Channel Estimation of MIMO Relay Systems With Multiple Relay Nodes

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1 Receved September 21, 2017, accepted November 12, 2017, date of publcaton November 20, 2017, date of current verson December 22, Dgtal Object Identfer /ACCESS Channel Estmaton of MIMO Relay Systems Wth Multple Relay Nodes ZHIQIANG HE 1,2, (Member, IEEE, JIAOLONG YANG 1, XIAODAN WANG 1, YANG LIU 1, AND YUE RONG 3, (Senor Member, IEEE 1 Key Laboratory of Unversal Wreless Communcaton, Mnstry of Educaton, Bejng Unversty of Posts and Telecommuncatons, Bejng , Chna 2 Key Laboratory of Underwater Acoustc Communcaton and Marne Informaton Technology, Mnstry of Educaton, Xamen Unversty, Xamen , Chna 3 Department of Electrcal and Computer Engneerng, Curtn Unversty, Bentley, WA 6102, Australa Correspondng author: Yue Rong (y.rong@curtn.edu.au Ths work was supported n part by the Natonal Natural Scence Foundaton of Chna under Grant and n part by the Australan Research Councl s Dscovery Projects fundng scheme under Grant DP ABSTRACT In ths paper, we consder the channel estmaton problem for multple-nput multple-output wreless relay communcaton systems wth multple relay nodes. In partcular, all ndvdual channel matrces of the frst-hop and second-hop lnks are estmated at the destnaton node by applyng the supermposed channel tranng algorthm, where tranng sequences are supermposed at the relay nodes to assst the estmaton of the relays-destnaton channel matrces. To mprove the performance of channel estmaton, we consder the estmaton error nherted from the second-hop channel estmaton and develop a new mnmal mean-squared error-based algorthm to estmate the frst-hop channel matrces. Furthermore, we derve the optmal power allocaton and tranng sequences at the source and relay nodes. Numercal examples demonstrate a better performance of the proposed supermposed channel tranng algorthm. INDEX TERMS Channel estmaton, MIMO relay, MMSE, multple relay nodes, power allocaton, supermposed tranng. I. INTRODUCTION Multple-nput multple-output (MIMO relay communcaton systems have attracted much research nterest recently due to ther capablty n mprovng the relablty and coverage of wreless systems 1, 2. Many studes have been carred out on the transcever optmzaton of amplfyand-forward (AF MIMO relay systems. The optmal relay precodng matrx whch maxmzes the source-destnaton mutual nformaton (MI of a two-hop MIMO relay system was developed n 3 and 4. In 5 and 6, the relay precodng matrx was desgned to mnmze the mean-squared error (MSE of the sgnal waveform estmaton. A unfed framework was developed n 7 and 8 to jontly optmze the source and relay precodng matrces for a broad class of objectve functons. The works n 3 8 focused on MIMO relay systems wth a sngle relay node at each hop. Systems wth multple parallel relay nodes have also attracted great nterests 9, 10. Behbahan et al. 9 developed the optmal relay precodng matrces wth multple relay nodes. In 10, jont source and relay precodng matrces optmzaton has been nvestgated wth power constrant at the output of the second-hop channel consderng both lnear and nonlnear recevers. It can be seen that for MIMO relay systems dscussed n 3 10, the knowledge of the nstantaneous channel state nformaton (CSI s requred for desgnng the optmal lnear recever at the destnaton node and optmzng the system through precodng matrces desgn. Nevertheless, n real relay communcaton systems, the nstantaneous CSI s unknown at the destnaton node, and therefore, needs to be estmated. Lolou and Vberg 11 and Lolou et al. 12 developed a least-squares (LS based channel estmaton algorthm for MIMO relay systems and further mproved ts performance by the weghted LS (WLS approach. A two-stage channel tranng method was developed n 13, where the optmal tranng sequence at the source and relay nodes was derved. In 14, a parallel factor (PARAFAC analyss based MIMO relay channel estmaton algorthm was proposed. Supermposed channel tranng algorthms were developed n 15 and 16 for two-way MIMO relay systems, and n 17 for tme-varyng MIMO relay systems, where a tranng sequence s supermposed at the relay node to estmate VOLUME 5, IEEE. Translatons and content mnng are permtted for academc research only. Personal use s also permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton

2 the CSI of frst-hop and second-hop channel at the destnaton nodes. The channel estmaton algorthms n were developed for MIMO relay systems wth a sngle relay node at each hop 3 8. Obvously, channel estmaton becomes more challengng for systems wth multple parallel relay nodes as more unknowns need to be estmated. In 18, LS and maxmum lkelhood (ML based channel estmators were derved based on the parameterzaton of the estmaton problem. The ML estmaton technque was used to estmate the bass expanson model (BEM weghtng coeffcents n 19. However, the optmzaton of tranng sequence, whch may further mprove the performance of the estmators, was not dscussed n 18 and 19. In ths paper, we apply the prncple of supermposed channel tranng to estmate all ndvdual frst-hop and secondhop channel matrces of MIMO relay systems wth multple parallel relay nodes. Note that the knowledge of both the source-relay and relay-destnaton channels s requred at the destnaton node for developng the optmal recever 9, 10. In the proposed algorthm, the channel estmaton process s completed n two tme blocks. At the frst tme block, the source node broadcasts tranng sequence to all relay nodes, whle at the second tme block, each relay node transmts lnearly amplfed receved sgnals as well as ts own tranng sequence. Then all ndvdual channel matrces of the frsthop and second-hop lnks can be effcently estmated at the destnaton node. We derve the optmal structure of the tranng sequences that mnmze the sum MSE of channel estmaton and optmze the power allocaton between the tranng sequences at the source and relay nodes. We consder the estmaton error nherted from the estmaton of the second-hop channel matrces and develop a new mnmal MSE (MMSE- based algorthm to estmate the frst-hop channel matrces, for whch no effcent method was proposed n 15. Moreover, compared wth 15 and 16, the optmzaton problems n ths paper are more challengng to solve, as more channel tranng sequences need to be optmzed. Numercal examples demonstrate a better performance of the proposed supermposed channel tranng algorthm compared wth the conventonal two-stage channel estmaton approach for MIMO relay systems wth multple relay nodes. The rest of ths paper s organzed as follows. The system model of a two-hop MIMO relay communcaton system wth multple relay nodes s presented n Secton II. The supermposed channel tranng algorthm s developed n Secton III, where the optmal tranng sequences and power allocaton at the source and relay nodes are derved. In Secton IV, we present a new MMSE-based algorthm to estmate the frsthop channel matrces consderng the error of the second-hop channel estmaton. Secton V shows numercal smulaton results to demonstrate the performance of the proposed algorthms. Fnally, conclusons are drawn n Secton VI. The followng notatons are used throughout the paper. Vectors and matrces are represented by lower case and upper case bold letters, respectvely; Superscrpts ( T, ( H, and ( 1 denote transpose, Hermtan transpose, and matrx nverson, respectvely; denotes the matrx Kronecker product 20; tr( stands for the matrx trace; vec( denotes the vectorzaton operator whch stacks all column vectors of a matrx on top of each other; I n denotes an n n dentty matrx; 1 s a vector wth all one entres; bdag and dag( stand for a block dagonal and dagonal matrx, respectvely; and E denotes the statstcal expectaton. II. SYSTEM MODEL We consder a two-hop MIMO relay communcaton system whch conssts of one source node, K parallel relay nodes, and one destnaton node as llustrated n Fg. 1. The source and the destnaton nodes have N s and N d antennas, respectvely, and the th relay node has N antennas. For = 1,..., K, H denotes the N N s frst-hop channel matrx from the source node to the th relay node and G s the N d N secondhop channel matrx from the th relay node to the destnaton node. Due to ts mert of smplcty, we consder the AF relay scheme at each relay. FIGURE 1. Block dagram of a two-hop MIMO relay communcaton system wth multple parallel relay nodes. The channel tranng process s mplemented n two tme blocks. At the frst tme block, the source node sends an N s T channel tranng matrx S to all relay nodes, where T s the length of the tranng sequence. The sgnals receved at the th relay node can be wrtten as Y = H S + V, = 1,..., K (1 where Y and V are the N T receved sgnal matrx and nose matrx at the th relay node, respectvely. At the second tme block, the source node s slent, and the th relay node amplfes ts receved sgnal wth an amplfyng factor α > 0 and supermposes ts own tranng matrx T. Therefore, the sgnal matrx transmtted by the th relay node s gven by X = αy + T, = 1,..., K. (2 The receved sgnal at the destnaton node can be wrtten as Y d = K G X + V d ( VOLUME 5, 2017

3 where V d s the N d T addtve Gaussan nose matrx at the destnaton node. We assume that the channel matrces H and G satsfy the well-known Gaussan-Kronecker model 21, where H and G are complex Gaussan random matrces wth H CN (0, T s R, G CN (0, C R d (4 where T s and R denote the N s N s and N N covarance matrx at the transmt and receve sde of H, respectvely, whle C and R d denote the N N and N d N d covarance matrx at the transmt and receve sde of G, respectvely. In other words, from (4 we have H = A H,w B H s, G = A d G,w K H (5 where A A H = R, B s B H s = T T s, A da H d = R d, K K H = C T, = 1,..., K, H,w and G,w are N N s and N d N complex Gaussan random matrces wth ndependent and dentcally dstrbuted (..d. zero mean and unt varance entres. We assume that H,w and G,w, = 1,..., K, are statstcally ndependent of each other. We also assume that the knowledge on the channel covarance matrces s avalable at the destnaton node. By substtutng (1 and (2 nto (3, we can rewrte the receved sgnal at the destnaton node as K Y d = G α(h S + V + T + V d = αms + K G T + V (6 where V = K αg V +V d s the total nose matrx at the destnaton node and M = K G H can be vewed as the compound source-destnaton channel matrx. It can be seen from (6 that the frst-hop channel H s multpled wth the second-hop channel G, whch makes t dffcult to estmate both H and G n one step. Thus, we propose an effcent twostep channel estmaton algorthm to estmate the second-hop channel G frst based on T, and then estmate the frst-hop channels. III. SUPERIMPOSED CHANNEL TRAINING FOR THE SECOND-HOP CHANNEL ESTIMATION In ths secton, the supermposed channel tranng prncple s appled to estmate the channel matrces M and G, = 1,..., K. Moreover, we desgn the optmal tranng matrces S and T, = 1,..., K, and the relay amplfyng factor α to mnmze the sum MSE of the channel estmaton. The man dea of the supermposed channel tranng algorthm s to use T to estmate the second-hop channel matrces G, = 1,..., K. By ntroducng the egenvalue decompostons (EVDs of T T s = U s s U H s, CT = U U H, = 1,..., K we have B H s = s 1 2 s U H s, KH = 1 2 U H, = 1,..., K (7 where s and are arbtrary N s N s and N N untary matrx, respectvely. Based on (5, the receved sgnal at the destnaton node (6 can be rewrtten as where M Y d = α M S + K G H K G T + V (8 S U H s S, H H U s, = 1,..., K G G U, T U H T, = 1,..., K. (9 In the followng, we develop an algorthm to estmate M and G n (8. In Secton IV, we wll present the algorthm to estmate the frst-hop channels H from (8. It wll be shown n Secton IV that the MSE of estmatng H ncreases wth the MSE of estmatng M and G. Thus, t s mportant to optmze the tranng matrces S and T, = 1,..., K, to mnmze the MSE of estmatng H. Applyng the dentty of vec(abc = (C T Avec(B 20, we can rewrte (8 n vector form as α y d = S T I Nd, T T 1 I N d,..., T T K I N d T m T, g T 1,..., gt K + v = Lγ + v (10 where α L S T I Nd, T T 1 I N d,..., T T K I N d T γ m T, g T 1,..., gt K y d vec(y d, m vec( M v vec(v, g vec( G, = 1,..., K. In (10, γ s the vector of unknown wth a dmenson of Q N d (N s + K N, L has a dmenson of T N d Q, and there s T N s + K N. Thanks to ts computatonal smplcty, a lnear estmator s appled at the destnaton node to estmate γ as ˆγ = W H y d (11 where ˆγ denotes the estmaton of γ and W s the weght matrx of the lnear estmator. Usng (11, the MSE of channel estmaton s gven by MSE 1 = Etr(( ˆγ γ ( ˆγ γ H = tr((w H L I Q R γ (W H L I Q H + W H R v W (12 where R γ = Eγ γ H and R v = Evv H are the covarance matrx of γ and v, respectvely. VOLUME 5,

4 Lemma 1: R γ and R v are gven by ( K R v = I T α tr(c T R d + I Nd (13 R γ = bdag s br d, 1 R d,..., K R d (14 where b K tr(r C T. Proof: See Appendx A. It s well-known that (12 s mnmzed by the lnear MMSE estmator 22 gven by ( 1LRγ W = LR γ L H + R v. (15 By substtutng (15 back nto (12 and applyng the matrx nverson lemma of (A + BCD 1 = A 1 A 1 B(DA 1 B + C 1 1 DA 1 we can obtan the MSE of estmatng γ as MSE 1 = tr ( (R 1 γ + L H R 1 v L 1. (16 The transmsson power consumed by the source node s gven by tr(ss H = tr( S S H. (17 Usng (2 and Lemma 3 n Appendx A, the transmsson power of the th relay node can be calculated as αetr(h SS H H H + I N + tr(t T H = αn + α tr( s S S H tr(r + tr( T T H. (18 From (16-(18, the optmal amplfyng factor α and the optmal tranng matrces can be desgned through solvng the followng optmzaton problem mn α, S, T 1,..., T K tr ( (R 1 γ + L H R 1 v L 1 (19 s.t. tr( S S H p s, α > 0 (20 αn + α tr( s S S H tr(r + tr( T T H p, = 1,..., K (21 where p s s the transmsson power avalable at the source node and p s the transmsson power at the th relay node. Usng 15, Th. 1, the optmal S and T as the soluton to the problem (19 (21 satsfy the equatons below for, j = 1,..., K S S H = s, T T H = (22 S T H = 0, T T H j = 0, = j (23 where s and are N s N s and N N dagonal matrx, respectvely, wth non-negatve dagonal elements. Based on (22 and (23, the optmal structure of the tranng sequences s gven by S = U s 1 2 s s, T = U 1 2, = 1,..., K (24 where, = s, 1,..., K, s an N T sem-untary matrx satsfyng the followng equatons H = I N, H j = 0,, j = s, 1,..., K, = j. (25 We would lke to note that sem-untary satsfyng (25 can be easly obtaned, for nstance, from the normalzed dscrete Fourer transform (DFT matrces. Based on (22 and (23, t can be seen that the MSE matrx (R 1 γ + L H R 1 v L 1 n (16 becomes block dagonal under the optmal S and T, ndcatng that the estmaton errors of m T, g T, = 1,..., K, are uncorrelated. By ntroducng the EVD of R d = U d d U H d, the problem (19-(21 can be rewrtten as ( K mn tr (D ds + α s D v 1 + (D d + D v 1 α,{ } (26 s.t. tr( s p s (27 α > 0, 0, = s, 1,..., K (28 αn + αtr( s s tr(r (29 + tr( p, = 1,..., K (30 where { } = {, = s, 1,..., K} and D ds = 1 s (b d 1 (31 D d = 1 1 d, = 1,..., K (32 ( K 1. D v = αtr(c T d + I Nd (33 Usng (31-(33, the problem (26-(29 can be equvalently converted to the followng optmzaton problem wth scalar varables mn α,{σ,m } s.t. N s N s N d ( n=1 N + 1 bλ s,m λ d,n + ασ s,m d v,n K N d ( n=1 1 1 λ,m λ d,n + σ,m d v,n 1 (34 σ s,m p s (35 ( Ns α λ s,m σ s,m tr(r + N N + σ,m p, = 1,..., K (36 α > 0, σ,m 0, m = 1,..., N, = s, 1,..., K (37 where {σ,m } = {σ,m, m = 1,..., N, = s, 1,..., K}, λ d,n s the nth dagonal element of d, λ,m, σ,m, = s, 1,..., K, are the mth dagonal element of and, respectvely, and ( K 1 d v,n αtr(c T λ d,n + 1. (38 For a gven α, the problem (34 (37 s a convex optmzaton problem wth respect to {σ,m }. Ths s because VOLUME 5, 2017

5 when α s fxed, b, λ s,m, λ d,n, d v,n and λ,m are known varables, and (34 s monotoncally decreasng and convex wth respect to {σ,m }. Moreover, the constrants n (35 and (36 are lnear nequalty constrants when α s fxed. Thus, wth fxed α, the problem (34 (37 wth respect to {σ,m } s a convex optmzaton problem, whch can be effcently solved by the Lagrange multpler method. The Karush-Kuhn-Tucker (KKT optmalty condtons 23 of the problem (34 (37 are gven below. Frstly, the gradent condtons are gven by N d n=1 N d αd v,n K ((bλ s,m λ d,n 1 + ασ s,m d v,n 2 = µ s + µ e,m (39 m = 1,..., N s d v,n n=1 ((λ,m λ d,n 1 + σ,m d v,n 2 = µ m = 1,..., N, = 1,..., K (40 where e,m = αλ s,m tr(r, and µ 0, = s, 1,..., K, are the Lagrange multplers. Secondly, the complementary slackness condtons are gven by N s µ s (p s σ s,m = 0 µ (p α ( Ns = 1,..., K. λ s,m σ s,m tr(r + N σ,m = 0 N When α and µ, = s, 1,..., K, are fxed, the nonnegatve {σ s,m } can be obtaned through the b-secton search, as the left-hand sde (LHS of (39 and (40 are monotoncally decreasng functons of σ s,m and σ,m, respectvely. To obtan µ s and µ, we can apply an outer b-secton search loop snce the LHS of (35 s a decreasng functon of σ s,m, and the LHS of (36 s an ncreasng functon of σ s,m and σ,m. Meanwhle, n (39, σ s,m monotoncally decreases wth respect to µ s and µ, and σ,m s a monotoncally decreasng functon of µ n (40. When α also becomes a varable n the optmzaton, the problem (34-(37 s not a convex optmzaton problem. But n ths case, the followng lemma s n order. Lemma 2: The object functon (34 subjectng to constrants (35-(37 s a unmodal functon wth respect to α. Proof: See Appendx B. For a unmodal functon, ts mnmal value can be effcently obtaned by usng the golden secton search (GSS 24. The procedure of applyng the GSS technque to fnd the optmal α s descrbed n Table I, where φ > 0 s the reducton factor, denotes the absolute value, and ε s a postve constant close to 0. It s shown n 24 that the optmal φ s 1.618, also known as the golden rato. It can be observed from Table I that at each teraton, the GSS method reduces the nterval contanng the optmal α to tmes of nterval at the precedng teraton. Thus, TABLE 1. Algorthm I: Procedure of fndng the optmal α n the Problem (34 (37. the length of the nterval contanng the optmal α after the nth teraton s Ɣ n = n Ɣ 0 24, where Ɣ 0 = α u α l s the length of the ntal feasble nterval. Therefore, the number of teratons requred to acheve the desred accuracy ε n Table 1 s gven by N = ln ε ln Ɣ 0 ln 0.618, where denotes the celng operator. The value of ε can be determned by consderng the complexty-performance tradeoff. In the smulatons, we choose ε = 10 3 and a l, a u = 0, 1. Thus, Ɣ 0 = 1 and N = 15. The unmodalty of (34 wth respect to α subjectng to (35-(37 wthn bounds a l, a u = 0, 1 s llustrated n Fg. 7 n Appendx B. IV. MMSE-BASED FIRST-HOP CHANNEL ESTIMATION In ths secton, we propose an MMSE-based algorthm to estmate the frst-hop channel matrces consderng the error of the second-hop channel estmaton n Secton III. After estmatng the second-hop channels G, = 1,..., K, ˆ G T can be subtracted from (8 as Ȳ d = Y d K ˆ G T = αg H S + K ( G ˆ G T + V = αĝ H S + V (41 where H = H T 1,..., H T K T and V = α(g Ĝ H S + K ( G ˆ G T + V (42 s the total nose matrx ncludng the channel estmaton errors. By applyng the vectorzaton operaton to both sdes of (41, we have ȳ d = ( α S T Ĝ h + v = L 2 h + v (43 where ȳ d = vec(ȳ d, h = vec( H, v = vec( V, and L 2 = α S T Ĝ. From (43, an MMSE estmate of h can be obtaned by ˆ h = W H 2 ȳd (44 where W 2 = (L 2 R h LH 2 + R v 1 L 2 R h, R h = E h h H, and R v = E v v H. From (44, the MSE of estmatng h s gven VOLUME 5,

6 FIGURE 2. Example 1: NMSE versus p at varous α wth N d = N = 4 and ρ = 0.8. by ( (R 1 MSE 3 = tr h + LH 2 R 1 v L 1 2 (45 where the dervaton of R h and R v s shown n Appendx C. Note that n (45, we take nto account the equvalent estmaton error nherted from the second-hop channel estmaton,.e., R v. Therefore, the total MSE of the frst-hop channel matrces estmaton depends on the accuracy of the second-hop channel matrces estmaton. In other words, the total MSE n (45 ncreases wth the ncreasng of the MSE of the second-hop channel estmaton. V. NUMERICAL EXAMPLES In ths secton, we study the performance of the proposed supermposed channel tranng algorthm for MIMO relay systems wth multple relay nodes through numercal smulatons. We smulate a MIMO relay system wth K = 2 relay nodes whch have the same number of antennas as the source node,.e., N s = N = N, = 1, 2, and we choose the smallest T possble,.e., T = N s + K N = 3N. The optmal tranng matrces for the supermposed channel tranng method are generated accordng to (24. In partcular, the sem-untary matrces n (25 are generated based on the normalzed DFT matrces as s m,n = 1 2πmn j 3N e 3N, 1 m,n = 1 2π(m+Nn j e 3N, 3N 2 m,n = 1 2π(m+2Nn j e 3N, 3N m = 1,..., N, n = 1,..., 3N. The channel covarance matrces have the wdely used exponental Toepltz structure 21 such that T s = R = C = ρ m n, = 1, 2, m, n = 1,..., N, and R d = ρ m n, m, n = 1,..., N d, where ρ s the correlaton coeffcent wth magntude ρ < 1. Wthout loss of generalty, we consder ρ = 0.8 and ρ = 0.2 for the hgh and low channel correlaton scenaros, respectvely. We assume that all nodes have the same amount of transmsson power as p s = p 1 = p 2 = p. In all smulaton examples, the normalzed MSE (NMSE of channel estmaton at the destnaton node s computed. In the frst example, we compare the NMSE performance of the supermposed channel tranng algorthm at varous α. Fg. 2 shows the NMSE of the proposed algorthm versus p at varous α wth N d = N = 4 and ρ = 0.8. The curve of the optmal α s obtaned by applyng the GSS method n the proposed supermposed channel tranng algorthm to fnd the optmal α for each value of p. It can be observed from Fg. 2 that the GSS technque s an effcent method to fnd the optmal α snce the optmal α curve constantly has the lowest MSE level for all values of p. Furthermore, we observe from Fg. 2 that the optmal α vares wth respect to p, whch means that usng a fxed α s strctly suboptmal. In partcular, for p between 10dB and 30dB, the NMSE wth α = 0.2 s close to the NMSE usng the optmal α. However, the curve assocated wth α = 0.07 has a lower NMSE than the one assocated wth α = 0.2 when p = 5dB. In addton, for other smulaton examples (e.g. dfferent N and ρ, the NMSE wth α = 0.2 mght not be close to the NMSE usng the optmal α. FIGURE 3. Example 2: NMSE versus p at varous N wth ρ = 0.8 and N d = N. In the second example, we set N d = N and nvestgate the NMSE performance of the proposed supermposed channel tranng algorthm usng the optmal α for varous smulaton scenaros. A comparson of the proposed algorthm has been made wth the conventonal two-stage MMSE-based channel tranng algorthm, where random orthogonal plot sequences are adopted to estmate channel matrces wth equally dstrbuted transmsson power at the relay node between two stages 15. Fg. 3 shows the NMSE performance of both methods wth ρ = 0.8 for N = 2 and N = 4, whle Fg. 4 demonstrates the performance of two methods wth ρ = 0.2. It can be observed from Fgs. 3 and 4 that the two algorthms have smlar performance when p < 10dB, whle at a hgh power level, the supermposed channel tranng algorthm yelds much smaller estmaton error than the conventonal two-stage channel estmaton method. Moreover, the relatonshp between the curves almost stays the same when the correlaton coeffcent ρ changes, ndcatng that the VOLUME 5, 2017

7 FIGURE 4. Example 2: NMSE versus p at varous N wth ρ = 0.2 and N d = N. proposed algorthm s effcent n both scenaros of hgh and low channel correlaton. We also observe from Fgs. 3 and 4 that for both algorthms the NMSE ncreases wth N, as more unknowns need to be estmated wth a larger number of antennas. In the thrd example, we study the NMSE performance of the algorthm proposed n Secton IV whch retreves the ndvdual CSI of the frst-hop channels. Snce the frst-hop channel estmaton s based on the second-hop one, and the error n second-hop estmaton s propagated to the frsthop channel estmaton, we apply the proposed supermposed channel tranng algorthm n Secton III to estmate the second-hop channels n ths example, whch has a better NMSE performance than the conventonal two-stage algorthm. Fg. 5 shows the NMSE performance of ths method wth ρ = 0.2 for varous N and N d. It can be seen that wth fxed N d, the NMSE ncreases wth N, as more unknowns need to be estmated. As expected, for a gven N, the NMSE decreases wth N d snce more observatons are avalable. FIGURE 5. Example 3: NMSE of the frst-hop channel estmaton versus p at varous N d and N wth ρ = 0.2. FIGURE 6. Example 3: NMSE of the frst-hop channel estmaton versus p at varous N d and N wth ρ = 0.8. Fg. 6 demonstrates the performance of ths approach wth ρ = 0.8. Smlar to Fg. 5, t can be seen from Fg. 6 that a better MSE performance can be acheved by settng N d = N 1 + N 2 nstead of N d = N 1 = N 2 = N. VI. CONCLUSIONS In ths paper, we have appled the supermposed channel tranng approach to MIMO relay communcaton systems wth multple relay nodes. The channel matrces of both the frst-hop and the second-hop lnks can be effcently estmated usng the proposed algorthms. The optmal tranng sequences and power allocaton at the source and relay nodes are derved. Furthermore, a new MMSE-based estmator has been developed to retreve the frst-hop channel nformaton. A better performance of the proposed supermposed channel tranng algorthm has been shown through numercal examples. The channel tranng algorthm proposed n ths paper can be readly extended to two-way MIMO relay networks wth multple relay nodes. APPENDIX A PROOF OF LEMMA 1 In order to derve R v and R γ, here we ntroduce the followng lemma. Lemma 3 25: For H CN (0,, there s EHAH H = tr(a T and EH H AH = tr( A T. To derve R v, we frst calculate the covarance matrx of v m usng (4 and Lemma 3, where v m denotes the mth column of V K E v m v H m = E αg v,m v H,m GH + v d,m v H d,m K = E αg G H + I Nd = K αtr(c T R d + I Nd (46 VOLUME 5,

8 where v,m and v d,m are the mth columns of V and V d, respectvely. As v m, m = 1,..., T, are ndependent, R v can be wrtten as ( K R v = I T αtr(c T R d + I Nd. Now let us calculate R γ. From (5, (7, and (9, the mth column of M and G can be wrtten as m m = λ 1 2 s,m K G A H,w π s,m, g,m = λ 1 2,m A d G,w π,m where π s,m and π,m are the mth column of s and, respectvely, λ s,m and λ,m are the mth dagonal element of s and, respectvely. Then smlar to (46, the covarance matrx of m m and g,m are gven by K E m m m H m = λ s,me G A H,w π s,m π H s,m HH,w AH G H K = λ s,m tr(r C T R d = λ s,m br d E g,m g H,m = λ,me A d G,w π,m π H,m GH,w AH d = λ,m R d. As m m, m = 1,..., N s and g,m, m = 1,..., N, = 1,..., K, are ndependent, we have R γ = bdag s br d, 1 R d,..., K R d. APPENDIX B PROOF OF LEMMA 2 Denotng a m,n 1/ ( bλ s,m λ d,n, cm ασ s,m, and g,m,n 1/ ( λ,m λ d,n, we can rewrte the problem (34 (37 as mn α,c,{σ } N s N d n=1 N + 1 a m,n + c m d v,n K N d n=1 1 g,m,n + σ,m d v,n (47 s.t. 1 T c αp s (48 z T c + 1T σ p αn, = 1,..., K (49 α > 0, c m 0, m = 1,..., N s (50 σ,m 0, m = 1,..., N, = 1,..., K (51 where c c 1,..., c Ns T, z tr(r λ s,1,..., λ s,ns T, σ σ,1,..., σ,n T, and {σ } = {σ, = 1,..., K}. For a very small α, the value of (47 s manly governed by the constrant n (48, snce the constrants n (49 s not actve compared wth the constrant (48 when α s small. As α ncreases from a small value, the feasble regon determned by (48 expands, whch contrbutes to the decrease n the value of (47. On the other hand, when α s large, the value of (47 s manly affected by the constrants n (49, as the constrant (48 s not actve compared wth those n (49 for a large value of α. As α decreases from a large value, the feasble regon specfed by (49 expands, resultng n a decreasng of (47. Now let us consder the effect of α on d v,n. Note that d v,n n (38 s monotoncally decreasng when α ncreases, and the value of (47 ncreases wth decreasng d v,n. Takng the two effects above nto account, we can draw the followng concluson regardng the value of (47 wth respect to α. When α ncreases from a very small postve number, the value of (47 starts to decrease as the potental decrease of (47 caused by the relaxed feasble regon (48 domnates the potental ncrease of (47 due to the decreasng d v,n. The value of (47 keeps decreasng as α ncreases tll a turnng pont where the decreasng of d v,n starts to domnate the effect of relaxed feasble regon (48. After ths turnng pont, the value of (47 wll monotoncally ncrease wth an ncreasng α. Therefore, the objectve functon (34 subjectng to (35-(37 s proved to be a unmodal functon wth respect to α. To valdate the analyss above, we plot the MSE value (34 versus α n Fg. 7 at varous levels of p s. Here we set p 1 = p 2 = 20dB and the other smulaton setups are the same as those n Fg. 2. It can be clearly seen from Fg. 7 that the objectve functon (34 subjectng to the constrants (35 (37 s a unmodal functon of α. FIGURE 7. NMSE versus α at varous p s wth p 1 = p 2 = 20dB. APPENDIX C CALCULATION OF R h AND R v From (5 and (7, we have H = H U s = A H,w s 1 2 s. (52 Usng (52, the covarance matrx of the pth column of H, denoted as h,p, s gven by E h,p h H,p = λ s,pe A H,w π s,p π H s,p HH,w AH = λ s,p R. (53 Then the covarance matrx of h p = h T 1,p,..., h T K,p T, the pth column of H, can be obtaned based on (53 as R h p = bdag λ s,p R 1, λ s,p R 2,..., λ s,p R K. ( VOLUME 5, 2017

9 As h = h T 1,, h T N s T, we have R h = E h h H = bdag R h 1, R h 2,..., R h Ns. Now we start to calculate R v = E v v H. From (42, the correlaton matrx of the mth and nth columns of V, m, n = 1,..., T, s gven by E v m v H n = αe H s m s H n H H H K + E t,m t H,n H + Ev m v H n (55 where = ( 1,..., K, = U H, = G ˆ G, = 1,..., K, ˆ G s the estmate of G, s m, t,m, and v m are the mth columns of S, T, and V, respectvely. From (46, we obtan K Ev m v H n = αtr(c T R d + I Nd, m = n; 0, m = n. 2 Wth t,m = 1 φ,m, where φ,m s the mth column of, the (e, f -th entry of the second term n (55 can be calculated as K E t,m t H,n H = = K K e,f E (δ,1,e,..., δ,n,e 1 2 φ,m φ H,n 1 ( 2 H δ,1,f,..., δ,n,f tr( 1 2 φ,m φ H,n 1 2 D,e,f (56 where δ,p,q s the qth entry of δ,p = g,p ˆ g,p, p = 1,..., N, q = 1,..., N d, g,p and ˆ g,p denote the pth columns of G and ˆ G, respectvely, and D,e,f s gven by (δ,1,f H ( D,e,f = E,..., δ,n,f δ,1,e,..., δ,n,e = dag(eδ,1,e δ,1,f,..., Eδ,N,eδ,N,f = dag (,1 e,f,...,,n e,f. (57 Here,p = E δ,p δ H,p = E ( g,p ˆ g,p ( g,p ˆ g H,p ( K 1 = (λ,p R d 1 + σ,p αtr(c T R d + I Nd 1. (58 In other words, D,e,f n (57 s a dagonal matrx whose pth dagonal entry, p = 1,..., N, s the (e, f -th entry of,p n (58. Fnally, the frst term n (55 can be calculated as K E H s m s H n H K H H = E H s m s H n H H H = where we used the fact that K E H s m s H n H H H = s H n K s s m E R H (59 E H s m s H n H H = EA H,w s 1 2 s s m s H n 1 2 s H s HH,w AH = s H n s s m R. To derve E R H, ts (e, f -th entry, e, f = 1,..., N d, can be calculated as E R H e,f (δ,1,e ( = E,..., δ,n,e U H H R U δ,1,f,..., δ,n,f = tr(u H R U D,e,f. (60 REFERENCES 1 Y. Fan and J. Thompson, MIMO confguratons for relay channels: Theory and practce, IEEE Trans. Wreless Commun., vol. 6, no. 5, pp , May L. Sangunett, A. A. D Amco, and Y. Rong, A tutoral on the optmzaton of amplfy-and-forward MIMO relay systems, IEEE J. Sel. Areas Commun., vol. 30, no. 8, pp , Sep X. Tang and Y. Hua, Optmal desgn of non-regeneratve MIMO wreless relays, IEEE Trans. Wreless Commun., vol. 6, no. 4, pp , Apr O. Muñoz-Medna, J. Vdal, and A. Agustín, Lnear transcever desgn n nonregeneratve relays wth channel state nformaton, IEEE Trans. Sgnal Process., vol. 55, no. 6, pp , Jun W. Guan and H. Luo, Jont MMSE transcever desgn n non-regeneratve MIMO relay systems, IEEE Commun. Lett., vol. 12, no. 7, pp , Jul Y. Rong, Lnear non-regeneratve multcarrer MIMO relay communcatons based on MMSE crteron, IEEE Trans. Commun., vol. 58, no. 7, pp , Jul Y. Rong, X. Tang, and Y. Hua, A unfed framework for optmzng lnear nonregeneratve multcarrer MIMO relay communcaton systems, IEEE Trans. Sgnal Process., vol. 57, no. 12, pp , Dec Y. Rong and Y. Hua, Optmalty of dagonalzaton of mult-hop MIMO relays, IEEE Trans. Wreless Commun., vol. 8, no. 12, pp , Dec A. S. Behbahan, R. Merched, and A. M. Eltawl, Optmzatons of a MIMO relay network, IEEE Trans. Sgnal Process., vol. 56, no. 10, pp , Oct A. Todng, M. R. A. Khandaker, and Y. Rong, Jont source and relay optmzaton for parallel MIMO relay networks, EURASIP J. Adv. Sgnal Process., vol. 2012, p. 174, Aug P. Lolou and M. Vberg, Least-squares based channel estmaton for MIMO relays, n Proc. Int. ITG Workshop Smart Antennas, Feb. 2008, pp P. Lolou, M. Vberg, and M. Coldrey, Effcent channel estmaton technques for amplfy and forward relayng systems, IEEE Trans. Commun., vol. 60, no. 11, pp , Nov T. Kong and Y. Hua, Optmal desgn of source and relay plots for MIMO relay channel estmaton, IEEE Trans. Sgnal Process., vol. 59, no. 9, pp , Sep VOLUME 5,

10 Z. He et al.: Channel Estmaton of MIMO Relay Systems Wth Multple Relay Nodes 14 Y. Rong, M. R. A. Khandaker, and Y. Xang, Channel estmaton of dualhop MIMO relay system va parallel factor analyss, IEEE Trans. Wreless Commun., vol. 11, no. 6, pp , Jun C. W. R. Chong, Y. Rong, and Y. Xang, Channel tranng algorthms for two-way MIMO relay systems, IEEE Trans. Sgnal Process., vol. 61, no. 16, pp , Aug C. W. R. Chong, Y. Rong, and Y. Xang, Channel estmaton for two-way MIMO relay systems n frequency-selectve fadng envronments, IEEE Trans. Wreless Commun., vol. 14, no. 1, pp , Jan C. W. R. Chong, Y. Rong, and Y. Xang, Channel estmaton for tmevaryng MIMO relay systems, IEEE Trans. Wreless Commun., vol. 14, no. 12, pp , Dec H. Mehrpouyan, S. D. Blosten, and B. Ottersten, Smultaneous estmaton of mult-relay MIMO channels, n Proc. IEEE Global Commun. Conf., Atlanta, GA, USA, Dec. 2013, pp L. Canonne-Velasquez, H. Nguyen-Le, and T. Le-Ngoc, Cascaded doubly-selectve channel estmaton n mult-relay AF OFDM transmssons, n Proc. IEEE Int. Conf. Commun., Ottawa, ON, Canada, Jun. 2012, pp J. W. Brewer, Kronecker products and matrx calculus n system theory, IEEE Trans. Crcuts Syst., vol. CAS-25, no. 9, pp , Sep D. S. Shu, G. Foschn, M. Gans, and J. Kahn, Fadng correlaton and ts effect on the capacty of multelement antenna systems, IEEE Trans. Commun., vol. 48, no. 3, pp , Mar S. M. Kay, Fundamentals of Statstcal Sgnal Processng: Estmaton Theory, vol. 1. Englewood Clffs, NJ, USA: Prentce-Hall, S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge, U.K.: Cambrdge Unv. Press, A. Antonou and W.-S. Lu, Practcal Optmzaton: Algorthms and Engneerng Applcatons. Sprng Street, NY, USA: Sprnger, A. Gupta and D. Nagar, Matrx Varate Dstrbutons. London, U.K.: Chapman & Hall, XIAODAN WANG receved the B.E. and M.Sc. degrees n nformaton and communcaton engneerng from the Bejng Unversty of Posts and Telecommuncatons, Chna, n 2014 and 2017, respectvely. She majors n new physcal layer technology appled n 5G, wth an emphass on channel estmaton and dfferent schemes of channel codng and nonorthogonal multple access. ZHIQIANG HE (S 01 M 04 receved the B.E. degree n sgnal and nformaton processng and Ph.D. degree (Hons. n sgnal and nformaton processng from the Bejng Unversty of Posts and Telecommuncatons, Chna, n 1999 and 2004, respectvely. Snce 2004, he has been wth the School of Informaton and Communcaton Engneerng, Bejng Unversty of Posts and Telecommuncatons, where he s currently a Professor and the Drector of the Center of Informaton Theory and Technology. Hs research nterests nclude sgnal and nformaton processng n wreless communcatons, networkng archtecture and protocol desgn, machne learnng, and underwater acoustc 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 He was a Post-Doctoral Researcher wth the Department of Electrcal Engneerng, Unversty of Calforna, Rversde, from 2006 to Snce 2007, he has been wth the Department of Electrcal and Computer Engneerng, Curtn Unversty, Bentley, WA, Australa, where he s currently a Professor. Hs 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. He has publshed over 150 journal and conference papers n these areas. Dr. Rong was a recpent of 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 He was an Edtor of the IEEE WIRELESS COMMUNICATIONS 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 a TPC Member for the IEEE ICC, WCSP, IWCMC, EUSIPCO, and ChnaCom. He s an Assocate Edtor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING. JIAOLONG YANG receved the B.E. degree n nformaton engneerng from the Bejng Unversty of Posts and Telecommuncatons, Bejng, Chna, n 2015, where he s currently pursung the M.Sc. degree n electroncs and telecommuncaton engneerng. Hs research nterests are manly on wreless communcatons YANG LIU receved the M.Sc. degree n nformaton and communcaton engneerng from the Bejng Unversty of Posts and Telecommuncatons, Bejng, Chna, n Hs research focuses on frequency-doman equalzaton algorthms, and the results have been publshed at IC-NIDC. VOLUME 5, 2017