Diversity Combining of Signals with Different Modulation Levels in Cooperative Relay Networks 1

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1 Dversty Combnng of gnas wth Dfferent oduaton Leves n Cooperatve Reay Networks Akram Bn edq and Ham Yankomerogu Broadband Communcatons and Wreess ystems BCW Centre Department of ystems and Computer Engneerng Careton Unversty Ottawa Ontaro K 5B6 Canada e-ma: {akram ham}@sce.careton.ca Abstract- In dgta cooperatve reayng sgnas from the source-destnaton and reay-destnaton nks are combned at the destnaton to acheve spata dversty. These sgnas do not necessary beong to the same moduaton scheme due to the varyng channe quates of the two nks. In ths paper we present nove and ow compexty schemes for dversty combnng of sgnas wth dfferent moduaton eves. We start by deveopng the optmum souton as a maxmum kehood detector LD. Due to ts hgh compexty we propose two other recever structures that we refer to as soft-bt maxmum kehood detector BLD and soft-bt maxmum rato combner BRC. The proposed schemes are smpe bt-by-bt detectors and ony. db nferor to the LD n performance. The BLD provdes ony margna performance gan over BRC through the computaton of the condtona probabty densty functons of the soft-bts. Consequenty the BRC s a more attractve and practca souton. The performance of BRC s compared to that of seecton combnng whch s the current approach n the terature for combnng sgnas wth dfferent moduatons. The BRC aong wth ts smpcty outperforms seecton combnng by amost db wthout bandwdth oss or the need for extra channe state nformaton. The BRC scheme can be vewed as a more genera form of the cassca maxmum rato combner RC. I. INTRODUCTION Next generaton wreess networks requre hgh data rates to meet the demand of newy emerged appcatons. The use of reays n the wreess network was shown to be a key technoogy to acheve these requrements []. Reays can be cassfed nto dgta and anaogy reays. Anaog reays ampfy and forward AF the receved sgna whe dgta reays decode and forward DF a regenerated verson of the receved sgna. In ths work dgta reayng s consdered as t s the focus of most of the next generaton wreess networks standards such as IEEE 8.6 []. Two key strateges proposed for ncreasng the data rate n reay networks are cooperatve reayng [][] and adaptve moduaton and codng AC. Whe the former s used to mprove the quaty of the nks by combnng the sgnas receved from the base staton B and the reay statons the atter s used to optmze the transmsson rate accordng to the channe condtons. In [5] and [6] t s shown that the average throughput of the wreess network can be sgnfcanty ncreased by combnng the two strateges. When AC s utzed the sgnas reachng the user termna UT from B and reay staton R don t necessary beong to the same moduaton yet they contan the same nformaton bts. In order to acheve spata dversty for sgnas wth dfferent moduaton eves seecton combnng rather than maxmum rato combnng RC was utzed because t was not known yet how to do RC for sgnas wth dfferent moduaton eves [5][6]. To the best of the authors knowedge the optma technque of combnng sgnas wth dfferent moduaton schemes s not nvestgated yet and ths s the gap that ths paper aspres to f. The need for such a technque arses from the nature of reay networks that mpose dfferent channe condtons on dfferent nks. For exampe snce the reays are fxed and can be nstaed at strategc ocatons reabe ne-of-sght LO nks can be estabshed between B and R and hence arger consteatons can be used to acheve hgh data rates. However because of the mobty of the users the nks from R to UT are not necessary as reabe so smaer consteatons can be used to ensure reabe transmsson. In order to acheve spata dversty at UT t s mperatve to estabsh an optma dversty combnng scheme for dfferent moduatons. The man focus of the paper w be on fxedreay networks. Nevertheess the extenson to the case of nomadc or mobe reays s straght forward. The rest of the paper s organzed as foows. In ecton II the system mode s presented. The optma detector s deveoped n ecton III. ecton IV expans the proposed schemes. muaton resuts of the proposed schemes are presented n ecton V foowed by concusons n ecton VI. Notatons: CNσ represents a crcuar symmetrc compex Gaussan random varabe wth zero mean and varance σ ; for a random varabe X X = E{X } denotes ts mean; f x s the condtona probabty densty functon X y= c of X evauated at x gven that y=c; for a compex number C Re{C} and Im{C} denote the rea and magnary parts of C respectvey; C * s the compex conugate of C; g x; σ denotes the Gaussan probabty densty functon PDF wth mean and varance g x; σ = / σ π exp x /σ. σ.e. Ths work s supported by an Ontaro Graduate choarshp OG. A concse verson of ths work was presented n the Wreess Word Research Forum eetng WWRF.

2 II. YTE ODEL We consder a muthop network of L transmttng nodes comprsng of L- Rs and a B and a recevng UT a havng a snge antenna. Ths ayout s shown n Fg.. The Rs are used to assst a UT at the ce-edge or otherwse sufferng from poor channe condtons. The Rs decode the sgnas they receve from B and forward them to the UT. The transmttng nodes transmt on L orthogona channes.e. they do not nterfere wth each other. For smpcty we consder tme dvson mutpe access TDA to nsure orthogona transmsson from a the nodes. The transmttng node B or R where {...L-} uses Quadrature Amptude oduaton QA wth Gray codng. Each QA symbo carres K bts where K =og and s the th moduaton eve. Wthout oss of generaty the QA consteaton has an average energy per bt equa to unty. The frame s dvded nto L sub-frames.e. one sub-frame for each transmttng node. A sub-frames contan the same sequence of C bts denoted by {s s...s C }. The th sub-frame conssts of T QA symbos each denoted by where {...T -}. The symbo contans the bt sequence {s K+ s K+... s +K }. Note that dfferent nodes w be assgned dfferent number of symbos dependng on ther moduaton schemes.e. T =C/K. nce T s an nteger C must be a common mutpe of {K K... K L- }. Wthout oss of generaty C w be used as the Least Common utpe LC of {K K... K L- }. In the frst sub-frame B broadcasts T QA symbos to a Rs and the UT. nce the Rs can be nstaed at strategc ocatons LO transmsson between B and the Rs can be acheved. Therefore the Rs can reaby decode the sgnas wth neggbe errors [5]. In the th sub-frame R forwards T symbos to UT usng QA moduaton. The frame structure s depcted n Fg.. B T symbos C... R R R R L- Fgure. ystem ode bts T ub-frame "" B T symbos C... UT bts T ub-frame "" R In the th sub-frame and the th symbo the receved sgna at UT s r and gven by r = α + n. The compex addtve whte Gaussan nose AWGN s represented by n and t s modeed as CN N /. The channe coeffcent between the transmttng node and UT s denoted by α and t captures the effects of both path oss and sma scae fadng due to mutpath propagaton. Fat sow fadng s consdered.e. the channe does not change for the whoe sub-frame. It s assumed that α s are known at the recever and modeed as ndependent CN σ wth σ = E{ α }. If fu channe state nformaton CI s avaabe at B optmzng the moduaton eves for a the transmttng nodes can mprove the end to end throughput sgnfcanty. uch optmzaton s studed extensvey n [5] and [6] and w not be repeated here. The nstantaneous sgna to nose rato NR of the nk from node to UT s γ = α / N and the average NR s γ = σ / N. After recevng a the sub-frames UT can utze the sgnas receved from L ndependent branches and acheves spata dversty. Remarks: - For mathematca competeness the most genera case of L- reays s consdered. However gven that each reay requres an orthogona channe t w be dffcut n practce to have more than two reays due to the mted rado resources. - For smpcty channe codng s not consdered n ths paper. However the proposed scheme works equay we as ong as a transmttng nodes use the same channe codng scheme. III. OPTIAL DETECTOR A. -QA moduaton wth Gray codng In ths research we focus on square -QA moduatons wth Gray codng snce these moduaton schemes are the most popuar schemes n wreess networks []. For a gven sequence of bts {s s...s K } where s {-} we present a nove mathematca mode that maps these K bts nto a Gray-coded -QA symbo. The mode s gven by s s... s = χ + β d K K / K / where d s an arbtrary constant used to contro the energy of the consteaton. For convenence the average energy per bt s fxed to unty by settng [7]: T... T symbosc bts ub-frame "" R Fg.. Frame tructure og d = T L symbos C bts L L L L L... L TL ub-frame "L-" R L-

3 The coeffcents χ and β can be computed recursvey as K / K / foow: sk / k = χk = k sk / k + χk + < k K / sk k = βk = k sk k + βk + < k K / The same mode derved above can be used to generate dfferent Gray-coded consteatons by ether re-abeng the bts or by negatng the sgn of the bts. The foowngs are exampes of -QA 6-QA and 6-QA Gray-coded symbos: s s = sd s d 6 s s s s s s + d s s = d s s s s s s = s s s s s s + + d Ths mode s essenta for the LD whch s descrbed n the next secton. B. axmum Lkehood Detector LD The optmum souton to the probem at hand s the LD. For addtve whte Gaussan nose the LD reduces to a mnmum dstance cassfer [7]. Consequenty the LD decdes on the sequence [ sˆ... sˆ C ] that satsfes the foowng crteron: [ sˆ... sˆ ] = C L T s... sc = = + + d arg mn r α s s... s K+ K+ + K where s s... s s gven by. K+ K+ + K Athough the LD acheves the optmum performance t has exorbtant computatona compexty. For exampe to perform the decodng n the case where = K = =6 K = and =6 K =6 the LD decodes C=LC{6}= bts onty. Ths requres computatons whch s ceary too compex. For ths reason we are motvated to provde practca schemes wth much reduced compexty and comparabe performance to the LD. The proposed recever schemes are descrbed n the subsequent secton. IV. THE PROPOED RECEIVER CHEE A. oft-bt axmum Lkehood Detector BLD uboptmum schemes are proposed to overcome the compexty of the LD. In the LD the compexty arses from the fact that dfferent moduaton schemes carry dfferent number of bts per symbo. As a resut bt-by-bt or symboby-symbo decodng s not possbe. A trva remedy to ths probem s to decode the bts from each nk and perform dversty combng on the hard bts. However t s found out through smuaton that ths souton doesn t acheve fu dversty because of the ost soft nformaton n the hard decodng. In order to avod osng the soft-bt nformaton we propose to map the receved -QA soft-symbo r nto K soft-bts. Then decodng can be preformed on the soft-bts whch s a bt-by-bt detecton rather than detectng a sequence of bts onty. To extract soft-bts from a soft-symbo the ogarthm of kehood rato LLR can be used. For Gray-coded -QA schemes descrbed by the LLR can be we approxmated by the foowng recursve expresson [8][9]: * Re{ α r } k = K k + K d α K+ k < k K + k= 6 * K Im{ α r } k = + K k+ K d α K+ k + < k K where s K + k s the K +k th soft-bt generated from the receved soft-symbo r. Note that mutpyng the receved soft-symbo by the conugate of the channe coeffcent s necessary to undo the phase rotaton caused by the channe. The BLD whch s bt-by-bt detector performs maxmum kehood detecton on the soft-bts. It decdes on the bt ŝ for {...C-} accordng to the foowng crteron: sˆ = fs sl s L fs sl s s s = > = L 7 sˆ = otherwse nce the symbos receved from dfferent nodes are statstcay ndependent the ont condtona PDF s the mutpcaton of the margna condtona PDFs. Hence equaton 6 reduces to: L L sˆ = f fs s = > fs s = = = 8 sˆ = otherwse From equaton 6 t s easy to see that the margna condtona PDFs of s K + k s the same as that of k for {...T -}. Thus t suffces to provde the expresson of the condtona PDFs of k for k {...K }. The condtona PDF of the soft-bts can be obtaned from 6 by usng the foowng propertes that appy to any random varabe X and constant c: fx x + fx x x f X x = x < f X + c x = fx x c The condtona PDFs of the soft-bts are gven by: f k sn=± x = K / ± gx ; nm α d α / N k= K m= K f k s x η n k fs k s x η n k k x η =± + =± + < 9 k K < k x > ηk K f / k K s x k K n=± <

4 K k + where ηk = d α and n {...K }. The eements of the matrces + and are tabuated n Tabe I for the moduaton schemes descrbed by. Note that the detecton rue n ths case ddn t smpfy to a mnmum dstance cassfer as n the case of the LD because the condtona PDFs of the soft-bts are not Gaussan. B. oft-bt axmum Rato Combner BRC To avod the computaton of the condtona PDF's and have a scheme that has compexty comparabe to the cassca RC we propose BRC. The BRC weghts the soft-bts accordng to ther reabtes and adds them n a way smar to RC hence the name BRC. The dfference n reabtes comes as a resut of two factors. Frst the soft-symbos consequenty the softbts experence dfferent channe condtons. nce the softbts are aready weghted accordng to ther channe condtons by the defnton n 6 there s no need to weght them agan n the combnng. econd the reabtes of the soft-bts vary accordng to ther moduaton eves. Thus we propose to weght the soft-bts wth d whch s gven by. The ratona behnd choosng d n such a manner s to scae the soft-bts that are generated from denser consteatons wth ess weght as they are more vunerabe to nose and vce versa. Consequenty the BRC decdes on the bt ŝ for {...C} accordng to the foowng crteron: L sˆ = f d s > = sˆ = otherwse. V. IULATION REULT In ths secton we present smuaton resuts of our schemes. The condtona PDFs for the soft-bts generated from a soft 6-QA symbo.e. =6 are shown n Fg. and Fg.. In both cases the theoretca condtona PDFs gven by 9 are n good agreement wth the smuaton resuts and t s evdent that the condtona PDFs don t foow a Gaussan dstrbuton for ow NR. To compare the performance of BLD BRC and LD we consder reay networks wth L= snge reay and L= two reays. For smpcty we assume the average channe condtons to be the same n a the nks.e. γ = γ for {...L-}. Tabe II shows the oss n NR of the BLD and BRC schemes compared to the optmum LD scheme for dfferent scenaros. It s cear that both BLD and BRC have very cose performance to the LD scheme degradaton s ess than. db. The oss n NR was measured at a of - whch s a reasonabe vaue for uncoded schemes. Nevertheess t was observed that the oss n NR vanshes at very ow. The BLD provdes neggbe performance gan compared to the BRC at the expense of the computaton of the condtona PDFs of the soft-bts. For these reasons throughout ths dscusson we w focus on BRC snce t s very smpe to mpement wth neggbe degradaton compared to the LD and BLD schemes. Fg. 5 and Fg. 6 show the performance of BRC for combnng sgnas wth dfferent moduaton schemes for snge and two reays respectvey. It s cear that a curves n Fg. 5 decay orders of magntude per decade dversty order of and a curves n Fg. 6 decay orders of magntude per decade dversty order of. Consequenty BRC acheves fu dversty dversty order of L. Note that the resuts for the case where = BPK were not shown snce both BPK and QPK have the same performance. In Fg. 7 we compare the performance of BRC wth the conventona seecton combnng n a snge reay network L=. We consder the cases where the average NR n the B-UT nk γ s ess than the average NR n the R -UT nk γ by 5 and 5 db. Ths represents practca scenaros where UT s coser to R than B. For a cases our scheme outperforms conventona seecton combnng by amost db. For the sake of presentaton we use the case where = and =6; however the same gan has been observed for dfferent set of moduatons. It s nterestng to note that ths s cose to the gan that cassca RC provdes over seecton combnng n the case of combnng sgnas wth the same moduaton eves. VI. CONCLUION In the current terature dversty combnng of sgnas wth dfferent moduaton eves n cooperatve reay networks has been addressed by means of seecton combnng whch s far from optma. In ths paper we have proposed optma as we as near-optma and ow compexty dversty combnng schemes of sgnas wth dfferent moduaton eves. The optmum detector s deveoped as an LD detector. To overcome the compexty of the LD we have proposed BLD and BRC whch are smpe bt-by-bt detectors and are ony. db nferor to the optma LD n performance. The BLD provdes ony margna performance gan over BRC through the computaton of the condtona PDFs of the soft-bts. Consequenty the BRC s a more attractve practca souton. The BRC aong wth ts smpcty outperforms seecton combnng by amost db wthout any bandwdth oss and wthout the need for extra channe state nformaton. The proposed scheme can be vewed as a more genera form of the cassca RC and t s an essenta scheme to ncorporate cooperatve dversty wth AC n next generaton wreess networks. REFERENCE [] B. H. Wake R. Pabst D. chutz P. Herhod H. Yankomerogu. ukheree H. Vswanathan. Lott W. Zrwas. Doher H. Aghvam D. D. Faconer and G. P. Fettwes Reay-based depoyment concepts for wreess and mobe broadband rado IEEE Commun. ag. vo. no. 9 pp ep.. [] Wmax Forum obe WAX Part I: A Technca Overvew and Performance Evauaton June 6.

5 [] N. Laneman D. Tse and G. Worne Cooperatve dversty n wreess networks: Effcent protocos and outage behavor IEEE Trans. On Informaton Theory vo. 5 pp. 6 8 Dec.. [] J. Boyer D. D. Faconer and H. Yankomerogu uthop dversty n wreess reayng channes IEEE Trans. Commun. vo. 5 no. pp. 8-8 Oct.. [5] B. Can H. Yankomerogu F. Onat E. Carvaho and H. Yomo Effcent Cooperatve Dversty chemes and Rado Resource Aocaton for IEEE 8.6 IEEE WCNC 8 arch Apr 8 Las Vegas UA. [6]. Hares H. Yankomerogu and B. Hashem "Dversty- and AC adaptve moduaton and codng-aware routng n TDA peer-to-peer muthop networks" IEEE GLOBECO -5 December an Francsco CA UA. [7] J. G. Proaks Dgta Coomuncatons. cgraw-h. [8]. Le Goff A. Gaveux and C. Berrou Turbo-codes and hgh spectra effcency moduaton Proc. ICC'9 New Oreans LA ay 99. [9] V. Aue and R. Nuessgen. ethod for generatng soft-bt nformaton from Gray coded sgnas. U.. Patent 967 ay.. Tabe I. The matrces oduaton Leve + and + for dfferent -QA consteatons. + = = muaton γ = db Theoretca γ = db muaton γ = db Theoretca γ = db muaton γ = db Theoretca γ = db γ = db γ = db γ = db 5d 6 d 6 d 6 d 6 d 6 Fgure. Condtona PDF of gven that s = s = generated from a 6-QA symbo for dfferent channe condtons. = = = =6 = =6 =6 =6 =6 =6 =6 =6 + 6 = = = 5 7 = Tabe II. Loss n NR db at = - of BLD and BRC compared to the optmum LD. cenaro BLD BRC = =6.. = = =6 =6.6.8 = = =6..69 = = =6..7 = =6 = =6 =6 =6..5 =6 =6 = γ db Fgure 5. performance of dversty combnng usng BRC for L=. = = = = = =6 = = =6 = =6 =6 = =6 =6 =6 =6 =6 =6 =6 =6 =6 =6 =6 =6 =6 = muaton γ = db Theoretca γ = db muaton γ = db Theoretca γ = db muaton γ = db Theoretca γ = db γ = db γ db Fgure 6. performance of dversty combnng usng BRC for L=. =6 = BRC =6 = eecton Combnng.8.6. γ = db γ = db γ =γ 5 db γ =γ 5 db. γ =γ 5 db 5d d d d d 5d 7d Fgure. Condtona PDF of generated from a 6-QA symbo gven that s = s = for dfferent channe condtons γ db Fgure 7. performance comparson of eecton Combnng wth BRC.