Join Transmier-Receiver UWB Rake Design in hepresenceofisi Nazlı Güney 1, Hakan Deliç 1, 1 Wireless Communicaions Laboraory Deparmen of Elecrical and Elecronics Engineering Boğaziçi Universiy Bebek 3434 Isanbul, Turkey E-mail: {naz,delic}@bounedur Geer Leus Faculy of EEMCS Delf Universiy of Technology 68 CD Delf, The Neherlands E-mail: gjleus@udelfnl Absrac In his paper, he performance of a ime-division duplex ulra-wideband sysem ha has a ransmier/receiver pair of rake combining srucures is opimized, where he oal number of rake fingers o be deployed a he ransmier and he receiver is fixed I is shown ha here exiss an opimum disribuion of fingers beween he wo srucures for sysems wih inersymbol inerference, which maximizes he signal-oinerference plus noise raio Depending on he oal number of rake fingers and/or pos-rake fingers, ie, hose a he receiver, he opimum placemen of pos-rake fingers changes as he simulaion resuls demonsrae Index Terms pre-rake, rake, ulra-wideband (UWB), inersymbol inerference I INTRODUCTION Ulra-wideband (UWB) sysems have a much lower fading margin han narrowband sysems, since he wide sysem bandwidh enables fine resoluion in ime of he received mulipahs [1] Experimenal invesigaions of he wideband indoor channel lying in he -8 GHz band confirm his fac and indicae ha rake receivers can achieve significan diversiy gains by collecing he oal signal energy disribued over a large number of pahs [] Furhermore, for imedivision duplex (TDD) sysems i is possible o move he rake srucure from he receiver o he ransmier by making use of he channel informaion esimaed in he reverse link To be specific, he ransmier scales and delays he original ransmied signal in such a way ha he operaion of mulipah combining is already performed when he signal arrives a he receiver, in which case he receiver may use he convenional design ha unes o a single pah [3], [4] Pre-raked ransmission has iniially been proposed in [3] for code division muliple access sysems, where is performance is shown o be equivalen o he convenional rake receiver over single-user links wih moderae daa raes Laer sudies conduced for UWB sysems in paricular, such as [5], indicae ha pre-raking may lead o suppressed inersymbol inerference (ISI) wih high daa raes, since he maximal raio combining (MRC) sage of rake recepion, which booss inerference energy a he receiver, is avoided This desirable propery of pre-raking is accompanied by a larger number of mulipahs a he receiver han ha exiss for he channel impulse response beween he ransmier and he receiver When properly exploied, hose pahs allow an improved performance compared o he pre-rake or pos-rake, ie, he rake srucure a he receiver, only sysems The pre/pos-rake srucure in [6], for insance, uses an MRC pos-rake o collec all of he mulipahs Moving one sep fuher, he eigenprecoder proposed in [7] has is pre- and pos-rake weighs deermined joinly o maximize he signal-o-noise raio (SNR) a he receiver The principal raio combining (PRC) pre/pos-rake wih a flexible number of pre-rake fingers [8] is a variaion of he eigenprecoder The PRC pre/pos-rake has o have an unlimied number of pos-rake fingers for opimum performance, since he received SNR increases wih he number of pre-rake fingers, which in urn requires a larger number of pos-rake fingers An infeasible number of pre- and pos-rake fingers may have o be deployed in oal a he ransmier and he receiver for UWB channels ha end o be exremely frequency-selecive The problem of how o disribue a fixed oal number of rake fingers beween he ransmier and he receiver was addressed previously in [9] Specifically, i is shown ha in he absence of ISI, here is an opimum number of pre-rake fingers, and hus pos-rake vecors, ha maximizes he received SNR when he pos-rake srucure combines he firs arriving pahs In his paper, i is demonsraed ha he presence of ISI resuls in a ransmier/receiver design ha does no allow a closed form expression for he pre- and pos-rake vecors, bu raher an ieraive algorihm has o be used o arrive a he rake vecors ha are opimum in he sense ha hey maximize he signal-o-inerference plus noise raio (SINR) of he decision saisic Moreover, simulaions indicae ha depending on he oal number of rake and/or pre-rake fingers, he opimum placemen of he pos-rake fingers becomes a difficul problem, which may even require an exhausive search in some cases (eg, as in [1] for a minimum mean-square error pos-rakeonly sysem), where combining he firs arriving pahs does no exhibi sufficien performance II UWB SYSTEM MODEL Considering a sysem which uses binary phase shif keying (BPSK) for daa modulaion, he signal ransmied afer pre- 978-1-444-515-/9/$5 9 IEEE
rake processing wih F fingers is given by s() = E b i= F 1 b i k= v k w r ( it b β k ), where E b is he bi energy, b i { 1, 1} is he ih bi, w r () is he ransmied pulse, and v k and β k are he gain and delay of he kh pre-rake finger, respecively The ransmied signal propagaes hrough he UWB channel, whose impulse response is modeled as L 1 h() = α l δ( τ l ), l= where α l and τ l denoe he gain and delay of he lh mulipah, respecively, and δ( ) is he Dirac dela funcion Accordingly, he signal received over a single-user link is r() = E b i= b i L 1 l= F 1 α l k= v k w rec ( it b β k τ l )+n() (1) a he receiver anenna oupu, where he received uni energy pulse w rec (), which has duraion T p, is differen from w r () due o he differeniaion effec of he anenna, and n() is an addiive whie Gaussian noise (AWGN) componen wih wo-sided power specral densiy N / Inspecion of (1) reveals ha since he pre-rake finger delays, β k = τ L 1 τ k, are obained from he channel impulse response, for τ l = lt p he number of mulipah componens a he receiver is L + F 1 Represening he received signal a he oupu of he correlaor ha is mached o he mh mulipah by r i,m = itb +(m+1)t p it b +mt p r()w rec ( it b mt p )d, m =,,L+ F, he received vecor for he ih bi is wrien as r i =[r i, r i,l+f ] T The final decision for he ih bi, where he pos-rake vecor w =[w w L+F ] T is used o perform rake combining a he receiving side, is made as in ˆbi =sgn { w T r i } () III TRANSCEIVER DESIGN The aim in his work is o maximize he SINR for sysems wih ISI, which perform ransmier/receiver rake processing under he consrain ha he oal number of rake fingers is bounded by some number, F : F + F r F where F r is he number of pos-rake fingers Such a limi may be desirable o conrol he cos, complexiy and size of he TDD ransceiver Alhough he number of mulipahs creaed in response o pre-raked ransmission is L+F 1, and hence he number of pos-rake fingers may be as large as ha as indicaed in (), we invesigae he case where F r <L+F 1 Noe ha one should really expec F >F r from a pracical sand-poin In he following, while he lengh of he pre-rake vecor is he same as he number of pre-rake fingers such ha v =[v F 1 v ] T, he vecor w r represens he pos-rake vecor of lengh L + F 1 ha has F r nonzero elemens High daa rae ransmission inroduces ISI o he decision saisic for he ih bi as in ξ = E b b i wr T Hv + I E b b i+m wr T H m v + wr T n i, m (3) where ξ = wr T r i is he decision saisic before hresholding, I = (L + F 1)/χ and χ = T b /T p The elemens of n i are he noise samples n i,m = itb +(m+1)t p it b +mt p n()w rec ( it b mt p )d, which are zero-mean Gaussian random variables ha have a variance of N / In (3) he channel marix for he desired signal componen is he (L + F 1) F convoluion marix α H = α 1 α α L 1 α α L 1 The channel marices in (3) ha describe he signals due o inerfering bis are derived from H as [ ] H( χm +1:L + F 1, :) H m =, m <, χm F [ ] H m = χm F, m >, H(1 : L + F 1 χm, :) where H(r 1 : r,c 1 : c ) denoes a submarix of H ha conains rows r 1 hrough r and columns c 1 hrough c, and d1 d is an all-zero marix of dimension d 1 d Wih independen and equiprobable bis, he SINR of ξ, which is denoed by, is expressed as wr T Hv N E b wr T w r + I wr T H m v (4) m Because he pos-rake vecor has F r non-zero elemens, i can be wrien as w r = S T r, (5)
where is an F r 1 vecor and S r is an F r (L + F 1) selecion marix ha deermines which pahs, ie, elemens of r i, are combined a he receiver The pahs ha arrive firs, in he middle and a he end are combined by using he marices given respecively by S r = [I Fr Fr (L+F F r 1)], (6) S r = [ Fr F 1 I Fr Fr (L F r)], (7) S r = [ Fr (L+F F r 1) I Fr ], (8) where I M represens an M M ideniy marix By subsiuing (5) in (4) and observing ha S r S T r = I Fr, anoher expression is obained for : = w T r S r Hv T S r S T r N E b + I T S r H m v m T S r Hv T N E b + I T S r H m v m When ISI is presen, he join ransmier-receiver rake design ha has a limied oal number of rake fingers is he soluion o he problem max,v w T r ( w T r S r Hv v T H T S T r N E b I Fr + I S r H m v v T H T ms T r m ) (9) The pre- and pos-rake vecors ha saisfy (9) are opimal in he sense ha hey maximize he SINR of he decision saisic Defining E b /N and he marices C v S r Hv v T H T S T r, D v 1 I I Fr + S r H m v v T H T ms T r, I i is recognized ha he expression in (9) is in he form of he generalized Rayleigh quoien, wt r C v T D v The pos-rake vecor ha maximizes for a given value of v (also C v and D v ) is he principal eigenvecor (corresponding o he larges eigenvalue) of he generalized eigenvalue problem (C v, D v ), which requires solving for ha saisfy C v = D v This resul follows from a generalizaion of he Rayleigh-Riz heorem, and if he inverse of D v exiss, is he principal eigenvecor of D 1 v C v [11] In his case, since C v is of rank one, he principal eigenvecor is D 1 v S r Hv corresponding o he eigenvalue v T H T S T r D 1 v S r Hv As a resul, he opimal pos-rake vecor for a given v is w op r = D 1 v S r Hv, (1) and he relaed SINR is given by v T H T S T r D 1 v S r Hv (11) Le v op be he soluion o he opimizaion problem v op = arg max v T H T S T r D 1 v S r Hv (1) v This is a very srucured problem when here is ISI, since D v conains erms involving v I canno be posed as an eigenvalue problem, which would lead o a closed form expression for v op, and he Rayleigh-Riz heorem is no applicable However, under he consrains T /P w = v T v =1,he opimum pre-rake vecor v for a given is obained by rewriing as T S r Hv P w v T v + I T S r H m v m v T H T S T r = T S r Hv ( v T P w I F + ) I H T ms T r T S r H m v m Defining C w H T S T r T S r H, D w P I w I F + H T ms T r T S r H m, m he pre-rake vecor ha maximizes for a given is found as he principal eigenvecor of he marix D 1 w C w, which is v op = D 1 w H T S T r (13) This also resuls in he opimizaion problem op = arg max T S r HD 1 w H T S T r (14) The soluions in (1) and (13) are coupled, and independen closed form expressions for v op and op argeing he ISIlimied scenarios do no exis However, i is possible o solve for (v op, op ) joinly using an ieraive procedure similar o he approach in [1], which is iniialized wih he pre- and pos-rake vecors for he ISI-free case In he absence of ISI, v op and op correspond o he principal eigenvecors of he marices H T S T r S r H and S r HH T S T r, which follow from (1) and (14) for D v =(1/ )I Fr and D w =(P w / )I F, respecively The soluion algorihm can be described as follows 1) Se he iniial value of o, eg, he opimal ISIfree soluion, which is he principal eigenvecor of S r HH T S T r ) Using obained previously, compue D 1 w H T S T r, which becomes he new v 3) Updae o D 1 v S r Hv based on he laes value of v from sep 4) Evaluae according o he expression in (11) 5) If he change in compared o he oucome of he previous ieraion exceeds a prese hreshold, compue one more ieraion by repeaing he procedure in seps -5 Oherwise sop he algorihm The ieraions are guaraneed o converge so ha reaches a maximum because seps and 3 above boh increase he SINR
IV NUMERICAL RESULTS The resuls for disribuing a oal fixed number of rake fingers opimally beween he ransmier and he receiver so as o maximize he SINR of he decision saisic, ξ, are presened in his secion Simulaions have been performed for he lineof-sigh (LOS) CM1 UWB channel model in [], where he 1 channel realizaions { creaed are normalized o uniy in L 1 } heir average energy as E l= α l =1 The opimal selecion of he finger posiions of he posrake, which maximizes he SINR of ξ for F = L +1 and T b > (L 1)T p is he problem addressed in Fig 1 The oal number of fingers, F, in Fig 1 is L +1 so ha F r = L when F =1, and similarly F r =1when F = L Also, he condiion T b > (L 1)T p ensures ha here is no ISI in he figure, where, in oher words, is he SNR of ξ Then is he maximum eigenvalue of he marix S r HH T S T r or equivalenly H T S T r S r H scaled by, which is obained by evaluaing (1) or (14) for D v = (1/ )I Fr or D w = (P w / )I F, respecively In addiion o he pos-rake vecors ha selec pahs arriving firs, in he middle and a he end as defined in (6)-(8), he all-pos-rake which has S r = I L+F 1, ie, F r = L + F 1, is included in Fig 1 While he all-pos-rake srucure is equivalen o he eigenprecoder in [7] for F = L, he PRC pre/pos-rake design in [8], which considers arbirary F corresponds o he curve all in Fig 1 Thus, i is possible o compare he performance of he proposed design wih hose in previous works using Fig 1 I is observed from he figure ha for all F he highes is obained by he PRC/pos-rake scheme using all pos-rake fingers, where increases wih F, which makes he eigenprecoder he opimum srucure in erms of maximizing However, in order o operae a a high when F is consrained o he lengh of he channel impulse response, i is bes o disribue he oal number of rake fingers almos equally beween he ransmier and he receiver if he firs arriving pahs are combined a he receiver For a large number of rake fingers a he ransmier, he figure suggess ha selecing he pahs ha arrive in he middle as in (7) improves As he wo curves for he firs and middle arriving pahs fail o be coninuous a large values of F,iis likely ha here is a beer way o selec he finger posiions for he pos-rake han hese wo Noe ha he form of (7) is a generalizaion of he pre-rake only srucure ha has a peak a he F h pah of he composie channel impulse response wih L + F 1 pahs if he pre-rake vecor of lengh F consiss of he channel coefficiens, α l [13] The / values in Fig suppor he previous claim ha a more sysemaic way of selecing he pahs a he receiver o form he pos-rake fingers is necessary for he proposed ransmier/receiver srucure, and ha inuiion may no be enough For especially F =9he idea of combining he firs arriving pahs loses is opimaliy, where he pahs ha arrive in he middle as in (7) should be combined o obain a larger value Displayed in Fig 3 are he maximum SINR values achiev- / 35 3 5 15 1 all 5 firs middle end 4 6 8 1 1 F Fig 1 The normalized SINR values, /, agains F for F = L +1, T b > (L 1)T p, ie, no ISI, and differen posiions of he pos-rake fingers / 3 5 15 1 5 5 1 15 5 3 35 F Fig The normalized SINR values, /, agains F for F =9, 17, 33 and T b > (L + F 1)T b, ie, no ISI Wihou markers: Firs arriving pahs (6) Wih markers: Middle arriving pahs (7) Solid curve: All-pos rake able in db for F =33and various values of E b /N, where T b = 1 ns and T p = 1 ns The firs arriving pahs are combined a he receiver using he marix in (6) I is observed ha he effecs of ISI are less of a problem when E b /N is low, where in paricular he ISI makes he accurae selecion of he opimal (F, F r ) pair less criical as shown more clearly in Fig 4 for E b /N = db Ye anoher observaion is ha in general he opimal (F, F r ) pair does no change wih ISI, which has been validaed by unrepored simulaions covering oher cases Finally, hese wo figures obained using he ieraive algorihm described boh poin o he fac ha moving he rake srucure from he receiver o he ransmier helps miigae ISI as advocaed previously in [5] In Fig 3 and
(db) 3 5 15 1 5 increasing E /N b wih ISI wihou ISI 5 5 1 15 5 3 35 F Fig 3 The SINR values for he ransmier-receiver design ha combines he firs arriving pahs, ie, uses (6), agains F for F =33,whereE b /N =, 1, db, T b =1ns and T p =1ns 45 4 35 3 5 15 1 5 wih ISI wihou ISI 5 1 15 5 3 35 F Fig 4 The SINR values for he ransmier-receiver design ha combines he firs arriving pahs, ie, uses (6), agains F for F =33,whereE b /N = db, T b =1ns and T p =1ns 4, wih F =1we observe a lowering of he value when ISI is presen However, for F =3he effec of ISI on is insignifican, where F =3and F =1correspond o he pre-rake only and pos-rake only sysems, respecively V CONCLUSION The maximum SINR values achievable by a ransmier/receiver pair ha employs rake combining boh a he ransmier and he receiver wih a limied oal number of rake fingers, F, is invesigaed in a search for he opimum ransceiver design Alhough he design wih he pos-rake srucure ha combines he firs arriving pahs has a sufficien performance a larger values of F, simulaions sugges ha here migh be beer choices for he pahs o be combined when F is lower The complexiy of he calculaion of he pre- and pos-rake vecors for sysems ha have ISI requires an ieraive algorihm as opposed o he ISI-free case which admis closed form expressions However, i is shown ha he opimum disribuion of he rake fingers beween he ransmier and he receiver does no change significanly when ISI is presen Thus he opimizaion of he finger selecion process for he pos-rake srucure of sysems wihou ISI should provide valuable insigh for how sysems wih ISI should be designed ACKNOWLEDGMENT This work was suppored by TÜBİTAK EEEAG under gran number 15E34 N Güney was also suppored by TÜBİTAK BDP H Deliç was also suppored in par by he TU Grans Program of Delf Universiy of Technology The ideas, views and conclusions expressed herein do no represen he official view of TÜBİTAK REFERENCES [1] M Z Win and R A Scholz, On he robusness of ulra-wide bandwidh signals in dense mulipah environmens, IEEE Communicaion Leers, vol, pp 51 53, February 1998 [] J R Foerser, Channel modeling sub-commiee repor (final), ech rep p815-/368r5-sg3a, IEEE P815 Working Group for Wireless Personal Area Neworks (WPANs), December [3] R Esmailzadeh, E Sourour, and M Nakagawa, Prerake diversiy combining in ime-division duplex CDMA mobile communicaions, IEEE Transacions on Vehicular Technology, vol 48, pp 795 81, May 1999 [4] K Usuda, H Zhang, and M Nakagawa, Pre-rake performance for pulse based UWB sysem in a sandardized UWB shor-range channel, in Proceedings of IEEE Wireless Communicaions and Neworking Conference, March 4, vol, pp 9 95 [5] M Jun and T Oh, Performances of pre-rake combining ime hopping UWB sysem, IEEE Transacions on Consumer Elecronics, vol 5, no 4, pp 133 137, November 4 [6] A N Barreo and G P Feweis, Performance improvemen in dsspread specrum cdma sysems using pre- and pos-rake, in Proceedings of Inernaional Zurich Seminar on Broadband Communicaions, February, pp 39 46 [7] R Irmer, A N Barreo, and G P Feweis, Transmier precoding for spread-specrum signals in frequency-selecive fading channels, in Proceedings of IEEE 3G Wireless, May 1, pp 939 944 [8] J-K Han, M-W Lee, and H-K Park, Principal raio combining for pre/pos-rake diversiy, IEEE Communicaions Leers, vol 6, pp 34 36, June [9] N Güney and H Deliç, Transmier-receiver rake diversiy combining for ulra-wideband communicaions, in Proceedings of IEEE AFRICON, Sepember 7, DOI 1119/AFRCON744165 [1] S Gezici, M Chiang, H V Poor, and H Kobayashi, Opimal and subopimal finger selecion algorihms for MMSE rake receivers in impulse radio ulra-wideband sysems, EURASIP Journal on Wireless Communicaions and Neworking, vol 6, DOI 11155/WCN/6/8449 [11] G Srang, Inroducion o Linear Algebra, Wellesley-Cambridge, 3 [1] R L Choi, R D Murch, and K B Leaief, MIMO CDMA anenna sysem for SINR enhancemen, IEEE Transacions on Wireless Communicaions, vol, pp 4 49, March 3 [13] S Imada and T Ohsuki, Pre-rake diversiy combining for UWB sysems in ieee 815 UWB mulipah channel, in Proceedings of Inernaional Workshop on Ulra Wideband Sysems, May 4, pp 36 4