Joint source and relay design for MIMO multi-relay systems using projected gradient approach

Transcription

1 Todng et al EURASIP Journal on Wreless Communcatons and Networkng 04, 04:5 RESEARC Open Access Jont source and relay desgn for MIMO mult-relay systems usng projected gradent approach Aprana Todng, Muhammad RA Khandaker and Yue Rong 3* Abstract In ths paper, we develop the optmal source precodng matrx and relay amplfyng matrces for non-regeneratve multple-nput multple-output MIMO relay communcaton systems wth parallel relay nodes usng the projected gradent PG approach We show that the optmal relay amplfyng matrces have a beamformng structure Explotng the structure of relay matrces, an teratve jont source and relay matrces optmzaton algorthm s developed to mnmze the mean-squared error MSE of the sgnal waveform estmaton at the destnaton usng the PG approach The performance of the proposed algorthm s demonstrated through numercal smulatons Keywords: MIMO relay; Parallel relay network; Beamformng; Non-regeneratve relay; Projected gradent Introducton Recently, multple-nput multple-output MIMO relay communcaton systems have attracted much research nterest and provded sgnfcant mprovement n terms of both spectral effcency and lnk relablty [-7] Many works have studed the optmal relay amplfyng matrx for the source-relay-destnaton channel In [,3], the optmal relay amplfyng matrx maxmzng the mutual nformaton MI between the source and destnaton nodes was derved, assumng that the source covarance matrx s an dentty matrx In [4-6], the optmal relay amplfyng matrx was desgned to mnmze the mean-squared error MSE of the sgnal waveform estmaton at the destnaton A few research has studed the jont optmzaton of the source precodng matrx and the relay amplfyng matrx for the source-relay-destnaton channel In [7], both the source and relay matrces were jontly desgned to maxmze the source-destnaton MI In [8,9], source and relay matrces were developed to jontly optmze a broad class of objectve functons The author of [0] nvestgated the *Correspondence: yrong@curtneduau 3 Department of Electrcal and Computer Engneerng, Curtn Unversty of Technology, Bentley, WA 60, Australa Full lst of author nformaton s avalable at the end of the artcle jont source and relay optmzaton for two-way MIMO relay systems usng the projected gradent PG approach The source and relay optmzaton for mult-user MIMO relay systems wth sngle relay node has been nvestgated n [-4] All the works n [-4] consdered a sngle relay node at each hop In general, jont source and relay precodng matrces desgn for MIMO relay systems wth multple relay nodes s more challengng than that for sngle-relay systems The authors of [5] developed the optmal relay amplfyng matrces wth multple relay nodes A matrxform conjugate gradent algorthm has been proposed n [6] to optmze the source and relay matrces In [7], the authors proposed a suboptmal source and relay matrces desgn for parallel MIMO relay systems by frst relaxng the power constrant at each relay node to the sum relay power constrants at the output of the second-hop channel and then scalng the relay matrces to satsfy the ndvdual relay power constrants In ths paper, we propose a jontly optmal source precodng matrx and relay amplfyng matrces desgn for a two-hop non-regeneratve MIMO relay network wth multple relay nodes usng the PG approach We show that the optmal relay amplfyng matrces have a beamformng structure Ths new result s not avalable n [6] It generalzes the optmal source and relay matrces desgn 04 Todng et al; lcensee Sprnger Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly credted

2 Todng et al EURASIP Journal on Wreless Communcatons and Networkng 04, 04:5 Page of 9 from a sngle relay node case [8] to multple parallel relay nodes scenaros Explotng the structure of relay matrces, an teratve jont source and relay matrces optmzaton algorthm s developed to mnmze the MSE of the sgnal waveform estmaton Dfferent to [7], n ths paper, we develop the optmal source and relay matrces by drectly consderng the transmsson power constrant at each relay node Smulaton results demonstrate the effectveness of the proposed teratve jont source and relay matrces desgn algorthm wth multple parallel relay nodes usng the PG approach The rest of ths paper s organzed as follows In Secton, we ntroduce the model of a non-regeneratve MIMO relay communcaton system wth parallel relay nodes The jont source and relay matrces desgn algorthm s developed n Secton 3 In Secton 4, we show some numercal smulatons Conclusons are drawn n Secton 5 Systemmodel In ths secton, we ntroduce the model of a two-hop MIMO relay communcaton system consstng of one source node, K parallel relay nodes, and one destnaton node as shown n Fgure We assume that the source and destnaton nodes have N s and N d antennas, respectvely, and each relay node has N r antennas The generalzaton to systems wth dfferent number of antennas at each relay node s straghtforward Due to ts mert of smplcty, a lnear non-regeneratve strategy s appled at each relay node The communcaton process between the source and destnaton nodes s completed n two tme slots In the frst tme slot, the N b N b N s modulated source symbol vector s s lnearly precoded as x = Bs, where B s an N s N b source precodng matrx We assume that the source sgnal vector satsfes E [ ss ] = I Nb,whereI n stands for an n n dentty matrx, s the matrx vector ermtan transpose, and E[ ] denotes statstcal expectaton The precoded vector x s transmtted to K parallel relay nodes The N r recevedsgnal vector at the th relay node can be wrtten as y r, = sr, x + v r,, =,, K, where sr, s the N r N s MIMO channel matrx between thesourceandtheth relay nodes and v r, s the addtve Gaussannosevectoratthe th relay node Inthesecondtmeslot,thesourcenodesslent,whle each relay node transmts the lnearly amplfed sgnal vector to the destnaton node as x r, = F y r,, =,, K, 3 where F s the N r N r amplfyng matrx at the th relay node The receved sgnal vector at the destnaton node can be wrtten as y d = rd, x r, + v d, 4 = where rd, s the N d N r MIMO channel matrx between the th relay and the destnaton nodes, and v d s the addtve Gaussan nose vector at the destnaton node v r v d F x N r N r y s B x x Ns sr, N r F N r rd, y W y Nd ŝ F K N r N r Source Relay Destnaton Fgure Block dagram of a parallel MIMO relay communcaton system

3 Todng et al EURASIP Journal on Wreless Communcatons and Networkng 04, 04:5 Page 3 of 9 Substtutng to 3 nto 4, we have y d = rd, F sr, Bs + rd, F v r, + v d = = rd F sr Bs + rd Fv r + v d s +ṽ, 5 [ ] T where sr T sr,, T sr,,, T sr,k s a KNr N s channel matrx between the source node and all relay nodes, rd [ ] rd,, rd,,, rd,k s an Nd KN r channel matrx between all relay nodes and the destnaton node, F bd[f, F,, F K ]sthekn r KN r block dagonal equvalent relay matrx, v r [ v T r,, vt r,,, ] T vt r,k s obtaned by stackng the nose vectors at all the relays, rd F sr B s the effectve MIMO channel matrx of the source-relay-destnaton lnk, and ṽ rd Fv r + v d s the equvalent nose vector ere, T denotes the matrx vector transpose, bd[ ] constructsablock- dagonal matrx We assume that all noses are ndependent and dentcally dstrbuted d Gaussan nose wth zero mean and unt varance The transmsson power consumed by each relay node 3 can be expressed as E [ tr x r, x [ r,] = tr F sr, BB sr, + I ] N r F, =,, K, 6 where tr stands for the matrx trace Usng a lnear recever, the estmated sgnal waveform vector at the destnaton node s gven by ŝ = W y d, where W s an N d N b weght matrx The MSE of the sgnal waveform estmaton s gven by [ ŝ ŝ ] MSE = tr E s s = tr W I Nb W I Nb + W CW, 7 where C s the equvalent nose covarance matrx gven by C = E [ ṽṽ ] = rd FF rd + I N d The weght matrx W whch mnmzes 7 s the Wener flter and can be wrtten as W = + C, 8 where denotes the matrx nverson Substtutng 8 back nto 7, t can be seen that the MSE s a functon of F and B and can be wrtten as [ ] MSE = tr I + Nb C 9 3 Jont source and relay matrx optmzaton In ths secton, we address the jont source and relay matrx optmzaton problem for MIMO mult-relay systems wth a lnear mnmum mean-squared error MMSE recever at the destnaton node In partcular, we show that optmal relay matrces have a general beamformng structure Based on 6 and 9, the jont source and relay matrces optmzaton problem can be formulated as [ mn tr I + Nb C ] 0 {F },B st tr BB P s tr [ F sr, BB sr, + I ] N r F P r,, =,, K, where {F } {F, =,, K}, s the transmt power constrant at the source node, whle s the power constrant at each relay node ere, P s > 0and P r, > 0, =,, K, are the correspondng power budget Obvously, to avod any loss of transmsson power n the relay system when a lnear recever s used, there should be N b mnkn r, N d The problem 0- s non-convex, and a globally optmal soluton of B and {F } s dffcult to obtan wth a reasonable computatonal complexty In ths paper, we develop an teratve algorthm to optmze B and {F } Frst, we show the optmal structure of {F } 3 Optmal structure of relay amplfyng matrces For gven source matrx B satsfyng, the relay matrces {F } are optmzed by solvng the followng problem: mn {F } [ ] tr I + Nb C 3 st tr F [ sr, BB sr, + I N r ] F Pr,, =,, K 4 Let us ntroduce the followng sngular value decompostons SVDs: sr, B = U s, s, V s,, rd, = U r, r, V r,, =,, K, 5 where s, and r, are R s, R s, and R r, R r, dagonal matrces, respectvely ere, R s, rank sr, B, R r, rank rd,, =,, K, and rank denotes the rank of a matrx Based on the defnton of matrx rank, R s, mnn r, N b and R r, mnn r, N d The followng theorem states the structure of the optmal {F } Theorem Usng the SVDs of 5, the optmal structure of F as the soluton to the problem 3-4 s gven by F = V r, A U s,, =,, K, 6 where A s an R r, R s, matrx, =,, K Proof See Appendx

4 Todng et al EURASIP Journal on Wreless Communcatons and Networkng 04, 04:5 Page 4 of 9 The remanng task s to fnd the optmal A, =,, K From 3 and 3 n Appendx, we can equvalently rewrte the optmzaton problem 3-4 as [ mn tr I Nb + V s, s, A r, U r, {A } = K U r, r, A A r, U r, + I N d = ] U r, r, A s, V s, 7 = st tr A s, + I R s, A Pr,, =,, K 8 Both the problem 3-4 and the problem 7-8 have matrx optmzaton varables owever, n the former problem, the optmzaton varable F s an N r N r matrx, whle the dmenson of A s R r, R s,, whch may be smaller than that of F Thus, solvng the problem 7-8 has a smaller computatonal complexty than solvng the problem 3-4 In general, the problem 7-8 s non-convex, and a globally optmal soluton s dffcult to obtan wth a reasonable computatonal complexty Fortunately, we can resort to numercal methods, such as the projected gradent algorthm [8] to fnd at least a locally optmal soluton of 7-8 Theorem Let us defne the objectve functon n 7 as f A Its gradent f A wth respect to A can be calculated by usng results on dervatves of matrces n [9] as E S + D R G E S, =,, K, 9 f A = R M where M, R, S, D, E,andG are defned n Appendx Proof See Appendx In each teraton of the PG algorthm, we frst obtan à = A s n f A by movng A one step towards the negatve gradent drecton of f A,wheres n > 0sthe step sze Snce à mght not satsfy the constrant 8, we need to project t onto the set gven by 8 The projected matrx Ā s obtaned by mnmzng the Frobenus norm of Ā à accordng to [8] subjectng to 8, whch can be formulated as the followng optmzaton problem: mn Ā tr Ā à Ā à 0 st tr Ā s, + I Rs, Ā Pr, Obvously, f trã s, + I Rs, à P r,,thenā = à Otherwse, the soluton to the problem 0- can be obtaned by usng the Lagrange multpler method, and the soluton s gven by [ Ā = à λ + IRs, + λ s,], where λ>0sthe soluton to the non-lnear equaton of [ tr à λ + IRs, + s,] λ s, + I Rs, [ λ + I Rs, + λ s,] à = P r, Equaton can be effcently solved by the bsecton method [8] The procedure of the PG algorthm s lsted n Algorthm, where n denotes the varable at the nth teraton, δ n and s n are the step sze parameters at the nth teraton, denotes the maxmum among the absolute value of all elements n the matrx, and ε s a postve constant close to 0 The step sze parameters δ n and s n are determned by the Armjo rule [8], e, s n = s s a constant through all teratons, whle at the nth teraton, δ n s set to be γ m n ere, m n s the mnmal non-negatve nteger that satsfes the followng nequalty f A n+ f A n αγ ntr m Ān f,whereα and γ are A n A n constants Accordng to [8], usually α s chosen close to 0, for example, α [0 5,0 ], whle a proper choce of γ s normally from 0 to 05 3 Optmal source precodng matrx Wth fxed {F }, the source precodng matrx B s optmzed by solvng the followng problem: Algorthm Procedure of applyng the projected gradent algorthm to solve the problem 7-8 Intalze the algorthm at a feasble A 0 for =,, K;setn = 0 For =,, K, Compute the gradent of 7 f A n ; Project à n = A n s n f A n to obtan Ā n ; Update A wth A n+ = A n Ān + δ n A n 3 If max A n+ A n ε, then end Otherwse, let n := n + andgotostep mn B [INb tr + B B ] 3 st tr BB P s, 4 tr F sr, BB sr, F P r,, =,, K, 5

5 Todng et al EURASIP Journal on Wreless Communcatons and Networkng 04, 04:5 Page 5 of 9 where sr F rd rd FF rd + I N d rd F sr,and P r, P r, tr F F, =,, K Letusntroduce BB, and a postve sem-defnte PSD matrx X, wth X I Ns + wherea B means that A B s a PSD matrx By usng the Schur complement [0], the problem 3-5 can be equvalently converted to the followng problem: mn tr X N s + N b 6 X, X INs st I Ns I Ns + 0, 7 tr P s, 0, 8 tr F sr, sr, F P r,, =,, K 9 The problem 6-9 s a convex sem-defnte programmng SDP problem whch can be effcently solved by the nteror pont method [0] Let us ntroduce the egenvalue decomposton EVD of = U U, where s a R R egenvalue matrx wth R = rank IfR = N b,thenfrom = BB,wehave B = U IfR > N b, the randomzaton technque [] can be appled to obtan a possbly suboptmal soluton of B wth rank N b IfR < N b, t ndcates that the system channel cannot support N b ndependent data streams, and thus, n ths case, a smaller N b should be chosen n the system desgn Now, the orgnal jont source and relay optmzaton problem 0- can be solved by an teratve algorthm as shown n Algorthm, where m denotes the varable at the mth teraton Ths algorthm s frst ntalzed at a random feasble B satsfyng At each teraton, we frst update {F } wth fxed B and then update B wth fxed {F } Note that the condtonal updates of each matrx may ether decrease or mantan but cannot ncrease the objectve functon 0 Monotonc convergence of {F } and B towards at least a locally optmal soluton follows drectly from ths observaton Note that n each teraton of ths algorthm, we need to update the relay amplfyng matrces accordng to the procedure lsted n Algorthm at a complexty order of O K Nd 3 + N3 r + b N3 and update the source precodng matrx through solvng the SDP problem 6-9 at a N complexty cost that s at most O s + K + 35 usng nteror pont methods [] Therefore, the per-teraton computatonal complexty order of the proposed algorthm s O K Nd 3 + N3 r + b N3 + N s + K + 35 The overall complexty of ths algorthm depends on the number of teratons untl convergence, whch wll be studed n the next secton Algorthm Procedure of solvng the problem 0- Intalze the algorthm at a feasble B 0 satsfyng constrant ; set m = 0 For fxed B m, obtan {F } m by solvng the problem 7-8 usng the PG algorthm 3 Update B m+ by solvng the problem 6-9 wth known {F } m 4 If B m+ B m ε, then end Otherwse, let m := m + andgotostep 4 Smulatons In ths secton, we study the performance of the proposed jontly optmal source and relay matrx desgn for MIMO mult-relay systems wth lnear MMSE recever All smulatons are conducted n a flat Raylegh fadng envronment where the channel matrces have zero-mean entres wth varances σ s /N s and σ r /KN r for sr and rd, respectvely For the sake of smplcty, we assume P r, = P r, =,, K The BPSK constellatons are used to modulate the source symbols, and all noses are d Gaussan wth zero mean and unt varance We defne SNR s = σ s P skn r /N s and SNR r = σ r P rn d /KN r as the sgnal-to-nose rato SNR for the source-relay lnk and the relay-destnaton lnk, respectvely We transmt 000N s randomly generated bts n each channel realzaton, and all smulaton results are averaged over 00 channel realzatons In all smulatons, we set N b = N s = N r = N d = 3, and the MMSE lnear recever n 8 s employed at the destnaton for symbol detecton In the frst example, a MIMO relay system wth K = 3 relay nodes s smulated We compare the normalzed MSE performance of the proposed jont source and relay optmzaton algorthm usng the projected gradent JSR- PG algorthm n Algorthm, the optmal relay-only algorthm usng the projected gradent ORO-PG algorthm n Algorthm wth B = P s /N s I Ns,andthenave amplfy-and-forward NAF algorthm Fgure shows the normalzed MSE of all algorthms versus SNR s for SNR r = 0 db Whle Fgure 3 demonstrates the normalzed MSE of all algorthms versus SNR r for SNR s fxed at 0 db It can be seen from Fgures and 3 that the JSR- PG and ORO-PG algorthms have a better performance than the NAF algorthm over the whole SNR s and SNR r range Moreover, the proposed JSR-PG algorthm yelds the lowest MSE among all three algorthms The number of teratons requred for the JSR-PG algorthm to converge to ε = 0 3 n a typcal channel realzaton are lsted n Table, where we set K = 3 and SNR r = 0 db It can be seen that the JSR-PG algorthm converges wthn several teratons, and thus, t s realzable wth the advancement of modern chp desgn

6 Todng et al EURASIP Journal on Wreless Communcatons and Networkng 04, 04:5 Page 6 of NAF Algorthm ORO PG Algorthm JSR PG Algorthm Normalzed MSE SNRs db Fgure Example Normalzed MSE versus SNR s wth K = 3, SNR r = 0 db In the second example, we compare the bt error rate BER performance of the proposed JSR-PG algorthm n Algorthm, the ORO-PG algorthm n Algorthm, the suboptmal source and relay matrx desgn n [7], the one-way relay verson of the conjugate gradent-based source and relay algorthm n [6], and the NAF algorthm Fgure4dsplaysthesystemBERversusSNR s for a MIMO relay system wth K = 3 relay nodes and fxed SNR r at 0 db It can be seen from Fgure 4 that the proposed JSR-PG algorthm has a better BER performance than the exstng algorthms over the whole SNR s range In the thrd example, we study the effect of the number of relay nodes to the system BER performance usng the JSR-PG and ORO-PG algorthms Fgure 5 dsplays the system BER versus SNR s wth K =, 3, and 5 for fxed SNR r at 0 db It can be seen that at BER = 0,for both the ORO-PG algorthm and JSR-PG algorthm, we can acheve approxmately 3-dB gan by ncreasng from K = tok = 5 It can also be seen that the performance gan of the JSR-PG algorthm over the ORO-PG algorthm ncreases wth the ncreasng number of relay nodes 0 0 NAF Algorthm ORO PG Algorthm JSR PG Algorthm Normalzed MSE SNRr db Fgure 3 Example Normalzed MSE versus SNR r wth K = 3, SNR s = 0 db

7 Todng et al EURASIP Journal on Wreless Communcatons and Networkng 04, 04:5 Page 7 of 9 Table Iteratons requred untl convergence n the JSR-PG algorthm SNR s db Iteratons Conclusons In ths paper, we have derved the general structure of the optmal relay amplfyng matrces for lnear non-regeneratve MIMO relay communcaton systems wth multple relay nodes usng the projected gradent approach The proposed source and relay matrces mnmze the MSE of the sgnal waveform estmaton The smulaton results demonstrate that the proposed algorthm has mproved the MSE and BER performance compared wth exstng technques Appendces Appendx Proof of Theorem Wthout loss of generalty, F can be wrtten as F = [ V r, V r, ] [ ] [ ] A X U s, Y Z U, =,, K, s, 30 where V r, V r, = I Nr V r, V r,, s, U U s, = INr U s, U s,,suchthat V r, [ V r,, V r,] and Ūs, [ U s,, U ] s, are N r N r untary matrces Matrces A, X, Y, Z are arbtrary matrces wth dmensons of R r, R s,, R r, N r R s,, N r R r, R s,, N r R r, N r R s,,respectvely Substtutng 5 and 30 back nto 3, we obtan that rd, F sr, B = U r, r, A s, V s, and rd,f F rd, = U r, r, A A + X X r, U r, Thus, we can rewrte 3 as [ MSE tr I Nb + V s, s, A r, U r, = K U r, r, A A = + X X r, U r, + I N d ] U r, r, A s, V s, 3 = It can be seen that 3 s mnmzed by X = 0 Rr, N r R s,, =,, K Substtutng 5 and 30 back nto the left-hand sde of the transmsson power constrant 4, we have tr [ F sr, BB sr, + I ] N r F = tr A s, + I Rs, A + Y s, + I Rs, Y 3 +X X + Z Z, =,, K From 3, we fnd that X = 0 Rr, N r R s,, Y = 0 Nr R r, R s,,andz = 0 Nr R r, N r R s, mnmze the power consumpton at each relay node Thus, we have F = V r, A U s,, =,, K 0 0 BER NAF Algorthm ORO PG Algorthm Algorthm [7] Algorthm [6] JSR PG Algorthm Fgure 4 Example BER versus SNR s wth K = 3, SNR r = 0 db SNRs db

8 Todng et al EURASIP Journal on Wreless Communcatons and Networkng 04, 04:5 Page 8 of BER ORO PG Algorthm K= ORO PG Algorthm K=3 ORO PG Algorthm K=5 JSR PG Algorthm K= JSR PG Algorthm K=3 JSR PG Algorthm K=5 Fgure 5 Example 3 BER versus SNR s for dfferent K,SNR r = 0 db SNRs db Appendx Proof of Theorem Let us defne Z K j=,j = U r,j r,j A j s,j V s,j and Y Kj=,j = U r,j r,j A j A j r,j U r,j + I Nd Then,f A can be wrtten as [INb f A = tr + Z + V s, s, A r, U r, Y + U r, r, A A r, U r, Z + U r, r, A s, V ] s, 33 Applyng I Nb +A C A =INb A AA +C A, 33 can be wrtten as f A = tr I Nb Z +V s, s, A r, U r, [ Z + U r, r, A s, V s, Z +V s, s, A r, U r, 34 + Y + U r, r, A A r, U r, Z + U r, r, A s, V s, ] Let us now defne E Z + U r, r, A s, V s,, K Y + U r, r, A A r, U r,,andg E E + K Wecan rewrte 34 as f A = tr I Nb E G E = tr I Nb E E G 35 The dervatve of f A wth respect to A s gven by f A = tr E E G A A Z + U r, r, A s, V s, E = tr G E E G A + Y + U r, r, A A r, U r, tr E G U r, r, A s, V s, A 36 Defnng M G E E G, R U r, r,, S s, V s,,andd A r, U r,, we can rewrte 36 as f A A = tr M Z + U r, r, A s, V s, E A + M Y + U r, r, A D T E G s, U r, r, V T s, 37 = tr M R A S E + M R A D A T E G R S T 38 Fnally, the gradent of f A s gven by f A f A = A = M R T S E = R M E S + D T TS +M R T D T E G T R R G E S where stands for complex conjugate 39

9 Todng et al EURASIP Journal on Wreless Communcatons and Networkng 04, 04:5 Page 9 of 9 Competng nterests The authors declare that they have no competng nterests Acknowledgements The work of Yue Rong was supported n part by the Australan Research Councl s Dscovery Projects fundng scheme project number DP4003 The frst author Aprana Todng would lke to thank the gher Educaton Mnstry of Indonesa DIKTI and the Paulus Chrstan Unversty of Indonesa UKI-Paulus of Makassar, Indonesa, for provdng her wth a PhD scholarshp at Curtn Unversty, Perth, Australa Author detals Department of Electrcal Engneerng, Unverstas Krsten Indonesa Paulus, Jln Pernts Kemerdekaan No 8 Daya, Makassar 9043, Indonesa Department of Electronc and Electrcal Engneerng, Unversty College London, Gower Street, London WCE 7JE, UK 3 Department of Electrcal and Computer Engneerng, Curtn Unversty of Technology, Bentley, WA 60, Australa 0 S Boyd, L Vandenberghe, Convex Optmzaton Cambrdge Unversty Press, Cambrdge, 004 P Tseng, Further results on approxmatng nonconvex quadratc optmzaton by semdefnte programmng relaxaton SIAM J Optm 4, Y Nesterov, A Nemrovsk, Interor Pont Polynomal Algorthms n Convex Programmng, SIAM, Phladelpha, 994 do:086/ Cte ths artcle as: Todng et al: Jont source and relay desgn for MIMO mult-relay systems usng projected gradent approach EURASIP Journal on Wreless Communcatons and Networkng 04 04:5 Receved: 5 May 04 Accepted: September 04 Publshed: 5 September 04 References B Wang, J Zhang, A øst-madsen, On the capacty of MIMO relay channels IEEE Trans Inf Theory 5, X Tang, Y ua, Optmal desgn of non-regeneratve MIMO wreless relays IEEE Trans Wreless Commun 6, O Muñoz-Medna, J Vdal, A Agustín, Lnear transcever desgn n nonregeneratve relays wth channel state nformaton IEEE Trans Sgnal Process 55, W Guan, Luo, Jont MMSE transcever desgn n non-regeneratve MIMO relay systems IEEE Commun Lett, G L, Y Wang, T Wu, J uang, Jont lnear flter desgn n mult-user cooperatve non-regeneratve MIMO relay systems EURASIP J Wreless Commun Netw 009, Y Rong, Lnear non-regeneratve multcarrer MIMO relay communcatons based on MMSE crteron IEEE Trans Commun 58, Z Fang, Y ua, JC Koshy, Jont source and relay optmzaton for a nonregeneratve MIMO relay, n Proc IEEE Workshop Sensor Array Mult-Channel Sgnal Process Waltham, WA, USA, 4 July 006, pp Y Rong, X Tang, Y ua, A unfed framework for optmzng lnear non-regeneratve multcarrer MIMO relay communcaton systems IEEE Trans Sgnal Process 57, Y Rong, Y ua, Optmalty of dagonalzaton of mult-hop MIMO relays IEEE Trans Wreless Commun 8, Y Rong, Jont source and relay optmzaton for two-way lnear non-regeneratve MIMO relay communcatons IEEE Trans Sgnal Process 60, MRA Khandaker, Y Rong, Jont transcever optmzaton for multuser MIMO relay communcaton systems IEEE Trans Sgnal Process 60, MRA Khandaker, Y Rong, Jont source and relay optmzaton for multuser MIMO relay communcaton systems, n Proceedngs of the 4th Internatonal Conference on Sgnal Processng and Communcaton Systems Gold Coast, Australa, 3 5 Dec 00 3 Wan, W Chen, Jont source and relay desgn for multuser MIMO nonregeneratve relay networks wth drect lnks IEEE Trans Veh Technol 6, J Zeng, Z Chen, L L, Iteratve jont source and relay optmzaton for multuser MIMO relay systems, n Proceedngs of the IEEE Vehcular Technology Conference Quebec Cty, QC, Canada, 3 6 Sept 0 5 AS Behbahan, R Merched, AM Eltawl, Optmzatons of a MIMO relay network IEEE Trans Sgnal Process 56, C-C u, Y-F Chou, Precodng desgn of MIMO AF two-way multple-relay systems IEEE Sgnal Process Lett 0, A Todng, MRA Khandaker, Y Rong, Jont source and relay optmzaton for parallel MIMO relay networks EURASIP J Adv Sgnal Process 0, DP Bertsekas, Nonlnear Programmng, nd edn, Athena Scentfc, Belmont,999 9 KB Petersen, MS Petersen, The Matrx Cookbook dk/pubdb/pphp?374 Accessed 9 Sept 04 Submt your manuscrpt to a journal and beneft from: 7 Convenent onlne submsson 7 Rgorous peer revew 7 Immedate publcaton on acceptance 7 Open access: artcles freely avalable onlne 7 gh vsblty wthn the feld 7 Retanng the copyrght to your artcle Submt your next manuscrpt at 7 sprngeropencom