Performance Analysis of Two-Way Relaying with Non-Coherent Differential Modulation

Transcription

1 004 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 0, NO. 6, JUNE 0 erformance Analyss of Two-Way Relayng wth Non-Coherent Dfferental Modulaton We Guan and. J. Ray u, Fellow, IEEE Abstract Ths wor focuses on a two-way denose-andforward relayng system usng non-coherent Dfferental Bnary hase-shft eyng DBS modulaton, whch has the welldefned relay denosng functon when channel state nformaton s unnown. We frst desgn the relay denosng functon and source decoders usng Maxmum elhood M prncples for the general case wth parallel relays. As the M denosng functon s hard to manpulate, we approxmate t as a mult-user detector followed by a physcal layer networ codng encoder and obtan the closed-form relay decodng error. For the snglerelay case, we show that the M source decoder s actually equvalent to the typcal DBS decoder for the relay-source channel and thus derve the exact end-to-end Bt Error Rate BER. To mnmze the average BER, we also nvestgate the power allocaton problem by use of asymptotc analyss at hgh Sgnal-to-Nose Rato SNR. We show that the optmal source power s nversely proportonal to the square root of the channel gan of the source-relay channel, and the optmal relay power decreases wth SNR. For the mult-relay case, though the exact analyss s ntractable, we develop upper bound and lower bound on BER and show that the dversty order s exactly. Index Terms Two-way relayng, dfferental modulaton, BER, dversty order, power allocaton. I. INTRODUCTION WIreless channel has negatve effects on sgnal propagaton n terms of channel fadng and path loss, and cooperatve communcatons, whch s asssted by a set of fxed or moble relay nodes, can elegantly overcome these shortcomngs by provdng dstrbuted spatal dversty and mang a more effcent use of transmt power []. As the termnals generally cannot transmt and receve on the same channel smultaneously due to hardware lmtatons, most of the recent lteratures focus on half-duplex relayng protocol, such as Amplfy-and-Forward AF and Decode-and-Forward DF [][3]. The DF relays frst decode the source nformaton and then forward a re-encoded sgnal, whle the AF relays just amplfy the receve sgnals subject to relay power constrants. However, the above half-duplex relayng protocol wll nevtably reduce the channel use. Ths s because both of the AF and DF relays use two tme phases to delver only one nformaton unt, whch ntroduces a pre-log factor on the spectral effcency [4]. The tradtonal selectve relayng protocol [][5] can partally recover such rate loss, as the relay nodes are actve only when necessary and thus Manuscrpt receved October, 00; revsed Aprl 5, 0; accepted Aprl 8, 0. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was R. Nabar. W. Guan and. J. Ray u are wth the Department of Electrcal and Computer Engneerng, Unversty of Maryland, College ar, MD 074, USA e-mal: {wguan, jrlu}@umd.edu. Dgtal Object Identfer 0.09/TWC save the redundant channel use. More recently, Two-Way Relayng TWR, where the two source nodes exchange ther nformaton at the same tme wth the help of the relay nodes, has drawn lots of attenton due to ts potental to fully recover the rate loss resulted from half-duplexng. There are generally two nds of TWR protocols dependng on the number of used tme phases,.e., Two-hase TWR -TWR and Three-hase TWR 3-TWR. In 3-TWR, two source nodes send nformaton to the relays successvely over the frst two phases, and the relays broadcast a mxture of the receved sgnal durng the thrd Broadcastng BC phase. The 3-DF-TWR s frst proposed n [6] for the sngle-relay case, where the relays perform Networ Codng NC [7] at bt-level through exclusve-or operatons. NC s power effcent n that the relays only need to send a sngle symbol, based on whch both sources can unquely decode the nformaton from the other end by use of ts own sde nformaton. It also shows there the 3-DF-TWR can acheve a maxmum throughput gan of 3 over the tradtonal one-way relayng that requres a total of 4 phases to complete the same nformaton exchange. In the 3-AF-TWR proposed n [8], the relay forwards a weghted sum of the sgnals receved n the frst two phases. By properly choosng the weghts, lower Bt Error Rate BER can be acheved than the tradtonal AF relayng. The -TWR maes one more step toward channel use savngs by lettng the two sources transmt smultaneously n a sngle Multple Access MA phase. Early wor on - AF-TWR and -DF-TWR can be found n [4], whch shows great enhancement on sum-rates from an nformaton theoretc vewpont. However, the proposed jont decode-and-forward protocol s hard to realze n practce wthout any specal multple access technque, as the source sgnals already combne n the ar and decodng them separately results n large BER. Wth such concerns, [9] proposes a new Denose-and-Forward DNF protocol, where the relays apply a denosng functon to map the receve sgnal nto another quantzed symbol that can be used by each source node to unquely decode the symbol transmtted from the other end. A smlar scheme called hyscal-ayer Networ Codng NC s proposed n [0], where the condton to guarantee one-to-one mappng between NC and NC s also gven. ater wor [] shows that -DNF-TWR has hgher sum-rates than -AF-TWR and -DF-TWR; [] derves the closed-form BER of - DNF-TWR wth coherent BS modulaton; and [8] and [3] reveal that -TWR generally has hgher sum-rates, whereas 3-TWR enjoys lower BER nstead. Whle TWR opens a door to mprove spectral effcency, most of the exstng wor [8], [9] and []-[3] all assume /$5.00 c 0 IEEE

2 GUAN and IU: ERFORMANCE ANAYSIS OF TWO-WAY REAYING WITH NON-COHERENT DIFFERENTIA MODUATION 005 that the termnals have full nowledge of Channel State Informaton CSI, whch on the other hand s hard to acqure n a fast-fadng envronment [3]. Such concerns motvate the non-coherent modulaton schemes, whch have been wdely examned for the tradtonal one-way relayng. For example, [4] develops the Maxmum elhood M decoders for DF relayng wth non-coherent bnary frequency-shft eyng modulaton and shows that the dversty order s roughly half of the number of relays; for sngle-relay case, [5] desgns the selectve relayng protocol and analyzes the correspondng BER; fnally, [6]-[8] propose relay selecton methods for the mult-relay case and show that the full dversty order can be acheved. In a lmted number of lteratures about TWR usng non-coherent modulaton, [9] desgns the non-coherent decoder for mnmum-shft eyng sgnals and valdates the throughput gan on a software rado testbed; [0] gves the symbol error rate for -AF-TWR wth relay selecton; and [] desgns a set of non-coherent decoders for both -AF- TWR and -DNF-TWR usng dfferental modulaton,. As summarzed above, -DNF-TWR wth non-coherent modulaton can beneft from both the hgh spectral effcency and reduced channel estmaton overhead. However, early wor on DNF protocol focuses manly on AWGN channels [9][0]. Although the decoder desgn ssue n the fadng channel has been partly addressed n [] for the sngle-relay system, the performance s manly evaluated through smulaton and there are no dscussons about resource allocaton. The adaptve denosng functon desgn n fadng channels s recently gven n [] where perfect nowledge about CSI s assumed. When CSI s unnown, how to optmze the denosng functon s stll an open problem except for the specal case wth BS modulaton. Indeed, a thorough nvestgaton nto such partcular scheme could not only shed some lght on the denosng functon desgn for the more general system usng hgher-order non-coherent modulatons, but also help to resolve other wreless networ desgn problems le relay deployment and resource allocatons, and such concerns motvate the current wor. Specfcally, we focus on a -DNF-TWR system usng non-coherent Dfferental BS DBS modulaton wth parallel relays. We frst derve the relay denosng functon and source decoder usng M prncples, and then proceed to analyze the correspondng decodng error at each termnal. As t s hard to manpulate the M denosng functon drectly, we approxmate t as a Mult-User Detector MUD followed by a NC encoder and obtan the closed-form relay decodng error. For the sngle-relay case, we reveal the equvalence between the M source decoder and the typcal DBS decoder for the relay-source channel, based on whch we obtan the exact end-to-end BER. We further nvestgate the power allocaton problem so as to mnmze the average system BER by use of asymptotc analyss, and show that the optmal source power s nversely proportonal to the square root of the channel gan of the source-relay channel, and the optmal relay power decreases wth Sgnal-to-Nose Rato SNR. For the mult-relay case, though the exact analyss s ntractable, we develop upper bound and lower bound on BER and show that the dversty order s exactly. We valdate all our results by computer smulatons. Fg.. r r s s.... r MA hase System model of -DNF-TWR. BC hase The rest of ths paper s organzed as follows: In Secton II, we descrbe the system model and desgn the relay denosng functon and source decoders. For sngle-relay case, exact error performance s gven n Secton III, where we also formulate the power allocaton problem. Then n Secton IV we analyze the dversty order of a mult-relay system, and we provde smulaton results n Secton V. Fnally some conclusons are gvennsectonvi. Notatons: Boldface lowercase letter a and boldface uppercase letter A represent vector n column form and matrx, respectvely. a and A represent the Eucldean norm of a vector a and the determnant of a square matrx A, respectvely., T and H stand for conjugate, transpose and conjugate transpose, respectvely. We shall use abbrevaton..d. for ndependent and dentcally dstrbuted, and denote Z CN μ, σ as a crcularly symmetrc complex Gaussan random varable Z wth..d. real part and magnary part N μ, σ.wedefne sgnx= f x>0 and 0 otherwse. Fnally, the probablty of an event A and the robablty Densty Functon DF of a random varable Z are denoted by A and fz, respectvely. II. SYSTEM MODE Consder a narrow-band -DNF-TWR system shown n Fg., where two sources S and S want to exchange nformaton wth the help of parallel relays. At the begnnng of the MA phase, the th =, source frst generates a sequence of..d uncoded BS symbols b n {, } of length, where n=,,..., s the symbol ndex. These raw symbols are then re-encoded through dfferental modulaton,.e., x n=x n b n for n=,,..., wth x 0= beng the reference symbol. The two sources then send the whole bloc of dfferentally encoded symbols smultaneously to all the relays durng MA phase. To facltate demonstratons, we defne a sequence of auxlary symbols bn=b n b n {, } for n=,,..., to ndcate whether the two raw BS symbols have the same sgns or not. Note that because each source nows ts own symbol, ths common nformaton bn s suffcent for both sources to decode the symbol from the other end. At the end of MA phase, the nth n=0,,..., symbol

3 006 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 0, NO. 6, JUNE 0 Σ Δ b n=,b n= Σ Δ b n=,b n= Σ Δ b n=,b n= Σ b n=,b n= = Σ,r = N 0, +, +I + N 0, +, Î = Σ,r = N 0, +, +I N 0, +, Î = Σ 3,r = N 0, +, +I + N 0,, Î Δ = Σ 4,r = N 0, +, +I + N 0,, Î 3 receved at the th =,,..., relay s then y n = s h MA, x n+ s h MA, x n+w MA n, where s =α s the th =, source power, s the total power and α [0, ] stands for the correspondng source power rato. h MA, 0,σ CN, s the ndependent channel coeffcent from the th =, source to the th =,,..., relay durng MA phase, where σ, s the channel gan. Here we assume that the channels reman unchanged wthn one bloc of length + ;however,no termnals now such CSI so as to elmnate the channel estmaton overhead. Fnally, w MA n CN 0,N 0 s the ndependent Addtve Whte Gaussan Nose AWGN at the th =,,..., relay wthn the nth n=0,,..., symbol nterval durng MA phase. Wth DNF protocol[9][0], the th =,,..., relay just maps the nth n=,..., receve symbol to another BS symbol ˆb r n {, } that can be used by each source to unquely decode the symbol transmtted from the other end. Here ˆb r n {, } can be regarded as an estmate of the auxlary symbol b n, so the relay denosng functon s actually equvalent to the decoder for bn. As no CSI s avalable, we use the sngle-symbol M decoder smlar to that proposed n [] throughout ths wor,.e., ˆbr n = arg max f y n bn, bn {,} where y n = y n,y n T s the vector of two consecutve receve symbols. It s easy to show that gven b n and b n, y n bn,b n CN 0, Σ b n,b n, where the condtonal covarance matrces are gven by 3 on the top of ths page. Here, = s σ, N 0 =α σ, s the channel SNR from the th =, source to the th =,,..., relay, = N 0 s the system SNR, and 0, Î 0 = I = 0 0 are two constant matrces. Based on the law of total probablty, the condtonal DF of y n can be expressed as f y n bn = f y n b n,b n. b n b n=bn 4 After some manpulatons, we can re-wrte the M decoder as ˆbr n =sgn ln lrf y n bn, 5 For the followng sngle-symbol decodng, we only requre that the channels eep unchanged wthn the nterval of two consecutve symbols. The quas-statc assumpton s just to smplfy the notatons. Actually bn s unformly dstrbuted. So the M decoder s equvalent to the maxmum a posteror decoder. The same argument holds for the followng source decoder. where lrf y n bn = g y n, Σ,r +g y n, Σ,r g y n, Σ 3,r +g y n, Σ 4,r 6 s the elhood Rato Functon RF of y n condtoned on bn, and g y, Σ = π Σ exp y H Σ y 7 s the DF of y CN 0, Σ. After decodng, the th =,..., relay re-encode {ˆbr n} nto n= t n=t n ˆb r n for n=,,..., through dfferental modulaton wth t 0=0 beng the reference symbol. Durng BC phase, all relays broadcast ther own dfferentally re-encoded symbols together through a set of orthogonal channels. It s worth notng that as the relays have no CSI, the typcal transmt dversty technque s unavalable here. So the co-channel relayng wll not brng any dversty gan compared to the sngle-relay case, as the broadcast sgnals would be randomly combned n the ar. In practce, the orthogonal relayng can happen n a fxed relay system where all relays operate on ts own dedcated channel. At the end of BC phase, the th =, sourcewll recevefromthe th =,..., relay r, n = r h BC, t n+w, BC n, n=0,,...,, 8 where r =β s the th =,..., relay power and β [0, ] stands for the correspondng relay power rato. h BC, 0,σ CN, s the ndependent channel coeffcent from the th =,..., relay to the th =, source durng BC phase, and we assume h BC, and h MA, are ndependent but have the same channel gan, whch s determned by the dstance between two termnals. However, all the results n ths wor can be easly extended to the more general case wth dfferent channel gans. Fnally, w, BCn CN 0,N 0 s the ndependent AWGN on the th =,,..., relaysource channel at the th =, source wthn the nth n=0,,..., symbol nterval durng BC phase. As explaned before, each source only needs to detect bn so as to decode the symbol from the other end. For example, f the decoded symbol for bn at source s ˆb s n=, then b n can be decoded as ˆb,s n=b n, otherwse ˆb,s n= b n f ˆb s n=. Agan we assume the th =, source uses the sngle-symbol M decoder based on the observatons {r, n} =,.e., ˆbs n = arg max f {r, n} bn {,} = bn, 9 where r, n= r, n,r, n T s the vector of two consecutve receve symbols from the th =,,...,

4 GUAN and IU: ERFORMANCE ANAYSIS OF TWO-WAY REAYING WITH NON-COHERENT DIFFERENTIA MODUATION 007 f {r, n} = bn = = ˆbr n {,} f r, n ˆb r n ˆbr n bn lrf r, n bn = g r, n, Σ,s g r, n, Σ,s F,r + g M,r +g r, n, Σ,s M,r r, n, Σ,s F,r 3 lrf y n bn = Σ3,r Σ,r cosh N 0, +, y Σ H nî y n,r cosh N 0,, y Σ HnÎ y n 3,r Σ,r Σ 3,r exp Σ,r Σ 3,r N 0, +, + y n 7 relay, and r, n ˆbr n CN 0, Σ ˆbr n,s where the condtonal covarance matrces are gven by Σ ˆbr Δ n=,s = Σ,s = N 0, +I + N 0, Î Σ ˆbr Δ, n=,s = Σ,s = N 0, +I N 0, Î 0 where, = r σ, N 0 =β σ, s the channel SNR from the th =,,..., relaytotheth =, source. As the sgnals from dfferent relays are ndependent condtoned on bn, we can rewrte the jont DF n 9 as gven on the top of next page, where we use the law of total probablty and the fact r, n s ndependent wth bn condtoned on ˆb r n. Based on, the M source decoder 9 can be smplfed to ˆbs n =sgn ln lrf r, n bn, = where lrf r, n bn gven on the top of ths page s the RF of r, n condtoned on bn, and M,r = ˆbr n = bn = = lrf y n bn bn =, 4 F,r = ˆbr n =bn = = lrf y n bn > bn = 5 are two nds of condtonal decodng error at the th =,,..., relay. The calculaton of these two terms s postponed to the next secton. Note that as both the relay decoder and source decoder 9 depend only on the secondorder statstcs of all channels, whch reman unchanged over tme, the whole system can beneft from a great reducton on channel estmaton overheads. III. ERFORMANCE ANAYSIS: SINGE-REAY CASE In ths secton, we wll examne the error performance of the proposed relay decoder and source decoder 9 for the sngle-relay case. Wthout loss of generalty, we assume only the th {,,..., } relay s actvated to assst the nformaton exchange between two sources. To optmze the end-to-end error performance, we shall also nvestgate the power allocaton problem. A. Relay Decodng Error By use of the law of total probablty, we can wrte the relay decodng error as Δ= ˆbr n = bn e,r = M,r + F,r, 6 where M,r and F,r are two nds of condtonal decodng error defned n 4 and 5, and both of them are related wth lrf y n bn. After substtutng 7 nto 6 and dong some manpulatons, we have 7 on the top of ths page, where coshx= ex +e x s the hyperbolc cosne functon. As t s really hard to analyze the error probablty based on the above RF, we use the followng approxmaton to facltate the analyss coshx max ex,e x = ex, 8 whch s qute tght when x s not too small. After such approxmaton, only exponental terms are left wth the exponent beng a quadratc form of y n, whch s analytcally tractable. After substtutng 8 bac nto 7, we wll arrve at lrf y n bn max g y n, Σ,r,gy n, Σ,r max g y n, Σ 3,r,gy n, Σ 4,r. 9 Now f we use 9 nstead n 5, t s easy to see that ths suboptmal decoder s actually a MUD ˆb,r n, ˆb,r n =arg max f y n b n,b n 0 b n {,},=, followedbyancencoderˆb r n=ˆb,r n ˆb,r n. That s, the relay frst jontly decodes the BS symbols b n and b n, and then maps the decoded symbols to a sngle BS symbol ˆb r n as an estmate of the ndcator symbol bn. As we shall see n the smulaton secton, ths suboptmal relay decoder wors almost as well as the M decoder 5 n all cases. The reason s that the two DFs of b n,b n correspondng to the same bn are actually well separated. As a result, the M regon of bn s very close to the drect unon of the ndvdual M regons of b n,b n, whch leads to the max operaton n 9.

5 008 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 0, NO. 6, JUNE 0 M,r = b n b n= a + b ˆy, n +a b ˆy, n a b ˆy, n +a + b ˆy, n ln th & ln th b n,b n b = 4,,, +, + mn,,,, +, + N 0, +, +, +, + h t,t,a,b,= 4abt t a t t b t + t exp t + t ln a 3 7 To characterze the error performance, let us frst calculate M,r. After substtutng 9 nto 4 and mang some manpulatons, we have on the top of ths page, where 4,,, +, + a = N 0, +, +, +, +, th =, +, +, +, +, 4 and b s gven n 3 on the top of ths page, and we defne ŷ n =ˆy, n, ˆy, n T = y n 5 as an auxlary random vector. Snce y n bn,b n CN 0, Σ b n,b n, ˆy, n and ˆy, n are actually ndependent exponental random varables condtoned on b n and b n. Therefore, 0 can be easly measured as M,r = h u,,u,,a,b, th, 6 where h t,t,a,b, gven n 7 on the top of ths page s a functon wth fve parameters, and u, = N 0, +, +,u, =. 8 N 0 In a smlar manner, we can show that F,r = h u 3,,u 4,,a,b, th, 9 where u 3, = N 0, +,u 4, = N 0, Fnally, pluggng 6 and 9 bac nto 6 leads to the closed-form relay decodng error. B. Source Decodng Error When there s only one actve relay n the system, the th =, source decoder can be reduced to ˆbs n=sgn ln lrf r, n bn = sgn ln lrf r, n ˆb r n Δ = ˆb r,s n, 3 where g r, n, Σ,s lrf r, n ˆb r n = 3 g r, n, Σ,s s the RF of r, n condtoned on ˆb r n. Note that the decoder on the second lne of 3 s actually a typcal noncoherent DBS decoder [3, Eqn.4-4-3] for the pontto-pont channel from the th {,,..., } relay to the th =, source, whose output ˆb r,s n s an estmate of the decoded symbol ˆb r n at the correspondng relay. Wth such equvalence relaton at hand, we can wrte the source decodng error as ˆbs n = bn = ˆbr,s n = bn where Δ = e,s = M,s + F,s, 33 M,s = ˆbs n = bn = = ˆbr,s n = bn =, 34 F,s = ˆbs n =bn = = ˆbr,s n =bn = 35 are two nds of condtonal decodng error at the th =, source, and we use the relaton ˆb s n=ˆb r,s n n After expandng 34 by use of the law of total probablty, we have 36 on the top of next page, where we use n a the fact that ˆb r,s n s ndependent of bn condtoned on ˆbr n, and n b we rely on the fact that the two nds of condtonal decodng error of a typcal non-coherent DBS decoder are equal and are gven by [3, Eqn.4-4-6] ˆbr,s n =ˆb r n = = ˆbr,s n = ˆb r n = Δ = D,s =, In a smlar way, we can derve F,s = D,s F,r + D,s F,r. 38 luggng 36 and 38 bac nto 33 we have e,s = D,s e,r + D,s e,r, 39 whch s the end-to-end BER at the th =, source.

6 GUAN and IU: ERFORMANCE ANAYSIS OF TWO-WAY REAYING WITH NON-COHERENT DIFFERENTIA MODUATION 009 M,s = a = ˆbr n {,} ˆbr n {,} ˆbr,s n = ˆb r n,bn = ˆbr n bn = ˆbr,s n = ˆb r n ˆbr n bn = b = D,s M,r + D,s M,r 36 η, = σ, + σ, 4σ, σ, mn σ,,σ, + σ, + σ, 4σ, ln σ, σ, σ, σ, + σ, 46 C. ower Allocaton Havng the closed-form BER 39, we are about to nvestgate the power allocaton among the two sources and the sngle relay so as to mnmze the average system BER, whch can be formulated as mn e = e,s + e,s s.t. α + α + β =, 0 α,α,β. 40 However, t s generally hard to drectly manpulate the exact BER 39, and the optmal soluton can only be derved through exhaustve search. In order to obtan one smple closed-form soluton, we choose to examne the asymptotc BER at hgh SNRs,.e.,. After some approxmatons, we can derve from 6, 9 and 37 M,r cm,r,c M,r = mnα σ,,ασ, F,r df,r ln d F,r,d F,r = ασ, +ασ, α α σ,. 4 σ, D,s e q D,s,qD,s = β,=, σ, After pluggng these approxmatons bac nto 6 and 39, we can obtan the asymptotc average BER at hgh SNRs,.e., c M,r + d F,r ln, 4 d F,r + q D,s + q D,s where we neglect the hgher-order terms. There are several observatons here. Frstly, t s easy to see that the BER s domnated by F,r, whch scales as ln at hgh SNRs. Therefore, more power should be allocated to the sources n order to reduce the relay decodng error. Secondly, the BER of the drect transmsson wth non-coherent DBS modulaton scales as [3, Eqn.4-4-8], whch decreases faster than the domnant error term F,r at hgh SNRs. In other words, -DNF-TWR s comparatvely not preferred than drect transmsson when SNR s ncreasng, and our smulaton results would show ths fact later. Fnally, t can be observed that F,r > M,r when source power s fxed and SNR s suffcently hgh. Ths s because t s relatvely easer to decode bn when the two source symbols have the same sgns, n whch case the two consecutve observatons y n and y n would have smlar envelopes at hgh SNRs. Now let us proceed to solve 40 by use of the asymptotc expresson 4. Note that the frst two terms n 4 depend only on source power rato α and α whle the last two terms only depend on β. So the optmzaton problem 40 can be resolved n two steps. In the frst step, we fx β and see to fnd the optmal source power,.e., d F,r c M,r + d F,r ln mn d F,r ln d F,r s.t. α + α = β, 0 α,α β. 43 where we neglect the term c M,r because t s much smaller than ln at hgh SNRs. Note that the functon φ x =x ln x s ncreasng when x<e, whch s the case for suffcently large. Therefore, t s equvalent to mnmzng d F,r nstead n 43, whose optmzer s { α opt = β α opt = β σ, σ, +σ, σ,. 44 σ, +σ, Clearly, the optmal source power s nversely proportonal to the square root of the channel gan of the correspondng source-relay channel. That s, more power should be allocated to the source that s far away from the relay, otherwse ts sgnal would be shadowed by that from the other end durng MA phase, whch ncreases the relay decodng error. Therefore, the above source power allocaton scheme actually provdes an elegant way to resolve the near-far problem. Next, f we plug 44 nto 4, t leads to an objectve functon that only nvolves the relay power coeffcent β. After some manpulatons, the second optmzaton problem can be formulated as mn η, β + η, β, s.t. 0 β, 45 where η, s gven n 46 on the top of last page and η, = σ, + σ, 4σ,. 47 σ, Note that we neglect the term β wthn the log functon n 46 when dervng the objectve functon n 45, as t s generally much smaller than at hgh SNRs. The optmzer of 45 can be easly derved as β opt = η, η, + η,. 48 It can be shown that β opt s a decreasng functon of SNR, whch concdes wth our prevous analyss that more power

7 00 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 0, NO. 6, JUNE 0 e,s U = { } ˆbU n = b n = s + ˆbU n =b n = s = { } DU s + D U b n = + s + b n = ˆbr,s n = b n = M,r + c D,s M,r + q D,s ˆbr,s n =b n = F,r + q D,s D,s + d F,r ln DU s + b n = + D U s + b n = + D U s = + D U s = + m D U s l D U s d F,r q l D,s + c M,rl q m D,s + d F,rm ln d F,rm should be allocated to the source as SNR s ncreasng. Another observaton s that the power allocaton coeffcents depend only on the channel gans and system SNR, whch are statc gven the nter-node dstances. The relay node can thus estmate these statstcs at the very begnnng of the transmsson by smply measurng the ncomng data power, and then feeds bac the calculated power allocaton coeffcents to the two source nodes. As such nformaton exchange s performed only once, the assocated overhead s actually neglgble. IV. ERFORMANCE ANAYSIS: MUTI-REAY CASE In ths secton, we shall turn our focus to the mult-relay case. However, the exact end-to-end BER analyss based on the M source decoder s not tractable due to the non-lnearty of the decson metrc. Alternatvely, we see to characterze the dversty order of the BER performance at hgh SNRs, whch reveals how the system performances mprove wth the number of relays. Followng s the man concluson of ths secton. roposton: The dversty order of -DNF-TWR wth non-coherent DBS modulaton s { log + e,s d = lm =, sodd log, s even =, 49 where e,s s the decodng error at the th =, source, and s the number of relays. The above result s somewhat counter-ntutve, as the dversty order s only about half of the number of relays. Such performance penalty s due to error propagaton, as the relays are assumed to forward whatever they decode wthout any error correcton. To prove ths, we see to fnd an upper bound and a lower bound on BER and show that they actually share the same dversty order as 49. A. BER Upper Bound In ths sub-secton, we would derve an upper bound on BER, the dversty order of whch provdes a lower bound on d n 49. Note that the M source decoder s optmum n the sense of mnmzng the decodng error, thus any suboptmal source decoder would lead to a strctly hgher BER. So we smply nvestgate a post-combnng decoder, where the th =, source frst apples the sngle-relay decoder 3 on each relay-source channel and obtans a set of estmates {ˆbr,s n}, and then feeds these estmates = nto a combner whose output s {,f Ds ˆbU s n = U > DU s, f D U s DU s, 50 { } where Ds U = m ˆb rm,s n = wth the complement set beng D s U. Now we shall analyze the BER of ths decoder at hgh SNRs. When s odd, the decson rule 50 s equvalent to ˆbU s n =f D U s +. So the decodng error at the th =, source can be wrtten n a smlar way as 33, whch s gven n 5 on the top of ths page. Note that the decodngs on dfferent relay-source channels are ndependent condtoned on bn, and the condtonal decodng errors at hgh SNRs for the th =,,..., branch can be derved from 36, 38 and 4 as gven n 5 and 53 on the top of last page. Therefore, we have 54 and 55 on the top of last page, where we neglect the hgher-order terms of. Clearly, e,s U has a dversty order of + n ths case as both of the two components have the same dversty orders. The case when s even can be characterzed n a smlar way. Now the decson rule 50 s reduced to ˆb U s n =f D U s +, and the decodng error s gven n 56 on the top of ths page. From 5 and 53, we can obtan the two nds of condtonal decodng error at hgh SNRs as 57 and 58 gven on the top of ths page. As the decodng error 56 s domnated by 57, ts dversty order s actually. Combnng these two cases would lead to the dversty order d gven n 49. B. BER ower Bound In ths sub-secton, we would nstead derve a lower bound on BER, the dversty order of whch provdes an upper bound

8 GUAN and IU: ERFORMANCE ANAYSIS OF TWO-WAY REAYING WITH NON-COHERENT DIFFERENTIA MODUATION 0 U e,s = { DU s D U s b n = } D U b n = + s +b n = q l D,s + c M,rl D U s +b n = U e,s = e,s = { D s + + D s + b n = D + s b n = { D s D s U = l D U s Ds U = + m Ds U b n = + Ds D b n = + s D s b n = D s +b n = + + q m D,s + d F,rm ln d F ln d F d F,rm } b n = 58 6 c + M 6 + } +b n = c M 65 d F ln + 66 d F on d n 49. Here we use a smlar technque proposed n [4]. Specfcally, we shall mae the followng two deal assumptons,.e., The relay-source channel s dstorton free,.e., r, n = t n, such that both sources can now the decoded symbols } {ˆbr n at all relays through dfferental = demodulaton; All relays have the same decodng ablty as the best relay,.e., M = mn M,r cm {,,...} where c M = df mn c M,r,and F = mn F,r {,,...} {,,...} where d F = mn d F,r. {,,...} Note that both of these two assumptons brng postve ln d F contrbutons to system performances, therefore helpng to lower the BER. e 9, the sngle-symbol M decoder at the th =, source can be wrtten as } ˆb s n = arg max {ˆbr f n bn bn {,} = D = sgn M s ln + D M s ln 59 F F { } where Ds = m ˆb rm n = wth the complement set beng D s. At hgh SNRs, both M and F approach 0 and, so the above decson rule s reduced to ln M ln F ˆb s n = {,f D s > D s, f D s D s, 60 whch s smlar to 50. So the error analyss can be done n the same way as we dd n the last sub-secton, and we shall sp some tedous ntermedate steps and drectly gve the fnal results. When s odd, the decodng error at the th =, source at hgh SNRs s gven n 6 63 on the top of ths page. Otherwse when s even, the decodng error at the th =, source at hgh SNRs s gven n on the top of ths page. Comparng 5 58 wth 6 66, we can observe that the two BER bounds have exactly the same dversty order as 49, thus completng the proof. V. NUMERICA RESUTS In ths secton, we present smulaton results for - DNF-TWR system usng non-coherent DBS modulaton. Throughout our smulatons, we use the path loss model σ = d 4, where σ s the channel gan and d s the dstance between two termnals. For smplcty, we normalze the dstance between two sources to, and we always place the relays on the lne connectng two sources. In all cases, BER refers to the average decodng error at source and source. Wthout specal explanaton, the transmt power s always equally splt among all termnals. We frst examne the performance of a sngle-relay system, where d,r and d,r are the dstances between the relay and two sources, respectvely. In Fg., we compare the BER of dfferent relay decoders wth the theoretcal results. The suboptmal relay decoder refers to the MUD followed by a NC encoder. It can be observed that there s almost no dfference between the M decoder and the suboptmal one, and both of them concde wth the theoretcal results gven

9 0 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 0, NO. 6, JUNE d r :d r =0.: SNR=0dB Optmal Suboptmal Equal SNR=5dB BER d r :d r =0.5:0.5 BER M Suboptmal Theoretcal Approxmaton SNR db 0. SNR =0dB d r Fg.. BER performances versus SNR. Fg. 4. BER performances wth power allocaton versus relay placement. 0 - d r :d r =0.: SNR=5dB SNR=5dB SNR=5dB 0. BER 0 - Optmal Suboptmal Equal d r :d r =0.4: SNR db Fg. 3. BER performances wth power allocaton versus SNR S Fg. 5. Comparson of -DNF-TWR and drect transmsson. Colored areas correspond to where -DNF-TWR can acheve lower BER. S by 39. Besdes, the asymptotc BER 4 s tght when SNR s suffcently hgh, e.g., when 5dB for d,r :d,r =0.:0.8 and when 5dB for d,r :d,r =0.5:0.5. The tghtness for the latter case s due to the hgh channel gans of both the sourcerelay channels, whch mae t easer to satsfy the hgh SNR assumpton. Then n Fg. 3 and Fg. 4, we proceed to study the benefts of power allocaton. The optmal scheme s found through exhaustve search, and the suboptmal one refers to that gven by 44 and 48 derved through asymptotc analyss. As we can see, the suboptmal scheme performs almost as well as the optmal scheme n most cases. From Fg. 4, we can observe some slght performance degradaton when the SNR s low and the relay s far from source. Ths s because the channel SNR from source to the relay s so low that the hgh SNR assumpton s not fully effectve on that channel. Compared wth equal power allocaton, about db SNR gan can be observed n Fg. 3 when d,r :d,r =0.:0.9. Such performance gan s dmnshng as the relay moves to the halfway between two sources, n whch case the equal power allocaton s nearoptmal. We also compare the -DNF-TWR wth drect transmsson usng the same modulaton scheme n Fg. 5. To do ths, we locate the two sources at 0.5, 0 and 0.5, 0, respectvely. We then compare the BER of these two systems at each grd on a square plane, and the colored areas correspond to where -DNF-TWR can acheve lower BER. To farly compare the performances, we splt the power equally between two sources for the drect transmsson; as for the -DNF-TWR, we use a mxed power allocaton scheme that frst determnes the source power rato by 44 and then fnds the optmal relay power through one-dmensonal search so as to reduce the tme complexty. As we can see, the preferred relay locatons are always concentrated around the halfway between two sources, otherwse the -DNF-TWR cannot beneft from the hgh channel gans resulted from the shorter source-relay dstances. Another observaton s that the preferred relay locatons actually shrn as SNR s ncreasng. Ths concdes wth our analyss n Secton III.C that drect transmsson s more preferred at hgh SNRs. Fnally n Fg. 6 and Fg. 7 we nvestgate the mult-relay

10 GUAN and IU: ERFORMANCE ANAYSIS OF TWO-WAY REAYING WITH NON-COHERENT DIFFERENTIA MODUATION 03 BER True BER BER lower bound BER upper bound =4 = SNR db = Fg. 6. BER performances wth multple relays all relays are at halfway between two sources. BER = = =3 =4 TxR x TxR x SNR db Fg. 7. BER performances wth multple relays all relays are equspaced between two sources. scenaro. We frst locate all relays at halfway between two sources, n whch case they should have the same decodng ablty. As we can see from Fg. 6, both of the BER bounds are tght n all cases, and they have the same slopes as we showed before. In Fg. 7, we further compare -DNF-TWR wth the typcal receve dversty system usng one transmt antenna and receve antenna TxRx, whch s well nown to have a dversty order of [3, Eqn.4-4-8]. It s clear that the dversty order of the system havng relay or relays s as TxRx, and the system havng 3 relays or 4 relays has a dversty order of as TxRx, whch valdates our proposton 49. It should be mentoned that as all relays operate on orthogonal channels, addng more relays would reduce the spectral effcency. Snce the dversty gan s acheved at a double loss of spectral effcency, t s better to deploy only a small number of relays n practcal systems so as to trade off these two performance measures. VI. CONCUSION AND FUTURE WOR In ths wor, we have developed M decoders for - DNF-TWR system usng non-coherent DBS modulaton and analyzed the correspondng error performances. For the sngle-relay case, the closed-form BER s obtaned after approxmatng the M relay decoder as the MUD followed by a NC encoder, and a near-optmal power allocaton s derved based on asymptotc analyss at hgh SNRs. For the multrelay case wth parallel relays, though the exact analyss s ntractable, we prove that the dversty order s exactly by developng proper bounds on BER performances. Future wor may focus on denosng functon desgn of -DNF- TWR system usng hgher-order non-coherent modulatons, whch s stll an open problem. One may also nvestgate the selectve relayng protocol to recover the full dversty order. REFERENCES [] R. abst, B. H. Wale, D. C. Schultz,. Herhold, H. Yanomeroglu, S. Muherjee, H. Vswanathan, M. ott, W. Zrwas, M. Dohler, H. Aghvam, D. D. Falconer, and G.. Fettwes, Relay-based deployment concepts for wreless and moble broadband rado, IEEE Commun. Mag., vol. 4, no. 9, pp , Sep [] J. N. aneman, D. N. C. Tse, and G. W. Wornell, Cooperatve dversty n wreless networs: effcent protocols and outage behavor, IEEE Trans. Inf. Theory, vol. 50, no., pp , Dec [3]. J. R. u, A.. Sade, W. Su, and A. wasns, Cooperatve Communcatons and Networng. Cambrdge Unversty ress, 008. [4] B. Ranov and A. Wttneben, Spectral effcent protocols for halfduplex fadng relay channels, IEEE J. Sel. Areas Commun., vol. 5, no., pp , Feb [5] A. S. Ibrahm, A.. Sade, W. Su, and. J. R. u, Cooperatve communcatons wth relay selecton: when to cooperate and whom to cooperate wth? IEEE Trans. Wreless Commun., vol. 7, no. 7, pp , July 008. [6]. arsson, N. Johansson, and.-e. Sunell, Coded b-drectonal relayng, n roc. IEEE VTC-Sprng, vol., pp , May 006. [7] R. Ahlswede, N. Ca, S.-Y. R., and R. W. Yeung, Networ nformaton flow, IEEE Trans. Inf. Theory, vol. 46, no. 4, pp. 04-6, July 000. [8] R. H. Y. oue, Y. H., and B. Vucetc, ractcal physcal layer networ codng for two-way relay channels: performance analyss and comparson, IEEE Trans. Wreless Commun., vol. 9, no., pp , Feb. 00. [9]. opovs and H. Yomo, The ant-pacets can ncrease the achevable throughput of a wreless mult-hop networ, n roc. IEEE Internatonal Conference on Communcaton, pp , June 006. [0] S. Zhang, S. C. ew, and.. am, hyscal-layer networ codng, n ACM MOBICOM, os Angeles, Sep []. opovs and H. Yomo, hyscal networ codng n two-way wreless relay channels, n roc. IEEE Internatonal Conference on Communcaton, pp , June 007. [] E. C. Y. eh, Y.-C. ang, and Y.. Guan, ower control for physcallayer networ codng n fadng envronments, n roc. IEEE ersonal, Indoor and Moble Rado Commun., pp. -5, Sep [3]. opovs and H. Yomo, Wreless networ codng by amplfy-andforward for b-drectonal traffc flows, IEEE Commun. ett., vol., no., pp. 6-9, Jan [4] D. Chen and J. N. aneman, Modulaton and demodulaton for cooperatve dversty n wreless systems, IEEE Trans. Wreless Commun., vol. 5, no. 7, pp , July 006. [5] T. Hmsoon, W.. Srwongparat, W. F. Su, and. J. R. u, Dfferental modulaton wth threshold-based decson combnng for cooperatve communcatons, IEEE Trans. Sgnal rocess., vol. 55, no. 7, pp , July 007. [6] T. Hmsoon, W.. Srwongparat, W. Su, and. J. R. u, Dfferental modulaton for multnode cooperatve communcatons, IEEE Trans. Sgnal rocess., vol. 56, no. 7, pp , July 008. [7] J. H. Yuan, Y. H., and. Chu, Dfferental modulaton and relay selecton wth detect-and-forward cooperatve relayng, IEEE Trans. Veh. Technol., vol. 59, no., pp. 6-68, Jan. 00.

11 04 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO. 0, NO. 6, JUNE 0 [8] Y.. Zhu,. Y. am, and Y. Xn, Dfferental modulaton for decodeand-forward multple relay systems, IEEE Trans. Commun., vol. 58, no., pp , Jan. 00. [9] S. att, S. Gollaota, and D. atab, Embracng wreless nterference: analog networ codng, n ACM SIGCOMM 007, Aug [0]. Y. Song, G. Hong, B.. Jao, and M. Debbah, Jont relay selecton and analog networ codng usng dfferental modulaton n two-way relay channels, IEEE Trans. Veh. Technol., vol. 59, no. 6, pp , July 00. [] T. Cu, F. F. Gao, and C. Tellambura, Dfferental modulaton for two-way wreless communcatons: a perspectve of dfferental networ codng at the physcal layer, IEEE Trans. Commun., vol. 57, no. 0, pp , Oct [] T. oe-ano,. opovs, and V. Taroh, Optmzed constellatons for two-way wreless relayng wth physcal networ codng, IEEE J. Sel. Areas Commun., vol. 7, no. 5, pp , June 009. [3] J. roas, Dgtal Communcatons, 4th edton. McGraw-Hll, 00. We Guan receved the B.S. n Electrcal Engneerng and Fnance double degree n 006, and M.S. wth hghest honor n Electrcal Engneerng n 009, both from Shangha JaoTong Unversty, Shangha, Chna. Now he s a h.d. student n the Department of Electrcal and Computer Engneerng at Unversty of Maryland, College ar. Hs current research nterests are n the areas of wreless communcatons and networs, ncludng cooperatve communcatons and networ codng. He receved the st rze n the 8th Natonal hyscs Contest, Shangha, and the A. James Clar School of Engneerng Dstngushed Graduate Fellowshp from Unversty of Maryland, College ar n J. Ray u F 03 s named a Dstngushed Scholar-Teacher of Unversty of Maryland, College ar, n 007, where he s Chrstne m Emnent rofessor of Informaton Technology. He serves as Assocate Char of Graduate Studes and Research of Electrcal and Computer Engneerng Department and leads the Maryland Sgnals and Informaton Group conductng research encompassng broad aspects of wreless communcatons and networng, nformaton forenscs and securty, multmeda sgnal processng, and bomedcal engneerng. Dr. u s the recpent of numerous honors and awards ncludng IEEE Sgnal rocessng Socety Techncal Achevement Award and Dstngushed ecturer. He also receved varous teachng and research recogntons from Unversty of Maryland ncludng unversty-level Inventon of the Year Award; and oole and ent Senor Faculty Teachng Award and Outstandng Faculty Research Award, both from A. James Clar School of Engneerng. An ISI Hghly Cted Author n Computer Scence, Dr. u s a Fellow of IEEE and AAAS. Dr. u s resdent-elect and was Vce resdent - ublcatons of IEEE Sgnal rocessng Socety. He was the Edtor-n-Chef of IEEE Sgnal rocessng Magazne and the foundng Edtor-n-Chef of EURASI Journal on Advances n Sgnal rocessng.