Linear Minimum Mean - Square - Error Transceiver Design for Amplify - and - Forward Multiple Antenna Relaying Systems

sHj } = 0Lk,Lj when k 6= j and E{sksHk } = ILk . With separate precoder Tk for different mobile terminals, the received signal at the relay station is r = HBRTs+ η, (4.1) where HBR denotes the NB×NR channel matrix between the BS and relay station, T = [T1, · · · , TK ], s = [sT1 , · · · , sTK ]T and the vector η denotes the additive Gaussian noise with zero mean and covariance matrix Rη. The power constraint at the BS is given by ∑ k Tr(TkT H k ) ≤ Ps, where Ps is the maximum transmit power. At the relay station, before retransmission the signal r is multiplied with a for- warding matrix W under a power constraint Tr(WRrW)H ≤ Pr, where Pr is the maximum transmit power at the relay station and Rr is the covariance matrix of the received signal r: Rr = HBRTT HHHBR +Rη. (4.2) Finally, at the kth mobile terminal, the received signal yk is yk = HRM,kWHBRTs+HRM,kWη + vk, (4.3) where matrix HRM,k is the NR × NM,k channel matrix between the relay station and the kth mobile terminal, and vk is the additive Gaussian noise at the kth mobile terminal with zero mean and covariance matrix Rvk . 77 At each mobile terminal, an equalizer Gk is employed to detect the data. The mean-square-error (MSE) of data detection at the kth terminal is MSEk(Gk,W,Tk) =E{‖Gkyk − sk‖2} =Tr(Gk(HRM,kWRrW HHHRM,k +Rvk)G H k )− Tr(GkHRM,kWHBRTk) − Tr((GkHRM,kWHBRTk)H) + Tr(ILk). (4.4) Now defining y = [yT1 , · · · , yTK ]T, HRM = [HTRM,1, · · · , HTRM,K ]T, v = [vT1 , · · · , vTK ]T, and G = diag{[G1, · · · , GK ]}, the sum MSE can be written as MSED(G,W,T) = K∑ k=1 MSEk(Gk,W,Tk) =Tr(G(HRMWRrW HHHRM +Rv)G H)− Tr(GHRMWHBRT) − Tr((GHRMWHBRT)H) + Tr(IL), (4.5) where L = ∑K k=1 Lk and Rv = diag{[Rv1 , · · · , RvK ]}. Therefore, the downlink transceiver design optimization problem can be formu- lated as min G,W,T MSED(G,W,T) s.t. Tr(TTH) ≤ Ps Tr(WRrW H) ≤ Pr G = diag{[G1, · · · , GK ]}. (4.6) The optimization problem (4.6) is a nonconvex optimization problem for T, W and G, and there is no closed-form solution. This challenge remains even for the special case of multiuser MIMO systems [39–41] where only single hop transmis- sion is involved. However, notice that when two out of the three variables are fixed, 78 the optimization problem (4.6) for the remaining variable is a convex problem, and thus can be solved. Therefore, an iterative algorithm alternating the design of three variables can be employed. 4.2.2 Proposed iterative algorithm (1) Equalizer design at the destination When T and W are fixed, the optimization problem (4.6) is an unconstrained convex quadratic optimization problem for G. Furthermore, since the structure of G is block diagonal, the design of individual Gk are decoupled. Therefore, the necessary and sufficient condition for the optimal solution is ∂ ∑ k MSEk(Gk,W,Tk) ∂G∗k = 0Lk,NM,k , (4.7) and the optimal equalizer for the kth mobile terminal can be easily shown to be Gk = (HRM,kWHBRTk) H(HRM,kWRrW HHHRM,k +Rvk) −1. (4.8) (2) Forwarding matrix design at the relay station When T and G are fixed, the optimization problem (4.6) is a constrained convex optimization problem for the variable W, and the Karush-Kuhn-Tucker (KKT) con- ditions are the necessary and sufficient conditions for the optimal solution [54]. The KKT conditions of the optimization problem (4.6) with respective to W are [58] HHRMG HGHRMWRr + λWRr = (HBRTGHRM) H (4.9) λ(Tr(WRrW H)− Pr) = 0, λ ≥ 0, (4.10) Tr(WRrW H) ≤ Pr, (4.11) where λ is the Lagrange multiplier. 79 Based on the first KKT condition (4.9), the optimal forwarding matrix W can be written as W = (HHRMG HGHRM + λI) −1(HBRTGHRM)HR−1r , (4.12) where the value of λ is computed using (4.10) and (4.11). Since λ also appears in W, (4.10) and (4.11) depends on λ in a nonlinear way and there is no closed-form solution. Below, we propose a low complexity method to solve (4.10) and (4.11). First, notice that in order to have (4.10) satisfied, either λ = 0 or Tr(WRrWH) = Pr must hold. If λ = 0 also makes (4.11) satisfied, λ = 0 is a solution to (4.10) and (4.11). On other hand, if λ = 0 does not make (4.11) satisfied, we have to solve Tr(WRrWH) = Pr. It can be proved that [71] when T and G are fixed, the function f(λ) = Tr(WRrWH) is a decreasing function of λ and the range of λ must be within 0 ≤ λ ≤ √ Tr(ER−1r EH) Pr (4.13) where E = ∑ k{(HBRTkGkHRM,k)H}. Therefore, λ can be efficiently com- puted by one-dimension search, such as bisection search or golden search. Since Tr(WRrW H) = Pr is a stronger condition than Tr(WRrWH) ≤ Pr, (4.11) is satisfied automatically in this case. In summary, λ is computed as λ = 0 if f(0) ≤ PrSolve f(λ) = Pr using bisection algorithm Otherwise . (4.14) (3) Precoder design at the BS When W and G are fixed, the optimization problem (4.6) can be straightfor- wardly formulated as the following convex quadratic optimization problem for the 80 precoder T min T Tr(NH0 T HA0TN0) + 2R{Tr(BH0 T)}+ c0 s.t. Tr(NH1 T HA1TN1) + 2R{Tr(BH1 T)}+ c1 ≤ 0, Tr(NH2 T HA2TN2) + 2R{Tr(BH2 T)}+ c2 ≤ 0, (4.15) where the corresponding parameters are defined as A0 = H H BRW HHHRMG HGHRMWHBR, A1 = I, A2 = H H BRW HWHBR, BH0 = −GHRMWHBR, B1 = B2 = 0, N0 = N1 = N2 = IL, c0 = Tr(RηW HHHRMG HGHRMW) + Tr(IL) + Tr(GRvG H)) c1 = −Ps, c2 = Tr(WRηWH)− Pr. (4.16) Notice that the objective function and the constraints are of the same form. Using the property Tr(AB) = vecH(AH)vec(B) and the property of Kronecker product, we can write (l = 0, 1, 2) Tr(NHl T HAlTNl) =Tr(N H l T HA H 2 l A 1 2 l TNl) =vecH(A 1 2 l TNl)vec(A 1 2 l TNl) =vecH(T)(N∗l ⊗A H 2 l )(N T l ⊗A 1 2 l )vec(T), (4.17) where the first equality is based on the fact that Al’s are positive semidefinite matri- ces. Furthermore, we can also write Tr(BHl T) = vec H(BHl )vec(T). Putting these two results into (4.15) and after introducing an auxiliary variable t [72], (4.15) is 81 equivalent to the following optimization problem min T,t t s.t. vecH(T)(N∗0 ⊗A H 2 0 )(N T 0 ⊗A 1 2 0 )vec(T) ≤ t− 2R{vecH(BH0 )vec(T)} vecH(T)(N∗1 ⊗A H 2 1 )(N T 1 ⊗A 1 2 1 )vec(T) ≤ −c1 − 2R{vecH(BH1 )vec(T)} vecH(T)(N∗2 ⊗A H 2 2 )(N T 2 ⊗A 1 2 2 )vec(T) ≤ −c2 − 2R{vecH(BH2 )vec(T)}. (4.18) Since c0 does not affect the optimization problem, it has been neglected in (4.18). With the Schur complement lemma [70], the optimization problem (4.18) can be further reformulated as the following semi-definite programming (SDP) problem [72] min T,t t s.t.  I (NT0 ⊗A 120 )vec(T) ((NT0 ⊗A 1 2 0 )vec(T)) H −2R{vecH(B0)vec(T)}+ t  º 0  I (NTl ⊗A 12l )vec(T) ((NTl ⊗A 1 2 l )vec(T)) H −2R{vecH(Bl)vec(T)} − cl  º 0, l = 1, 2. (4.19) The precoder at the BS is designed by solving this SDP problem using standard numerical algorithms such as interior-point polynomial algorithms [58, 72]. 4.2.3 Summary and Initialization In summary, the downlink beamforming matrices are computed iteratively. Since in each iteration, the MSE monotonically decreases, the iterative algorithm is guar- anteed to converge to at least a local optimum. For initialization, identity matrices can be chosen as initial values due to its simplicity and better performance com- pared to randomly generated initial matrices [40,41,45]. On the other hand, we can 82 also use a suboptimal design by viewing the downlink dual-hop AF MIMO relay cellular networks as a combination of conventional point-to-point MIMO system in the first hop, and multiuser MIMO downlink system in the second hop. More specifically, for the first hop, the linear minimum mean-square-error (LMMSE) precoder T at BS and equalizer W1 at relay station can be jointly designed us- ing the point-to-point water-filling solution given in [23]. For the second hop, the precoder W2 at relay station and equalizer G at mobile terminals can be designed using the beamforming algorithm for multiuser MIMO systems proposed in [27]. Based on the results of W1 and W2, the forwarding matrix at relay station equals to W = W1W2. We refer this suboptimal algorithm as ‘separate LMMSE transceiver design’. It will be shown in Simulation section that the convergence speed using the second initialization is better than that of the first one. Finally, the iterative design procedure is formally given by Algorithm 1 With initialG0, W0 andT0, the algorithm proceeds iteratively and in each iteration: (1) G is updated using (4.8); (2) W is updated using (4.12) and (3.94); (3) T is updated by solving (4.19). The algorithm stops when ‖MSEID − MSEI+1D ‖ ≤ TD, where MSEID is the total MSE in the Ith iteration and TD is a threshold value. 4.3 Uplink Transceiver Design 4.3.1 System model and analogy with downlink design In this section we will focus on transceiver design for uplink, as shown in Fig. 4.1b. In uplink, there are Lk data streams to be transmitted from the kth mobile 83 terminal to the BS, and the signal from the kth mobile terminal is denoted as sk. Without loss of generality, it is assumed that the transmitted data streams are inde- pendent: E{sksHj } = 0Lk,Lj when k 6= j and E{sksHk } = ILk . At the kth mobile terminal, the transmit signal sk is multiplied by a precoder matrix Pk under a power constraint Tr(PkPHk ) ≤ Ps,k, where Ps,k is the maximum transmit power at the kth mobile terminal. The received signal x at the relay station is the superposition of signals from different terminals through different channels and is given by x = HMRPs+ n, (4.20) where HMR , [HMR,1 · · · HMR,K ], P , diag{[P1, · · · ,PK ]}, s , [sT1 · · · sTK ]T, with HMR,k being the NR ×NM,k channel matrix between the kth mobile terminal and relay station, and n is the additive Gaussian noise at the relay station with zero mean and covariance matrix Rn. Since the data transmitted from different mobile terminals are independent, the correlation matrix of x equals to Rx = HMRPP HHMR +Rn. (4.21) At the relay station, the received signal x is multiplied with a linear forwarding matrix F, with a power constraint Tr(FRxFH) ≤ Pr, where Pr is the maximum transmit power at the relay station. Finally, the received signal at the BS is y = HRBFHMRPs+HRBFn+ ξ, (4.22) where HRB is the NB ×NR channel matrix between the relay station and BS, and ξ is the additive zero mean Gaussian noise with covariance Rξ. 84 When a linear equalizer B is adopted at the BS, the total MSE of the detected data is MSEU(B,F,P) =E{‖By − s‖2} =Tr(B(HRBFRxF HHHRB +Rξ)B H)− Tr(BHRBFHMRP) − Tr((BHRBFHMRP)H) + Tr(IL), (4.23) where L = ∑K k=1 Lk is the total number of data streams. Finally, the optimization problem for transceiver design in the uplink case is formulated as min B,F,P MSEU(B,F,P) s.t. Tr(PkP H k ) ≤ Ps,k, k = 1, · · · , K Tr(FRxF H) ≤ Pr P = diag{[P1, · · · , PK ]}. (4.24) Comparing (4.24) with the downlink problem (4.6), it can be seen that the two problems are in the same form, except that i) there are individual constraints on Pk in (4.24) instead of a sum constraint on the corresponding Tk in (4.6), and ii) the diagonal structure constraint is on precoder instead of equalizer. However we can still employ the iterative algorithm developed in the previous section for this uplink transceiver design problem. More specifically, for equalizer B design, the problem is an unconstrained convex optimization problem and the optimal solution can be directly computed from the derivative of the objective function. For for- warding matrix F design, the problem is a convex quadratic optimization problem with only one constraint. In this case, the optimal solution can be solved based on KKT conditions. Finally, for precoder P design, the problem is a convex quadratic optimization with multiple constraints, which can be transformed into a standard 85 SDP problem. Notice that a SDP problem can handle any number of linear ma- trix inequality constraints and the diagonal structure of P does not affect the SDP problem. Although the optimization problem (4.24) can be solved using an iterative algo- rithm alternating the three variables B, F and P, this solution provide little insight into the nature of the problem. Below we consider the fully loaded or overloaded MIMO systems in which the number of independent data streams from mobile ter- minals is greater than or equal to the number of its antennas, i.e., NM,k ≤ Lk [73], [74]. The solution is found to be insightful and includes several existing algorithms for conventional AF MIMO relay or multiuser MIMO as special cases. Remark 1: The proposed Algorithm 1 in the previous section can be applied to a more general case when there are multiple BSs, relay stations and mobile terminals. This case corresponds to the cooperation of multiple cells, where multiple BSs communicate with mobile terminals via multiple relay stations. The details can be found in [75]. 4.3.2 Uplink transceiver design for fully loaded or overloaded sys- tems First, we reduce the number of variables of the optimization problem. Noticing that there is no constraint on B, the optimal B satisfies ∂MSEU(B,F,P)/∂B∗ = 0L,NB , and the optimal equalizer at the BS can be written as a function of forwarding matrix and precoder matrix. Therefore B = (HRBFHMRP)H(HRBFRxFHHHRB+ 86 Rξ) −1. Substituting this result into (4.23), the uplink MSE is simplified as MSEU(F,P) = Tr(IL)− Tr((HRBFHMRP)H(HRBFRxFHHHRB +Rξ)−1(HRBFHMRP)). (4.25) Based on the definition of Rx = HMRPPHHHMR +Rn, it can be expressed as Rx = R 1/2 n (R −1/2 n HMRPP HHHMRR −1/2 n + I︸ ︷︷ ︸ ,Ξ )R1/2n . (4.26) Now introducing F˜ = FR1/2n Ξ1/2, the MSE (4.25) becomes MSEU(F˜,P) =Tr(IL)− Tr((HRBF˜Ξ−1/2R−1/2n HMRP)H × (HRBF˜F˜HHHRB +Rξ)−1(HRBF˜Ξ−1/2R−1/2n HMRP)). (4.27) Thus the uplink transceiver design optimization problem (4.24) is rewritten as min F˜,P MSEU(F˜,P) s.t. Tr(PkP H k ) ≤ Ps,k, k = 1, · · · , K Tr(F˜F˜H) ≤ Pr P = diag{[P1, · · · , PK ]}. (4.28) Unfortunately, the optimization problem (4.28) is still nonconvex for F˜ and P, and thus there is no closed-form solution. However, notice that if either F˜ or P is fixed, the optimization problem is convex with respect to the remaining variable. Therefore, an iterative algorithm which designs F˜ and P alternatively, is proposed as follows. (1) Design F˜ when P is fixed 87 From (4.27), it is noticed that F˜ appears both inside and outside of the inverse operation. In order simplify the objective function, we use the following variant of matrix inversion lemma CH(CCH +D)−1C = I− (CHD−1C+ I)−1. (4.29) Taking C = HRBF˜ and D = Rξ, the MSE (4.27) can be reformulated as [71] MSEU(F˜,P) =Tr((Ξ −1/2R−1/2n HMRP)(Ξ −1/2R−1/2nr HMRP) H × (F˜HHHRBR−1ξ HRBF˜+ I)−1) + Tr((PHHHMRR−1n HMRP+ I)−1). (4.30) Now, F˜ only appears inside the matrix inverse. If P is fixed, the last term of (4.30) is independent of F˜, and the optimization problem (4.28) becomes min F˜ Tr((Ξ−1/2R−1/2n HMRP)(Ξ −1/2R−1/2n HMRP) H︸ ︷︷ ︸ ,Θ (F˜HHHRBR −1 ξ HRB︸ ︷︷ ︸ ,M F˜+ I)−1) s.t. Tr(F˜F˜H) ≤ Pr. (4.31) Based on eigen-decomposition, Θ = UΘΛΘUHΘ and M = UMΛMU H M, and defin- ing ΛF˜ , UHMF˜UΘ, (4.32) the optimization problem (4.31) can be simplified as min ΛF˜ Tr(ΛΘ(Λ H F˜ ΛMΛF˜ + I) −1) s.t. Tr(ΛF˜Λ H F˜ ) ≤ Pr. (4.33) Without loss of generality, the diagonal elements of ΛΘ and ΛM are assumed to be arranged in decreasing order. The closed-form solution of (4.33) can be shown to 88 be [71] ΛF˜ =  [( 1√ µf Λ˜ −1/2 M Λ˜ 1/2 Θ − Λ˜ −1 M )+]1/2 0L,NR−L 0NR−L,L 0NR−L,NR−L  , (4.34) where Λ˜Θ and Λ˜M are the L×L principal submatrices of ΛΘ and ΛM, respectively. The scalar µf is the Lagrange multiplier which makes Tr(ΛF˜Λ H F˜ ) = Pr hold. Based on (4.32) and (4.34), the optimal F˜ can be recovered as F˜ = UM,L [( 1√ µf Λ˜ −1/2 M Λ˜ 1/2 Θ − Λ˜ −1 M )+]1/2 UHΘ,L, (4.35) where UM,L and UΘ,L are the first L columns of UM and UΘ, respectively. Finally, the optimal F is given by F = F˜Ξ−1/2R−1/2n . (2) Design P when F˜ is fixed Since Ξ in (4.27) depends on P, the MSE expression in (4.27) is a compli- cated function of P, direct optimization of P seems intractable. However, based on the property of trace operator Tr(DC) = Tr(CD), the total MSE (4.27) can be reformulated as MSEU(F˜,P) =Tr(IL)− Tr((HRBF˜)H(HRBF˜F˜HHHRB +Rξ)−1 ×HRBF˜)(Ξ−1/2R−1/2n HMRPPHHHMRR−1/2n︸ ︷︷ ︸ =Ξ−I Ξ−1/2)) =Tr(IL)− Tr((HRBF˜)H(HRBF˜F˜HHHRB +Rξ)−1HRBF˜)(INR −Ξ−1)). (4.36) Substituting the definition of Ξ into (4.36), the MSE can be further rewritten as MSEU(F˜,P) =Tr((HRBF˜) H(HRBF˜F˜ HHHRB +Rξ) −1(HRBF˜)︸ ︷︷ ︸ ,Π (R−1/2n HMRPP HHHMRR −1/2 n + INR) −1) + Tr(IL)− Tr((HRBF˜)H(HRBF˜F˜HHHRB +Rξ)−1(HRBF˜)), (4.37) 89 where P only appears inside of the inverse operation. As the last two terms of (4.37) are independent of P, the optimization problem for P is min P Tr(Π(R−1/2n HMRPP HHHMRR −1/2 n + INR) −1) s.t. Tr(PkP H k ) ≤ Ps,k k = 1, · · · , K P = diag{[P1, · · · , PK ]}. (4.38) With the definitions of HMR and P, HMRPP HHHMR = K∑ k=1 {HMR,k PkPHk︸ ︷︷ ︸ ,Qk HHMR,k}. (4.39) Putting (4.39) into (4.38), the optimization problem becomes min Qk Tr(Π(R−1/2n K∑ k=1 {HMR,kQkHHMR,k}R−1/2n + INR)−1) s.t. Tr(Qk) ≤ Ps,k, k = 1, · · · , K, Qk º 0. (4.40) Using the Schur-complement lemma [55], the optimization problem (4.40) can be further formulated as a standard SDP optimization problem [72] min X,Qk Tr(X) s.t.  X Π1/2 Π1/2 R −1/2 n ∑ k{HMR,kQkHHMR,k}R−1/2n + INR  º 0 Tr(Qk) ≤ Ps,k, k = 1, · · · , K Qk º 0. (4.41) The SDP problems can be efficiently solved using interior-point polynomial algo- rithms [58]. 90 In summary, when NM,k ≤ Lk, the uplink transceiver design alternates be- tween the design of F˜ in (4.35) and Qk in (4.41). The algorithm stops when ‖MSEIU − MSEI+1U ‖ ≤ TU , where MSEIU is the total MSE in the Ith iteration and TU is a threshold value. After convergence, Pk = Q1/2k , F = F˜Ξ−1/2R−1/2n and B = (HRBFHMRP)H(HRBFRxFHHHRB +Rξ) −1. We refer the algorithm in this section as Algorithm 2. Remark 2: In case NM,k > Lk, there is an additional constraint Rank{Qk} ≤ Nk in (4.40). In this case, as rank constraints are nonconvex, transition from (4.40) to (4.41) involves a relaxation on the rank constraint. Then the objective function of (4.41) is a lower bound of that of (4.40). However, this problem seems to be common to all multiuser MIMO uplink beamforming [39, 43]. Notice that when NM,k ≤ Lk, there is no relaxation involved. 4.3.3 Special cases Notice that (4.35) has a more general form than the water-filling solution in tra- ditional point-to-point MIMO systems. On the other hand, (4.41) is a SDP problem frequently encountered in multiuser MIMO systems. In particular, they include the following existing algorithms as special cases. • If HRB = IL and Rξ = 0L,L, we have Π = IL in (4.40), and the SDP optimiza- tion problem (4.41) reduces to that of the uplink multiuser MIMO systems [43], [39]. Therefore, they have the same solution. • Substituting K = 1 and P = IL1 into (4.35), it reduces to the solution proposed for LMMSE joint design of relay forwarding matrix and destination equalizer in AF MIMO relay systems without source precoder [30]. • Notice that when there is only one mobile terminal (K = 1), the optimiza- tion problem (4.38) is in the same form as (4.31). Defining HHMRR −1 n HMR = 91 UMRΛMRU H MR, and Π = UΠΛΠU H Π, a closed-form solution can be derived us- ing the same procedure as for F˜, and we have P = UMR,L [( 1√ µp Λ˜ −1/2 MR Λ˜ 1/2 Π − Λ˜ −1 MR )+]1/2 (4.42) where the Λ˜MR and Λ˜Π are the L × L principal submatrices of ΛMR and ΛΠ, respectively, and the matrix UMR,L is the first L columns of UMR. The scalar µp is the Lagrange multiplier which makes Tr(PPH) = Ps,1 hold. In this case, the solution given by (4.42) corresponds to the source precoder design for AF MIMO relay systems with single user [31]. • Furthermore, substituting HRB = IL and Rξ = 0L,L into (4.42), it becomes the closed-form solution for LMMSE transceiver design in point-to-point MIMO systems [23]. 4.4 Simulation Results and Discussions In this section, we investigate the performance of the proposed algorithms for downlink and uplink. In the simulations, there is one BS, one relay station and two mobile terminals. For each mobile terminal, two independent data streams will be transmitted in the uplink (or received in the downlink) simultaneously. For each data stream, 10000 independent QPSK symbols are transmitted. The elements of MIMO channels between BS and relay station and between relay station and mobile terminals are generated as independent complex Gaussian random variables with zero mean and unit variance. Each point in the following figures is an average of 500 independent channel realizations. In order to solve SDP problems, the widely used optimization matlab toolbox CVX is adopted [76]. The thresholds for terminating the iterative algorithms are set at TD = TU = 0.0001. 92 0 2 4 6 8 10 12 14 16 18 20 10−1 100 Iteration number M SE Initialization is identity matrices Initialization is sperate LMMSE design P s /ση 2 =10dB and P r /σ v 2 =20dB P s /ση 2 =20dB and P r /σ v 2 =20dB Figure 4.2 The convergence behavior of the proposed Algorithm 1 when NB = 4, NR = 4 and NM,k = 2 with 2 users. First, let us focus on the downlink. In downlink, the noise covariance matrices at relay station and mobile terminals are Rη = σ2ηINR and Rv1 = Rv2 = σ 2 vINM , respectively. We define the first hop SNR at the relay station as Ps/ση2, and the second hop SNR at mobile terminals as Pr/σ2v . Fig. 4.2 shows the convergence behavior of the proposed Algorithm 1 for downlink with different second hop SNR at mobile terminals when NB = 4, NR = 4, NM,k = 2. Both initializations with identity matrices and the separate LMMSE design are shown. It can be seen that the proposed algorithm converges quickly, within 20 iterations. Furthermore, the convergence speed with separate LMMSE design as initialization is faster than that with identity matrices. It can also be seen that the two initializations result in the same MSE after convergence. Fig. 4.3 compares the total data MSEs of the proposed Algorithm 1 and several suboptimal algorithms versus the first hop SNR Ps/σ2η . The second hop SNR at 93 0 5 10 15 20 25 30 10−1 100 P s /ση 2 (dB) M SE Equalization at terminals only Equalization at both relay and mobile terminals Separate LMMSE transceiver design for two hops The proposed Algorithm 1 Figure 4.3 Total MSEs of detected data of the proposed Algorithm 1 and suboptimal algorithms, when NB = 4, NR = 4, NM,k = 2 and Pr/σ2v=20dB. mobile terminals is fixed to be 20dB. The number of antennas is set as NB = 4, NR = 4 and NM,k = 2. The suboptimal algorithms under consideration are •Direct amplify-and-forward, in which the precoder T at BS and forwarding matrix W at relay are proportional to identity matrices. At mobile terminals, LMMSE equalizer for the combined first hop and second hop channel is adopted to recover the signal [30]. • The first hop channel is equalized at relay and then the second hop channel is equalized at mobile terminals, both with LMMSE equalizers. • Separate LMMSE design proposed for initialization of Algorithm 1. From Fig. 4.3, it can be seen that as there is no precoder design at BS for the first two suboptimal algorithms, the data streams at different terminals cannot be effi- ciently separated by linear equalizers, resulting in poor performances. The separate LMMSE transceiver design has a much better performance. On the other hand, the 94 0 5 10 15 20 25 30 10−2 10−1 100 P s /ση 2 (dB) M SE The proposed Algorithm 1 without precoder design The proposed Algorithm 1 NB=6, NR=4, NM,k=2 NB=6, NR=6, NM,k=2 Figure 4.4 Total MSEs of detected data of the proposed Algorithm 1 with and without precoder design. proposed Algorithm 1 has the best performance among the four algorithms. The gap between the MSEs of the separate LMMSE design and that of Algorithm 1 is the performance gain obtained by additional iterations. As the proposed Algorithm 1 involves a computational expensive SDP for the precoder T design, it is of great interest to investigate how much degradation would result from skipping the precoder design. Fig. 4.4 compares the total data MSEs of the proposed Algorithm 1 and the same algorithm but fixing the precoder T ∝ I. The second hop SNR at mobile terminals Pr/σ2v is fixed to be 20dB. From Fig. 4.4, it can be seen that a properly designed precoder significantly improves the system performance when the first hop SNR is high. Without the precoder, the data MSEs exhibit error floors at much lower Ps/σ2η . On the other hand, we can also see that increasing the number of antennas at the relay station greatly improves the system performance, as it simultaneously increases the diversity gain of the two hops. 95 0 1 2 3 4 5 6 7 8 9 10 10−1 100 Iteration number M SE P s /σ n 2 =10dB and P r /σξ 2 =20dB P s /σ n 2 =20dB and P r /σξ 2 =20dB Figure 4.5 The convergence behavior of Algorithm 2 for uplink when NB = 4, NR = 4 and NM,k = 2. Now, let us turn to the results in the uplink. In uplink case, the noise covariance matrices at relay station and BS are Rn = σ2nINR and Rξ = σ 2 ξINB , respectively. We define the fist hop SNR at the relay station as Ps/σn2, where Ps = ∑K k=1 Ps,k. The second hop SNR at the BS is defined as Pr/σ2ξ . Fig. 4.5 shows the convergence behavior of the proposed Algorithm 2 for uplink when NB = 4, NR = 4 and NM,k = 2. Notice that in this case, at each mobile terminal the number of antennas equals to that of the data streams, and Algorithm 2 involves no relaxation. The initialization is identity matrices. It can be seen that Algorithm 2 converges very fast, indicating its superior performance. Fig. 4.6 shows the total data MSEs of the proposed Algorithm 2 and suboptimal algorithms, when NB = 4, NR = 4, NM,k = 2 and the SNR at relay station Ps/σ2n is fixed to be 20dB. The suboptimal algorithms are similar to those for the downlink. In particular, we consider 96 0 5 10 15 20 25 30 10−1 100 P r /σξ 2 (dB) M SE Equalization at BS only Equalization at both relay station and BS Separate LMMSE transceiver design for two hops The proposed Algorithm 2 Figure 4.6 Total MSEs of detected data of Algorithm 2 and suboptimal algorithms, when NB = 4, NR = 4, NM,k = 2 and Ps/σ2n=20dB. • Equalization of the equivalent two-hop channel is applied only at the BS. • Equalization is applied at relay station for the mobile-to-relay channel, and also at BS for the relay-to-BS channel. • Separate LMMSE design. The first hop is considered as a traditional multiuser MIMO uplink system, and the beamforming matrices are designed using the algo- rithms in [41] and [43]. The second hop is considered as a point-to-point MIMO system, and the beamforming matrices are designed using the result in [23]. From Fig. 4.6, it can be seen that the performance of the proposed Algorithm 2 is better than other suboptimal algorithms. However, as the signals from different terminals are cooperatively detected at BS, the gaps between the performance of the suboptimal algorithms from that of Algorithm 2 is much smaller compared to their counterparts in downlink. When Lk < NM,k in the uplink, strictly speaking, Algorithm 2 involves a re- laxation, and its performance is not guaranteed. However, a simple variation of 97 0 2 4 6 8 10 12 14 16 18 20 10−2 10−1 100 P s /σ n 2 (dB) M SE Algorithm by Guan The proposed Algorithm 2 The proposed Algorithm 1 NB=4, NR=4, NM,k=4 NB=4, NR=6, NM,k=4 Figure 4.7 Total MSEs of the detected data of the Algorithm 1, Algorithm 2 with relaxation and the algorithm proposed in [30]. Algorithm 1 can be used for transceiver design in this case. Fig. 4.7 shows the total data MSEs of Algorithm 1 for uplink and Algorithm 2 with rank relaxation, when Lk = 2 and NM,k = 4. The SNR at BS is fixed at Pr/σ2ξ=20dB. The joint relay forwarding matrix and destination equalizer design in [30] is also shown for com- parison. It can be viewed as a design without source precoders at mobile terminals. From Fig. 4.7, it can be seen that Algorithm 1 and Algorithm 2, which involve the joint design of precoder, forwarding matrix and equalizer perform better than the algorithm in [30]. This indicates the importance of source precoder design in AF relay cellular networks. Furthermore, although Algorithm 2 involves a relaxation, its performance is still satisfactory, and is close to that of Algorithm 1. Finally, it can also be concluded that increasing the number of antennas at relay station can greatly improve the performance of uplink transceiver design for all algorithms. 98 4.5 Conclusions In this chapter, LMMSE transceiver design for amplify-and-forward MIMO re- lay cellular networks has been investigated. Both uplink and downlink cases were considered. In the downlink, precoder at base station, forwarding matrix at relay station and equalizer at mobile terminals were jointly designed by an iterative algo- rithm. On the other hand, in the uplink case, we demonstrated that in general the transceiver design problem can be solved by an iterative algorithm with the same structure as in the downlink case. Furthermore, for the fully loaded or overloaded uplink systems, a novel transceiver design algorithm was derived and it includes several existing algorithms for conventional point-to-point or multiuser systems as special cases. Finally, simulation results were presented to show the performance advantage of the proposed algorithms over several suboptimal schemes. 99 Chapter 5 Conclusions and Future Research 5.1 Conclusions In this thesis, the joint design of linear relay forwarding matrix and destination equalizer for dual-hop single-user AF MIMO relay systems with Gaussian random channel uncertainties in both hops was first considered. The data MSE formula at the destination averaged over the random channel uncertainties was first derived. In order to minimize the average MSE, two robust design algorithms were pro- posed: an iterative algorithm with guaranteed convergence and a closed-form solu- tion with a mild relaxation. In the general case, the iterative algorithm has a better performance but a higher complexity. Although a mild relaxation is required for the general case, the closed-form solution was shown to be optimal when the column correlation matrix of the channel estimation error in the second hop is an identity matrix. Furthermore, we proceed to design robust linear transceiver for AF MIMO- OFDM relay systems in which relay forwarding matrix and destination equalizer were jointly designed based on MMSE criterion. The linear channel estimators and the corresponding averaged MSE expressions over channel estimation errors were first derived. Then a general solution for optimal forwarding and equalizer matrices 100 was proposed. When the channel estimation errors are uncorrelated, the optimal solution is in closed-form, and it also includes several existing transceiver design results as special cases. On the other hand, when channel estimation errors are correlated, a practical algorithm was introduced. Finally, LMMSE transceiver design for AF multiple-antenna relaying cellular networks was investigated, in which multiple mobile terminals communicate with the BS via a relay station. As all nodes are equipped with multiple antennas, pre- coder at source, forwarding matrix at relay and equalizer at destination were jointly designed to minimize the total MSEs of detected data at the destination. Both down- link and uplink have been considered. It is found that the downlink and uplink transceiver design problems are in the same form, and iterative algorithms with the same structure can be used to solve the design problems. For the specific cases of fully loaded or overloaded uplink systems, a novel algorithm is derived and several existing algorithms can be considered as its special cases. 5.2 Future Research Directions There are some possible directions for the future research based on the results given in this thesis. In this thesis, for the linear robust transceiver design, we focus on minimizing an averaged MSE over channel estimation errors. For robust signal processing design, another criterion for robust design is worst-case or min-max. This criterion aims at minimizing the objective function at the worst case in an uncertain region which is usually norm-bounded. The worst case robust transceiver design for AF MIMO relay systems also has great meaning, which can guarantee the worst case performance. 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Ye, “CVX: Matlab Software for Disciplined Con- vex Programming,” available at: http : //www.stanford.edu/boyd/cvx/, V.1.0RC3, Feb. 2007. 109 Vita July 2005 B.Eng. in Information Countermeasure, Xidian University September 2006–Present Ph.D. candidate, Department of Electrical and Electronic Engineering, The University of Hong Kong He can be reached at the email address xingchengwen@gmail.com. 110 List of Publications Journal Articles: 1. Chengwen Xing, Shaodan Ma and Yik-Chung Wu, “On Low Complexity Robust Beamforming with Positive Semi-Definite Constraints,” IEEE Trans- actions on Signal Processing vol. 57, no. 12, pp. 4942–4945, Dec. 2009. 2. Chengwen Xing, Shaodan Ma and Yik-Chung Wu, “Robust Joint Design of Linear Relay Precoder and Destination Equalizer for Dual-Hop Amplify-and- Forward MIMO Relay Systems,” IEEE Transactions on Signal Processing vol. 58, no. 4, pp. 2273–2283, Apr. 2010. 3. Xiao Li, Chengwen Xing, Yik-Chung Wu and S.C. Chan, “Timing Estima- tion and Re-synchronization for Amplify-and-Forward Communication Sys- tems,” IEEE Transactions on Signal Processing vol. 58, no. 4, pp. 2218– 2229, Apr. 2010. 4. Chengwen Xing, Shaodan Ma, Yik-Chung Wu and Tung-Sang Ng, “Transceiver Design at Relay and Destination for Dual-Hop Non-regenerative MIMO- OFDM Relay Systems Under Channel Uncertainties,” submitted to IEEE Transactions on Signal Processing, revised June 2010. 5. Chengwen Xing, Minghua Xia, Shaodan Ma and Yik-Chung Wu, “Linear MMSE Beamforming Design for Amplify-and-Forward Multi-Antenna Re- laying Cellular Networks,” under preparation. 6. Chengwen Xing, Shaodan Ma and Yik-Chung Wu, “Robust Joint Transceiver Design for Dual-Hop AF MIMO Relay Systems Using Weighted MMSE Cri- terion,” under preparation. 111 Articles in Refereed Conference Proceedings: 1. Chengwen Xing, Shaodan Ma and Yik-Chung Wu, “Iterative LMMSE Transceiver Design for Dual-Hop AF MIMO Relay Systems Under Channel Uncertain- ties,” Proceedings of IEEE Personal, Indoor and Mobile Radio Communica- tions Symposium (PIMRC’2009), 2009, Japan. 2. Chengwen Xing, Shaodan Ma and Yik-Chung Wu, “Bayesian Robust Lin- ear Transceiver Design for Dual-Hop Amplify-and-Forward MIMO Relay Systems,” Proceedings of IEEE Global Communications Conference (Globe- Com’2009),, 2009, U.S.A. 3. Chengwen Xing, Shaodan Ma, Yik-Chung Wu and Tung-Sang Ng, “Ro- bust Beamforming for Amplify-and-Forward MIMO Relay Systems Based on Quadratic Matrix Programming,” Proceedings of IEEE International Con- ference on Acoustics, Speech, and Signal Processing (ICASSP’2010), 2010, U.S.A. 4. Chengwen Xing, Shaodan Ma, Yik-Chung Wu, Tung-Sang Ng, and H. Vin- cent Poor,“Linear Transceiver Design for Amplify-and-Forward MIMO Re- lay Systems under Channel Uncertainties,” IEEE Wireless Communications and Networking Conference (WCNC’2010), 2010, Australia. 5. Shaodan Ma, Chengwen Xing, Yijia Fan, Yik-Chung Wu, Tung-Sang Ng, and H. Vincent Poor, “Iterative Transceiver Design for MIMO AF Relay Net- works with Multiple Sources” accepted by IEEE Milcom 2010 (Invited Pa- per).