IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 66, NO. 10, OCTOBER

Transcription

1 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 66, NO. 10, OCTOBER Robust MMSE Transcever Desgn for Nonregeneratve Multcastng MIMO Relay Systems Lenn Gopal, Member, IEEE, Yue Rong, Senor Member, IEEE, and Zhuquan Zang, Member, IEEE Abstract In ths paper, we nvestgate the transcever desgn for nonregeneratve multcastng multple-nput multple-output (MIMO relay systems, one transmtter broadcasts common message to multple recevers wth the ad of a relay node. The transmtter, relay, and recevers are all equpped wth multple antennas. We assume that the true (unknown channel matrces have Gaussan dstrbuton, wth the estmated channels as the mean value, and the channel estmaton errors follow the well-known Kronecker model. We frst develop an teratve robust algorthm to jontly desgn the transmtter, relay, and recever matrces to mze the mal mean-squared error (MSE of the sgnal waveform estmaton among all recevers. Then, we derve the optmal structure of the relay precodng matrx and show that the MSE at each recever can be decomposed nto the sum of the MSEs of the frst-hop and second-hop channels. Based on ths MSE decomposton, we develop a smplfed transcever desgn algorthm wth a low computatonal complexty. Numercal smulatons demonstrate the mproved robustness of the proposed transcever desgn algorthms aganst the msmatch between the true and estmated channels. Interestngly, compared wth the teratve algorthm, the smplfed transcever desgn has only neglgble performance loss wth sgnfcantly reduced computatonal complexty. Index Terms Mnmum mean-squared error (MMSE, multcastng, nonregeneratve MIMO relay, robustness. I. INTRODUCTION IN MANY practcal communcaton systems, one source node transmts common nformaton to multple recevers smultaneously. These systems are referred to as multcast broadcastng or multcastng systems. Recently, multcastng systems have attracted much research nterest, due to the ncreasng demand for moble applcatons such as locaton based vdeo broadcastng and streag meda. The wreless channel has the multcast broadcastng nature, makng t sutable for multcastng applcatons. owever, the wreless system performance may be degraded due to the channel fadng and shadowng effects. By deployng multantenna and beamforg technques at the transmtter and recevers, the channel shadowng effect can be mtgated [2]. Next Manuscrpt receved March 16, 2017; accepted May 5, Date of publcaton May 11, 2017; date of current verson October 13, Ths work was supported by the Australan Research Councl s Dscovery Projects fundng scheme (DP The revew of ths paper was coordnated by Dr. T. Tanguch. Ths paper was presented n part at the 22nd Internatonal Conference on Telecommuncatons, Sydney, NSW, Australa, Apr , (Correspondng author: Yue Rong. L. Gopal and Z. Zhang are wth the Department of Electrcal and Computer Engneerng, Curtn Unversty, Mr 98009, Malaysa (e-mal: lenn@ curtn.edu.my; zqzang@curtn.edu.my. Y. Rong s wth the Department of Electrcal and Computer Engneerng, Curtn Unversty, Bentley, WA 6102, Australa (e-mal: y.rong@curtn.edu.au. Dgtal Object Identfer /TVT generaton wreless standards such as WMAX m and 3GPP LTE-Advanced have already ncluded technologes whch enable better multcastng solutons based on mult-antenna and beamforg technques [3]. Due to the nonconvex nature of the problem, desgnng the optmal transmt beamforg vector for multcastng s dffcult n general. Capacty lmts of mult-antenna multcastng channel have been studed n [4], and the channel spatal correlaton effect on the channel capacty has been nvestgated n [5]. In [6], transmt beamforg vectors for physcal layer multcastng have been desgned wth the assumpton that the channel state nformaton (CSI s avalable at the transmtter. In the multcastng systems [3] [8], sngle-antenna has been assumed at the recevers. Recently multcastng systems wth mult-antenna recevers have been nvestgated n [9] [11]. In the case of long dstance between the transmtter and recevers, relay node s necessary to effcently mtgate the pathloss of wreless channel. In [12], a cooperatve protocol for multcastng systems wth multple transmt antennas s proposed wth the assumpton that the users are equpped wth sngle antenna. A two-hop multple-nput multple-output (MIMO relay multcastng system has been proposed n [13] one transmtter multcasts common message to multple recevers wth the ad of a relay node, and the transmtter, relay, and recevers are all equpped wth multple antennas. It s also assumed n [13] that the true CSI of all channels s avalable at the relay node. owever, n practcal communcaton systems, the exact CSI s not avalable, and therefore, has to be estmated. There s always msmatch between the true and estmated CSI. ence, the performance of the algorthm n [13] wll degrade due to such CSI msmatch. In ths paper, we propose a transcever desgn algorthm for multcastng MIMO relay systems whch s robust aganst the CSI msmatch. Smlar to [13], the transmtter, relay, and recevers n the system are all equpped wth multple antennas. owever, dfferent to [13], the true channel matrces have Gaussan dstrbuton, wth the estmated channels as the mean value, and the channel estmaton errors follow the well-known Kronecker model [14], [15]. We frst propose an teratve robust algorthm to jontly desgn the transmtter, relay, and recever matrces to mze the mal mean-squared error (MSE of the sgnal waveform estmaton among all recevers. Note that recently some transcever optmzaton problems n MIMO systems have been unfed nto a framework of matrx-monotonc optmzaton n [16]. A general robust lnear transcever desgn for mult-hop amplfy-and-forward MIMO relay systems has been proposed n [17]. Quadratc matrx programg has been appled n [18] for transcever optmzaton n MIMO relay systems. owever, the problem of robust transcever desgn for multcastng MIMO relay systems s not IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See standards/publcatons/rghts/ndex.html for more nformaton.

2 8980 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 66, NO. 10, OCTOBER 2017 Fg. 1. Block dagram of a two-hop nonregeneratve multcastng MIMO relay system. covered n [16] [18]. We would lke to menton that robust transcever desgn for sngle-user MIMO relay systems [14], [15], [19] [21], and multuser MIMO relay systems [22] can not be smply extended to multcastng MIMO relay systems. Due to the hgh computatonal complexty of the proposed teratve algorthm, we develop a smplfed robust transcever desgn scheme n whch we show that the MSE at each recever can be decomposed nto the sum of the MSEs of the frst-hop and second-hop channels. Ths extends the result of MSE decomposton [13] and [23] from MIMO relay systems wth perfect CSI to practcal MIMO relay systems wth mperfect CSI. Based on ths MSE decomposton, we develop a transcever desgn algorthm wth low computatonal complexty. We show that under some mld condtons, the transmtter and relay precodng matrces can be optmzed separately. In partcular, the transmtter precodng matrx optmzaton problem has a closed-form soluton, whle the relay precodng matrx can be optmzed through solvng a convex semdefnte programg (SDP problem [24]. Numercal smulatons demonstrate the mproved robustness of both proposed teratve and smplfed algorthms aganst the CSI msmatch. Interestngly, compared wth the teratve algorthm, the smplfed transcever desgn has only neglgble performance loss wth sgnfcantly reduced computatonal complexty. The rest of the paper s organzed as follows. The system model s presented n Secton II. In Secton III, the proposed robust transcever desgn algorthms are developed. Numercal examples are shown n Secton IV to verfy the performance of the proposed algorthms, and conclusons are drawn n Secton V. II. SYSTEM MODEL We consder a two-hop nonregeneratve MIMO relay multcastng system wth L recevers as shown n Fg. 1, the transmtter and relay have N S and N R antennas, respectvely. For smplcty, we assume that each recever has N D antennas. It s assumed that due to severe pathloss, there are no drect lnks between the transmtter and recevers. The data transmsson takes place over two tme slots. The receved sgnal at the relay durng the frst tme slot s gven by y r = 1 Fx + n 1 (1 x C N B 1 s the source sgnal vector satsfyng Exx = I N B, N B s chosen to satsfy N B (N S,N R,N D, 1 C N R N S s the MIMO channel matrx between the transmtter and the relay node, F C N S N B s the transmtter precodng matrx, and n 1 C N R 1 s the addtve nose vector at the relay node. ere ( denotes matrx ermtan transpose, E stands for statstcal expectaton, and I n represents the n n dentty matrx. At the second tme slot, the relay node lnearly precodes y r wth the relay precodng matrx G C N R N R, and broadcasts the lnearly precoded sgnal vector x r = Gy r (2 to all recevers. The receved sgnal vector at the th recever can be wrtten as y d, = 2, G 1 Fx + 2, Gn 1 + n 2,, = 1,,L (3 2, C N D N R s the MIMO channel matrx between the relay node and the th recever, n 2, C N D 1 s the addtve nose vector at the th recever. We assume that all noses are ndependent and dentcally dstrbuted (..d. wth zero mean and unt varance. In practce, channel 1 can be estmated at the relay node through channel tranng usng plot sgnals sent by the transmtter. Channel 2, s estmated at the th recever usng a tranng sequence sent from the relay node, and the estmated 2, s then sent from the th recever to the relay node. The optmal transmtter precodng matrx F and relay precodng matrx G are calculated at the relay node. Then the relay node forwards F to the transmtter and broadcasts F, G, and 1 to all recevers. In general, the nstantaneous CSI s requred for the optmal desgn of the precodng matrces F and G. owever, n practce, the exact CSI s not avalable due to channel estmaton errors. In fact, the exact CSI of 1 and 2, can be modeled as [15] 1 = Ĥ1 + Δ 1 (4 2, = Ĥ2, + Δ 2,, = 1,,L (5 Ĥ1 and Ĥ2, are the estmated transmtter-relay and relayth recever channel matrces, respectvely, Δ 1 and Δ 2, are the correspondng channel estmaton error matrces. We assume that Δ 1 and Δ 2, satsfy the Gaussan-Kronecker model as [15] ( Δ 1 CN 0, Σ 1 Ψ T 1 (6 ( Δ 2, CN 0, Σ 2, Ψ T 2,, = 1,,L (7 denotes the matrx Kronecker product, ( T stands for the matrx transpose, Σ 1 and Ψ T 1 are the row and column covarance matrces of Δ 1, respectvely, and smlarly, Σ 2, and Ψ T 2, are the row and column covarance matrces of Δ 2,, respectvely. We would lke to note that the Gaussan-Kronecker model s well-accepted to model the channel estmaton error matrces. As an example, t has been shown n [15] that the error matrces usng channel estmaton algorthms n [25] and [26] have the Gaussan-Kronecker structure. From (4 (7, the channel matrces 1 and 2, can be modeled as [15] 1 = Ĥ1 + Σ ω 1 Ψ (8 2, = Ĥ2, + Σ 1 2 2, ω 2, Ψ 1 2 2,, = 1,,L (9 ω 1 and ω 2, are complex Gaussan random matrces whose entres are..d. wth zero mean and unt varance. Lemma 1: [15], [27] For CN(Ĥ, Σ ΨT and any constant matrx A, there s E A = ĤAĤ + tr AΨ Σ (10

3 GOPAL et al.: ROBUST MMSE TRANSCEIVER DESIGN FOR NONREGENERATIVE MULTICASTING MIMO RELAY SYSTEMS 8981 tr denotes the matrx trace. At the th recever, a lnear recever wth the weght matrx W s appled to retreve the source sgnal vector x. ence, the estmated sgnal at the th recever can be expressed as x = W y d,, = 1,,L. (11 Usng (3 and (11, the MSE of the sgnal waveform estmaton at the th recever s gven by M (W, G, F = E tr (W y d, x(w y d, x = tr ( W 2, G 1 F I N B ( W 2, G 1 F I N B + W R n, W (12 R n, s the equvalent nose covarance matrx gven by R n, = 2, GG 2, + I N D. From (2, the transmsson power consumed by the relay node can be wrtten as P (G = Etrx r x r = trg( 1 FF 1 + I N R G. (13 III. PROPOSED ROBUST TRANSCEIVER DESIGN ALGORITMS It can be seen from (12 that f the exact 1 and 2, are unavalable at the recevers, t s mpossble to desgn W that optmzes M (W, G, F n (12. If we desgn W, F, and G based only on Ĥ1 and Ĥ2,, there can be a great performance degradaton due to the msmatch between 1, 2, and Ĥ1, Ĥ 2,. Instead of optmzng (12, we desgn W, F, and G to mze J (W, G, F E 1, 2, M (W, G, F the statstcal expectaton s carred out wth respect to 1 and 2,, wth the dstrbuton gven n (8 and (9. By applyng Lemma 1 n (10 to (12, we obtan J (W, G, F = tr I N B +W (Ĥ2, GΞG Ĥ 2, + tr GΞG Ψ 2, Σ2, +I N D W W Ĥ 2, GĤ1F F Ĥ 1 G Ĥ 2,W (14 Ξ = Ĥ1FF Ĥ 1 + trff Ψ 1 Σ 1 + I N R. (15 The detaled steps leadng to (14 are presented n Appendx A. Snce the true 1 s unknown, we consder the averaged transmsson power at the relay node, whch can be calculated from (13 as E 1 P (G = E 1 trg( 1 FF 1 + I N R G = trgξg. (16 In the proposed transcever desgn, we am to mze the mum of (14 among all recevers subjectng to the transmtter and relay power constrants, whch can be wrtten as the followng optmzaton problem F,G,W J (W, G, F (17 s.t. tr GΞG P r (18 trff P s (19 W = W,= 1,,L, (18 and (19 are the transmsson power constrants at the relay node and the transmtter, respectvely, and P r > 0, P s > 0 are the correspondng power budgets. A. Iteratve Transcever Desgn Algorthm The - problem (17 (19 s very hard to solve due to the complcated objectve functon (17. In the followng, we develop an teratve algorthm 1 to solve the problem (17 (19. Frstly, for any gven F and G, the optmal W that mzes the MSE functon (14 s the well known MMSE recever (Wener flter whch s gven by [28] W = F Ĥ 1 G Ĥ 2,(Ĥ2, GΞG Ĥ 2, + trgξg Ψ 2, Σ 2, + I N D 1,= 1,,L (20 ( 1 denotes the matrx nverse. Secondly, wth gven F and W, the problem of optmzng G can be wrtten as G,t 1 t 1 (21 s.t. J (G t 1, = 1,,L (22 trgξg P r (23 J (G s (14 wrtten as a functon of G wth fxed F and W as J (G = trw W + trw Ĥ 2, GΞG Ĥ 2,W + trgξg Ψ 2, trw Σ 2, W + N B trw Ĥ 2, GĤ1F + F Ĥ 1 G Ĥ 2,W. (24 Let us ntroduce for = 1,,L D 1 2 D 2 = Ĥ 2,W W Ĥ 2, + trw Σ 2, W Ψ 2,. Then (24 can be rewrtten as J (G = trw W + N B + trd 2 GΞG D 1 2 W Ĥ 2, GĤ1F F Ĥ 1 G Ĥ 2,W. (25 Let us ntroduce GΞG Φ, A B means that B A 0 s postve semdefnte (PSD. Usng (25, the problem (21 (23 can be equvalently rewrtten as the followng 1 Note that the teratve method provdes a system MSE and bt-error-rate (BER lower-bound whch s useful to judge the performance of the smplfed algorthm developed n the next subsecton.

4 8982 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 66, NO. 10, OCTOBER 2017 SDP problem G,Φ,t 1 t 1 (26 s.t. J (G, Φ t 1, = 1,,L (27 trφ P r (28 ( Φ G G Ξ 1 0 (29 J (G, Φ = tr W W + N B + tr D 2 ΦD 1 2 W Ĥ 2, GĤ1F F Ĥ 1 G Ĥ 2,W Thrdly, wth gven G and W, the problem of optmzng F can be wrtten as F,t 2 t 2 (30 s.t. J (F t 2, = 1,,L (31. trgξg P r (32 trff P s (33 J (F s (14 wrtten as a functon of F wth fxed G and W as J (F = tr Θ 2 FF Θ S tr W Ĥ 2, GĤ1F + F Ĥ 1 G Ĥ 2,W (34 ere Θ and S are gven n (86 and (87 n Appendx B, respectvely. The detaled steps leadng to (34 are presented n Appendx B. The left-hand sde of the constrant (32 can be rewrtten as trgξg = trĥ 1 G GĤ1FF. + trgσ 1 G trff Ψ 1 + trgg = tro 2 FF O trgg (35 Ĥ 1 G GĤ1 + trgσ 1 G Ψ 1 = O 1 2 O 2. Let us ntroduce O 2 FF O 1 2 X, FF Y. Usng (34 and (35, the problem (30 (33 can be rewrtten as the followng SDP problem F,X,Y,t 2 t 2 (36 s.t. J (F, Y t 2, = 1,,L (37 trx + trgg P r (38 ( X O 2 F F 0 O 1 2 I N B (39 try P s (40 ( Y F F 0 (41 I N B TABLE I PROCEDURE OF TE PROPOSED ITERATIVE TRANSCEIVER DESIGN ALGORITM 1 Intalze the algorthm wth F ( 0 = P s /N B [I N B, 0] T and G ( 0 = P r /trξ ( 0 I N R ;Setn = 0. 2 Update W ( n usng F ( n and G ( n as (20. 3 Update G ( n + 1 usng W ( n and F ( n by solvng the problem (26 (29. 4 Update F ( n + 1 usng W ( n and G ( n + 1 by solvng the problem (36 (41. 5 If ( J ( n J ( n + 1 / J ( n <ε, teraton ends; otherwse go to step (2. J (F, Y = tr Θ 2 YΘ S tr W Ĥ 2, GĤ1F+F Ĥ 1 G Ĥ 2,W The procedure of applyng the teratve algorthm to solve the problem (17 (19 s lsted n Table I, the superscrpt (n denotes the number of teratons, ε s a small postve number close to zero, and J (n stands for the value of (17 at the nth teraton. At each teraton, the complexty of solvng the SDP problem (26 (29 s O(NR 4 (L N R 2 (L N R 0.5, and the complexty order of solvng the problem (36 (41 s O((N S N B + 2NS 22 (L (N S + N B 2 (L (N S + N B 0.5. Thus, t can be seen from Table I that the teratve algorthm has a hgh computatonal complexty, as ts overall complexty s the per teraton complexty multply wth the number of teratons requred tll convergence. In the followng, we develop a smplfed algorthm for the MSE problem (17 (19 such that nearly optmal transmt and relay matrces can be desgned wth a sgnfcantly reduced computatonal complexty. B. Smplfed Transcever Desgn Algorthm In the followng, we propose a low computatonal complexty approach to optmze F and G. By substtutng (20 back nto (14, we have J (G, F = tr I N B F Ĥ 1 G Ĥ 2,(Ĥ2, GΞG Ĥ 2,. + trgξg Ψ 2, Σ 2, + I N D 1Ĥ 2, GĤ1F. (42 Then the problem (17 (19 can be wrtten as the followng optmzaton problem F,G J (G, F (43 s.t. tr GΞG P r (44 trff P s. (45 The followng theorem shows the optmal structure of G as the soluton to the problem (43 (45. Theorem 1: The optmal relay precodng matrx G for each transmtter-relay-recever lnk can be expressed as G = TW r = TF Ĥ 1 Ξ 1 (46

5 GOPAL et al.: ROBUST MMSE TRANSCEIVER DESIGN FOR NONREGENERATIVE MULTICASTING MIMO RELAY SYSTEMS 8983 W r F Ĥ 1 Ξ 1 can be vewed as the lnear MMSE recever at the relay node, and T s unknown and can be vewed as the precodng matrx at the transmt sde of the second-hop MIMO multcastng channel. Usng G n (46, the MSE of the estmated sgnal at the th recever (42 can be reformulated as the sum of two MSE functons J (T, F = tr (I N B +F Ĥ 1 Υ 1 Ĥ 1 F 1 [ + tr R 1 +T Ĥ 2, ( tr TRT ] 1Ĥ 1 Ψ 2, Σ2, + I N D 2, T (47 Υ = trff Ψ 1 Σ 1 + I N R (48 R = F Ĥ 1 (Ĥ1 FF Ĥ 1 + Υ 1Ĥ 1 F. (49 Proof: See Appendx C. Interestngly, Theorem 1 extends the MSE matrx decomposton from MIMO relay systems wth perfect CSI to two-hop MIMO relay systems wth mperfect CSI. In fact, the frst term n (47 s the MSE of the frst-hop sgnal waveform estmaton at the relay node whch s gven by E 1 Etr(W r y r x(w r y r x = E 1 tr(w r 1 F I NB (W r 1 F I NB +W r W r = trw r ΞWr W r Ĥ 1 F F Ĥ 1 Wr + I N B = tr (I N B +F Ĥ 1 Υ 1 Ĥ 1 F 1 whle the second term n (47 can be vewed as the ncrement of the MSE ntroduced by the second-hop. Usng (46, the power consumpton at the relay node (16 can be rewrtten as tr(trt. ence, the problem (43 (45 can be equvalently rewrtten as F,T J (T, F (50 s.t. trtrt P r (51 Usng the matrx nverson lemma [29] trff P s. (52 (A + BCD 1 = A 1 A 1 B(DA 1 B + C 1 1 DA 1 matrx R n (49 can be expressed as R = F Ĥ ( 1 Υ 1 Υ 1 Ĥ 1 F ( F Ĥ 1 Υ 1 Ĥ 1 F + I N B 1F Ĥ 1 Υ 1 Ĥ 1 F (53 = F Ĥ 1 Υ 1 Ĥ 1 F ( F Ĥ 1 Υ 1 Ĥ 1 F + I N B 1. (54 In the case of small CSI msmatch, for example, Ψ 1 = σei 2 N S and Σ 1 = σei 2 N R wth σe 2 1, σe 2 denotes the varance of channel msmatch, trff Ψ 1 Σ 1 s much smaller compared wth I N R. Then Υ can be approxmated as I N R. Consequently t can be seen from (54 that n ths case, R can be approxmated as I N B,fF Ĥ 1 Ĥ1F s much greater than I N B, whch can be satsfed wth a large P s [23]. Therefore, the problem (50 (52 can be approxmated as F,T tr(i N B + F Ĥ 1 Υ 1 Ĥ 1 F 1 + tr(i N B + T Ĥ 2,K 1 Ĥ 2, T 1 (55 s.t. trtt P r (56 trff P s (57 K = tr TT Ψ 2, Σ2, + I N D, = 1,,L. (58 We observe from the problem (55 (57 that the frst trace term n the objectve functon (55 does not depend on T, whle the value of the second trace term n (55 s not affected by F. Therefore, the problem (55 (57 can be decomposed nto the problem of optmzng F as F tr(i N B + F Ĥ 1 Υ 1 Ĥ 1 F 1 (59 s.t. trff P s (60 and the problem whch optmzes T as T tr(i N B + T Ĥ 2,K 1 Ĥ 2, T 1 (61 s.t. trtt P r. (62 Interestngly, we observed durng smulatons that keepng Υ n the optmzaton of F acheves a better system performance than treatng Υ = I N R, wthout ncreasng the complexty of optmzng F. Moreover, t can be seen from (48 and (58 that for multcastng MIMO relay systems wth the exact CSI knowledge, we have Υ = I N R and K = I N D, = 1,,L, Thus, n ths case, the problems (59 (60 and (61 (62 become non-robust transcever desgn problems n [13]. Therefore, the proposed robust transcever desgn extends the algorthm n [13] to the general practcal multcastng MIMO systems wth CSI msmatch. 1 Optmzaton of F: When Ψ 1 = σei 2 N S,.e., the columns of Δ 1 are uncorrelated, from (48 we have Υ = σetrff 2 Σ 1 + I N R. It can be easly shown that the optmal soluton of the problem (59 (60 must meet equalty at the constrant (60,.e., the optmal F should satsfy trff = P s. In ths case, Υ = σep 2 s Σ 1 + I N R does not depend on F, and the problem (59 (60 has a closed-form soluton as shown later. owever, for the general case of Ψ 1 σei 2 N S, Υ s a functon of F, whch makes the problem (59 (60 dffcult to solve. To overcome ths challenge, we apply the followng nequalty [15] tr FF Ψ 1 Ps λ M (Ψ 1 (63 λ M ( stands for the mal egenvalue of a matrx. From (63, an upper-bound of (59 s gven by tr(i N B + F Ĥ 1 Υ 1 Ĥ 1 F 1 tr(i N B +F Ĥ 1 (P s λ M (Ψ 1 Σ 1 +I N R 1 Ĥ 1 F 1. (64 Interestngly, the equalty n (64 holds when Ψ 1 = σei 2 N S,as n ths case λ M (Ψ 1 =σe. 2 Based on the dscusson above, let us ntroduce A Ĥ 1 (P s λ M (Ψ 1 Σ 1 +I N R 1 Ĥ 1.

6 8984 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 66, NO. 10, OCTOBER 2017 The problem (59 (60 s modfed to the followng transmtter precodng matrx optmzaton problem F tr(i N B + F AF 1 (65 s.t. trff P s. (66 Let us ntroduce the egenvalue decomposton (EVD of A as A = U A Λ A U A, the dagonal elements of Λ A are sorted n a decreasng order. It can be shown that the soluton to the problem (65 (66 s gven by F = U A,1 Λ 1 2 F (67 U A,1 contans the leftmost N B columns of U A assocated wth the largest N B egenvalues and Λ F s a dagonal matrx. Usng (67, the problem (65 (66 can be wrtten as the followng constraned optmzaton problem wth scalar varables λ F, s.t. N B = λ F, λ A, (68 N B λ F, P s (69 =1 λ F, 0, = 1,,N B (70 λ F, and λ A,, = 1,,N B, are the th dagonal elements of Λ F and Λ A, respectvely, and λ F, = λ F,1,, λ F,NB. The problem (68 (70 has the well-known water-fllng soluton as ( + λ F, = 1 λ A, λ A, μ 1, = 1,,N B (x + = (x, 0, and μ>0 satsfes the nonlnear equaton of N B =1 1 λ A, ( λ A, μ 1 + = Ps. 2 Optmzaton of T: Smlar to the technque used n optmzng F, wehavetr TT Ψ 2, Pr λ M (Ψ 2,. Let us ntroduce B Ĥ 2,(P r λ M (Ψ 2, Σ 2, + I N D 1 Ĥ 2,, = 1,,L. The problem (61 (62 can be modfed to the followng problem T tr(i N B + T B T 1 (71 s.t. trtt P r. (72 Usng the matrx dentty of tr(i m + A m n B n m 1 = tr(i n + B n m A m n 1 + m n, the - problem (71 (72 can be wrtten as ( tr I N R + B 1 2 QB N B N R (73 Q s.t. tr ( Q P r (74 Q 0 (75 Q = TT. Note that for the case of N R >N B (when T s a tall matrx, to facltate solvng the problem (73 (75, the constrant of rank(q =N B s removed n the problem (73 (75. Nevertheless, T can be obtaned from Q as T = cu q,1 Λ 1 2 q,1, U q Λ q U q s the EVD of Q, Λ q,1 contans the N B largest egenvalues of Q, U q,1 contans the egenvectors assocated wth the largest N B egenvalues, and c = P r /tru q,1 Λ q,1 U q,1 s a scalar to meet equalty n the power constrant (74. Let us ntroduce a real valued slack varable ρ and PSD matrces Z wth ( I N R + B 1 2 QB 2 1 Z,= 1,,L.Byusng the Schur complement [24], the problem (73 (75 can be equvalently rewrtten as ρ (76 ρ,q,z s.t. tr(z ρ, = 1,,L (77 tr ( Q P r (78 ( Z I N R 0, = 1,,L I N R I N R +B 1 2 QB 2 (79 Q 0 (80 Z = Z 1,, Z L. The problem (76 (80 s a convex SDP problem and can be solved by the convex programg toolbox CVX [30]. In the proposed smplfed transcever desgn, the orgnal transmtter and relay matrces optmzaton problem (43 (45 s decomposed nto two MSE mzaton problems. The computatonal complexty of optmzng the relay precodng matrx s governed by solvng the SDP problem (76 (80, whch has an order of O((L NR 6.5. The computaton of optmzng the transmtter precodng matrx nvolves perforg matrx EVD and calculatng the power loadng parameters, whose complexty are neglgble compared wth that of solvng the problem (76 (80. Thus, the overall complexty order of the proposed smplfed desgn s O((L NR 6.5, whch s the same as that of the non-robust transcever desgn n [13], and much lower than that of the teratve transcever desgn algorthm n Secton III-A. IV. SIMULATION RESULTS In ths secton, we nvestgate the performance of the proposed robust transcever desgn algorthm for multcastng MIMO relay systems through numercal smulatons. We smulate a two-hop nonregeneratve MIMO relay multcastng system. The nformaton-carryng symbols are modulated usng the QPSK constellatons. The sgnal-to-nose ratos (SNRs of the frsthop and second-hop channels are defned as SNR 1 = P s /N S and SNR 2 = P r /N R, respectvely. Unless explctly mentoned, we set SNR 1 = 30 db and N B = N S = N R = N D = 4. In the smulatons, the correlaton matrces of the channel estmaton errors are modeled as [15] [Ψ 1 ] m,n = α m n, m,n = 1,,N s [Ψ 2, ] m,n = α m n, m,n = 1,,N R, = 1,,L [Σ 1 ] m,n = σ 2 eβ m n, m,n = 1,,N R [Σ 2, ] m,n = σ 2 eβ m n, m,n = 1,,N D, = 1,,L 0 α, β 1 are correlaton coeffcents, and σ 2 e denotes the varance of the estmaton error.

7 GOPAL et al.: ROBUST MMSE TRANSCEIVER DESIGN FOR NONREGENERATIVE MULTICASTING MIMO RELAY SYSTEMS 8985 Fg. 2. Example 1: Convergence rate of the proposed robust-ite algorthm at varous SNR 1. SNR 2 = 20 db, σe 2 = 0.02, L = 2, and α = β = 0. Fg. 3. Example 2: BER versus SNR 2 wth dfferent number of antennas. σe 2 = 0.02, L = 2, and α = β = 0. The estmated channel matrces Ĥ1 and Ĥ2, are generated based on the followng dstrbutons ( Ĥ 1 CN 0, 1 σ2 e σ 2 e ( Ĥ 2, CN 0, 1 σ2 e σe 2 Σ 1 Ψ T 1 Σ 2, Ψ T 2,, = 1,,L. We compare the performance of the proposed robust teratve algorthm (robust-ite, the proposed smplfed robust transcever desgn (robust-sim wth 1 The algorthm n [13] usng only the estmated CSI Ĥ1, Ĥ 2,, = 1,,L(denoted as the non-robust algorthm. 2 The algorthm n [13] wth the true CSI 1, 2,, = 1,, L(denoted as the TCSI algorthm, whch provdes a lower bound of the system performance. 3 The robust decomposton approach (robust-da proposed n [22]. In the frst smulaton example, we study the convergence rate of the proposed teratve algorthm at varous levels of SNR 1. We set SNR 2 = 20 db, L = 2, σ 2 e = 0.02, α = β = 0, and the robust-ite algorthm s ntalzed wth ether scaled dentty or random matrces. Fg. 2 llustrates the normalzed MSE (NMSE of the proposed robust-ite algorthm versus the number of teratons. It can be seen from Fg. 2 that the NMSE always decreases and reduces to fxed values after a few teratons. Interestngly, the number of teratons requred tll convergence ncreases SNR 1. Moreover, t can be observed from Fg. 2 that the convergence speed of the robust-ite algorthm wth the scaled dentty matrx ntalzaton s faster than that of random matrx ntalzaton. ence, for the rest of the smulatons, the scaled dentty matrx s chosen to ntalze the robust-ite algorthm. In the second smulaton example, we study the BER performance of the proposed algorthms wth dfferent number of antennas at the transmtter, relay and recevers. In ths example, we set L = 2, σ 2 e = 0.02, and α = β = 0. Fg. 3 shows the BER of four algorthms versus SNR 2. It can be seen from Fg. 3 that wth N S = N D = 4, the system BER of the proposed robust algorthms approaches ence, wth a fxed N B,the BER performance of the proposed robust algorthms mproves Fg. 4. Example 3: NMSE versus SNR 2 at varous σ 2 e. L = 2andα = β = 0. when N S and N D ncrease, as the system spatal dversty gan ncreases wth the number of antennas. Moreover, t can be seen from Fg. 3 that the BER mprovement of both proposed algorthms over the robust-da method s larger wth N S = N D = 4 compared wth N S = N D = 2. In the thrd smulaton example, we study the performance of the proposed algorthms at varous values of estmaton error varance σe. 2 In ths smulaton, the number of recevers s set as L = 2 and the correlaton coeffcents are set as α = β = 0 (.e., Ψ 1 = I N S, Ψ 2, = I N R, Σ 1 = σei 2 N R and Σ 2, = σei 2 N D. Fg. 4 shows the NMSE of fve algorthms versus SNR 2. It can be observed from Fg. 4 that over the whole range of SNR 2, the proposed robust algorthms sgnfcantly outperform the non-robust algorthm. Both proposed algorthms have a better MSE performance than the robust-da algorthm. It can also be seen from Fg. 4 that for all three robust and non-robust algorthms, the system MSE decreases when σe 2 s reduced. For ths example, the BER yelded by the fve algorthms tested versus SNR 2 s shown n Fg. 5. It can be clearly seen from Fg. 5 that the three robust algorthms yeld much lower BER compared wth the non-robust algorthm. As σe 2 decreases, the system BER of all four algorthms decreases. We also observe

8 8986 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 66, NO. 10, OCTOBER 2017 Fg. 5. Example 3: BER versus SNR 2 at varous σ 2 e. L = 2andα = β = 0. Fg. 7. β = 0. Example 4: BER versus SNR 2 at varous α. L = 2, σ 2 e = 0.005, and Fg. 6. Example 4: NMSE versus SNR 2 at varous α. L = 2, σ 2 e = 0.005, and β = 0. Fg. 8. Example 5: NMSE versus SNR 2 at varous β. L = 2, σ 2 e = 0.005, and α = 0. from Fg. 5 that both proposed transcever desgn algorthms have a better BER performance than the robust-da algorthm. It can be observed from Fgs. 4 and 5 that the proposed robust- ITE and robust-sim algorthms acheve very close NMSE and BER performance to that of the approach n [13] wth true CSI. It can also be seen from Fgs. 4 and 5 that the MSE and BER yelded by the proposed smplfed desgn s very close to that of the proposed teratve algorthm. In the followng smulaton examples, we focus on the smplfed transcever desgn, as t has neglgble performance loss but wth sgnfcantly reduced computatonal complexty, whch makes t more attractve for practcal multcastng MIMO relay systems. In the fourth example, we nvestgate the performance of the proposed smplfed algorthm at varous values of the correlaton coeffcent α wth β = 0. In ths example, the number of recevers s set as L = 2 and the estmaton error varance s chosen as σ 2 e = Fgs. 6 and 7 demonstrate the MSE and BER performance of the three algorthms tested versus SNR 2, respectvely. It can be seen from Fgs. 6 and 7 that the proposed robust-sim algorthm provdes better MSE and BER performances than the non-robust and robust-da algorthms for all the α tested. Both robust algorthms outperform the non-robust algorthm. Moreover, the MSE and BER yelded by all three algorthms ncrease wth α. Ths s due to the fact that as α ncreases, the correlaton among the elements of channel matrces ncrease, leadng to the loss of spatal dversty. In the ffth example, we study the performance of the proposed robust-sim algorthm at varous values of the correlaton coeffcent β wth α = 0, L = 2, and σ 2 e = The MSE and BER performances of all three algorthms are shown n Fgs. 8 and 9, respectvely. Smlar to Fgs. 6 and 7, t can be seen from Fgs. 8 and 9 that the proposed robust-sim algorthm outperforms the non-robust and robust-da algorthms. It can also be observed that the MSE and BER of the three algorthms ncrease wth the ncreasng correlaton coeffcent β. In the sxth smulaton example, we study the performance of the proposed robust-sim algorthm at varous number of recevers L. In Fg. 10, we compare the MSE performance of the proposed robust-sim algorthm wth the TCSI and robust- DA algorthms at varous L as a functon of SNR 2 wth σ 2 e = and α = β = 0. It can be noted from Fg. 10 that as we ncrease the number of recevers, the MSE of the three algorthms ncreases. Ths s reasonable snce t s more lkely to fnd a worse relay-recever channel among the ncreased number of users and we choose the worst-user MSE as the objectve functon. Moreover, t can be seen from Fg. 10 that the proposed robust-sim algorthm outperforms the robust-da algorthm at both L = 2 and L = 6.

9 GOPAL et al.: ROBUST MMSE TRANSCEIVER DESIGN FOR NONREGENERATIVE MULTICASTING MIMO RELAY SYSTEMS 8987 both L = 2 and L = 6. From Fgs. 10 and 11 we can see that the proposed robust-sim algorthm has a very closed MSE and BER performance to the TCSI algorthm for both L = 2 and L = 6. Fg. 9. α = 0. Example 5: BER versus SNR 2 at varous β. L = 2, σ 2 e = 0.005, and V. CONCLUSION In ths paper, we have addressed the challengng ssue of robust transcever optmzaton for two-hop multcastng MIMO relay systems when there s msmatch between the true and estmated channel matrces. The true channel matrces are assumed as Gaussan random matrces wth the estmated channels as the mean value, and estmaton error follows the well-known Kronecker model. In the proposed teratve algorthm, the transmtter, relay, and recever matrces are desgned to mze the mal MSE of the sgnal waveform estmaton among all recevers. Due to the hgh computatonal complexty of the proposed teratve algorthm, we develop a smplfed transcever desgn n whch we show that the MSE at each recever can be decomposed nto the sum of the MSEs of the frst-hop and second-hop channels. Based on the proposed MSE decomposton, a transcever optmzaton algorthm wth low computatonal complexty has been developed. Numercal smulatons demonstrate that the proposed transcever desgn algorthms have an mproved robustness aganst the msmatch between the true and estmated channels than exstng algorthms. APPENDIX A DERIVATION OF (14 Fg. 10. Example 6: NMSE versus SNR 2 at varous L. σ 2 e = and α = β = 0. Takng the statstcal expectaton of (12 wth respect to 1 and 2,,wehave E 1, 2, M (W, G, F = tr W E 2, 2, GE 1 1 FF 1 G 2, W W E 1, 2, 2, G 1 F F E 1, 2, 1 G 2, W + I N B + W ( E 2, 2, GG 2, + IN D W. (81 Usng the dstrbuton n (4, (6 and applyng Lemma 1, we obtan E 1 1 FF 1 = Ĥ 1 FF Ĥ 1 + trff Ψ 1 Σ 1. (82 By substtutng (82 back nto (81, we have E 1, 2, M (W, G, F = tr ( W E 2, 2, GΞG 2, + IN D W Fg. 11. Example 6: BER versus SNR 2 at varous L. σ 2 e = and α = β = 0. For ths example, the BER performance of the TCSI, robust- SIM, and robust-da algorthms s demonstrated n Fg. 11. It can be noted from Fg. 11 that smlar to Fg. 10, the BER of the three algorthms ncrease wth the number of recevers and t can be notced from Fg. 11 that the proposed robust-sim algorthm has a lower BER compared wth the robust-da algorthm for W Ĥ 2, GĤ1F F Ĥ 1 G Ĥ 2,W + I N B. (83 Based on the dstrbuton n (5, (7 and applyng Lemma 1, we obtan E 2, 2, GΞG 2, = Ĥ2,GΞG Ĥ 2, + tr GΞG Ψ 2, Σ2,. (84 By substtutng (84 back nto (83, we obtan (14.

10 8988 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 66, NO. 10, OCTOBER 2017 APPENDIX B DERIVATION OF (34 By substtutng (15 back nto (14 and usng the dentty of trab = trba, (14 can be rewrtten as J (F = trff Ĥ 1 G Ĥ 2,W W Ĥ 2, GĤ1 + trw Ĥ 2, GΣ 1 G Ĥ 2,W trff Ψ 1 + trw Σ 2, W (trff Ĥ 1 G Ψ 2, GĤ1 + trff Ψ 1 trgσ 1 G Ψ 2, + trgg Ψ 2, + trw W +N B + trw Ĥ 2, GG Ĥ 2,W trw Ĥ 2, GĤ1F + F Ĥ 1 G Ĥ 2,W. (85 By ntroducng for = 1,,L and Θ 1 2 Θ 2 = Ĥ 1 G Ĥ 2,W W Ĥ 2, GĤ1 + trw Ĥ 2, GΣ 1 G Ĥ 2,W Ψ 1 + trw Σ 2, W (Ĥ 1 G Ψ 2, GĤ1 + trgσ 1 G Ψ 2, Ψ 1 (86 S = trw Σ 2, W trgg Ψ 2, + trw W + trw Ĥ 2, GG Ĥ 2,W + N B (87 (85 can be rewrtten as (34. APPENDIX C PROOF OF TEOREM 1 Let us ntroduce P tr GΞG Ψ 2, Σ2, + I N D, = 1,,L. (88 The MSE functon n (42 can be rewrtten as J (G, F = tri N B F Ĥ 1 G Ĥ 2,(Ĥ2, G(Ĥ1FF Ĥ 1 + Υ G Ĥ 2, + P 1Ĥ 2, GĤ1F. (89 By ntroducng G GΥ 1 2, 1 Υ 1 2 Ĥ1, 2, P 1 2 Ĥ 2, (90 we can rewrte (89 as J ( G, F = tri N B F ( 1 G 2, 2, G( 1 FF 1 + I N R G 2, + I N D 1 2, G 1 F = tr [ I N B +F 1 G 2, ( 2, G G 2, + I N D 1 2, G 1 F ] 1 (91 the matrx nverson lemma (53 has been appled to obtan (91. Usng (90, the constrant (44 can be rewrtten as tr G( 1 FF 1 + I N R G P r. (92 Based on (45, (91, and (92, the MMSE-based transcever optmzaton problem for the th recever can be wrtten as F, G J ( G, F (93 s.t. tr G( 1 FF 1 + I N R G P r (94 trff P s. (95 It can be shown smlar to [23] that the optmal G as the soluton to the problem (93 (95 can be wrtten as G = TF 1 ( 1 FF 1 + I N R 1 (96 and the objectve functon (93 can be decomposed nto two MSE terms as J (T, F = tr(i N B + F 1 1 F 1 + tr( R 1 + T 2, 2, T 1 (97 R = F 1 ( 1 FF 1 + I N R 1 1 F. (98 Substtutng G and 1 n (90 and Υ n (48 nto (96, we have G = TF Ĥ 1 Ξ 1, whch proves (46. By Substtutng 1 n (90 nto (98, we obtan R = F Ĥ 1 (Ĥ1FF Ĥ 1 + Υ 1 Ĥ 1 F = R n (49. Moreover, by substtutng (46 nto (88, we have P = tr TF Ĥ 1 Ξ 1 Ĥ 1 FT Ψ 2, Σ2, + I N D = tr TRT Ψ 2, Σ2, + I N D. (99 Thus, from (90 and (97 we have J (T, F = tr(i N B +F Ĥ 1 Υ 1 Ĥ 1 F 1 + tr(r 1 + T Ĥ 2,P 1 Ĥ 2, T 1. (100 Fnally, by substtutng (99 nto (100, we prove (47. REFERENCES [1] L. Gopal, Y. Rong, and Z. Zang, Smplfed robust desgn for nonregeneratve multcastng MIMO relay systems, n Proc. 22nd Int. Conf. Commun., Sydney, Australa, Apr , 2015, pp [2] D. Tse and P. Vswanath, Fundamentals of Wreless Communcaton. Cambrdge, U.K.: Cambrdge Unv. Press, [3] M. A. Khojastepour, A. Saleh-Golsefd, and S. Rangarajan, Towards an optmal beamforg algorthm for physcal layer multcastng, n Proc. IEEE Inf. Theory Workshop, 2011, pp [4] N. Jndal and Z.-Q. Luo, Capacty lmts of multple antenna multcast, n Proc. IEEE ISIT, Seatle, WA, USA, Jul , 2006, pp [5] S. Y. Park, D. J. Love, and D.. Km, Capacty lmts of mult-antenna multcastng under correlated fadng channels, IEEE Trans. Commun., vol. 58, no. 7, pp , Jul [6] N. D. Sdropoulos, T. N. Davdson, and Z.-Q. Luo, Transmt beamforg for physcal-layer multcastng, IEEE Trans. Sgnal Process., vol.54, no. 6, pp , Jun [7] E. Chu and V. Lau, Precodng desgn for mult-antenna multcast broadcast servces wth lmted feedback, IEEE Syst. J., vol. 4, no. 4, pp , Dec

11 GOPAL et al.: ROBUST MMSE TRANSCEIVER DESIGN FOR NONREGENERATIVE MULTICASTING MIMO RELAY SYSTEMS 8989 [8] E. Matskan, N. D. Sdropoulos, Z.-Q. Luo, and L. Tassulas, Effcent batch and adaptve approxmaton algorthms for jont multcast beamforg and admsson control, IEEE Trans. Sgnal Process., vol. 57, no. 12, pp , Dec [9] M. A. Khojastepour, A. Khajehnejad, K. Sundaresan, and S. Rangarajan, Adaptve beamforg algorthms for wreless lnk layer multcastng, n Proc. 22nd Int. Symp. Personal, Indoor Moble Rado Commun., Toronto, Canada, Sep , 2011, pp [10] M. Kalszan, E. Pollaks, and S. Stańczak, Effcent beamforg algorthms for MIMO multcast wth applcaton-layer codng, n Proc. IEEE Int. Symp. Inf. Theory, St. Petersburg, Russa, Jul. 31 Aug. 5, 2011, pp [11] S. Sh, M. Schubert, and. Boche, Physcal layer multcastng wth lnear MIMO transcevers, n Proc. 42nd Annu. Conf. Inf. Sc. Syst., Prnceton, NJ, USA, Mar. 2008, pp [12]. Zhang, J. Jn, X. You, G. Wu, and. Wang, Cooperatve mult-antenna multcastng for wreless networks, n Proc. IEEE GLOBECOM, Mam, FL, USA, Dec. 6 10, [13] M. R. A. Khandaker and Y. Rong, Precodng desgn for MIMO relay multcastng, IEEE Trans. Wreless Commun., vol. 12, no. 7, pp , Jul [14] Y. Rong, Robust desgn for lnear non-regeneratve MIMO relays wth mperfect channel state nformaton, IEEE Trans. Sgnal Process., vol. 59, no. 5, pp , May [15] C. Xng, S. Ma, and Y.-C. Wu, Robust jont desgn of lnear relay precoder and destnaton equalzer for dual-hop amplfy-and-forward MIMO relay systems, IEEE Trans. Sgnal Process., vol. 58, no. 4, pp , Apr [16] C. Xng, S. Ma, and Y. Zhou, Matrx-monotonc optmzaton for MIMO systems, IEEE Trans. Sgnal Process., vol. 63, no. 2, pp , Jan [17] C. Xng, S. Ma, Z. Fe, Y. C. Wu, and. V. Poor, A general robust lnear transcever desgn for mult-hop amplfy-and-forward MIMO relayng systems, IEEE Trans. Sgnal Process., vol. 61, no. 5, pp , Mar [18] C. Xng, S. L, Z. Fe, and J. Kuang, ow to understand lnear mum mean-square-error transcever desgn for multple-nput-multple-output systems from quadratc matrx programg, IET Commun., vol. 7, pp , Aug [19] B. K. Chalse and L. Vandendorpe, Jont lnear processng for an amplfyand-forward MIMO relay channel wth mperfect channel state nformaton, EURASIP J. Adv. Sgnal Process., vol. 2010, 2010, Art. no [20] F.-S. Tseng, M.-Y. Chang, and W.-R. Wu, Jont Tomlnson-arashma source and lnear relay precoder desgn n amplfy-and-forward MIMO relay systems va MMSE crteron, IEEE Trans. Veh. Technol., vol. 60, no. 4, pp , May [21] Z. e, W. Jang, and Y. Rong, Robust desgn for amplfy-and-forward MIMO relay systems wth drect lnk and mperfect channel nformaton, IEEE Trans. Wreless Commun., vol. 14, no. 1, pp , Jan [22] J. Lu, F. Gao, and Z. Qu, Robust transcever desgn for mult-user multple-nput multple-output amplfy-and-forward relay systems, IET Commun., vol. 8, pp , Mar [23] Y. Rong, Smplfed algorthms for optmzng multuser mult-hop MIMO relay systems, IEEE Trans. Commun., vol. 59, no. 10, pp , Oct [24] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge, U.K.: Cambrdge Unv. Press, [25] L. Musavan, M. R. Nakh, M. Dohler, and A.. Aghvam, Effect of channel uncertanty on the mutual nformaton of MIMO fadng channels, IEEE Trans. Veh. Technol., vol. 56, no. 5, pp , Sep [26] M. Dng and S. D. Blosten, MIMO mum total MSE transcever desgn wth mperfect CSI at both ends, IEEE Trans. Sgnal Process., vol. 57, no. 3, pp , Mar [27] A. Gupta and D. Nagar, Matrx Varate Dstrbutons. London, U.K.: Chapman & all, [28] S. M. Kay, Fundamentals of Statstcal Sgnal Processng: Estmaton Theory. Englewood Clffs, NJ, USA: Prentce-all, [29] D. Bernsten, Matrx Mathematcs: Theory, Facts, and Formulas. Prnceton, NJ, USA: Prnceton Unv. Press, [30] M. Grant and S. Boyd, The CVX Users Gude, Release 2.1, Oct [Onlne]. Avalable: Lenn Gopal (M 06 receved the B.Eng. degree n electrcal and electroncs engneerng from Madura Kamaraj Unversty, Madura, Inda, n 1996, the M.Eng. degree n telecommuncatons engneerng from Multmeda Unversty, Cyberjeya, Malaysa, n 2006, and the Ph.D. degree from Curtn Unversty, Bentley, WA, Australa, n e s currently a Senor Lecture wth the Department of Electrcal and Computer Engneerng, Faculty of Engneerng and Scence, Curtn Unversty, Mr, Malaysa. s research nterests nclude sgnal processng for communcatons, wreless communcatons, and FPGA applcatons for wreless communcatons. Yue Rong (S 03 M 06 SM 11 receved the Ph.D. degree (summa cum laude n electrcal engneerng from the Darmstadt Unversty of Technology, Darmstadt, Germany, n e was a Postdoctoral Researcher wth the Department of Electrcal Engneerng, Unversty of Calforna, Rversde, CA, USA, from February 2006 to November Snce December 2007, he has been wth the Department of Electrcal and Computer Engneerng, Curtn Unversty, Bentley, WA, Australa, he s currently a Full Professor. s research nterests nclude sgnal processng for communcatons, wreless communcatons, underwater acoustc communcatons, applcatons of lnear algebra and optmzaton methods, and statstcal and array sgnal processng. e has publshed more than 140 journal and conference papers n these areas. Dr. Rong receved the Best Paper Award at the 2011 Internatonal Conference on Wreless Communcatons and Sgnal Processng, the Best Paper Award at the 2010 Asa-Pacfc Conference on Communcatons, and the Young Researcher of the Year Award of the Faculty of Scence and Engneerng at Curtn Unversty n e s an Assocate Edtor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING. e was an Edtor of the IEEE WIRELESS COMMUNICA- TIONS LETTERS from 2012 to 2014, a Guest Edtor of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS specal ssue on theores and methods for advanced wreless relays, and was a TPC Member for the IEEE ICC, WCSP, IWCMC, EUSIPCO, and ChnaCom. Zhuquan Zang (M 93 receved the B.Sc. degree from Shandong Normal Unversty, Jnan, Chna, the M.Sc. degree from Shandong Unversty, Jnan, Chna, and the Ph.D. degree n systems engneerng from the Australan Natonal Unversty, Canberra, Australa, n From 1993 to 1994, he was wth the Unversty of Western Australa, Perth, Australa, as a Research Assocate n the area of optmzaton, optmal control, and system dentfcaton. From 1994 to 2002, he was at the Australan Telecommuncatons Research Insttute, Curtn Unversty, Perth, Australa, frst as a Research Fellow and then as a Senor Research Fellow. From 2002 to 2005, he was at the Western Australan Telecommuncatons Research Insttute (A jont venture between Curtn Unversty and the Unversty of Western Australa, Perth, Australa. Durng ths perod, he was also afflated wth the Australan Telecommuncatons Cooperatve Research Centre. Snce early 2006, he has been wth the Faculty of Engneerng and Scence, Curtn Unversty, Malaysa. s man research nterests nclude the areas of constraned flter set desgn for bandwdth-effcent wreless multuser communcatons, wdeband waveform desgn for radar and sonar, and computatonally effcent optmzaton methods and ther applcaton to effectve multscale control scheme desgn, PAPR reducton n MIMO OFDM systems, and robust transcever desgn for nonregeneratve MIMO wreless communcaton systems.