Joint Source and Relay Optimization for Distributed MIMO Relay System

Transcription

1 th Aia-Pacific Confeence on Communication (APCC) 2nd 5th Octoe 2011 Sutea Haou Reot, Kota Kinaalu, Saah, Malayia Joint Souce and Relay Optimization fo Ditiuted MIMO Relay Sytem Apiana Toding Dept Electical & Compute Eng Cutin Univeity Bentley, WA 6102, Autalia Muhammad R A Khandake Dept Electical & Compute Eng Cutin Univeity Bentley, WA 6102, Autalia mkhandake@potgadcutineduau Yue Rong Dept Electical & Compute Eng Cutin Univeity Bentley, WA 6102, Autalia yong@cutineduau Atact In thi pape, we develop the optimal tanmit eamfoming vecto and the elay amplifying facto fo a multipleinput multiple-output (MIMO) elay communication ytem with ditiuted elay node Uing the optimal eamfoming vecto, an iteative joint ouce and elay eamfoming algoithm i developed to minimize the mean-quaed eo (MSE) of the ignal wavefom etimation Numeical imulation ae caied out to demontate the pefomance of the popoed joint ouce and elay eamfoming algoithm Index Tem Multiple-input multiple-output (MIMO), elay netwok, paallel elay, eamfoming x 1 x 2 H v f 1 f 2 Hd vd y 1 y 2 w ŝ x N f K y Nd I INTRODUCTION In ode to etalih a eliale wiele communication link, one need to compenate fo the effect of ignal fading and hadowing An efficient way to adde thi iue i to tanmit ignal though one o multiple elay [1]-[4] Intoducing multiple antenna at tanmitting and eceiving end, which we call multiple-input multiple-output (MIMO) elay communication ytem, can povide futhe impovement in tem of oth pectal efficiency and link eliaility [3]-[4] Many wok have tudied the optimal elay amplifying matix fo MIMO elay channel In [5] and [6], the optimal elay amplifying matix i deigned to maximize the mutual infomation (MI) etween the ouce node and the detination node, auming that the ouce covaiance matix i an identity matix In [7] and [8], the optimal elay amplifying matix wa deigned to minimize the mean-quaed eo (MSE) of the ignal wavefom etimation at the detination Howeve, few eeach have tudied the jointly optimal ouce pecoding matix and the elay amplifying matix fo the ouce-elay-detination channel In [9], the ouce covaiance matix and the elay amplifying matix wee jointly deigned to maximize the ouce-detination MI In [10], a unified famewok wa developed to jointly optimize the ouce pecoding matix and the elay amplifying matix fo a oad cla of ojective function All the wok [3]-[10] focu on MIMO elay ytem with a ingle elay node at each hop In [11] and [12], the optimal ouce and elay matice ae deigned fo a multihop MIMO elay netwok with eial elay MIMO elay ytem with multiple paallel elay node have een invetigated in [13]-[15] In [14], the autho invetigated ig 1 Souce Relay Detination Block diagam of a ditiuted MIMO elay communication ytem the jointly optimal tuctue of the ouce pecoding matix and the elay amplifying matice conideing a linea minimal MSE (MMSE) eceive at the detination In [15], a non-linea eceive i ued to deign the matice On the othe hand, a ditiuted elay netwok i invetigated in [16] whee multiple ue and elay, each having a ingle antenna, ae conideed In thi pape, we popoe a jointly optimal ouce and elay eamfoming algoithm which minimize the MSE of the ignal wavefom etimation fo ingle-antenna elay node in a MIMO elay communication ytem In contat to [13]-[15], whee the eceive powe containt at the detination node i conideed, we conide in thi pape the um tanmit powe containt thoughout all elay node The et of thi pape i oganized a follow The ytem model i decied in Section II In Section III, we tudy the jointly optimal ouce and elay algoithm Section IV how the imulation eult Concluion ae dawn in Section V II SYSTEM MODEL ig 1 illutate a two-hop MIMO elay communication ytem coniting of one ouce node, K paallel elay node, and one detination node We aume that the ouce and the detination node have N and N d antenna, epectively, wheea each elay node ha a ingle antenna Due to it meit of implicity, we conide the amplify-and-fowad cheme at each elay The communication poce etween the ouce /11/$ IEEE 604

2 and detination node i completed in two time lot In the fit time lot, the modulated ymol i linealy pecoded a x = (1) whee i an N 1 tanmit eamfoming vecto The pecoded vecto x i then tanmitted to the elay node The eceived ignal at the ith elay node can e witten a y,i = h,i x+v,i, i = 1,,K (2) whee h,i i the 1 N channel vecto etween the ouce and the ith elay node, y,i and v,i ae the eceived ignal and the additive Gauian noie at the ith elay node, epectively In the econd time lot, the ouce node i ilent, while each elay node tanmit the amplified ignal to the detination node a x,i = f i y,i, i = 1,,K (3) wheef i i the amplifying coefficient at theith elay node The eceived ignal vecto at the detination node can e witten a y d = h d,i x,i +v d (4) whee h d,i i the N d 1 channel vecto etween the ith elay and the detination node, y d and v d ae the eceived ignal and the additive Gauian noie vecto at the detination node, epectively Sutituting (1)-(3) into (4), we have y d = (h d,i f i h,i +h d,i f i v,i )+v d whee we define = H d H +H d v +v d (5) H [ h T,1,h T,2,,h T,K H d [h d,1,h d,2,,h d,k ] diag ( [f 1,f 2,,f K ] T ) v [ v,1,v,2,,v,k ] T Hee( ) T denote the matix (vecto) tanpoe, diag(a) tand fo a diagonal matix with the vecto a a the main diagonal and zeo elewhee, H i a K N channel matix etween the ouce node and all elay node,h d i an N d K channel matix etween all elay node and the detination node, and v i otained y tacking the noie tem at all the elay We aume that all noie ae independent and identically ditiuted (iid) with zeo mean and unit vaiance The diagam of the equivalent MIMO elay ytem decied y (5) i hown in ig 2 The eceived ignal vecto at the detination node can e equivalently witten a y d = h+ v whee we define h H d H a the effective channel vecto of the ouce-elay-detination link, and v H d v + v d a the equivalent noie vecto ] T v d N 1 K N K K Nd K N d 1 H + H d + w y x y d Souce Relay Detination ig 2 Block diagam of the equivalent MIMO elay ytem III OPTIMAL SOURCE AND RELAY DESIGN In thi ection we develop the optimal tanmit eamfoming vecto and the elay amplifying matix to minimize the MSE of the ignal wavefom etimation Uing a linea eceive, the etimated ignal wavefom vecto at the detination node i given y ŝ = w H y d (6) whee w i an N d 1 weight vecto, and ( ) H denote the matix (vecto) Hemitian tanpoe The MMSE appoach tie to find a weight vecto w that minimize the tatitical expectation of the ignal wavefom etimation given y MSE = E [ ŝ 2] (7) whee E[ ] denote tatitical expectation We aume that the ouce ignal atifie E[ 2 ] = 1 Sutituting (6) into (7), we find that the w which minimize (7) can e witten a w = ( h h H + C) 1 h (8) whee ( ) 1 denote the matix inveion, and equivalent noie covaiance matix given y v ŝ C i the C = H d H H H d +I Nd (9) Hee I n i an n n identity matix Sutituting (8) ack into (7), we otain the minimal MSE a a function of and, given y MSE = 1 h H ( h h H + C) 1 h (10) Applying the matix inveion lemma (A + BCD) 1 = A 1 A 1 B(DA 1 B+C 1 ) 1 DA 1, (10) can e witten a MSE = ( 1+ h H C 1 h) 1 (11) om (3), the total tanmiion powe conumed y all elay node can e expeed a t ( E[x x H ]) = t ( [ H H H H +I K] H ) (12) whee t( ) tand fo matix tace Uing (12), the joint ouce and elay optimization polem can e fomulated a min, MSE (13) t H P (14) t ( [ H H H H +I K] H ) P (15) whee (14) i the tanmit powe containt at the ouce node, and (15) i the um tanmit powe containt thoughout all elay node Hee P > 0 and P > 0 ae the coeponding 605

3 powe udget The polem (13)-(15) i highly nonconvex and a cloed-fom expeion of the optimal andi intactale In thi pape, we develop an iteative algoithm to optimize and A Optimal Relay acto o given eamfoming vecto atifying (14), we optimize the elay matix y olving the following optimization polem Let u intoduce min MSE (16) t t ( [ H H H H +I K] H ) P (17) h H (18) Sutituting (18) ack into (16)-(17), we can ewite the optimization polem a [ min 1+ hh H H H d (H d H H H d +I N d ) 1 ] 1(19) H d h t f i 2( h,i 2 +1 ) P (20) whee h,i tand fo the ith element of h Polem (19)-(20) i equivalent to max t h H H H H d (H d H H H d +I N d ) 1 H d h (21) f i 2( h,i 2 +1 ) P (22) Since the ojective function (21) i till a complicated function of, in the following, we optimize an uppe-ound of (21) The polem can e ewitten a max t h H H H H dh d h (23) f i 2( h,i 2 +1 ) P (24) Let f [f 1,f 2,,f K ] T denote the diagonal element of and define D diag( h ), o that h = D f (25) Now y utituting (25) in (23)-(24), we can expe the maximization polem a follow max f H D H H H dh d D f f (26) t f H Af P (27) whee A D D H +I K Defining f A 1 2f, polem (26)- (27) can e equivalently witten a max fh A H 2 D H H H d H dd A 1 2 f f t fh f P Intoducing Z A H 2 D H H H d H dd A 1 2, we otain max Z f f (28) t fh f P (29) The Lagangian of the optimization polem (28)-(29) can e witten a L = f H Z f +µ 1 ( f H f P ) (30) whee µ 1 0 i the Lagangian multiplie aociated with the containt (29) Taking the deivative of L with epect to f H and letting the eult e 0, it can e hown that the optimal f atifie the following equation Z f = µ 1 f Thu f = P eig(z), whee eig(z) tand fo the pincipal eigenvecto of Z B Joint Souce and Relay Optimization o a fixed, polem (13) - (15) can e expeed a elow to optimize the eamfoming vecto ( min 1+ H Ψ 1 ) 1 (31) t H P (32) whee we define H Ψ 2 P (33) Ψ 1 H H H H H d H dh Ψ 2 H H H H P P t( H ) The polem (31)-(33) i equivalent to max H Ψ 1 (34) t H P (35) H Ψ 2 P (36) The Lagangian function aociated with the polem (34)- (36) can e witten a L = H Ψ 1 +µ 2 ( H P )+µ 3 ( H Ψ 2 P ) (37) Hee µ 2 0 and µ 3 0 ae the Lagangian multiplie aociated with the containt (35) and (36), epectively The polem (34)-(36)can e olved y uing KKT condition [17] that can e expeed a L H = 0 (38) µ 2 ( H P ) = 0 (39) µ 3 ( H Ψ 2 P ) = 0 (40) H P (41) H Ψ 2 P (42) µ 2 0, µ 3 0 (43) 606

4 Now we et out to olve (38)-(43) If µ 2 > 0 and µ 3 = 0, then the optimization polem (34)-(36) can e ewitten a max H Ψ 1 (44) t H = P (45) Optimal Relay & Souce K=2 Optimal Relay & Souce K=3 Optimal Relay & Souce K=5 Thu we olve a = P eig(ψ 1 ) (46) Ifin (46) atifie the containt (36), then (46) i the optimal olution to the polem (34)-(36) Othewie, if µ 3 > 0 and µ 2 = 0, then the optimization polem (34)-(36) can e ewitten a Then we olve a max H Ψ 1 (47) t H Ψ 2 = P (48) = αeig(ψ 1 2 Ψ 1) (49) whee α = P /(eig H (Ψ 1 2 Ψ 1)Ψ 2 eig(ψ 1 2 Ψ 1)) If in (49) atifie the containt (35), then (49) i the optimal olution to the polem (34)-(36) inally, if µ 2 > 0 and µ 3 > 0, then we have fom (38) that Ψ 1 = µ 2 +µ 3 Ψ 2 which can e equivalently witten a ( I N + µ ) 1 3 Ψ 2 Ψ 1 = µ 2 (50) µ 2 Thu the optimal can e otained a = P eig((i N +λψ 2 ) 1 Ψ 1 ) (51) whee λ > 0 can e found y utituting (51) ack into (48) and olving the otained nonlinea equation Now the oiginal polem (13)-(15) can e olved in an iteative fahion In each iteation, we fit fix and update f y olving the polem (26)-(27) Then we update with fixed though olving the polem (34)-(36) The pocedue of the popoed iteative algoithm i lited in Tale I, whee ε i a mall poitive nume cloe to zeo Since each update may only educe o maintain, ut can not inceae the MSE, a monotonic convegence of thi iteative algoithm follow diectly fom thi oevation TABLE I PROCEDURE O SOLVING THE PROBLEM (13)-(15) BY THE PROPOSED ITERATIVE ALGORITHM 1) Initialize the algoithm with andom (0) ; Set n = 0 2) Solve the upolem (26)-(27) uing given (n) to otain f (n) 3) Solve the upolem (34)-(36) uing (n) to otain (n) 4) if (n+1) (n) ε, then end Othewie, let n := n+1 and go to tep 2 BER SNR (db) ig 3 Example 1 BER veu SNR with vaying K N = N d = 3, SNR = 20dB IV SIMULATIONS In thi ection, we tudy the pefomance of the popoed optimal joint ouce and elay eamfoming algoithm fo ditiuted MIMO elay ytem All imulation ae conducted in a flat Rayleigh fading envionment uing BPSK contellation, and the noie ae iid Gauian andom vaiale with zeo mean and unit vaiance The channel matice have zeo-mean entie with vaiance σ 2 /N and σ 2 /K fo H and H d, epectively We vay the ignal-to-noie atio (SNR) in the ouce-to-elay link SNR while fixing the SNR in the elayto-detination link SNR to 20dB We tanmit 1000 andomly geneated it in each channel ealization, and the it-eoate (BER) eult ae aveaged though 200 andom channel ealization In the fit example, we tudy the effect of the nume of elay to the ytem BER pefomance uing the popoed algoithm We chooe N = N d = 3 ig 3 how the BER pefomance with K = 2, 3, and 5 It can e een that at BER = 10 3, we achieve a 25-dB gain y inceaing fom K = 2 to K = 5 In the econd example, we imulate a ditiuted MIMO elay ytem with N = N d = 5 ig 4 how the BER pefomance with K = 2, 3, and 5 It can e een that at BER = 10 3, we achieve a 2-dB gain y inceaing fom K = 2 to K = 5 V CONCLUSIONS In thi pape, we have developed the optimal ouce and elay eamfoming vecto fo MIMO elay communication ytem with ditiuted elay node The popoed algoithm minimize the MSE of the ignal wavefom etimation Simulation eult demontate the pefomance of the algoithm ACKNOWLEDGMENT Thi wok wa uppoted unde Autalian Reeach Council Dicovey Poject funding cheme (poject nume DP , DP ) 607

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