Distributed Transmit Diversity in Relay Networks

Transcription

1 Dstrbuted Transmt Dversty n Relay etworks Cemal Akçaba, Patrck Kuppnger and Helmut Bölcske Communcaton Technology Laboratory ETH Zurch, Swtzerland Emal: {cakcaba patrcku boelcske}@nareeethzch Abstract We analyze fadng relay networks, a sngleantenna source-destnaton termnal par communcates through a set of half-duplex sngle-antenna relays usng a two-hop protocol wth lnear processng at the relay level A famly of relayng schemes s presented whch acheves the entre optmal dverstymultplexng DM tradeoff curve As a byproduct of our analyss, t follows that delay dversty and phase-rollng at the relay level are optmal wth respect to the entre DM-tradeoff curve, provded the delays and the modulaton frequences, respectvely, are chosen approprately I ITRODUCTIO Effcently utlzng the avalable dstrbuted spatal dversty n wreless networks s a challengng problem In ths paper, we consder fadng relay networks, a sngle-antenna sourcedestnaton termnal par communcates through a set of K half-duplex sngle-antenna relays We assume that there s no drect lnk between the source and the destnaton termnals and communcaton takes place usng a two-hop protocol over two tme slots The source termnal and the relays do not have any channel state nformaton CSI, and the destnaton termnal knows all channels n the network perfectly Prevous work: For setups smlar to that descrbed above, Laneman and Wornell propose space-tme coded cooperatve dversty protocols achevng full spatal dversty gan e, the dversty order equals the number of relay termnals For the setup consdered n ths paper, Jng and Hassb 2 analyze dstrbuted lnear dsperson space-tme codng schemes and show that a dversty order equal to the number of relay termnals can be acheved In 3, assumng the presence of a drect lnk between source and destnaton, Azaran et al show that an extenson to the mult-relay case of a protocol prevously ntroduced n 4 s dversty-multplexng DM tradeoff optmal Contrbutons: In ths paper, we are nterested n a class of smple relayng schemes whch s based on lnear processng at the relay level and hence converts the overall channel between the source and the destnaton termnal nto a tme, frequency or tme-frequency selectve sngle-nput sngle-output SISO channel Ths s attractve from an mplementaton pont-of-vew, as t allows to realze dstrbuted spatal dversty through the applcaton of standard forward error correcton codng over the resultng selectve-fadng SISO channel The class of relayng schemes analyzed n ths paper encompasses phase rollng 5, 6 and cyclc delay dversty 7 at the relay level In 7, t Ths research was supported by oka Research Center Helsnk, Fnland and by the STREP project o IST MEMBRAE wthn the Sxth Framework Programme of the European Commsson s concluded, through smulatons, that a K-relay cyclc delay dversty system can acheve a dversty gan of K In 5, t s demonstrated that phase-rollng at the relay level can acheve second-order dversty The contrbutons n ths paper can be summarzed as follows: We ntroduce a broad famly of relay transmt dversty schemes based on lnear processng at the relay level Whle the numercal results n 5, 7 are for the case of fxed rate e, the rate does not scale wth SR, we provde a suffcent condton on the proposed class of relay transmt dversty schemes to be optmal wth respect to wrt the entre DM-tradeoff curve as defned n 8 The tools used to prove DM-tradeoff optmalty are a method for computng the optmal DM-tradeoff curve n selectvefadng channels, ntroduced n 9, and a set of technques descrbed n 3 otaton: The superscrpts T,H and stand for transpose, conjugate transpose, and conjugaton, respectvely x represents the th element of the column vector x, and X,j stands for the element n the th row and jth column of the matrx X X Y denotes the Hadamard product of the matrces X and Y rankx stands for the rank of X TrX and X F denote the trace and the Frobenus norm of X, respectvely I s the dentty matrx 0 denotes the all zeros matrx of approprate sze We say that the square matrces X and Y are orthogonal to each other f X, Y = Tr XY H = 0 All logarthms are to the base 2 daga, a 2,, a denotes the dagonal matrx wth a on dagonal entry The dscrete Fourer transform DFT matrx F s defned as F ln = e j 2π l n X C 0, σ 2 stands for a crcularly symmetrc complex Gaussan random varable RV wth varance σ 2 Let the postve RV X be parametrzed by > 0 The exponental order of X n s defned as v = log X log f = g denotes exponental equalty, n, of the functons f and g, e, log f log g lm = lm log log The symbols,, > and < are defned analogously II SYSTEM MODEL Prelmnares: We consder a wreless network wth K + 2 sngle-antenna termnals, a source termnal S communcates wth a destnaton termnal D through a set of K half-duplex relay termnals R =, 2,, K For the sake of smplcty, we assume that there s no drect lnk between S and D The

2 channels S R, wth fadng coeffcent f, and R D, wth fadng coeffcent h, =, 2,, K, are d C 0, and reman constant over the tme-scale of nterest We defne the column vectors f = f f 2 f K T and h = h h 2 h K T Communcaton takes place over two tme slots In the frst tme slot, S transmts symbols consecutvely The relay termnals process the receved length- sequence usng a lnear transformaton as descrbed n the sgnal model below and transmt the result durng the second tme slot to D, whle S remans slent We assume that S and the relay termnals do not have CSI, as D knows f, h =, 2,, K perfectly For smplcty, we assume perfect synchronzaton of the entre network and gnore the mpact of shadowng and pathloss Throughout the paper, we assume that K Sgnal model: The vectors x, r, y C represent the transmtted sgnal, receved sgnal at R, and receved sgnal at D, respectvely The vector r s gven by r = f x + w, =, 2,, K denotes the average sgnal-to-nose rato SR for all lnks and w s the -dmensonal nose vector at R, wth d C 0, entres The w are ndependent across as well The transmtted sgnal x obeys the constrant E{x H x} = The relay termnal R apples a lnear transformaton accordng to G r, the matrx G satsfes G G H = + G r Ths I, scales the result and transmts the sgnal ensures that the per-relay transmt power per dmenson s gven by We emphasze that enforcng a per-relay transmt power of /K, whch leads to a total transmt power across relays of, does not change the man statements and conclusons n the remander of the paper The overall nput-output relaton reads K y = + h f G x + z 2 the effectve nose term z when condtoned on h s crcularly symmetrc complex Gaussan dstrbuted wth E{ z h} = 0 and E{ z z H h} = oi o = + + h 2 Snce we wll be nterested n the mutual nformaton MI between y and x under the assumpton that D knows all the channels n the network perfectly, we can dvde 2 by o to obtan the effectve nput-output relaton y = + + h 2 K h f G x + z 3 z when condtoned on h s a crcularly symmetrc complex Gaussan nose vector wth E{z h} = 0 and E{zz H h} = I In the remander of the paper, we shall be nterested n the case + h Wth 2 K h f G, we can now rewrte the nput- H eff = + h 2 output relaton 3 as ++ h 2 y = H eff x + z 4 A B denotes the lnk between termnals A and B III ACHIEVIG THE OPTIMAL DIVERSITY-MULTIPLEXIG TRADEOFF Under the assumptons stated n the prevous secton, t follows that the maxmum MI of the effectve channel n 4 s acheved by d Gaussan codebooks The correspondng MI s gven by Iy; x H eff = log + λ n H eff H H 2 eff 5 n=0 the factor /2 s due to the half-duplex constrant The DM-tradeoff realzed by a famly one at each SR of codebooks C r wth rate R = r log, r 0, /2, s gven by the functon log P e, r dr = lm log P e, r s the error probablty obtaned through maxmum lkelhood ML decodng We say that C r operates at multplexng gan r For a gven SR, the codebook C r C r contans 2r codewords x ext, we compute the optmal DM-tradeoff curve, as defned n 8, for the effectve channel H eff and provde a suffcent condton on the matrces G =, 2,, K n conjuncton wth a famly of codebooks C r r 0, /2 to be DM-tradeoff optmal Followng the framework n 8, we defne the probablty of outage at multplexng gan r and SR as P O, r Iy; x H eff < r log 6 Drectly analyzng 6 s challengng as closed-form expressons for the egenvalue dstrbuton of H eff do not seem to be avalable However, notng that Iy; x H eff I J y; x H eff 7 I J y; x H eff = 2 log + n=0 = 2 log + H eff 2 F λ n H eff H H eff we can resort to a technque developed n 9 to show that the DM-tradeoff correspondng to I J y; x H eff equals that correspondng to Iy; x H eff The sgnfcance of ths result les n the fact that the quantty H eff 2 F lends tself ncely to analytcal treatment In the followng, we wll need the K code dfference matrx defned as 8 Φ x = G x G 2 x G K x 9 x = x ˆx denotes the code dfference vector assocated wth the codewords x, ˆx Our man result can now be summarzed as follows Theorem : For the half-duplex relay channel n 4, the optmal DM-tradeoff curve s gven by dr = K 2r, r 0, /2 0

3 Let {G, G 2,, G K } be a set of transformaton matrces and C r a famly of codebooks such that for any codebook C r C r and any two codewords x, ˆx C r the condton rankφ x = K holds Then, the ML decodng error probablty satsfes P e, r = dr Proof: See Appendx A Dscusson: Theorem shows that the DM-tradeoff propertes of the half-duplex relay channel n 4 are equal to the cooperatve upper bound apart from the factor /2 loss, whch s due to the half-duplex constrant correspondng to a system wth one transmt and K cooperatng receve antennas ose forwardng at the relay level and the lack of cooperaton, hence, do not mpact the DM-tradeoff behavor, provded the matrces G and the famly of codebooks C r are chosen accordng to the condtons n Theorem Azaran et al 3, assumng the presence of a drect lnk between source and destnaton, show that extendng Protocol I n 4 to the mult-relay case by allowng only one relay to transmt n a gven tme slot yelds DM-tradeoff optmalty wrt the entre DM-tradeoff curve Our results show, however, that DM-tradeoff optmalty can be obtaned even f all relays transmt n all tme slots as long as the full-rank condton n Theorem s satsfed Another mmedate concluson that can be drawn from Theorem s that cyclc delay dversty 7 and phase-rollng 5, 6 at the relay level are optmal wrt the entre DM-tradeoff curve, provded the delays, the modulaton frequences and the codebooks are chosen approprately Ths can be seen as follows We start by notng that the cyclc delay dversty scheme 7 can be cast nto our framework by settng G = P P denotes the permutaton matrx that, when appled to a vector x, cyclcally shfts the elements n x up by postonswth {, = j P, P j = 2 0, j the condton rankφ x = K takes a partcularly smple form, namely F x k 0 for all k {, 2,, } To see ths note that rankφ x = rankfφ x and P = F H Λ F Λ = dag e jθ0, e jθ,, e jθ 3 wth θ n = 2πn ext, we have rankfφ x = rankσ l l 2 l K 4 Σ = dag F x, F x 2,, F x and l k = e jθk, k =, 2,,, =, 2, K As a consequence of 2, the columns of the matrx l l 2 l K are orthogonal and hence rankfφ x = K f Σ has full rank whch s the case f F x k 0 for all k {, 2,, } In the case of phase-rollng 5, 6, we have G = Λ Agan, the condton rankφ x = K takes a partcularly smple form, namely x k 0 for all k {, 2,, } The proof of ths statement follows by consderng Φ x drectly, puttng rankφ x nto the form of the rght-hand sde of 4 and applyng the remanng steps n the argument for the cyclc delay dversty case Whle the numercal results n 7, 5 are for the r = 0 case, our analyss reveals optmalty of cyclc delay dversty and phase-rollng for the entre DM-tradeoff curve, provded the codebooks satsfy the full-rank condton n Theorem We fnally note that cyclc delay dversty and phase-rollng are tme-frequency duals of each other n the sense that the lnear transformaton matrces for the two schemes obey G = FP F H Relaton to approxmately unversal codes 0: For the halfduplex relay channel nvestgated n ths paper, a famly of codes C r s DM-tradeoff optmal f µ mn > 2r 5 µ mn s the smallest egenvalue of Φ x H Φ x over all x = x ˆx wth x, ˆx C r Ths result follows mmedately from 29 n the proof of Theorem Based on 5, we can conclude usng the same arguments as n Sec IV A n 9 that any famly of codes C r satsfyng 5 wll also be approxmately unversal n the sense of 0, Th 3 Relaton to code desgn crtera for pont-to-pont case: We conclude our dscusson by pontng out that the condtons of Theorem guarantee DM-tradeoff optmalty n pont-to-pont multple-nput sngle-output systems as well IV COCLUSIOS We ntroduced a famly of lnear relay processng schemes achevng the optmal DM-tradeoff curve of half-duplex relay channels Cyclc delay dversty and phase-rollng were shown to be DM-tradeoff optmal specal cases Our analyss can readly be extended to account for the presence of a drect lnk between the source and the destnaton termnals Fnally, we note that the DM-tradeoff framework seems to be too crude to quantfy potental performance dfferences between relay transmt dversty schemes wth dfferent egenvalue spread of the Graman matrx of the G APPEDIX A PROOF OF THEOREM We start by notng that an upper bound on the DM-tradeoff curve can be obtaned by applyng the broadcast cut-set bound to the descrbed network and evaluatng the correspondng DM-tradeoff for d Gaussan codebooks It s shown n, 2 that the broadcast cut amounts to a pont-to-pont lnk wth a sngle transmt and K cooperatng receve antennas Takng nto account the factor /2 loss due to the half-duplex nature of the relay termnals, t follows mmedately from the results n 8 that the DM-tradeoff curve correspondng to the network analyzed n ths paper s upper-bounded by dr K 2r, r 0, /2 In the followng, we shall show that ths upper bound s achevable, despte the lack of cooperaton between the relay termnals, provded that, for every r 0, /2, C r C r satsfes

4 rankφ x = K for all x = x ˆx wth x, ˆx C r We start by notng that and the lower bound PJ A P P O, r PJ I J y; x H eff < r log 2 log + λ max h 2 < r log 9 2 log K + λ max f 2 h 2 < r log J = {H eff I J y; x H eff < r log } s defned as Jensen outage event Snce 2 log K + λ max v u < r log H eff 2 K K F = + h 2 Tr h f h j fj G G H the key steps 8 and 9 follow from the Raylegh-Rtz j j= theorem 3 and the fact that + K u + K for = + h 2 h H K h h = h f and the Graman K = Tr G G H Tr G K G H Tr G G H K Tr G K G H K we have PJ 2 log + h H K h + h 2 < r log In what follows, we wrte h 2 = u and f 2 = v u and v are RVs; the choce of ths transformaton wll become clear later Further, we defne the events A = {u, u 2,, u K, v, v 2,, v K u 0, v 0 {, 2,, K}} and the complementary event Ā as the event at least one u or v s negatve Usng the law of total probablty, we can wrte PJ A PJ A + P Ā P J Ā and bound PJ accordng to PA PJ A PJ PA PJ A + P Ā PA PJ A PJ PA PJ A 6 PJ A PJ PJ A 7 6 follows from the defnton of the u and the v, ther ndependence and by notng that P Ā decays exponentally fast log PA n The double nequalty 7 results from lm log = 0 We have thus shown that PJ J A ext, denotng the mnmum and maxmum egenvalue of K as λ mn and λ max, respectvely, we get the upper bound PJ A P 2 log + λ mn + K h 2 < r log 8 2 log + λ K mn f 2 h 2 < r log + K 2 log + λ K mn v u < r log + K u 0 =, 2,, K and > It can be shown that the full-rank condton on Φ x mples rankk = K and therefore λ mn > 0 We next defne the followng events { } B = u, v max v u > 0 { U = u, v 2 log + λ mn max v u } < r log + K { L = u, v 2 log + Kλ max max v u } < r log the max s taken over =, 2,, K n all three cases Wth these defntons, we arrve at PL PJ A PU 20 PL B + P L B PJ A PU B + P U B PL B PJ A PU B + P B 20 follows from K v u K max v u ow, we can expand PU B as PU B 0 < max v u < 2r + ɛ 2 K f 2 h 2 < P f 2 h 2 < K 2r ɛ K = F 2r ɛ F K 22 ɛ = log +K λ mn /log and F x = 2 x K 2 x wth K denotng the frst-order modfed Bessel functon of the second knd Further, we have P B = F K 23 for 22 and 23 we used the fact that the CDF of the product of two Raylegh dstrbuted RVs s gven by 2xK 2x for x > 0 4 In the ensung dscusson, all statements nvolvng r hold for r 0, /2 Combnng 22 and 23, we get PU B + P K B = F 2r ɛ = K 2r 24

5 the exponental equalty n 24 s proved usng a Taylor seres expanson of F /x around x = 0 and nvokng asymptotc propertes of logx 5, Eq To complete the proof, we establsh that PL B has the same exponental behavor n as PU B+P B Usng the same arguments as n 2-24, t readly follows that PL B = 0 < max v u < 2r ɛ 2 K F 2r+ɛ 2 F K = K 2r ɛ 2 = logkλmax log We have thus shown that and hence K 2r PJ J, r PJ K 2r = K 2r Snce P J, r P O, r as a result of 7, and snce the outage probablty s a lower bound to the error probablty acheved by any code 8, we have P J, r P O, r P e, r 25 Followng the approach ntroduced n 9, we now complete the proof of the theorem by dentfyng a famly of codes whch has P e, r J, r and hence results n a DM-tradeoff curve whch equals the Jensen DM-tradeoff curve derved above We start by wrtng P e, r J Perror J + P error, J PJ + P error, J ext, we upper-bound P error, J through the unon bound P error, J 2r P ˆx x, J 26 we used the fact that the codebook, C r, contans 2r codewords and P ˆx x, J denotes the maxmum parwse error probablty over all codeword pars and all channels n J for ML decodng Wth x = x ˆx, we have Pˆx x H eff = H = Q 2 H x F exp 4 H x 2 F exp 4 + K Φ x h 2 F exp 4 + K µ mn h 2 27 µ mn denotes the mnmum egenvalue of Φ x H Φ x and 27 follows from applyng the Raylegh-Rtz theorem Substtutng h 2 = K v u nto 27, we have P ˆx x, J { } E h J exp µ mn K v u 4 + K exp µ mn 4 + K 2r follows snce the event J requres that K v u 2r Fnally, nsertng 28 nto 26, we get P error, J 2r exp µ mn 4 + K 2r 29 The proof s complete snce Φ x has full rank for all x and for all codebooks n C r and hence µ mn > 0 whch mples that 29 decays exponentally n for all r 0, /2 Summarzng our results, we obtan P e, r PJ + P error, J PJ + 2r exp µ mn 4 + K 2r PJ J, r whch combned wth 25 yelds the desred result REFERECES J Laneman and G W Wornell, Dstrbuted space-tme-coded protocols for explotng cooperatve dversty n wreless networks, IEEE Trans Inf Theory, vol 49, no 0, pp , Oct Y Jng and B Hassb, Dstrbuted space-tme codng n wreless relay networks, IEEE Trans Wreless Comm, vol 5, no 2, pp , Dec K Azaran, H El Gamal, and P Schnter, On the achevable dverstymultplexng tradeoff n half-duplex cooperatve channels, IEEE Trans Inf Theory, vol 5, no 2, pp , Dec R U abar, H Bölcske, and F W Kneubühler, Fadng relay channels: Performance lmts and space-tme sgnal desgn, IEEE J Sel Areas Comm, vol 22, no 6, pp , Aug I Hammerström, M Kuhn, and A Wttneben, Cooperatve dversty by relay phase rotatons n block fadng envronments, n Proc Ffth IEEE Workshop on Sgnal Processng Advances n Wreless Communcatons SPAWC, July 2004, pp P Kuppnger, Transformaton of dstrbuted spatal nto temporal dversty by relay phase rotatons, MSc Thess, Imperal College London, Sept S B Slmane and A Osseran, Relay communcaton wth delay dversty for future communcaton systems, n Proc IEEE VTC Fall, Sept 2006, pp 5 8 L Zheng and D C Tse, Dversty and multplexng: A fundamental tradeoff n multple-antenna channels, IEEE Trans Inf Theory, vol 49, no 5, pp , May P Coronel and H Bölcske, Dversty-multplexng tradeoff n selectvefadng MIMO channels, n Proc IEEE ISIT, ce, France, June 2007, to appear 0 S Tavldar and P Vswanath, Approxmately unversal codes over slowfadng channels, IEEE Trans Inf Theory, vol 52, no 7, pp , July 2007 M Gastpar and M Vetterl, On the capacty of large Gaussan relay networks, IEEE Trans Inf Theory, vol 5, no 3, pp , March H Bölcske, R U abar, Ö Oyman, and A J Paulraj, Capacty scalng laws n MIMO relay networks, IEEE Trans Wreless Comm, vol 5, no 6, pp , Jun R A Horn and C R Johnson, Matrx Analyss ew York, Y: Cambrdge Press, J Salo, H El-Sallab, and P Vankanen, The dstrbuton of the product of ndependent Raylegh random varables, IEEE Trans Ant and Prop, vol 54, no 2, pp , Feb M Abramowtz and I Stegun, Handbook of Mathematcal Functons ew York: Dover, 965