Energy-Ef cient Antenna Sharing and Relaying for Wireless Networks

Transcription

1 Energy-Ef cien Anenna Sharing and Relaying for Wireless Neworks J. Nicholas Laneman and Gregory W. Wornell Research Laboraory of Elecronics Massachuses Insiue of Technology Cambridge, MA USA Absrac We develop energy-ef cien ransmission proocols for wireless neworks ha exploi spaial diversiy creaed by anenna sharing: coordinaed ransmission and/or processing by several disribued radios. We focus on singleuser ransmission and examine several possibiliies for he sraegy employed by he assising radio, or relay, including decoding and forwarding as well as amplifying and forwarding. In each case, we develop receivers based upon maximum-likelihood and/or maximum signal-o-noise raio crieria, relae heir srucures, and compare heir bi-error probabiliy performance by means of analysis and simulaions. We cas singlehop and mulihop rouing ino our framework for comparison purposes. All of our anenna sharing proocols offer diversiy gains over single- and muli-hop ransmission, and our resuls sugges ha lowcomplexiy amplifying and forwarding is energy-ef cien in spie of noise ampli caion a he relay. I. INTRODUCTION Relaying informaion over several poin-o-poin communicaion links is a basic building block of communicaion neworks. Such relaying is uilized in wired and wireless neworks o achieve higher nework conneciviy (broader coverage), ef- cien uilizaion of resources such as power and bandwidh, beer economies of scale in he cos of long-haul ransmissions (hrough raf c aggregaion), ineroperabiliy among neworks, and more easily manageable, hierarchical nework archiecures. In wireless neworks, direc ransmission beween widely separaed radios can be very expensive in erms of ransmied power required for reliable communicaion. High-power ransmissions lead o faser baery drain (shorer nework life) as well as increased inerference a nearby radios. As alernaives o direc ransmission, here are wo basic and frequenlyemployed examples of relayed ransmission for wireless neworks. In cellular seings, for example, neworks provide conneciviy beween low-power mobiles by providing local connecions o high-power basesaions ha are relayed via a wireline basesaion nework. In sensor neworks, and miliary bale eld communicaion neworks in general, he use of wireline infrasrucure is ofen precluded and he radios may be subsanially power consrained; for hese ad-hoc or peer-o-peer neworks, ransmissions can be relayed wirelessly. As hese examples sugges, relayed ransmission enliss wo or more radios o perform muliple ransmissions. The end-o-end ransmissions poenially incur higher delay, bu because he individual ransmissions are over shorer disances (in he wireless case), or over high-qualiy cabling (in he wireline case), he This work has been suppored in par by ARL Federaed Labs under Cooperaive Agreemen No. DAAL , and by NSF under Gran No. CCR as well as hrough an NSF Graduae Research Fellowship. x Relay 1 y 3 Source y 2 d 1,2 x 2 d 1,3 d 2,3 y 3 Desinaion Fig. 1. Example hree-radio (sub)nework for which relaying proocols, and especially, anenna sharing, or diversiy, proocols can be moivaed and developed. Indicaed are he ransmied signals x 1 and x 2, he received signals y 2, y 3, and y 3, and he radio separaions d i,j. power requiremens for reliable communicaion can be much lower. The basic relaying proocols described above are consruced from he sequenial use of poin-o-poin links, where he links are essenially viewed a he nework proocol layer; however, more general approaches are possible ha involve he coordinaion of boh he direc and relayed ransmissions, a he nework and lower proocol layers, and correspond o scenarios o which he classical relay channel model [1] applies. In his paper, we develop energy-ef cien relaying proocols ha creae and exploi spaial diversiy o comba fading due o mulipah propagaion, a paricularly severe form of inerference experienced in wireless neworks. To illusrae he main conceps, we consider he simple wireless nework depiced in Fig. 1. We focus speci cally on ransmissions from radio 1, called he source, o radio 3, called he desinaion, wih he possibiliy of employing radio 2 as a relay. A he physical layer, he desinaion receives poenially useful signals from all ransmiers ha are acive, and may combine muliple ransmissions of he same signal o reduce variaions in performance caused by signal fading, a echnique referred o broadly as spaial diversiy combining [2]. We refer o his form of spaial diversiy as anenna sharing, in conras o he currenly more convenional forms of spaial diversiy [3], because he radios essenially share heir anennas and oher resources o creae a virual array hrough disribued ransmission and signal processing. Afer developing a mahemaical model in Secion II for he nework in Fig. 1, we scrach he surface of he rich se of de- [1]

2 sign issues and opions ha arise in he conex of anenna sharing and relaying for wireless neworks. Secion III cass he basic relaying proocols, referred o as singlehop and mulihop ransmission, respecively, ino our framework, and explores a number of possibiliies for anenna sharing proocols, in erms of wha signals he source and relay joinly ransmi as well as how he relay and desinaion joinly process signals. Performance comparisons, and simulaion resuls in Secion IV, sugges ha anenna sharing ransmission proocols are capable of overcoming he noisy channels beween he disribued radio anennas o achieve diversiy gain and ouperform singlehop and mulihop ransmission in a variey of scenarios of ineres. II. SYSTEM MODEL In our model for he hree-radio wireless nework depiced in Fig. 1, narrowband ransmissions suffer he effecs of pah loss and a fading as arise in e.g., slow-frequency-hop neworks. Our analysis focuses on he case of slow fading o isolae he bene s of spaial diversiy alone; however, we emphasize a he ouse ha our resuls exend naurally o he kinds of highly mobile scenarios in which faser fading is encounered. Our baseband-equivalen, discree-ime channel model for he nework in Fig. 1 consiss of wo subchannels, orhogonal in, e.g., adjacen ime slos or frequencies. This decomposiion is necessary because pracical limiaions in radio implemenaion preven he relay from simulaneously ransmiing and receiving on he same channel. On he rs subchannel, he source ransmis a sequence x 1 [n], wih average sample energy 1, and he relay and desinaion receive signals y 2 [n] =a 1,2 x 1 [n]+z 2 [n], (1) y 3 [n] =a 1,3 x 1 [n]+z 3 [n], (2) respecively. On he second subchannel, he relay ransmis a sequence x 2[n], wih average sample energy 1, and he desinaion receives 1 y 3[n] =a 2,3 E2 x 2[n]+z 3[n]. (3) Here a i,j capures he effecs of pah loss and saic fading on ransmissions from radio i o radio j, E i is he ransmied energy of radio i, and z j [n] and z 3[n] model addiive receiver noise and oher forms of inerference. Saisically, we model he fading coef ciens a i,j as zeromean, muually independen complex joinly Gaussian random variables wih variances σa 2 i,j, and we model he addiive noises z j [n] and z 3[n] as zero-mean, muually independen, whie complex joinly Gaussian sequences wih variance N 0. We de ne he signal-o-noise raio (SNR) in each received signal as γ i,j = ai,j 2 E i /N 0 ; under he Rayleigh fading model, 1 We employ he noaion ( ) o disinguish he signals on he second subchannel from hose on he rs; he fading coef ciens a i,j are he same because he subchannels are assumed o be adjacen and he fading is a across frequency. y 3 [n-1] y 3 [n] w (x * 1- x ) 0 w (x 1- x ) * 0 2 Re{.} 2 Re{.} f(. ) Fig. 2. Desinaion receiver srucure. x 0 x 1 [n-1] x 1 he SNRs are independen exponenial random variables wih expeced values γ i,j =E[γi,j ]=σa 2 i,j E i /N 0. III. TRANSMISSION PROTOCOLS Wihin he physical layer framework described in Secion II, we examine several proocols ha suppor ransmission beween he source and desinaion. Each proocol consiss of a source modulaion forma, a relay processing/modulaion scheme, and a desinaion receiver srucure. For simpliciy of exposiion, we rea coherenly-deeced, consan-modulus binary ransmissions, so ha he source ransmied signal x 1 [n] is whie and akes values x 0 and x 1 wih equal probabiliy. To enable coheren deecion, he relay and desinaion receivers mus rs obain, via raining sequences in he proocol headers, accurae esimaes of he link fading coef ciens; in several scenarios, he desinaion also uilizes an esimae of γ 1,2. We assume hese esimaes are perfec in our preliminary analysis. All of our desinaion receiver srucures can be implemened as shown in Fig. 2. This combiner can be viewed as a generalized mached- ler, or maximum-raio combiner, suiably modi ed o he proocol. As we will see, qualiaive comparisons among he various ransmission proocols can be made by examining heir respecive weighs w and w as well as heir mappings f( ). A. Singlehop Transmission Singlehop ransmission, ofen referred o as singlehop rouing in he ad-hoc neworking communiy [4], consiss of direc ransmission beween he source and desinaion radios. In his case, he source ransmis x 1 [n], he relay ransmis x 2[n] =0, i.e., nohing, and he desinaion processes only (2). Minimum probabiliy of error (MPE) deecion corresponds o condiional MPE deecion for each value of he fading coef cien a 1,3. Since he inpu symbols are equally likely, condiional MPE deecion corresponds o condiional maximumlikelihood (ML) deecion; his can be implemened by he combiner in Fig. 2 wih any mapping f( ) and weighs w = a 1,3, w =0, (4) N 0 i.e., he desinaion ignores y 3[n]. Since he equivalen channel is condiionally Gaussian wih SNR γ 1,3, he condiional error [2]

3 probabiliy for singlehop ransmission can be obained from sandard Gaussian resuls [2] ) = Q ((1 ρ)γ 1,3, (5) P SH γ1,3 where Q() = 1 e s2 /2 ds, and ρ is a consan depending upon he modulaion forma. For example, coherenly- deeced BPSK has ρ = 1, while coherenly-deeced FSK has ρ =0. The average error performance of singlehop ransmission, P SH, follows by averaging (5) over he exponenial probabiliy densiy funcion for γ 1,3 ; he resul can be approximaed for large (average) SNR by [2] P SH 1, γ K γ 1,3 1, (6) 1,3 2π where K is anoher consan depending upon he modulaion forma. For example, coherenly-deeced BPSK has K =4, while coherenly deeced FSK has K =2. B. Mulihop Transmission The basic wireless relaying proocol qualiaively described in Secion I is called muli-hop rouing in he ad-hoc neworking communiy [4]. Muli-hop ransmission in our framework can be viewed as cascading singlehop ransmission beween he source and relay wih singlehop ransmission beween he relay and desinaion. Speci cally, he source ransmis x 1 [n], and he relay forms an esimae ˇx 1 [n] from (1). The relay ransmis his esimae as x 2[n] =ˇx 1 [n 1]. Finally, he desinaion forms an esimae ˆx 1 [n 1] of x 1 [n 1] from (3). The sample delay accouns for processing and (relaive) propagaion delay hrough he relay. As we will develop, ML deecion of x 1 [n] a he relay is preferable. We examine wo desinaion receivers for mulihop ransmission. The rs forms ML esimaes of he relay s ransmied signal x 2[n], and is useful for developing average error performance bounds. The second makes ML esimaes of he source ransmied sequence x 1 [n 1]. B.1 ML Deecion of x 2[n] Condiional ML deecion of x 2[n] corresponds o he singlehop ML deecor from Secion III-A, wih he rolls of y 3 [n 1] and y 3[n] swapped. Speci cally, he condiional ML deecor can be implemened as he combiner in Fig. 2 wih E2 w =0, w = a 2,3, f()=. (7) N 0 If he relay decision process can be modeled as a binarysymmeric channel (BSC) wih crossover probabiliy ɛ depending upon he SNR γ 1,2, he condiional error probabiliy in esimaing x 1 [n 1] a he combiner speci ed by (7) can be upper bounded by P MH γ1,2,γ 2,3 ɛ + P SH γ2,3, (8) where he rs erm arises from he even ha he relay makes a decision error, and he second erm arises from he even ha he desinaion makes a decision error given ha he relay does no. The resul (8) suggess ha MPE deecion of x 1 [n] a he relay is preferable. In his case, ɛ = P SH γ1,2, and he average error performance can be approximaed for large SNR by [2] P MH 1 + 1, γ K γ 1,2 K γ 1,2, γ 2,3 1. (9) 2,3 As we will see in Secion IV, his bound is igh in several regimes of ineres. B.2 ML Deecion of x 1 [n 1] Condiional ML deecion of x 1 [n 1] a he desinaion is somewha more involved, bu can also be implemened as a combiner in he form of Fig. 2. Again, assuming he relay decision process can be modeled as a BSC wih crossover probabiliy ɛ, some algebra shows ha he desinaion condiional ML deecor of x 1 [n 1] has [ ] w =0, w = a 2,3 E2 ɛ+(1 ɛ)e, f()=ln N 0 ɛe. (10) +(1 ɛ) The key sep in obaining (10) lies in he expanding he likelihood p(y 3 a 2,3, x 1 ) by averaging over wheher or no he relay makes a decision error, i.e., p (y 3 a 2,3, x 1 = x 0 )=(1 ɛ)p(y 3 a 2,3,ˇx 1 =x 0 ) +ɛp(y 3 a 2,3,ˇx 1 =x 1 ), for x = x 0, and similarly for x = x 1. The resuls (10) follows afer subsiuion of he condiional Gaussian likelihoods, aking he log-likelihood raio, and algebraic simpli caions. Limiing argumens indicae, and Fig. 3 exhibis, ha he mapping f() in (10) essenially clips is inpu o he values ± ln[ɛ/(1 ɛ)] and is approximaely linear beween hese exremes for small. Forɛ<1/2, he mappings in (7) and (10) saisfy f() 0 for 0 and f() < 0 for <0; hence, heir symbol esimaes for uncoded ransmissions will be idenical. Consequenly, as we have seen previously, ML deecion a he relay is preferable, and he average error performance of his mulihop proocol should also be well approximaed by (9). Finally, we observe ha as ɛ 1/2, he mapping f() in (10) goes o 0, and if he desinaion xes ɛ =0in he deecor, i.e., i does no explicily ake ino accoun he uncerainy of he relay decisions, (10) reduces o (7). Alhough apparenly irrelevan for uncoded mulihop ransmission, we will see in Secion III-C ha he clipping propery of f() in (10) is signi can in he conex of diversiy ransmission. More generally, while clipping he mached- ler oupu may be irrelevan for uncoded ransmissions, i is imporan for coded sysems (wih symbol-by-symbol deecion employed a he relay), because i limis he conribuion of any one symbol s log-likelihood, e.g., branch meric in a Vierbi algorihm, o he sum of he log-likelihoods, e.g., pah meric in a Vierbi algorihm. [3]

4 f() Fig. 3. Combiner mapping f() from (10). Successively higher dashed curves (for >0) correspond o ɛ =10 1,10 2,...,10 4, respecively. For comparison, he solid curve corresponds o he linear mapping f() =. C. Diversiy Transmission wih Decoding Relay Our rs diversiy ransmission proocol combines singlehop and mulihop ransmission o creae and exploi spaial diversiy. Speci cally, our proocol for diversiy ransmission wih a decoding relay consiss of he following. The source ransmis x 1 [n] o boh he relay and desinaion on he subchannel (1) and (2). The relay forms an esimae of x 1 [n] from is received signal y 2 [n], and ransmis his esimae, delayed by one sample o accoun for processing and (relaive) propagaion delay, as x 2[n] on subchannel (3). As hroughou Secion III-B, we assume he relay decision process can be modeled as a BSC wih crossover probabiliy ɛ, and based upon reasoning similar o ha discussion, we employ ML deecion a he relay. The desinaion esimaes x 1 [n 1] from boh is received signals (2) and (3). When suiably combined, he chances of boh signals exhibiing deep fading is reduced; herein lies he diversiy bene. The challenge in his seing is o design a deecor ha can overcome he effecs of uncerainy in he relay decisions and sill exploi he available spaial diversiy. C.1 ML Deecion of x 1 [n 1] Combining he resuls of Secion III-A and Secion III-B, condiional ML deecion of x 1 [n 1] from boh (2) and (3) can be implemened as he combiner in Fig. 2 wih [ ] w = a 1,2, w = a 2,3 E2 ɛ+(1 ɛ)e, f()=ln N 0 N 0 ɛe. +(1 ɛ) (11) Here he clipping effec of f() in (11) is more imporan han i was for uncoded mulihop ransmission. The nonlineariy in f() increasingly reduces, wih increasing ɛ, he conribuion of he diversiy branch hrough he relay. If he desinaion assumes he relay decisions are always correc, hen (11) wih ɛ 0 becomes a convenional maximumraio combiner E2 w = a 1,2, w = a 2,3, f()=. (12) N 0 N 0 This combiner, hough mismached in general, performs reasonably well for small ɛ. Similar o he bound in Secion III- B.1, we can upper bound he average error performance of he deecor corresponding o (12), again using large SNR approximaions from [2], by P DD K γ 1,2 K 2, γ γ 1,3 γ 1,2, γ 1,3, γ 2,3 1. (13) 2,3 The rs erm in (13) arises from he even ha he relay makes a decision error, and he second erm arises from he even ha he desinaion makes a decision error given ha he relay does no, corresponding o a convenional ransmi anenna diversiy scenario [3]. This bound is only useful for approximaing he performance of he ML deecor (11) in channel environmens for which γ 1,2 is especially large, e.g., when he relay is very close o he source. C.2 Maximum SNR Deecor As an alernaive design crierion, we deermine he receiver ha maximizes he SNR of he slicer inpu. To arrive a his max SNR receiver, we examine he relay decision ˇx 1 = x 1 + e, where e is a random variable capuring he effecs of decision errors. A few calculaions yield { ɛ(x 1 x 0 ) if x 1 = x 0 E[e x 1 ]=, ɛ(x 0 x 1 ) if x 1 = x 1 σ 2 e = ɛ(1 ɛ) x 1 x 0 2. Leing x 1 = x 1 +E[e x 1 ], and ẽ = e E[e x 1 ], he relay esimae ˇx 1 = x 1 + ẽ can be viewed as equally-likely symbols drawn from he consan-magniude consellaion x 1 =(1 ɛ)x 1 +ɛx 0, x 0 =(1 ɛ)x 0 +ɛx 1, (14) plus an addiive noise ha is uncorrelaed wih x 1, having mean zero and variance σ 2 e. Thus, he wo signals received by he desinaion may be wrien as y 3 [n 1] = a 1,3 x 1 [n 1] + z 3 [n 1], y 3[n] =a 2,3 E2 ( x[n 1] + ẽ[n 1]) + z 3[n]. (15) Observing ha ( x 1 x 0 )=(1 2ɛ)(x 1 x 0 ), he maximum SNR desinaion receiver, a mached- ler for (15), can be implemened as he combiner in Fig. 2 wih w = a 1,3, w = a 2,3 E2 (1 2ɛ) N 0 a 2,3 2 E 2 σe 2, + N 0 (16) f() =. [4]

5 Examining w in (16) more closely, we see ha i consiss of he maximum-raio combiner weigh w in (12) followed by he linear mapping f() = (1 2ɛ) γ 2,3 σe 2. (17) +1 For ɛ 1/2, (17) goes o zero, indicaing ha he maximum SNR deecor ignores he received signal y 3 jus as he ML deecor de ned by (11). Like he ML deecor, (16) converges o he maximum-raio combiner (12) for ɛ 0. We conclude his secion by noing ha [5] develops resuls similar o (11) and (16) in he conex of cellular neworks. Speci cally, he linear deecor in [5] corresponds o he combiner in Fig. 2 wih E2 w = a 1,3, w =λ a 2,3, f()=; N 0 N 0 he parameer λ is chosen numerically o minimize he condiional error probabiliy of his linear deecor. D. Diversiy Transmission wih Amplifying Relay In he previous secion, we explored several desinaion deecion algorihms assuming he relay employed ML deecion. If we consrain he relay o employ linear processing, i.e., amplifying, alernaive ransmission proocols resul. We migh expec his consrain o induce excessive noise ampli caion, bu, as he simulaion resuls in Secion IV sugges, a desinaion ML deecor designed for an amplifying relay can be quie compeiive, and perhaps even ouperform he ransmission proocols from he previous secion, when he relay is close o he desinaion. Our proocol for diversiy ransmission wih an amplifying relay consiss of he following. The source ransmis x 1 [n] o boh he relay and desinaion on he subchannel (1) and (2). The relay ransmis an ampli ed (and delayed) version of is received sequence, i.e., x 2[n] =βy 2 [n 1] on he subchannel (3). To decode symbol x 1 [n 1], he desinaion processes is wo received signals y 3 [n 1] = a 1,3 x 1 [n 1] + z 3 [n 1], y 3[n] =a 2,3 β(a 1,2 x 1 [n 1] + z 2 [n 1]) + z 3[n]. (18) The desinaion condiional ML deecor of x 1 [n 1] from (18) can be implemened as he combiner in Fig. 2 wih w = a 1,3, w = a 2,3β a1,2 N 0 ( a 2,3 2 β 2, +1)N 0 f()=. (19) To saisfy is oupu power consrain, he relay ampli er can operae a a maximum gain saisfying β 2 = E 2 a 1,2 2 E 1 + N 0, (20) where we allow he gain o depend upon he fading realizaion a 1,2 from he source o he relay. Subsiuing (20) ino (19), we see ha he channel is condiionally Gaussian wih SNR ha can be manipulaed ino he form γ 1,3 + γ eq, where γ 1 eq = γ 1 1,2 + γ 1 2,3 + γ 1 1,2 γ 1 2,3. (21) The condiional bi-error probabiliy can be readily compued using sandard Gaussian resuls, yielding ( ) = Q (1 ρ)[γ 1,3 +γ eq ]. (22) P DA γ1,3,γ 1,2,γ 2,3 Noe ha he condiional error probabiliy (22) exhibis a sum of SNRs as we migh expec in a diversiy scenario. Examining (21), we see ha γ eq <γ min = min {γ1,3,γ 2,3 }. (23) Since γ 1,3 and γ 2,3 are independen exponenial random variables in our model, heir minimum is also exponenial wih expeced value saisfying γ 1 min = γ 1 1,3 + γ 1 2,3, (24) analogous o a parallel combinaion of resisances in circui heory. Since Q() is decreasing in, (23) gives ( ) Q (1 ρ)[γ 1,3 +γ min ] (25) P DA γ1,3,γ 1,2,γ 2,3 Finally, averaging (25) over he exponenial densiy funcions for γ i,j, we obain a lower bound on he average error performance of diversiy ransmission wih an amplifying relay. Using he large SNR approximaions from [2], we obain P DA 3 K 2 γ 1,3 γ min, γ 1,3, γ min 1. (26) In addiion o he lower bound provided by (26), we can esimae he average error performance P DA by compuing sample averages of independen realizaions of (22), or by Mone Carlo simulaion of he sysem. IV. PERFORMANCE SIMULATIONS To compare performance of he ransmission proocols, we examine a nework wih coordinaes normalized by he disance d 1,3 beween he source and desinaion radios. In hese coordinaes, he source can be locaed a (0, 0), and he desinaion can be locaed a (1, 0), wihou loss of generaliy. Due o space consideraions, we limi our scope o scenarios wih he relay locaed a (l, 0) for l 0, i.e., he relay is very close o he source; l =1/2,i.e., he relay is halfway beween he source and desinaion; and l 1, i.e., he relay is very close o he desinaion. The fading variances σa 2 i,j can be assigned using wireless pah-loss models based on he nework geomery [6]; here, we [5]

6 uilize models of he form σa 2 i,j d ν i,j, where d i,j is he disance from mobile i o mobile j, and ν is a consan whose value, as esimaed from eld experimens, lies in he range 3 ν 5. Due o space consideraions, we repor resuls for ν =4, a value ypical of urban environmens. To normalize for he oal nework energy E per ransmied bi, we se E 1 = E for singlehop ransmission, and E 1 = E 2 = E/2 for all oher ransmission proocols. We plo he simulaed average error performance agains he singlehop average SNR. More generally, we can consider power allocaions of he form E 1 = αe and E 2 =(1 α)e, and selec he parameer α o minimize a variey of nework performance crieria. Appropriae rae or bandwidh normalizaion of he resuls is beyond he scope of his paper. Figs. 4 6 show simulaed performance resuls of he various ransmission proocols for uncoded BPSK ransmissions, i.e., x 0 = 1 and x 1 =+1, for relay locaions (0.1, 0), (0.5, 0), and (0.9, 0), respecively. The bounds in (9) and (26) are also shown, as dashed and dashed-doed lines, respecively, o demonsrae how well hey can approximae sysem performance. Aside from he apparen diversiy gains (decrease in slope on a log scale) for he anenna sharing proocols, mulihop and anenna sharing proocols exhibi power gain (shif he curve o he lef) on he order of 3(ν 2) db for he relay locaed halfway beween he source and desinaion. Noe ha his power gain is speci c o our pah-loss models. Somewha surprisingly, diversiy ransmission wih an amplifying relay appears o perform comparably, if no beer, han he diversiy ransmission schemes wih a decoding relay. Characerizing his relaionship more compleely in various regimes will be addressed in fuure work. We noe ha our resuls can, in principle, be exended naurally o muliple relays, wheher employed serially or in parallel. While our analysis has been carried ou sricly a he physical layer of he nework in Fig. 1, obaining he gains demonsraed in his paper requires a re-examinaion of he nework proocol sack, a leas hrough he radiional physical and medium-access conrol (MAC) layers, o provide he coordinaion funcions required by our anenna sharing ransmission proocols. REFERENCES [1] Thomas M. Cover and Joy A. Thomas, Elemens of Informaion Theory, John Wiley & Sons, Inc., New York, [2] John G. Proakis, Digial Communicaions, McGraw-Hill, Inc., New York, Third ediion, [3] Aradhana Narula, Michell D. Tro, and Gregory W. Wornell, Performance limis of coded diversiy mehods for ransmier anenna arrays, IEEE Trans. Inform. Theory, vol. 45, no. 7, pp , Nov [4] Fouad A. Tobagi, Modeling and performance analysis of mulihop packe radio neworks, Proc. IEEE, vol. 75, no. 1, pp , Jan [5] Andrew Sendonaris, Elza Erkip, and Behnaam Aazhang, A novel diversiy scheme for increasing he capaciy of cellular sysems, Submied o IEEE Trans. on Communicaions, [6] Theodore S. Rappapor, Wireless Communicaions: Principles and Pracice, Prenice-Hall, Inc., Upper Saddle River, New Jersey, Average Error Performance Single Hop Mulihop Div. w/dec, ML Div. w/dec, max SNR Div. w/amp Singlehop Average SNR (db) Fig. 4. Simulaed performance of he ransmission proocols for ν =4and normalized geomeries wih he relay locaed a (0.1, 0), i.e., close o he source. Average Error Performance Single Hop Mulihop 10 5 Div. w/dec, ML Div. w/dec, max SNR Div. w/amp Singlehop Average SNR (db) Fig. 5. Simulaed performance of he ransmission proocols for ν =4and normalized geomeries wih he relay locaed a (0.5, 0), i.e., halfway beween he source and desinaion. Average Error Performance Single Hop Mulihop 10 5 Div. w/dec, ML Div. w/dec, max SNR Div. w/amp Singlehop Average SNR (db) Fig. 6. Simulaed performance of he ransmission proocols for ν =4and normalized geomeries wih he relay locaed a (0.9, 0), i.e., close o he desinaion. [6]