Robust transceiver design for AF MIMO relay systems with column correlations

Transcription

1 Ttle Robust transcever desgn for AF MIMO relay systems wth column correlatons Author(s) Xng, C; Fe, Z; Wu, YC; Ma, S; Kuang, J Ctaton The 0 IEEE Internatonal Conference on Sgnal rocessng, Communcatons and Computng (ICSCC), X'an, Chna, 4-6 September 0. In IEEE ICSCC roceedngs, 0, p. -6 Issued Date 0 URL Rghts IEEE Internatonal Conference on Sgnal rocessng, Communcatons and Computng roceedngs. Copyrght IEEE.

2 Robust Transcever Desgn for AF MIMO Relay Systems wth Column Correlatons Chengwen Xng, Zesong Fe, Yk-Chung Wu, Shaodan Ma and Jngg Kuang School of Informaton and Electroncs, Bejng Insttute of Technology, Bejng, Chna Emal: {zesongfe, Department of Electrcal and Electronc Engneerng, The Unversty of Hong Kong, Hong Kong Emal: Abstract In ths paper, we nvestgate the robust transcever desgn for dual-hop amplfy-and-forward (AF) MIMO relay systems wth Gaussan dstrbuted channel estmaton errors. Ag at maxmzng the mutual nformaton under mperfect channel state nformaton (CSI), source precoder at source and forwardng matrx at the relay are jontly optmzed. Usng some elegant attrbutes of matrx-monotone functons, the structures of the optmal solutons are derved frst. Then based on the derved structure an teratve waterfllng soluton s proposed. Several exstng algorthms are shown to be specal cases of the proposed soluton. Fnally, the effectveness of the proposed robust desgn s demonstrated by smulaton results. I. INTRODUCTION Cooperatve communcaton s one of the key parts of the future communcaton protocols, as the deployment of relays can mprove lnk equalty, extend coverage range and mtgate nference. In general, there are varous relay strateges whch are casted nto three man categores,.e., amplfy-andforward (AF), decode-and-forward (DF) and compressed-andforward (CF). Among these relayng strateges, AF strategy whch has the lowest complexty s most sutable for practcal mplementaton. It s also well-establshed that adoptng multple antennas has a potental to mprove overall wreless system performance. In order to reap both benefts promsed by cooperatve communcaton and mult-nput mult-output (MIMO) systems, lnear transcever desgn for AF MIMO relayng systems has been wdely researched n [] [5]. Generally, speakng there are two man knds of crtera for transcever desgn: capacty maxmzaton and mean-square-error (MSE) mzaton. Jont desgn of relay forwardng matrx and destnaton equalzer for mzng MSE s dscussed n [] and [3]. Furthermore, jont desgn of source precoder relay forwardng matrx and destnaton equalzer for mzng MSE s nvestgated n [5]. The capacty maxmzaton transcever desgn has been dscussed n [], [5]. In most of prevous works, channel state nformaton (CSI) s assumed to be perfectly known. However, ths assumpton cannot be met n practce. Channel estmaton errors are always nevtable and drastcally degrades system performance. It Ths research work was supported n part by Ercsson and Sno- Swedsh IMT-Advanced and Beyond Cooperatve rogram under Grant No.008DFA780. s well known that robust desgns can reduce or mtgate the negatve effects ntroduced by mperfect CSI. Ths s also the motvaton of our work. In ths paper, we jontly optmze source precoder matrx and relay forwardng matrx for mutual nformaton maxmzaton under channel estmaton errors. Based on the propertes of matrx-monotone functons, the optmal structure of robust transcevers s derved. Then, an teratve water-fllng soluton s proposed. Fnally, the numercal result shows the performance advantage of the proposed robust desgn. The followng notatons are used throughout ths paper. Boldface lowercase letters denote vectors, whle boldface uppercase letters denote matrces. The notaton Z H denotes the Hermtan of the matrx Z, and Tr(Z) s the trace of the matrx Z. The notaton Z / s the Hermtan square root of the postve sem-defnte matrx Z, such that Z / Z / = Z and Z / s also a Hermtan matrx. For a rectangular dagonal matrx Λ, Λ denotes the man dagonal elements are n decreasng order and Λ denotes the man dagonal elements are n ncreasng order. For two Hermtan matrces, C ર D means that C D s a postve sem-defnte matrx. The symbol λ (Z) represents the th largest egenvalue of Z. II. SYSTEM MODEL AND ROBLEM FORMULATION A. Transmtted and Receved Sgnals In our work, a dual-hop AF MIMO relay system s nvestgated, n whch there s one source wth N S antennas, one relay wth M R receve antennas and N R transmt antennas, and one destnaton wth M D antennas. Because of deep fadng, the drect lnk between the source and destnaton s not taken nto account. At the frst hop, the source transmts data to the relay. The receved sgnal at the relay s denoted as x = H sr s + n () where H sr s the MIMO channel matrx between the source and the relay, and s the precoder matrx at the source. The vector s s the N data vector transmtted by the source wth the covarance matrx R s = E{ss H } = I N. Furthermore, n s the addtve Gaussan nose vector wth correlaton matrx R n = σ I MR. At the relay, the receved sgnal x s multpled by a forwardng matrx F. Then the resultant sgnal s transmtted

3 to the destnaton. The receved sgnal y at the destnaton can be wrtten as y = H rd FH sr s + H rd Fn + n, () where H rd s the MIMO channel matrx between the relay and the destnaton, and n s the addtve Gaussan nose vector at the second hop wth covarance matrx R n = σi MD.In order to guarantee the transmtted data s can be recovered at the destnaton, t s assumed that N S, M R, N R, and M D are greater than or equal to N [3]. When channel estmaton errors are taken nto account, the dual-hop channels read as H sr = H sr +ΔH sr, H rd = H rd +ΔH rd, (3) where H sr and H rd are the channel estmates and ΔH sr and ΔH rd are the correspondng estmaton errors wth zero-mean Gaussan dstrbuted entres. Addtonally, the estmaton errors are ndependent wth each other as the channels are separately estmated. Referrng to estmaton errors, the followng wdely used Kronecker structure s adopted [6], [7] ΔH sr = Σ / sr H W,sr Ψ / sr ΔH rd = Σ / rd H W,rdΨ / rd, (4) where the entres of H W,sr and H W,rd are dentcal and ndependent dstrbuted (..d.) wth zero mean and unt varance. The column correlaton matrces (Ψ sr and Ψ rd ) and the row correlaton matrces (Σ sr and Σ rd ) are detered by tranng sequences and channel estmators [8]. To the best of the authors knowledge, for a general case even for a pontto-pont MIMO system, there s no closed-form soluton. In ths paper, we focus on the case wth column correlatons only.e., Σ sr = α I, Σ rd = α I, (5) as ths case corresponds to a practcal lnear mum mean square error (LMMSE) channel estmator [8]. B. roblem Formulaton At the destnaton, a lnear equalzer G s adopted to detect the data vector s. The mean-square-error (MSE) matrx s E{(Gy s)(gy s) H }, where the expectaton s taken wth respect to random data, channel estmaton errors, and nose. In [9], t s shown that E{(Gy s)(gy s) H } = G( H rd FR x F H H H rd + K )G H + I N ( H H H srf H H H rdg H ) (G H rd F H sr ), (6) where matrces R x and K are defned as R x E{xx H } = H sr H H H sr + K K Tr( H Ψ sr )Σ sr + R n K Tr(FR x F H Ψ rd )Σ rd + R n. (7) It s obvous that R x s the covarance matrx of the receved sgnal at the relay. Usng lnear Bayesan theory, the LMMSE equalzer at the destnaton equals to G =( H rd F H sr ) H ( H rd FR x F H H H rd + K ), (8) based on whch the MSE matrx n (6) s rewrtten as Φ MSE (F, ) =I ( H rd F H sr ) H ( H rd FR x F H H H rd + K ) ( H rd F H sr ) (9) Capacty s one of the most mportant and wdely used performance metrcs for transcever desgns. Gven the receved plots n channel estmaton as y and y, the channel capacty s denoted as I(s; y y, y ), whch s the condtonal mutual nformaton based on known y and y [7]. To the best of our knowledge, the exact capacty for MIMO channels wth estmaton errors at both ends s largely open even for pontto-pont MIMO systems [7]. To proceed, a common logc s to derve and use bounds.e., lower bound or upper bound. Snce we am to maxmze channel capacty, lower bound s more meanngful than upper bound. In Appendx A t has been proved that log Φ MSE (F, ) I(s; y y, y ). (0) Ths s a wdely-establshed lower bound [7] and becomes tghter as estmaton errors are smaller. As a result, the robust transcever desgn for maxmzng mutual nformaton s formulated as F, log Φ MSE(F, ) s.t. Tr(FR x F H ) r Tr() s. () Based on the defnton of R x n (7), R x s a functon of. In order to smplfy the analyss, we defne a new varable F FK / (K / H sr H H H srk / + I /, () Π ) based on whch FR x F H = F F H and the two constrants nvolved n () become ndependent. Meanwhle, the MSE matrx n (9) s correspondngly rewrtten as Φ MSE ( F, ) =I ( H rd FΠ / K / K / H H rd + K ) ( H rd FΠ / H sr ) H ( H rd F F H H sr ). (3) Fnally, the optmzaton problem for the robust desgn becomes F, log Φ MSE ( F, ) s.t. Tr( F F H ) r Tr() s. (4) In the followng, the optmal solutons of (4) wll be dscussed n detals. III. OTIMAL SOLUTIONS In our work, we nvestgate the optmzaton problem (4) from matrx-monotone functon vewpont. The dea of utlzng the propertes of matrx-monotone functons to desgn MIMO transcevers has been address n [0]. In ths paper, we extend ths dea to robust transcever desgns for a AF MIMO relayng system.

4 Defnton : A matrx-monotone functon s defned as g( ) whch maps a matrx varable from a subsect of postve semdefnte matrces to a real number. If g( ) s a monotoncally decreasng matrx-monotone functon on postve sem-defnte matrces, t satsfes A ર B ર 0 g(a) g(b). (5) On the other hand, when g( ) a monotoncally ncreasng matrx-monotone, t means g( ) s a monotoncally decreasng matrx-monotone functon [0]. In the followng we focus our attenton on a knd of optmzaton problems wth a decreasng matrx-monotone functon as objectve, whch s formulated as ( g X H H H HX ) X η x s.t. Tr(XX H ) η x =Tr(XX H Ψ)α + σn. (6) Solvng the optmzaton problem (6), two mportant solutons are derved and are the bass for the followng dervatons. Concluson : The optmal soluton of (6) satsfes Tr(XX H )=Tr[XX H (α Ψ + σni)]/η x =. (7) Defnng the untary matrx V H and rectangular dagonal matrx Λ H based on the followng sngular value decomposton H(α Ψ + σni) / = U H Λ H VH H wth Λ H, (8) the optmal soluton of the optmzaton problem (6) has followng structure X opt = η x (α Ψ + σni) / V H Λ X U H Ξ wth Λ T XΛ T HΛ H Λ X, and η x = σn/{ αtr[vh(α H Ψ + σni) / Ψ(α Ψ + σni) / V H Λ X Λ T X]}, (9) where U Ξ s an untary matrx and Λ X s a rectangular dagonal matrx wth real dagonal elements. roof: See Appendx B. Concluson : For a complex matrx A and a postve defne matrx N, based on followng egen-decomposton AN A H = U ANA Λ ANA VANA H wth Λ ANA (0) when the objectve functon s g ( X H H H ) HX/η x =log A H (X H H H HX/η x + I) A + N () the untary matrx U Ξ n (9) equals to U Ξ = U ANA. () Defnng N X = {Rank(H H H), Rank(AA H )}, Λ x n (9) has the structure of [ ] Λ Λ x = x 0, (3) 0 0 where Λ x s a N X N X dagonal matrx. roof: See Appendx C. A. The structure of optmal F In ths secton the structure of optmal F s derved. Based on the matrx nverson lemma, the MSE matrx n (3) can be rewrtten as Φ MSE ( F, ) =(Π / K / (Π / K / A H sr ) H ( F H H H rdk H rd F + I) H H sr )+( H H srk H sr + I) N based on whch for any gven the optmzaton problem wth respect to F becomes as F log A H ( F H H H rdk H rd F + I) A + N s.t. Tr( F F H ) r K =Tr( F F H Ψ rd )Σ rd + σn I. (4) Defnng untary matrces U and V based on the followng sngular value decomposton K / H sr = U Λ V H wth Λ, (5) we have the followng egen-decomposton A N AH = U Λ Λ T U H wth Λ Λ T. (6) Together wth the followng sngular value decomposton H rd (α r Ψ rd + σn ) / = U rd Λ rd Vrd H wth Λ rd, and based on Conclusons and, the optmal F has the followng structure F = η f (α r Ψ rd + σ n I) / V rd,n Λ F UH,N wth Λ F Λ rd Λ F and η f = σn /{ α Tr[Vrd,N(α H r Ψ rd + σn I) / Ψ rd (α r Ψ rd + σ n I) / V rd,n Λ F]}, (7) where Λ F s a N N dagonal matrx. Meanwhle, for the optmal F the followng constrant s fulflled Tr( F F H )=Tr[ F F H (α r Ψ rd + σn I)]/η f = r. (8) B. The structure of optmal In the followng, t wll be proved that gven the structure of F the optmzaton problem for s the same as that for F. Usng the optmal structure of F, wehave ( F H H H rdk H rd F + I) =(U,N Λ F Λ rd Λ F UH,N + I). Usng the followng substtuton Λ ( Λ F Λ rd Λ F + I), (9) and the matrx nverson lemma agan, the MSE matrx can be reformulated as Φ MSE( F, ) =V (I Λ ) / V H ( H H H srk H sr + I) Notce that Σ rd = α I. V (I Λ ) / V H + V Λ V H. (30) A F N F

5 Therefore, the optmzaton problem wth respectve to s equvalent to log A H F (H H H srk H sr + I) A F + N F s.t. Tr( H ) s, K =Tr( H Ψ sr )Σ sr + σ n I. (3) Based on the defntons of A F and N F, t can be derved that has the followng egen-decomposton AH F A F N F A F N F AH F = V Λ F Λ rd Λ F VH. (3) Together wth followng sngular value decomposton, H sr (α s Ψ sr + σ n I) / = U sr Λ sr V H sr (33) and usng Conclusons and 3, the optmal has the followng structure = η p (α s Ψ sr + σ n I) / V sr,n Λ V H wth Λ Λ Λ sr and η p = σn /{ α Tr[Vsr,N H (α s Ψ sr + σn I) / Ψ sr (α s Ψ sr + σ n I) / V sr,n Λ ]}, (34) where Λ a N N dagonal matrx. Consderng that as there are no constrants on V, V can be an arbtrary N N untary matrx. Based on Concluson, t can be concluded that the optmal satsfes Tr( H )=Tr[ H (α s Ψ sr + σ n I)]/η p = s. (35) Substtutng (34) nto (5), t can be derved that U,N = U sr,n and then the optmal structure of F s F = η f (α r Ψ rd + σ n I) / V rd,n Λ F UH sr,n. (36) Based on the optmal structure gven by (34) and (36) and wth regard to the fact that η p and η f are detered by Λ and Λ F, respectvely, the left unknown varables are only Λ and Λ F. C. roposed Solutons for Λ F and Λ Based on (8) and (35), the optmzaton problem (4) also equals to F, s.t. log Φ MSE ( F, ) Tr[ F F H (α r Ψ rd + σ n I)]/η f = r Tr[ H (α s Ψ sr + σ n I)]/η p = s. (37) Furthermore, wth the followng dagonal matrces Λ sr =dag{λ sr, } Λ rd =dag{λ rd, } Λ F =dag{f } Λ =dag{p } (38) 3 Notce that Σ sr = α I. and substtutng (34) and (36) nto (37), the optmzaton problem (37) can be rewrtten as f,p s.t. N f log λ rd, + p λ sr, + (p = λ sr, +)(f λ rd, +) f = r p = s. (39) Wth respectve to the fact the problem (39) s nherently nonconvex and dffcult to solve, an teratve water-fllng soluton s proposed n ths paper. When p s are fxed, f s can be computed as + p f λ sr, + = (p λ sr, ) + 4p λ sr, λ rd, μ f λ rd, λ rd, (40) where μ f 0 s the Lagrange multpler whch makes f = r. On the other hand, when f s are fxed p s can be computed as + f p λ rd, + = (f λ rd, ) + 4f λ rd, λ sr, μ p λ sr, λ sr, (4) where μ p 0 s the Lagrange multpler whch makes p = s. Specal cases: Several exstng algorthms can be consdered as specal cases of our proposed soluton. When CSI s perfectly known and = I, the proposed soluton for F reduces to that n []. When CSI s perfectly known, the proposed soluton for and F reduces to that gven n [5]. When the second hop channel s an dentty matrx and noseless, the proposed soluton for source precoder desgn reduces to that gven n [7]. IV. SIMULATION RESULTS AND DISCUSSIONS In ths secton, smulaton results are presented to assess the performance of the proposed algorthm and for the purpose of comparson, the algorthm based on the estmated channel only (wthout takng the channel errors nto account) [5]. In the followng, we consder an AF MIMO relay system where the source, relay and destnaton are equpped wth same number of antennas,.e., N S = M R = N R = M D = 4. The channels H sr and H rd are randomly generated accordng to..d. Gaussan dstrbuton. To estmate the channels, a practcal LMMSE estmaton algorthm s adopted [8]. For the tranng sequence D, a famous exponental correlaton matrx s used to descrbe the correlaton matrx of D,.e., DD H R ρ where [R ρ ] j = ρ j. As a result, Σ sr = Σ rd = I and Ψ sr = Ψ rd = (I 4 +SNR EST R α ) where SNR EST s the sgnal-to-nose rato (SNR) n channel estmaton process [8] 4. In the smulaton, for data transmsson process the SNR 4 The detaled dervaton s gven n [8]

6 Sum rate (Bts/Hz/S) The proposed robust desgn The algorthm based on estmated CSI only /SNR =0.004 EST /SNR =0.0 EST /SNR EST = SNR (db) Fg.. Sum rates of dfferent algorthms when ρ =0.5. where ŝ s an arbtrary estmate of s ncludng LMMSE estmate. It should be ponted out that as CSI s not perfectly known, ŝ = s. Notce that s s the condtonal mean and thus we have E{(s s)( s ŝ) H } = 0 (44) based on whch the rght hand sde of (43) also equals to E{(s s)(s s) H } + E{( s ŝ)( s ŝ) H } = E{(s s + s ŝ)(s s + s ŝ) H } = E{(s ŝ)(s ŝ) H }. (45) Substtutng (45) nto (43), we have Cov(s y, y, y ) E{(s ŝ)(s ŝ) H } = Φ MSE (F, ). (46) It s also well-known that wth fxed covarance matrx, Gaussan dstrbuton has the maxmum entropy. Therefore, t s concluded that at relay s defned as s /σ, and the SNR at destnaton s defned as r /σ. For smplcty, t s also assumed that s /σ = r /σ. Each pont n the followng fgure s an average of 0000 ndependent channel realzatons. Fg. shows the sum rates of dfferent algorthms ncludng the proposed robust desgn and ts counterpart based on estmated CSI only when ρ = 0.5. It can be seen that the performance of the proposed robust desgn s always better than that of the desgn based estmated on CSI only. Furthermore, as the channel estmaton SNR decreases the performance gan of the robust desgn becomes larger. H(s y, y, y ) E{log πecov(s y, y, y ) } log πeφ MSE (F, ), based on whch, an lower bound of the capacty (4) s I(y; s y, y ) (47) V. CONCLUSIONS Robust mutual nformaton maxmzaton transcever desgn H(s) log πeφ MSE (F, ) = log Φ MSE (F, ). (48) The fnal equalty comes from the fact R s = I. for dual-hop AF MIMO relay systems was nvestgated. Wth AENDIX B Gaussan dstrbuted channel errors the precoder at the source ROOF OF CONCLUSION and forwardng matrx at the relay were jontly desgned. The structures of the optmal solutons were derved frst, whch As g( ) s a matrx monotoncally decreasng functon, t dfferentates our work from the exstng works. For the unknown can be proven that for the optmal soluton the power constrant dagonal matrces, a well-known teratve water-fllng soluton was proposed. The smulaton result demonstrated the performance advantage of our robust desgn. s always actve,.e., Tr(XX H )= [0]. As a result, we have the followng relatonshp. η x = αtr(xx H Ψ)+σn AENDIX A = αtr(xx H Ψ)+σn Tr(XX H )/ LOWER BOUND OF CAACITY = Denotng y and y as the receved plots n the separate =Tr(XX H (α Ψ + σni))/. (49) dual hop channel estmatons, the capacty between the source and destnaton equals to From (49), the constrant of the optmzaton problem (6) equals to I(s; y y, y )=H(s y, y ) H(s y, y, y ) (4) Tr(XX H )=Tr[XX H (α Ψ + σni)]/η x =, (50) =H(s) based on whch the optmzaton problem (6) s equvalent to where H(s y, y ) denotes the condtonal entropy of s when ( y and y are known and H(s y, y, y ) s the condtonal g X H H H H ) X X entropy of s when y, y and y are known. As y and y are η x ndependent wth s, H(s y, y )=H(s). The second term on s.t. Tr[XX H (α Ψ + σni)]/η x =. (5) the rght hand sde of (4) wll be dscussed n the followng. Then, defnng a new varable Denotng s = E y,y,y {s} as the condtonal mean and based on the defnton of covarance matrx, the condtonal X =/ η x (α Ψ + σni) / X, (5) covarance matrx satsfes the optmzaton problem (5) s further reformulated as Cov(s y, y, y )=E{(s s)(s s) H ( ) } g X H (α Ψ + σni) / H H H(α Ψ + σni) / X E{(s s)(s s) H } + E{( s ŝ)( s ŝ) H } X (43) s.t. Tr( X X H )=. (53)

7 For any gven X, based on the followng sngular decompostons H(α Ψ + σni) / X = V Ξ Λ Ξ U H Ξ wth Λ Ξ H(α Ψ + σni) / = U H Λ H VH H wth Λ H, (54) postve sem-defnte matrces M and N.e., M + N (λ (M)+λ (N)) [], we drectly have log A H (X H H H HX/η x + I) A + N log N + log[λ (AN A H )+λ (X H H H HX/η x + I)] there exsts a matrx X satsfyng X = V H Λ X U H Ξ (55) wth/bλ H Λ X = Λ Ξ (56) where Λ X s a dagonal matrx wth the same rank as Λ Ξ and b s a scalar whch makes Tr( X X H )= hold. Based on Lemma n [], the followng nequalty holds X H (α Ψ + σni) / H H H(α Ψ + σni) / X ર X H (α Ψ + σni) / H H H(α Ψ + σni) / X. (57) Together wth the fact that g( ) s a matrx monotoncally decreasng functon, the followng nequalty holds g( X H (α Ψ + σni) / H H H(α Ψ + σni) / X) g( X H (α Ψ + σni) / H H H(α Ψ + σni) / X). (58) Therefore, t s concluded that the optmal X has the structure gven by (55). Furthermore, based on the defnton of X (5), the optmal X has the followng structure X opt = η x (α Ψ + σ ni) / V H Λ X U H Ξ (59) where Λ X s a dagonal matrx. In (59) η x s unknown ether. In order to solve η x, substtute the structure of X n (59) nto the defnton of η x n (6), and then we get the followng equaton η x =Tr(XX H Ψ)α + σ n = η x αtr[v H X(α Ψ + σ ni) / Ψ(α Ψ + σ ni) / V X Λ X Λ T X]+σ n. (60) Ths s a smple lnear functon of η p, and η p can be easly solved to be η x = σ n/{ αtr[v H X(α Ψ + σ ni) / Ψ(α Ψ + σ ni) / V X Λ X Λ T X]}. (6) AENDIX C ROOF OF CONCLUSION The objectve functon n () can be reformulated as log A H (X H H H HX/η x + I) A + N =log N A H (X H H H HX/η x + I) AN + I =log N +log (X H H H HX/η x + I) AN A H + I =log N +log AN A H +(X H H H HX/η x + I) log (X H H H HX/η x + I) (6) where the second equalty s based on the fact that AB + I = BA + I. Usng the matrx nequalty that for two log[λ (X H H H HX/η x + I)]. (63) Together wth the optmal structure gven by Concluson, n order to make the equalty n (63) hold the followng equaton holds U Ξ = U ANA. (64) In lght of the fact that power s never loaded to the egenchannels wth zero magntudes [], the dagonal matrx Λ x has the followng structure Λ x = [ Λ x ], (65) where Λ x s a N X N X dagonal matrx and N X = {Rank(H H H), Rank(AA H )}. REFERENCES [] O. Munoz-Medna, J. Vdal, and A. Agustn, Lnear transcever desgn n nonregeneratve relays wth channel state nformaton, IEEE Trans. Sgnal rocess., vol. 55, no. 6, pp , June 007. [] X. Tang and Y. Hua, Optmal desgn of non-regeneratve MIMO wreless relays, IEEE Trans. Wreless Commun., vol. 6, no. 4, pp , Apr [3] W. Guan and H. Luo, Jont MMSE transcever desgn n nonregeneratve MIMO relay systems, IEEE Commun. Lett., vol., no. 7, pp , July 008. [4] R. Mo and Y. Chew, recoder desgn for non-regeneratve MIMO relay systems, IEEE Trans. Wreless Commun., vol. 8, no. 0, pp , Oct [5] Y. Rong, X. Tang, and Y. Hua, A unfed framework for optmzng lnear nonregeneratve multcarrer MIMO relay communcaton systems, IEEE Trans. Sgnal rocess., vol. 57, no., pp , Dec [6] X. Zhang, D.. alomar, and B. Ottersten, Statstcally robust desgn of lnear MIMO transcevers, IEEE Trans. Sgnal rocess., vol. 56, no. 8, pp , Aug [7] M. Dng and S. D. Blosten, Maxmum mutual nformaton desgn for MIMO systems wth mperfect channel knowledge, IEEE Trans. Infor. Theory, vol. 56, no. 0, pp , Oct. 00. [8] C. Xng, S. Ma, Y.-C. Wu, and T.-S. Ng, Transcever desgn for dual-hop nonregeneratve MIMO-OFDM relay systems under channel uncertantes, IEEE Trans. Sgnal rocess., vol. 58, no., pp , Dec. 00. [9] 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 rocess., vol. 58, no. 4, pp , Apr. 00. [0] E. Jorsweck and H. Boche, Majorzaton and Matrx-Monotone Functons n Wreless Communcatons. Now ublshers, 007. [] D.. alomar, J. M. Coff, and M. A. Lagunas, Jont Tx-Rx beamforg desgn for multcarrer MIMO channels: A unfed framework for convex optmzaton, IEEE Trans. Sgnal rocess., vol. 5, no. 9, pp , Sep [] A. W. Marshall and I. Olkn, Inequaltes: Theory of Majorzaton and Its Applcatons. New York: Academc ress, 979.