Performance of SDMA Systems Using Transmitter Preprocessing Based on Noisy Feedback of Vector-Quantized Channel Impulse Responses

Transcription

1 Performance of SDMA Systems Usng Transmtter Preprocessng Based on Nosy Feedback of Vector-Quantzed Channe Impuse Responses Du Yang, Le-Lang Yang and Lajos Hanzo Schoo of ECS, Unversty of Southampton, SO7 BJ, UK Te: ; Fax: E-ma: Abstract In ths contrbuton we nvestgate the performance of Spata Dvson Mutpe Access SDMA mutpe-nput mutpe-output MIMO systems usng transmtter preprocessng, when the channe knowedge requred for preprocessng s acqured by the recever and conveyed to the transmtter va nose feedback channes that may aso confct fadng Specfcay, n our system the MIMO channe mpuse responses CIRs are vector quantzed Then, the CIR magntudes and phases are conveyed to the transmtter va a feedback channe, whch s nose contamnated and may aso experence Rayegh fadng At the transmtter, the CIRs used for transmt preprocessng are recovered usng a soft estmator, whch s optmum n the mnmum mean-square error MMSE sense, and s mpemented based on the so-caed Hadamard soft-decodng prncpes Our study and smuaton resuts demonstrate that vector quantzaton combned wth soft-decodng consttutes an effcent technque of feedng back the CIRs from the recever to the transmtter However, t s aso known that the performance of the zero-forcng ZF or MMSE transmt preprocessng schemes s hghy senstve to the effect of quantzaton errors as we as to the feedback channe nduced errors I INTRODUCTION In recent years transmtter preprocessng technques have receved wde research attenton 4, snce the DownLnk DL mutuser nterference MUI may be mtgated by carryng out the requred sgna processng at the base-staton BS, where the assocated compexty ncrease s ess crtca than at the mobe Consequenty, power-effcent and ow-compexty mobe termnas MTs may be mpemented However, n order to carry out transmtter preprocessng at the BS for the sake of mtgatng the MUI, the BS has to have the knowedge of a DL channes n advance, e even before these channes are actuay encountered by the transmtted DL sgnang In tme-dvson dupex TDD systems, the knowedge of DL channes can be extracted from the UpLnk UL channes, snce n TDD mode the DL and UL TDD channes are recproca However, n wreess systems usng frequency-dvson dupexng FDD, the empoyment of transmtter preprocessng becomes much more chaengng n comparson to the TDD-type systems Ths s because n FDD-type wreess systems the UL and DL channes are usuay not recproca Therefore, n FDD-type systems the DL CIRs woud have to be estmated by the recever and expcty fed back from the MTs to the BS In contrast to 4, where dea knowedge of the DL CIRs was assumed, n ths contrbuton we consder the more reastc transmtter preprocessng scenaro, when the BS has mted and possby channe-nfested knowedge about the downnk CIRs, whch are conveyed through nose feedback channes experencng fadng When transmtter preprocessng s carred out under the above-mentoned reastc condtons, t s mportant to desgn effcent CIR-sgnang schemes and to nvestgate the mpact of the recovered mperfect CIR knowedge on the achevabe performance In ths contrbuton, reastc transmtter preprocessng usng error-prone CIR sgnang s nvestgated n the context of a SDMA system, whch supports ether snge or mutpe DL users We assume that the CIRs between the BS transmt antennas and the MT s receve antennas are vector quantzed VQ, where both the VQ codebook desgn and VQ code ndex assgnment are consdered The bts representng the quantzed CIRs are bnary phase-shft keyng BPSK moduated and conveyed to the BS transmtter through a fadng channe mpared by addtve Gaussan nose At the BS s transmtter we assume that the CIRs used for transmt preprocessng are recovered usng a soft-decodng scheme, n order to enhance the accuracy of CIR detecton Note that one of our man objectves n ths contrbuton s to quantfy the number of bts requred for representng the requred channe nformaton n the context of varous feedback channe scenaros, so that the resutant bt error rate BER performance remans cose to that acheved when the BS transmtter empoys perfect knowedge of the DL CIRs A Transmtter Preprocessng II PRELIMINARIES Let us consder a MIMO system empoyng N transmt antennas, whch transmt a vector of K symbos to the remote MT or MTs, where we assume that N K At the recever sde, there are a tota of K receve antennas, whch may beong to one or severa MTs Consequenty, when near transmtter preprocessng s apped, t can be shown that the K-ength vector y receved by the K antennas can y = H T Px + n, where x =x,x,,x K T contans the K transmtted symbos It s assumed that E x k = and x k s an ndependenty and dentcay dstrbuted d unform random varabe In n s the K-ength nose vector, hostng compex Gaussan random varabes havng a zero mean and a varance of σ /=/SNR per dmenson, where SNR represents the sgna-to-nose rato averaged over a the N transmsson channes In P s the N K transmtter preprocessng matrx, whch s gven by P =p,p,,p K, where p k represents a vector for preprocessng x k, k =,,K Furthermore, n the matrx H contanng the CIRs between the transmt and recever antennas can H =h,h,,h K h k =h k,h k,,h kn T,k=,,,K, 3 where h k s the CIR, whch s often referred to as the spata sgnature, correspondng to the kth receve antenna, where x k s detected In ths contrbuton three dfferent-compexty preprocessng schemes are consdered n our smuatons, whch are based on the prncpes of transmtter matched-fterng TMF, transmtter zeroforcng TZF and transmtter mnmum mean-square error TMMSE processng 4 Specfcay, n the context of TMF, the preprocessng matrx P s gven by 4 P = H β, 4 where β = dag {β,β,,β K} s a matrx of normazaton factors requred for mantanng a constant transmtted power, whch can be chosen to satsfy Tr PP H = K, 0-/$ IEEE 9

2 where TrX represents the trace of X In the context of TZF-asssted preprocessng, the preprocessng matrx P s gven by, 4 P = H H T H β 6 Fnay,the matrx P used for TMMSE-asssted preprocessng s gven by 4 P = H H T H + σ I K β 7 As seen by comparng 6 and 7, TZF carres out mutuser preprocessng wthout takng nto account the effects of the background nose, whe TMMSE jonty mnmzes the effects of the MUI and of the background nose Correspondngy, as shown n 3, 4, TZFasssted preprocessng suppresses the MUI at the cost of potentay ampfyng the background nose B Vector Quantzaton Let us ntay assume that the source {α n}, whch has to be quantzed, namey the MIMO CIR vector, can be modeed by a zeromean, statonary and ergodc vector process Ths MIMO CIR vector s encoded usng vector quantzaton, whch s descrbed as foows The VQ source encoder carres out the mappng E: R V I U,whch maps the V -dmensona unquantzed, e contnuous vaued source vector α n R V nto a fnte precson representaton I n I U,where we have I n = Eα n, whei U = {0,,,U } represents the U number of VQ centrods The encoder s mappng functon E of the V -dmensona MIMO CIR hyperspace R V,sothatwehaveα n R I n = Hence, the unquantzed MIMO CIR set R s quantzed nto the th s defned by a VQ partton {R } U =0 VQ regon or ce Let us defne the VQ-encoded centrods, {c } U =0 as c E α n I n = = E α n α n R The VQ-centrod set {c } U =0 s stored n the VQ-encoded codebook, whch has a sze of U entres, whe I U s referred to as the ndex of the th codeword Let b {, +},=0,,,V be the bts n the bnary representaton of the nteger VQ codebook ndex I U Whenb s transmtted va a feedback channe from a MT to the BS, the channe observaton for b at the BS s recever can y f =h f b +n f,=0,,,v, 8 where the superscrpt f ndcates the feedback channe, h f represents the Rayegh faded channe gan experenced by the th bt, whe n f denotes the correspondng nose sampe, whch s assumed to be a Gaussan dstrbuted random varabe wth zero-mean and a varance of /SNR f per dmenson Note that, n ths contrbuton we nvestgate the performance of transmt preprocessng schemes usng fnte-precson channe knowedge at the transmtter, when assumng that the eements of H n 3 are ether ndependent or correated n the spata-doman For the scenaro when the eements of H are ndependent and compex Gaussan dstrbuted, the MIMO channe s CIR matrx H can be ready generated wth the ad of the compex Gaussan dstrbuton By contrast, when the eements of the MIMO CIR matrx H are correated, n our smuatons the correated MIMO CIR matrx H s generated usng the approach proposed n 6 Ths approach creates the spatay correated MIMO channes usng the vrtua channe representaton VCR assocated wth the -dmensona Dscrete Fourer Transform DFT III MIMO CIR QUANTIZATION AND CODEBOOK DESIGN Durng our nta study we found that when the CIRs are vector quantzed, the phases and amptudes shoud be quantzed separatey usng two vectors The reason for the above observaton s that the phase and amptude are dstrbuted sgnfcanty dfferenty More expcty, the former s unformy dstrbuted, whe the atter s assumed to be Rayegh dstrbuted n ths contrbuton Furthermore, n our nta studes, the BER performance of SDMA systems usng transmtter preprocessng was found to be more senstve to phase errors than to amptude errors, hence the phase potentay requres a more accurate VQ Our codebooks are generated usng the generazed Loyd agorthm, whch s descrbed as foows: Step : Generate a suffcenty arge number of M emprca data sampes X =α,α,,α M,whereα s V -dmensona and represents ether the amptudes or phases of the MIMO CIR matrx H to be quantzed; Step : Set m = and randomy choose U coumns n the emprca data X as the nta codebook C m, whch can be expressed as C m = c m 0,c m,,c m U ; 9 Step 3 : Use Loyd procedure for updatng the codebook C m to generate C m+ as foows: a Assocate each coumn n X wth a codeword n C m Specfcay, a vector α t n X s assocated wth c m,f t satsfes d α t,c m <d α t,c m j ; for a j, where d, represents the Eucdean dstance of the arguments Provded that d α t,c m = d α t,c m j, then α t s assocated wth c m f j, otherwse, t s assocated wth c m j f >j; b A the unquantzed CIR vectors {α } that w be represented by the same VQ codeword, say c m,formavq ce referred to as R,=0,,,U ; c Based on the updated ces {R }, recompute the VQ centrods representng the quantzed CIRs n the context of a the VQ ces, and correspondngy update the codebook to C m+ = c m+ 0,c m+,,c m+ U ; Step 4 : Based on the updated codebook C m+, compute the VQ scheme s overa dstorton measured n terms of ts meansquare error MSE, whch s expressed as D m+ = M U α j c m+ ; 0 =0 α j R Step : If the reatve dstorton reducton D m+ D m acheved durng the mth VQ tranng step s suffcenty sma, curta the teratons and use C m+ as the codebook C = c 0,c,,c U Otherwse, set m + m and repeat the process from Step 3 After obtanng the codebook C =c 0,c,,c U, the ndces of the VQ codewords can be re-ordered, so that the quantzed CIR VQ codewords exhbtng smarty are represented by ndces havng smar bnary representatons When the codeword ndces are arranged n ths way, the effect of transmsson errors caused by nose and fadng mposed by the transmsson channes can be mnmzed Fgs and show the codeword dstrbutons for the amptudes and phases of two spatay correated MIMO channes havng a correaton coeffcent of ρ 08, when vector quantzaton based on the abovementoned agorthm s empoyed Both the amptudes and phases were quantzed nto 04 egtmate codewords Correated Rayegh fadng channes havng a correaton coeffcent of ρ 08 were assumed It can be observed that the codewords representng both the amptudes and phases tend to be ndeed cosey correated IV SOFT-DECODING ASSISTED MIMO CIR RECOVERY We assume that the VQ ndex, say I n = representng α n,and correspondng to a specfc CIR tap at a certan tme nstant s fed back to the BS s transmtter va a feedback channe When BPSK 0

3 Amptude of Channe Amptude of Channe Fg Iustraton of the 04 codewords for the amptudes of two spata correated MIMO channes havng a correaton coeffcent of ρ 08, when vector quantzaton s empoyed Phase of Channe Phase of Channe Fg Iustraton of the 04 codewords for the phases of two spatay correated MIMO channes havng a correaton coeffcent of ρ 08, when vector quantzaton s empoyed moduaton s assumed, the observatons for the VQ ndex I n = at the BS s transmtter can y f =h f b +n f,=0,,,v, where V s the number of bts requred by the bnary representaton of I n =, whch s expressed as b = {b V,,b 0}, h f represents the fadng gan, whe n f s the Gaussan nose Furthermore, provded that the phase assocated wth a specfc fadng gan h f s known, after removng the effects of the phase, can y f = h f b +n f,=0,,,v The receved CIRs of the MIMO channe are recovered wth the ad of a soft source decoder, whch s a non-near MMSE estmator 7 Its objectve s to compute the estmate ˆα n of α n, based on the observatons of y f for =0,,,V, whereα n represents ether a set of amptudes or a set of phases Let the vector { } y f = y f 0,yf,,yf V, 3 host the V observatons correspondng to α n Then, α n s estmated by the MMSE estmator accordng to 7 ˆα n = = U U E α n I n = j P I n = j y f c jp I n = j y f, 4 where c j = E X n I n = j, j =0,,,U, s the codeword or centrod of the jth VQ partton descrbed n Secton III In 4 P I n = j y f represents the a-posteror probabty of the VQ codebook ndex I n = j gven the observaton vector of y f,whch can P I n = j y f P I n = j p y f I n = j =, p y f where P I n = j s the a-pror probabty of the event I n = j,whch can be found accordng to the specfc channe statstcs encountered, whe p y f I n = j s the probabty densty functon PDF of the event, when the natura bnary representaton of I n = j s transmtted over the feedback channe Assumng that the consecutve sampes of the MIMO feedback channe s CIRs encountered durng the transmsson of the VQ bts are ndependent of each other, we can express t as where p y f as V p y f I n = j = y f =0 p y f b = b j, 6 b j s the probabty densty of observng, gven that b = b j was transmtted, whch can be expressed p f y b = b y j = exp πσ y h f σ b j, 7 where σ s the varance of the Gaussan nose Fnay, n p y f s the probabty densty of the observaton y f, whch can p y f U = P I n = j p y f I n = j 8 In our smuatons the so-caed Hadamard soft-decodng technque 8 s empoyed for mpementng the above-mentoned MMSE estmaton n procedure of 7 n terms of the ndvdua bts representng the transmtted VQ ndex 8 To be more specfc, wth the ad of the Hadamard soft-decodng technque of 7, the th VQ centrod can c = Tm, 9 where m s the th coumn of a U U Syvester-type Hadamard matrx M 8 To eaborate a tte further, et us express the natura bnary representaton of the nteger as b = {b V,,b 0} Then, t can be shown that the th coumn m n M can be generated as 8 m = b V b V 0 b 0 In 9 T s the Hadamard transform matrx whch s fuy determned by the encoder centrods and can 8 T = CM, U

4 where C s the codebook obtaned accordng to Secton III Upon substtutng 9 nto 4, the MMSE estmate ˆα n of α n can ˆα n = T U m jp I n = j y f Foowng a number of further steps and by referrng to Secton III, ˆα n can y ˆα n = T Rmq f, 3 r T mq y f where we have R m = Furthermore, q y f q y f = U =0 U P I n = m m T, r m = P I n = m 4 =0 n 3 can ˆbV ˆbV ˆb 0, where ˆb k,k=0,,,v represents the a-posteror estmaton of b k, whch s gven by ˆbk = E b k y f P b k =±=0 = P b k =+ y f P b k = y f = p p y f f y b y k =+ p f y b y k =, 6 y b k = can be ex- s gven by whe p f y b y k =+ and p f y pressed from 7 Furthermore p p y f = b=± y f p f y b y k = b 7 Let us now provde some performance resuts n the next secton V PERFORMANCE RESULTS AND CONCLUSIONS Fgs3 and 4 show the BER performance of the mutpe-nput snge-output MISO system usng four transmt antennas and a snge receve antenna, when communcatng over the four correated Rayegh fadng channes havng a correaton coeffcent of ρ 08 In ths exampe four specfc scenaros concernng the knowedge of the CIR used for transmt preprocessng were consdered The frst scenaro assumes that the transmtter empoys perfect CIR knowedge for preprocessng The second nvestgaton assumes that fnteprecson, but nstantaneous CIR knowedge s conveyed va an dea feedback channe mposng no nose and no fadng 3 By contrast, the thrd and fourth cases assume that nstantaneous, fnte-precson CIR knowedge s fed back from the recever to the transmtter va an mperfect feedback channe, whch s ether a Gaussan channe or an AWGN-contamnated Rayegh fadng channe In our smuatons characterzed n Fgs3 and 4 we assume that both the amptudes and phases of the non-dspersve CIR taps are deay estmated, both of whch are quantzed usng V =bts correspondng to U = 4096 VQ codewords Furthermore, for the feedback channe the sgnas receved by the BS s four receve antennas are coherenty dversty Bt Error Rate Quantzaton MSE Enveope: Phase: 0003 rad Perfect CIR Knowedge Idea Feedback Channe AWGN Feedback Channe Recever-dversty=4 SNR=dB AWGN Feedback Channe Recever-dversty=4 SNR=3dB AWGN Feedback Channe Recever-dversty=4 SNR=4dB Rayegh Feedback Channe Recever-dversty=4 SNR=0dB Rayegh Feedback Channe Recever-dversty=4 SNR=dB SNR per bt db Fg 3 BER versus SNR per bt performance of a MISO channe usng four transmt antennas and a snge receve antenna, when communcatng over ndependent Rayegh fadng MIMO channes, where bts are used for quantzng both the amptudes and phases, respectvey Bt Error Rate Quantzaton MSE Enveope: 0007 Phase: 0003 rad Perfect CIR Knowedge Idea Feedback Channe AWGN Feedback Channe wth Receve-dversty SNR=dB AWGN Feedback Channe wth Receve-dversty SNR=dB AWGN Feedback Channe wth Receve-dversty SNR=4dB Rayegh Feedback Channe wth Receve-dversty SNR=4dB Rayegh Feedback Channe wth Receve-dversty SNR=8dB SNR per bt db Fg 4 BER versus SNR per bt performance of a MISO channe usng four transmt antennas and a snge receve antenna, when communcatng over correated Rayegh fadng MIMO channes assocated wth ρ 08,where bts are used for quantzng both the amptudes and phases, respectvey combned, so that the CIRs can be estmated more reaby Observe from the resuts of Fg3, that f the four channes correspondng to the four transmt antennas are ndependent, usng U = 4096 codewords for the quantzaton of both the amptudes and phases s st nsuffcenty accurate As shown n Fg3, an approxmatey 0 db of SNR oss s experenced at the BER of 0 4 by VQ scheme By contrast, when the four transmsson channes are correated, as shown n Fg 4, the BER performance acheved over a wde SNR range s cose to that obtaned by usng perfect CIR knowedge for transmtter preprocessng From the resuts of Fgs3 and 4 we nfer that a Gaussan feedback channe havng a SNR of 4dB s suffcenty reabe for conveyng the CIR nformaton By contrast, for Rayegh fadng feedback channes, the SNR requred for reaby conveyng the channe nformaton s about db, when the four recevedversty channes are ndependenty fadng, as evdenced by Fg3 When the four channes are correated, the requred SNR becomes 8dB, as shown n Fg 4 The reason for ths s that uncorreated fadng resuts n a hgher dversty gan for the feedback channe than experenced n the correated fadng scenaro Furthermore, when comparng the resuts of Fgs3 to those of Fg 4, t can be observed that the MIMO system experencng ndependent fadng outperforms that experencng correated fadng Agan, ths s because a sgnfcant dversty gan oss s encountered due to the correaton among the MIMO transmsson channes Fgs and Fgs 6 show the BER performance of the MIMO system

5 Bt Error Rate Perfect CIR Knowedge Idea Feedback Channe AWGN Feedback Channe wth Receve-dversty SNR=3dB AWGN Feedback Channe wth Receve-dversty SNR=dB AWGN Feedback Channe wth Receve-dversty SNR=7dB Rayegh Feedback Channe wth Receve-dversty SNR=dB Rayegh Feedback Channe wth Receve-dversty SNR=30dB SNR per bt db Bt Error Rate Perfect TMF Idea TMF Perfect TZF Idea TMF Perfect TMMSE Idea TMMSE AWGN Feedback TZF SNR=4dB SNR per bt db Fg TZF: BER versus SNR per bt performance of a MIMO channe usng two transmt antennas, two receve antennas and supportng two users, when communcatng over correated Rayegh fadng channes assocated wth ρ 08, where 0 bts are used for quantzng both the amptudes and phases, respectvey, of each user Fg 7 Performance comparson of the TMF, TZF and soft decodng for vector quantzaton over nosy channes wth memory TMMSE asssted preprocessng schemes for the MIMO system usng 4 transmt antennas, 4 receve antennas and supportng 4 users, when communcatng over correated Rayegh fadng channes assocated wth ρ 08, where bts are used for quantzng both the amptudes and phases of each user Bt Error Rate Perfect CIR Knowedge Idea Feedback Channe AWGN Feedback Channe wth Receve-dversty SNR=dB AWGN Feedback Channe wth Receve-dversty SNR=7dB Rayegh Feedback Channe wth Receve-dversty SNR=dB Rayegh Feedback Channe wth Receve-dversty SNR=30dB SNR per bt db Fg 6 TMMSE: BER versus SNR per bt performance of a MIMO channe usng two transmt antennas, two receve antennas and supportng two users, when communcatng over correated Rayegh fadng channes assocated wth ρ 08,where0 bts are used for quantzng both the amptudes and phases, respectvey, of each user usng two transmt antennas, two receve antennas and supportng two users, whe empoyng the soft decodng aded recovery of the VQ ndces over nosy channes for ether the TZF Fg or for the TMMSE Fg 6 based preprocessng schemes mentoned n secton II, when communcatng over correated Rayegh fadng MIMO channes havng a correaton coeffcent of ρ 08 The resuts of Fgs and 6 demonstrate that as expected, VQ of the CIR resuts n a performance oss, when the downnk SNR s reatvey hgh When the downnk SNR s beow db, the performance oss mposed by the vector quantzaton of the CIR s hardy notceabe Fnay, n Fg 7 we compare the attanabe BER performance, when usng TMF, TZF and TMMSE-asssted transmtter preprocessng Specfcay, we assumed a MIMO system usng N t = 4 transmt antennas, N r = 4 receve antennas and supportng K = 4 users, when communcatng over correated Rayegh fadng channes havng a correaton coeffcent of ρ 08 The resuts seen n Fg 7 refect how senstve the BER performance acheved by the transmtter preprocessng schemes s to the non-dea CIR knowedge As the resuts seen n Fg 7 demonstrate, the TMF scheme acheves the worst BER performance among the preprocessng schemes consdered However, t s nonetheess a robust preprocessng scheme, hence ts performance does not degrade sgnfcanty upon ncreasng the channe estmaton error By contrast, both the TZF and TMMSE asssted preprocessng schemes are hghy senstve to the accuracy of the CIR knowedge, especay, n the reatvey hgh SNR regon, where a sgnfcant BER performance degradaton can be observed In concuson, we have nvestgated the achevabe performance of SDMA MIMO systems usng varous transmtter preprocessng schemes, when the CIR knowedge used for preprocessng s vector quantzed and conveyed va nose feedback channes from the MT s recevers to the BS s transmtter Our study demonstrated that vector quantzaton combned wth soft-decodng s an effcent technque of sgnang the CIR knowedge to the BS s transmtter However, our study aso shows that the performance of the TZF and TMMSE transmtter preprocessng schemes are hghy senstve to the non-dea CIR knowedge ACKNOWLEDGEMENT The author woud ke to acknowedge the fnanca assstance of the EPSRC of UK, and of the EU under the auspces of the Newcom and Phonx projects REFERENCES R Esmazadeh, E Sourour, and M Nakagawa, Prerake dversty combnng n tme-dvson dupex 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