorithms to sove an approximation of the probem The proposed agorithms have much smaer computationa compexity and signaing overhead than the iterative agorithm in [6] as anayzed and shown ater. i=j A. Optima Structure of Reay Ampifying Matrices By introducing N N 0 matrices T, =2,,,the foowing theorem estabishes the structure of the optima reay ampifying matrices, and demonstrates that the MSE matrix at the destination node can be decomposed into the sum of the MSE matrices at a reay nodes. TEOREM 1: The optima reay ampifying matrices have the foowing structure F = T A D, =2,,. 13 Using 13, the MSE matrix at the destination node can be equivaenty decomposed to E = I N0 +F 1 1 1F 1 + where =2 R +T T 14 R A D A, =2,,. 15 PROOF: See Appendix A. Interestingy, it can be seen from 13 that the optima reay ampifying matrices F, =2,,, can be decomposed into F = T W,whereW =A A + C A, = 2,,, is the weight matrix of the inear MMSE fiter for the received signa vector at the 1-th reay node given by y = A s+ v, and the inear fiter T wi be designed ater. The term R, + T T =2,,,in 14 is the increment of the MSE matrix introduced by the 1-th reay node as detaied in Appendix A. It is worth noting that R is in fact the covariance matrix of W y as R = W E[y y ]W. It can be seen from 14 that the effect of noise in the first hop is refected by I N0 in the first term. As SNR increases, F 1 1 1F 1 increases to infinity, and I N0 + F 1 1 1F 1 approaches 0N0 N 0. Simiary, the effect of noise in the th hop, =2,,,isrefected by R. Interestingy, as SNR increases, R approaches I N0, T T increases to infinity, and consequenty, R + T T approaches 0N0 N 0. By expoiting 13, the transmission power consumed by each reay node can be written as tr F D F =tr T A D D D A T =tr T R T, =2,,. 16 From 2 we have IN0 tr + F F 1 N u =tr I N1 + G i Q i G i + N 0 N 1 17 where Q i E[u i u i ]=B ib i, i =1,,N u,isthecovariance matrix of the signa transmitted by the ith user. Now by using 14-17, the probem can be equivaenty

4 RONG: SIMPIFIED AGORITMS FOR OPTIMIZING MUTIUSER MUTI-OP MIMO REAY SYSTEMS 2899 rewritten as {Q i},{t } s.t. N u tr I N1 + G i Q i G i + R + T T 18 =2 tr T R T p, =2,, 19 trq i q i, Q i ર 0, i =1,,N u 20 where {T } [T 2,, T ], {Q i } [Q 1,, Q Nu ],andર stands for the matrix positive semi-definiteness. B. Proposed Agorithm 1 Using the matrix inversion emma A+BCD = A A BDA B + C DA, it can be seen from 15 that R = A C C A A C A + I N0 A C A = A C A A C A, +I N0 =2,,. In the case of moderatey high SNR where A C A I N0, we can approximate R as I N0, = 2,,.In other words, in such case, the vaue of Q i, i =1,,N u, does not affect R, =2,,,andT does not affect R j, j = +1,,. This fact impies that the objective function 18 and the constraints in 19 are decouped with respect to the variabes {Q i } and {T }. Therefore, the probem can be approximated and decomposed into the foowing reay ampifying matrix optimization probem for each =2,, R tr T + T T 21 s.t. tr T R T p 22 and the source covariance matrices optimization probem N u tr I N1 + G i Q i G i 23 {Q i} s.t. trq i q i, Q i ર 0, i =1,,N u.24 In the foowing, we show that the probem has a water-fiing soution. et us introduce the eigenvaue decomposition EVD of = V Λ V and R = U Σ U, = 2,,, where the dimensions of V and Λ are N N, the dimensions of U and Σ are N 0 N 0,and the diagona eements in Λ and Σ are sorted in increasing orders. By introducing T T R 1 2, the probem can be rewritten as T tr R 1 2 IN0 + T T R s.t. tr T T p. 26 Using emma 2 in Appendix A, the soution to the probem in terms of the singuar vaue decomposition SVD of T is given by T = V,1 Ω U,whereV,1 contains the rightmost N 0 coumns of V. Thus the structure of the optima inear fiter T is given by T = V,1 Δ U, Δ = Ω Σ 1 2, =2,, 27 where Δ is an N 0 N 0 diagona matrix that remains to be optimized. Interestingy, it can be seen from 27 that at the 1- th reay node, the inear fiter T first performs beamforg to the direction of the eigenvectors of R, then it aocates power to N 0 streams through Δ,andfinay beamforms to the direction of the eigenvectors of. Substituting 27 back into 21-22, we find that the matrix-variabe optimization probem is converted to the foowing optima power oading probem with scaar variabes δ,1,,δ,n0 s.t. N 0 N 0 σ,i 1 + δ2,i λ,i 28 δ,iσ 2,i p 29 where δ,i,σ,i,λ,i, i =1,,N 0, denote the ith diagona eement of Δ, Σ, Λ, respectivey. Using the agrange mutipier method [14], it can be shown that the probem has a water-fiing soution given by δ 2,i = 1 λ,i λ,i μ σ,i 1 σ,i +, i =1,,N 0 where x + maxx, 0, and μ > 0 is the agrangian mutipier and the soution to the noninear equation of N0 σ,i λ,i += λ,i μ σ,i 1 σ p,i. Substituting 27 back into 13, the structure of the optima reay ampifying matrices is given by F = V,1 Δ U A D, =2,,. 30 Interestingy, athough 30 is derived under a high SNR assumption, this structure is in fact optima for the whoe SNR region as stated by the foowing theorem. TEOREM 2: The structure of the reay ampifying matrix given in 30 can be equivaenty written as F = V,1 Υ U,1, =2,,, which is optima for muti-hop MIMO reay systems as proved in [3]. ere Υ, =2,,, are N 0 N 0 diagona matrices, 1 F 1 = U 1 Γ 1 V1, = U Γ V, =2,,, are SVDs of 1 F 1 and with the diagona eements of Γ sorted in increasing orders, and U,1 contains the rightmost N 0 coumns of U. PROOF: See Appendix B. Finay, the source covariance matrices optimization probem can be soved as foows. By introducing a positive semi-definite PSD matrix X with X ર I N1 + Nu G iq i G i and using the Schur compement [14], the probem can be converted to the probem of {Q i},x s.t. trx 31 X IN1 I N1 I N1 + N u G iq i G ર 0 i 32 trq i q i, Q i ર 0, i =1,,N u.33

5 2900 IEEE TRANSACTIONS ON COMMUNICATIONS, VO. 59, NO. 10, OCTOBER 2011 TABE I PROCEDURE OF OPTIMIZING TE SOURCE AND REAY MATRICES. 1 Sove the SDP probem to obtain {Q i }. 2 For =2: Compute R ; Sove the probem to have T ;ObtainF as in 13. End. The probem is a convex semi-definite programg SDP probem which can be efficienty soved by the interiorpoint method [14]. Then the optima B i can be obtained as B i = U Q,i Λ 1 2 Q,i U b,whereu Q,i Λ Q,i U Q,i is the EVD of Q i, and U b is an arbitrary M i M i unitary matrix. We woud ike to mention that when the number of independent data streams transmitted by the ith user defined as N b,i is smaer than the number of transmit antennas M i, soving the probem might yied Q i whose rank is arger than N b,i.in such case, the randomization technique [15] can be appied to obtain a possiby suboptima soution of B i with rank N b,i. Nevertheess, in this paper, we focus on N b,i = M i, i = 1,,N u. Thus, B i obtained through soving the probem is optima. The procedure of optimizing a source precoding matrices and reay ampifying matrices is described in Tabe I. We woud ike to mention that in each iteration of the agorithm in [6], the compexity of updating the source precoding matrices is simiar to that of soving the probem Whie at each iteration of [6], an aternating power oading agorithm is appied to update the reay ampifying matrices, which has a higher computationa compexity than that of soving the probem Therefore, the computationa compexity of carrying out the procedure in Tabe I is ess than that of each iteration in [6]. Interestingy, the procedure in Tabe I can be carried out in a distributed manner where each reay node performs the necessary optimization procedure ocay. In particuar, the first reay node optimizes a source precoding matrices and sends back B i to user i. Thefirst reay node aso computes the optima F 2. Then at the th reay node, =2,, 1, the optima F +1 is computed based on +1, C,andA.TheCSIof +1 can be first estimated at the +1-th reay node through channe training [16], and then fed back to the th reay node. The knowedge of C and A is forwarded from the 1-th reay node. Note that due to its iterative nature, the agorithm in [6] requires centraized processing. Obviousy, compared with the centraized method, the distributed approach requires much ess information exchange and signaing overhead among different nodes, and thus, is preferred in practica reay systems. C. Proposed Agorithm 2 In this agorithm, the source precoding matrices are optimized by soving the probem owever, since R approaches I N0 as SNR increases, at a high SNR environment, the reay ampifying matrices optimization can be further simpified by substituting R in 21 and 22 with I N0.Then we have the foowing optimization probem for each T, =2,, T s.t. IN0 tr + T T 34 tr T T p. 35 Interestingy, 34 is in fact an upper-bound of 21. Appying emma 2 in Appendix A, the soution to the probem is given by T = V,1 Θ Π, =2,, 36 where Π can be any N 0 N 0 unitary matrix, and Θ is an N 0 N 0 diagona matrix. Substituting 36 back into 34-35, we find that the ith diagona eement of Θ is given by [ + ] 1 1 λ,i 2 θ,i = λ,i ν, i =1,,N 0.ereν > 0 is the soution to the noninear equation of + N 0 1 λ,i λ,i ν = p. Compared with the probem 21-22, the reay ampifying matrices designed by the probem has a smaer computationa compexity, since the atter agorithm does not need to compute R and its SVD. Comparing 36 with 27, we can choose Π = U = V1 which provides an optima structure of F. = 2,,, as proved in Appendix B. owever, in this case, a reay nodes need to know V 1,which increases the signaing overhead. For the reason of simpicity, we choose Π = I N0. Through numerica simuations in Section IV, we wi see that there is ony a negigibe increase in MSE and BER by using Π = I N0 instead of Π = V1. IV. NUMERICA EXAMPES In this section, we study the performance of the proposed mutiuser muti-hop MIMO reay design agorithms through numerica simuations. We simuate a fat Rayeigh fading environment where a channe matrices have entries with zero mean. In particuar, the variance of entries in G i is 1/M i, i =1,,N u, and the variance of entries in is 1/N, =2,,. A noises are compex circuary symmetric with zero mean and unit variance. We aso assume that p = P, = 2,,, q i = Q, i = 1,,N u. We woud ike to mention that in order to imize 10, each user shoud use its maxima power Q. In fact, 10 can be equivaenty rewritten as N u MSE = tr I N1 + Ψ G i B i B i G i N Ψ 1 +N 0 where ΨΨ = F i i i F i i=2 =2 i= 37 2 F i i + I N i F i. i= i= Assug that the ith user appies B i with tr B i B i = q i < Q, obviousy, the vaue of 37 can be reduced by using B i = α i Bi,whereα i = Q/ q i > 1. A simuation resuts are averaged over 5000 independent channe reaizations. The CVX convex optimization software

6 RONG: SIMPIFIED AGORITMS FOR OPTIMIZING MUTIUSER MUTI-OP MIMO REAY SYSTEMS MSE Exampe Proposed Agorithm 1 Proposed Agorithm 2 Proposed Agorithm 3 Iterative Agorithm Exampe P db Fig. 2. MSE versus P. Exampe 1: =2, N u =2, M 1 =3, M 2 =2, N 1 =8,andN 2 =7; Exampe 2: =3, N u =3, M =2,andN =8. BER Exampe Proposed Agorithm 1 Proposed Agorithm 2 Proposed Agorithm 3 Iterative Agorithm Exampe P db Fig. 3. BER versus P. Exampe 1: =2, N u =2, M 1 =3, M 2 =2, N 1 =8,andN 2 =7; Exampe 2: =3, N u =3, M =2,andN =8. package [17] is appied to sove the SDP probem For a exampes, we set Q =20dB and compare the performance of the agorithm described in Tabe I denoted as Proposed Agorithm 1, the agorithm where the reay ampifying matrices are designed by soving the probem using Π = V1 denoted as Proposed Agorithm 2, the agorithm of soving the probem with Π = I N0 denoted as Proposed Agorithm 3, and the optima iterative agorithm deveoped in [6] denoted as Iterative Agorithm. In our first exampe, we simuate a two-hop reay system with N u =2, M 1 =3, M 2 =2, N 1 =8,andN 2 =7. Fig. 2 shows the MSE performance of a agorithms versus P, and the system BER yieded by a agorithms with QPSK consteations are iustrated in Fig. 3 versus P. Our resuts ceary demonstrate that the Proposed Agorithms 1-3 ony have sighty higher MSE and BER than the Iterative Agorithm. Note that three proposed agorithms have a much smaer computationa compexity and signaing overhead than the Iterative Agorithm as anayzed in Subsection III-B. We aso observe from Figs. 2 and 3 that there is a significant performance improvement for a agorithms when P increases from 20dB to 25dB. The reason is that the MIMO reay system simuated is under-oaded in terms of data streams since N u M i = N 0 <N, =1,,. In such scenario, the system has a high spatia diversity order, which overcomes the saturation effect caused by the fixed Q. In MIMO reay systems with a ow spatia diversity order, one can observe the saturation effect as P increases when Q is fixed. We simuate muti-hop 3 mutiuser reay systems in the foowing two exampes. Since there are many parameters on the system setup for muti-hop reays, for simpicity, we consider reay systems where a users have the same number of antennas i.e., M i = M, i =1,,N u and a reay nodes and the destination node have the same number of antennas i.e., N = N, = 1,,N. The extension to systems where different nodes have different number of antennas is straight-forward. A three-hop =3 MIMO reay system is simuated in the second exampe with N u =3, M =2, and N = 8. Fig. 2 and Fig. 3 show the MSE and BER comparisons among four agorithms, respectivey. It can be seen that due to the approximation from the probem to the probem 34-35, the MSE and BER gaps between the Proposed Agorithm 2, the Proposed Agorithm 3, and the other two agorithm increase at ow to medium P.But the performance of the Proposed Agorithm 1 is very cose to that of the Iterative Agorithm. It can aso be observed from Figs. 2 and 3 that both the MSE and BER vaues of Exampe 1 are ower than those of Exampe 2, indicating that the agorithm yieding a ower MSE indeed guarantees a better BER performance. In the third exampe, a five-hop = 5 reay system is simuated with N u = 3, M = 2, and N = 9. The MSE and BER comparisons of four agorithms are shown in Fig. 4 and Fig. 5, respectivey. It can be ceary seen that the Proposed Agorithm 1 yieds amost the same BER as the Iterative Agorithm. It can aso be observed from Figs. 2-5 that there is ony a sma gap in both the MSE and BER performance between the Proposed Agorithm 2 and the Proposed Agorithm 3. The reason is that the performance of both agorithms depend not ony on Π, but aso on the power oading matrix Θ in 36. From the fact that the gap between the Proposed Agorithms 2 and 3 is smaer than that between the Proposed Agorithms 1 and 2, it shows that the choice of the power oading matrix pays a more important roe than that of Π. Since both the Proposed Agorithms 2 and 3 use the same Θ, the performance gap caused by using different Π is sma. Based on the simuation resuts and taking into account the compexity-performance tradeoff, the Proposed Agorithm 1 is most suitabe for practica mutiuser muti-hop MIMO reay systems. V. CONCUSIONS We addressed the issue of mutiaccess communication through muti-hop non-regenerative MIMO reays. It has been shown that the MSE matrix of the signa waveform estimation at the destination node can be decomposed into the sum of the MSE matrices at a reay nodes. Simpified source and reay optimization agorithms have been proposed which greaty

7 2902 IEEE TRANSACTIONS ON COMMUNICATIONS, VO. 59, NO. 10, OCTOBER 2011 MSE Proposed Agorithm 1 Proposed Agorithm 2 Proposed Agorithm 3 Iterative Agorithm P db Fig. 4. MSE versus P. Exampe 3: =5, N u =3, M =2, N =9. BER Proposed Agorithm 1 Proposed Agorithm 2 Proposed Agorithm 3 Iterative Agorithm P db Fig. 5. BER versus P. Exampe 3: =5, N u =3, M =2, N =9. reduce the computationa compexity and signaing overhead with ony a negigibe MSE and BER degradation. APPENDIX A PROOF OF TEOREM 1 To prove Theorem 1, we need the foowing two emmas. EMMA 1 [18, 9..1.h]: For two N N positive semidefinite matrices A and B with eigenvaues λ a,i and λ b,i, i = 1,,N, arranged in the same order, respectivey, it foows that trab N λ a,iλ b,n+1 i. EMMA 2: For matrices A, T, with dimensions of m n, m, andk, respectivey, where k,, m n, r rank n, andrankt =n, the soution to the probem of tr A I m + T T A s.t. tr TT p T 38 is given by T = V h,1 ΛU a in terms of the SVD of T. ere = U h Σ h Vh, A = U aσ a Va 39 are the SVDs of and A, with dimensions of U h, Σ h, V h, U a, Σ a,andv a being k k, k,, m n, n n, and n n, respectivey. The diagona eements of Σ h and Σ a are sorted in increasing orders, and V h,1 contains the rightmost n coumns of V h. PROOF: et us introduce the EVD of T T =[U x U x ]bd 0 m n m n, Λ x [U x U x ] 40 where the dimensions of U x, U x,andλ x are m m n, m n, andn n, respectivey, 0 p m denotes a p m matrix with a zero entries. Substituting 39 and 40 into 38, the objective function in 38 can be written as tr Σ a U a Ux I n + Λ x U x + U x U x Ua Σ a. 41 From emma 1 we know that 41 is imized if the diagona eements of Λ x are sorted in increasing order, and U x U a = Φ n, where Φ n stands for an arbitrary n n diagona matrix with unit-norm main diagona eements, i.e., [Φ n ] i,i =1, [Φ n ] i,j =0,i,j =1,,n,i = j. Without affecting the objective function in 38, we choose U x = U a. It wi be seen ater on that the power constraint in 38 is invariant to U x. From 40 we have T = V x Λ 1 2 x U a 42 where V x is a k n semi-unitary matrix with Vx V x = I n. Substituting the SVD of in 39 into 42 and eft mutipying by U h on both sides, we have [ ] [ ] 0k r r 0 k r r U ˆT = h, r V 0 r r Σ x Λ 1 2 x U a 43 h,r U h,r where ˆT = Vh T, Σ h,r is a r r diagona matrix containing the nonzero singuar vaues of, andu h, r and U h,r contain coumns of U h associated with the zero and nonzero singuar vaues of, respectivey. If k = = r, 43 hods if and ony if ˆT = Σ h,r U h,r V xλ 1 2 x U a. 44 If >k= r, then 43 hods if and ony if [ ] 0r r Σ h,r ˆT = U h,r V x Λ 1 2 x U a. 45 Finay, if k > r and > r, 43 is true if and ony if U h, r V x = 0 k r n and 45 hods. From 45, we see that in the atter two cases, there are many soutions for ˆT. We shoud choose ˆT that tr ˆT ˆT = trtt is imized. Such ˆT is the imum norm soution given by ˆT = [ 0 r r Σ ] T h,r U h,r V x Λ 1 2 x U a. 46 Interestingy, both 44 and 46 ead to the same transmission power, given by tr TT =tr Λ 1 2 x Vx U h,r Σ 2 h,r U h,rv x Λ 1 2 x. 47 Obviousy, 47 is invariant to U x. Using emma 1, we know that 47 is imized by U h,r V x = Φ r. Without oss of generaity, we choose V x = U h,r. From 44 and 46, we find that T = V h,1 Σ h,r Λ 1 2 x U a. Thus, we prove the optima structure of T with Λ = Σ h,r Λ 1 2 x.

8 RONG: SIMPIFIED AGORITMS FOR OPTIMIZING MUTIUSER MUTI-OP MIMO REAY SYSTEMS 2903 Now we start to prove Theorem 1. The MSE matrix 7 can be rewritten as E = I N0 A F F D F + I N F A 48 [ = I N0 A D D F F D +D ]A 49 = I N0 + A C A + A D F F D + D A 50 where the matrix inversion emma is used to obtain 48 and 50, and the identity B BCB + I B = C CB BC+C is appied to get 49. Interestingy, the first term in 50 is the MSE matrix associated with a MIMO reay constituted by the first 1 hops of the origina reay, and the second term in 50 is the increment of the MSE matrix introduced by the ast-hop of the origina reay. Moreover, since the first term in 50 is irreevant to F, the optimization probem in for F can be equivaenty written as D F F D +D A tr A 51 F s.t. tr F D F p. 52 By introducing F = F D 1 2, the probem can be rewritten as tr A 1 F D 2 F F D +I 1 2 A 53 s.t. tr F F p. 54 et us introduce the foowing EVD and SVD: = V Λ V, 1 D 2 A = U d Λ d Vd,whereΛ and V are N N matrices, the dimension of U d, Λ d, V d are N N 0, N 0 N 0, N 0 N 0, respectivey, and the diagona eements of Λ and Λ d are sorted in increasing order. It can be seen from emma 2 that the SVD of the optima F in is given by F = V,1 Λ f U d,whereλ f is the N 0 N 0 diagona singuar vaue matrix, and V,1 contains the rightmost N 0 coumns of V. With simpe manipuations, we obtain F = V,1 Λ f Λ d V d V dλ d U d = T A 1 D 2 where T V,1 Λ f Λ d V d. Therefore, we have F = T A D. Now by using the matrix inversion emma, the second term in 50 can be written as A A T T A A + D [ = A D D A A D A +T T ] A D A [ = T T + ] A D A. 55 Substituting 55 back into 50 and using 15, we obtain E = E + T T + R 56 where E = I N0 + A C A is the MSE matrix at the 1-th hop. Simiar to 50-56, we can show that the optima F is given by F = T A D, =2,, 1, and E = E + T T + R, =2,, 1.57 E 1 = I N0 + A = 1 C 1 A 1 IN0 + F F 1 By combining 56-58, we have E = I N0 + F F 1 + =2 R. + T T APPENDIX B PROOF OF TEOREM 2 The proof is conducted through mathematica induction. First, for =1,wehaveA 1 = 1 F 1, D 1 = 1 F 1 F I N1,andR 2 = F 1 1 1F 1 F I N 1 1 F 1.By using 1 F 1 = U 1 Γ 1 V1, we obtain that U 2 = V 1, A 1 D 1 = V 1 Γ 1,1 Γ 2 1,1 + I N 0 U,whereΓ 1,1 1,1 is an N 0 N 0 diagona matrix containing the nonzero singuar vaues of 1 F 1. Thus from 30 we have F 2 = V 2,1 Υ 2 U, 1,1 where Υ 2 Δ 2 Γ 1,1 Γ 2 1,1 + I N0 is an N 0 N 0 diagona matrix. The optimaity of the structure of such F 2 has been proven in [3] for muti-hop MIMO reay systems. Second, assug that for 2, the optima F j given in 30 can be written as F j = V j,1 Υ j U, j,1 j = 2,,, we now show that F +1 in 30 can be written as F +1 = V +1,1 Υ +1 U,1. In fact, from 2 5 we can write A = U,1 i= Γ i,1υ i Γ 1,1 V1, where Γ i,1 is an N 0 N 0 diagona matrix containing the argest N 0 singuar vaues of i. Then from 8 we j have D = U,1 j=1 i= Γ2 i,1 Υ2 i 1,1 Γ2 U,1 + I N, from 15 we get R +1 = 2 j V 1 i= Γ2 i,1 Υ2 i 1,1 j=1 Γ2 i= Γ2 i,1 Υ2 i Γ2 1,1 + V I N0 1, and thus U +1 = V 1. Finay, from 30 we have F +1 = V +1,1 Υ +1 U,1,where 2 Υ +1 = Δ +1 Γ i,1 Υ i Γ 1,1 i= j Γ 2 i,1 Υ2 i Γ2 1,1 + I N0. j=1 i= Therefore, based on the principe of mathematica induction, we have F = V,1 Υ U,1, =2,,. REFERENCES [1] X. Tang and Y. ua, Optima design of non-regenerative MIMO wireess reays," IEEE Trans. Wireess Commun., vo. 6, pp , Apr [2] Y. Rong, X. Tang, and Y. ua, A unified framework for optimizing inear non-regenerative muticarrier MIMO reay communication systems," IEEE Trans. Signa Process., vo. 57, pp , Dec [3] Y. Rong and Y. ua, Optimaity of diagonaization of muti-hop MIMO reays," IEEE Trans. Wireess Commun., vo. 8, pp , Dec [4] C.-B. Chae, T. Tang, R. W. eath Jr., and S. Cho, MIMO reaying with inear processing for mutiuser transmission in fixed reay networks," IEEE Trans. Signa Process., vo. 56, pp , Feb [5] Y. Yu and Y. ua, Power aocation for a MIMO reay system with mutipe-antenna users," IEEE Trans. Signa Process., vo. 58, pp , May 2010.

9 2904 IEEE TRANSACTIONS ON COMMUNICATIONS, VO. 59, NO. 10, OCTOBER 2011 [6] M. R. A. Khandaker and Y. Rong, Joint source and reay optimization for mutiuser MIMO reay communication systems," in Proc. 4th Intern. Conf. Signa Process. Commun. Sys., Dec [7] R. Zhang, C. C. Chai, and Y. C. iang, Joint beamforg and power contro for muti-antenna reay broadcast channe with QoS constraints," IEEE Trans. Signa Process., vo. 57, pp , Feb [8] M. Cao, X. Wang, S.-J. Kim, and M. Madihian, Muti-hop wireess backhau networks: a cross-ayer design paradigm," IEEE J. Se. Areas Commun., vo. 25, pp , May [9] V. Genc, S. Murphy, Y. Yu, and J. Murphy, IEEE j reaybased wireess access networks: an overview," IEEE Wireess Commun., vo. 15, pp , Oct [10] C. Song, K.-J. ee, and I. ee, MMSE based transceiver designs in cosed-oop non-regenerative MIMO reaying systems," IEEE Trans. Wireess Commun., vo. 9, pp , Juy [11] Y. Rong and M. R. A. Khandaker, On upink-downink duaity of mutihop MIMO reay channe," IEEE Trans. Wireess Commun., vo. 10, pp , June [12] D. P. Paomar, J. M. Cioffi, and M. A. agunas, Joint Tx-Rx beamforg design for muticarrier MIMO channes: a unified framework for convex optimization," IEEE Trans. Signa Process., vo. 51, pp , Sep [13] S. M. Kay, Fundamentas of Statistica Signa Processing: Estimation Theory. Prentice a, [14] S. Boyd and. Vandenberghe, Convex Optimization. Cambridge University Press, [15] P. Tseng, Further resuts on approximating nonconvex quadratic optimization by semidefinite programg reaxation," SIAM J. Optim., vo. 14, no. 1, pp , Juy [16] P. ioiou and M. Viberg, east-squares based channe estimation for MIMO reays," in Proc. IEEE WSA 2008, Feb. 2008, pp [17] M. Grant and S. Boyd, Cvx: Matab software for discipined convex programg webpage and software." Avaiabe: Apr [18] A. W. Marsha and I. Okin, Inequaities: Theory of Majorization and Its Appications. Academic Press, Yue Rong S 03-M 06-SM 11 received the B.E. degree from Shanghai Jiao Tong University, China, the M.Sc. degree from the University of Duisburg- Essen, Duisburg, Germany, and the Ph.D. degree summa cum aude from Darmstadt University of Technoogy, Darmstadt, Germany, a in eectrica engineering, in 1999, 2002, and 2005, respectivey. From Apri 2001 to October 2001, he was a research assistant at the Fraunhofer Institute of Microeectronic Circuits and Systems, Duisburg, Germany. From October 2001 to March 2002, he was with Nokia td., Bochum, Germany. From November 2002 to March 2005, he was a Research Associate at the Department of Communication Systems in the University of Duisburg-Essen. From Apri 2005 to January 2006, he was with the Institute of Teecommunications at Darmstadt University of Technoogy, as a Research Associate. From February 2006 to November 2007, he was a Postdoctora Researcher with the Department of Eectrica Engineering, University of Caifornia, Riverside. Since December 2007, he has been with the Department of Eectrica and Computer Engineering, Curtin University of Technoogy, Perth, Austraia, where he is now a Senior ecturer. is research interests incude signa processing for communications, wireess communications, wireess networks, appications of inear agebra and optimization methods, and statistica and array signa processing. Dr. Rong received the Best Paper Award at the 16th Asia-Pacific Conference on Communications, Auckand, New Zeaand, 2010, the 2010 Young Researcher of the Year Award of the Facuty of Science and Engineering at Curtin University, the 2004 Chinese Government Award for Outstanding Sef- Financed Students Abroad China, and the DAAD/ABB Graduate Sponsoring Asia Feowship Germany. e has co-authored more than 50 referred IEEE journa and conference papers. e is a Guest Editor of the IEEE JSAC specia issue on Theories and Methods for Advanced Wireess Reays to be pubished in 2012, a Guest Editor of the EURASIP JASP Specia Issue on Signa Processing Methods for Diversity and Its Appications to be pubished in 2012, and has served as a TPC member for IEEE ICC, IWCMC, and ChinaCom.