Interference suppression in the presence of quantization errors

Transcription

1 Forty-Seventh Annul Allerton Conference Allerton House, UIUC, Illinois, USA September - October, 009 Interference suppression in the presence of quntiztion errors Omr Bkr Mrk Johnson Rghurmn Mudumbi Upmnyu Mdhow Electricl Engineering nd Computer Sciences, UC Berkeley, Berkeley, CA 9470 Electricl nd Computer Engineering, University of Iow, Iow City, IA 54 Electricl nd Computer Engineering, UC Snt Brbr, Snt Brbr, CA 96 Emil:{ombkr, mjohnson}@eecsberkeleyedu, rmudumbi@engineeringuiowedu, mdhow@eceucsbedu Abstrct Multi-element ntenns offer the possibility of incresing the sptil reuse of wireless spectrum by nulling out interfering signls However, the interference suppression performnce is highly sensitive to smll errors in the gins pplied to the ntenn elements In this pper, we exmine in detil the effect of one specific source of error tht rises from quntizing rry weights We show tht simple pproch bsed on sclr quntiztion tht ignores the correltion of the quntiztion errors fils to fully utilize the interference suppression cpbility of the rry: the residul interference level does not decrese with the number of ntenns Unfortuntely, the optimum pproch to computing the weights involves vector quntiztion over spce tht grows exponentilly with the number of ntenns nd number of quntiztion bits, nd is therefore computtionlly intrctble We propose insted simple suboptiml method tht greedily optimizes the SIR, coefficient-by-coefficient Simultions show tht this greedy pproch provides substntil SIR gins over the nive pproch, with SIR growing polynomilly in the number of ntenns We derive nlyticl bounds tht indicte tht even lrger SIR gins (exponentil in the number of ntenns) re potentilly chievble, so tht finding trctble lgorithms tht improve upon our suboptiml pproch is n importnt open problem I INTRODUCTION The use of multiple ntenns offers the potentil of significntly improving the performnce nd cpcity of interferencelimited wireless communiction systems, with pplictions including sptil multiplexing over point-to-point link, multipcket reception, nd uncoordinted spectrum shring mong different networks Stndrd interference suppression techniques include bemforming using the zero-forcing (ZF) or liner minimum men-squred error (MMSE) criterion In modern mostly digitl trnsceiver rchitectures, bemforming is typiclly implemented digitlly, using quntized weight vectors In this pper, we exmine the effect of such coefficient quntiztion on interference suppression performnce, specificlly focusing on the performnce of quntized versions of ZF receive bemformer which steers nulls in the direction of the interfering signls Since the MMSE bemformer tends to the ZF bemformer t high signl-to-noise rtio (SNR), we expect our results to lso pply, t lest qulittively, to the Omr Bkr s reserch is sponsored by fellowship from King Abdullh University of Science nd Technology MMSE bemformer in regimes where interference, rther thn noise, is the dominnt impirment We find tht quntiztion of the bemforming weights cn drmticlly degrde interference suppression performnce if performed in nive fshion, which could severely limit our bility to exploit the lrge sptil multiplexing gins tht re theoreticlly vilble with multiple ntenns Specificlly, while the contribution of the desired signl to the bemformer output is highly robust to quntiztion error in the bemforming weights, fr more precise control of the weights is required to effectively steer nulls Indeed, it is cler tht, with probbility one, perfect nulling of even single interferer is no longer possible Wht we would still hope for, however, is tht the Signl to Interference Rtio (SIR) scles up s quickly s possible with N, the number of ntenns, nd degrdes s slowly s possible with K, the number of interferers Our min results re summrized s follows: ) A nive strtegy of coefficient-by-coefficient quntiztion of the ZF bemforming weights hs poor performnce: the SINR scles s N/K, s it would for sptil mtched filter Tht is, we obtin the combining gins for the desired signl, but incresing the number of degrees of freedom N does not led to ny improved scling in the residul interference power t the output of the bemformer ) We provide nlyticl estimtes tht indicte tht we should be ble to perform much better thn the nive strtegy An upper bound implies tht the SINR could potentilly grow exponentilly fst with N, while pessimistic estimte indictes tht the SINR should grow t lest s fst s N This motivtes the serch for improvements to the nive sclr quntiztion strtegy 3) We propose suboptiml vector quntiztion strtegy in which ech coefficient is optimized sequentilly to mximize SINR, strting from the nive solution This greedy sequentil strtegy yields substntil gins in SINR over the nive strtegy, demonstrting the polynomil growth of the kind predicted by the pessimistic nlyticl estimtes 4) We provide extensive simultion results for vrious strtegies While optiml vector quntiztion of the /09/$ IEEE 6

2 bemforming weights is computtionlly infesible for lrge N, exhustive serch for smll N is found to perform significntly better thn the greedy sequentil strtegy, possibly even leding to exponentil dependence on N Finding computtionlly efficient vector quntiztion lgorithms tht cn pproch the performnce of the optiml solution therefore remins n importnt open problem Relted Work To the best of our knowledge, this pper is the first systemtic exmintion of the effect of quntiztion of ntenn weights on interference suppression However, quntiztion in MIMO wireless systems hs been extensively studied in mny contexts in previous work The most notble exmple of this is the work on the design of codebooks in chnnels with limited chnnel-stte feedbck [], [], where the gol is to find trnsmit bemforming vectors tht mximize the desired signl contribution t the receiver This problem hs been shown to be relted to the problem of sphere-pcking in Grssmnin mnifold [3], nd the geometry of vectors on the multi-dimensionl hypersphere proves to be importnt to understnding the codebook design problem [4] Outline: In Section II, we describe simple interference suppression problem nd show tht nive sclr quntiztion incurs severe performnce penlty Section III contins nlyticl estimtes of the performnce tht cn be chieved with vector quntiztion, with optimistic estimtes predicting exponentil gins of SIR with N nd pessimistic estimtes predicting polynomil gins In Section IV, we describe n lgorithm tht cn significntly improve the performnce using greedy sequentil vector quntiztion strtegy Extensive simultion results re provided in Section V, while Section VI contins our conclusions II THE INTERFERENCE SUPPRESSION PROBLEM We begin by considering multi-ntenn receive bemforming system, where the gol is to mximize the rry gin in the direction of desired trnsmitter In this section, we nlyze the effect of errors in the choice of ntenn weights For clrity we focus on simple setup where receiver with N element ntenn rry receives the desired [ signl tht hs the (complex bsebnd) chnnel gin h 0 = h0 [], h 0 [], h 0 [N] ] nd n interfering signl with [ chnnel gin h = h [], h [], h [N] ], where the components of the chnnels h 0 nd h re drwn from iid complex Gussin distributions, ie Ryleigh fding This cn be redily generlized to multiple interferers nd to Ricin or Line-of-Sight (LOS) chnnels, nd we lter present some simultion results to show tht the sme ides extend to more generl scenrios Indeed, we give rguments tht show tht Ryleigh fding chnnel represents worst-cse scenrio when deling with quntized weights A similr nlysis lso pplies to trnsmit rry tht seeks to minimize its interference t given loctions We focus on the receive rry in this pper for clrity We will denote sclrs in lower cse, vectors in bold lower cse, nd mtrices in bold upper cse A Problem sttement We begin by presenting quick review of the theory behind the computtion of optiml bemforming weights The incoming signl t the input of the rry y[n] is the sum of the desired signl nd interference nd noise: y[n] = h 0 d[n] + h d [n] + v[n] where d[n] is the desired signl, d [n] is the interfering signl, nd v[n] is the white noise vector t the receiver with vrince For simplicity, we shll ssume tht the desired nd interfering signls hve the sme power, ie they re drwn from Ryleigh distributions with the sme vrince, which we set to unity without loss of generlity Using rry weights w, w =, the signl t the output of the rry will be w H y[n] nd the resulting output SINR is given by: SINR out = wh h 0 w H h + where ( ) H denotes the complex conjugte trnspose In the bsence of interference, the output signl to noise rtio (SNR) is mximized by choosing w opt = h 0 h 0, SNR opt = h 0 Note tht when verged over the fding chnnel gin h 0, the SNR vries s E [ SNR opt ] = E[ h 0 ] N In other words, the SNR increses linerly with the number of ntenns N Physiclly, lrger rry permits higher ntenn directivities which ccount for this increse Note tht the effective noise power w H v σv does not increse with N When n interferer is present, complete interference rejection cn be chieved by choosing bemforming weight vector w tht is the projection of the desired vector h 0 onto the subspce orthogonl to the interference vector h : () w opt = w projection = h 0 h H h h h H h 0 () If we let θ = cos ( h H h0 h 0 h ) be the ngle between the desired nd interference chnnel vectors 3, we cn write the SNR from the projection bemformer s SNR opt = h 0 sin θ When verged over the fding chnnel gins h 0, h, we still get liner increse in SNR with N: E [ SNR opt ] = E[ h 0 sin θ ] N 3 Note tht θ is n ngle in N-dimensionl spce, nd does not correspond to physicl direction (3) (4) 6

3 The finl inequlity in (4) is bsed on the observtion tht E [ sin ] θ, which is derived in Eqution (8) in Appendix I The projection-bsed bemformer () does not tke noise into ccount In generl, mximizing the output SINR does not necessrily require complete interference rejection; reducing the interference to the noise level my be sufficient Optimizing the output SINR leds to the Minimum Vrince Distortionless Response (MVDR) bemformer [5] If we define the noise+interference correltion mtrix R N+I = h h H + σνi N where I N is the N N identity mtrix, then the output SINR cn be mximized by choosing w opt : w opt = w MVDR = R N+I h 0 h H 0 R N+I h 0 The denomintor in (5) is normlizing fctor When the interference power is much lrger thn the noise power, both projection nd MVDR yield virtully identicl results This is typiclly the cse in bndwidth-limited wireless links nd we focus on this cse exclusively in this pper; therefore the resulting SINR is still pproximtely given by (3) nd its vrition with N by (4) B Effect of errors in weights In prctice the weights tht re ctully pplied to the rry elements my differ from the optiml desired weights from () for mny resons such s errors in chnnel estimtion, clibrtion errors nd quntiztion effects Let ŵ be the weight pplied to the rry We will ssume tht ŵ = (wopt+ w) (w opt+ w) where the rel nd imginry prts (in-phse nd qudrture) of ech component of the weight error vector w re iid zero men rndom vribles with vrince N σ w Remrk Since ny scled version of the vector w yields the sme SINR, we constrin w to be unit vector, ie w =, in order to eliminte this mbiguity In prctice, the weight pplied to ech ntenn element usully vries between fixed limits, eg (, ), nd this weight is quntized to fixed number of bits Thus the vrince of ech component of the quntiztion error vector w is lso fixed However, becuse of the scle invrince of the SINR, we choose to impose the unit vector constrint for nlyticl convenience (insted of the fixed dynmic rnge constrint) As result, the components of w scle s N, nd the vrince of the quntiztion noise lso scles inversely with N since quntiztion error vries proportionlly with the dynmic rnge of the elements of w itself In other words, E [ w ] = σw independent of N Intuitively, the errors w result in ŵ deviting from w opt by n ngle θ ( w = cos w optŵ ) H This devition will result in reduction in the signl strength in the desired direction s well s n increse in the interference power, since ŵ will no longer be orthogonl to the interference subspce The desired power is proportionl to cosθ w, nd the increse in interference (lekge) is proportionl to sinθ w (See Figure ) For smll ngles θ w, we cn use the stndrd pproximtions sinθ w θ w nd cosθ w This explins why nulls re more (5) sensitive thn peks to phse nd mplitude errors, since sinθ chnges more rpidly thn cosθ when θ is smll Fig The optimum bemforming vector w opt cn be viewed s projection of the desired signl onto the subspce orthogonl to the interference subspce The distorted bemforming vector ŵ opt cn be decomposed into two orthogonl components: ŵ opt = ŵopt +ŵ opt ŵ opt, which is prllel to w opt, represents the potentil loss in bemforming gin, nd is proportionl to cos θ ŵopt, which is orthogonl to wopt, represents the potentil lekge into the interference subspce, nd is proportionl to sinθ Tking the errors w into ccount, we cn updte () for the SNR in the bsence of interference s SNR = ŵh h 0 ŵ = wh opth 0 + w H h 0 SNR opt ( w ) (6) This implies tht the SNR verged over the fding still scles s N: E [ SNR ] α E [ ] SNR opt (7) with t most only constnt fctor loss α = E [ ( w ) ] ( σw ) due to the errors Now, let us consider the effect of the weight errors w on the zero-forcing bemformer in the presence of n interferer The SINR is given by ŵ H h 0 SINR = ŵ H h + ŵ σv h0 sinθ + w H h 0 = w H (8) h + ( + w )σv where θ is defined s in (3) The numertor of (8) when verged over the weight errors w nd the fding coefficients h 0 nd h is lower-bounded by N +σ w which scles linerly with N, nd the verged denomintor equls σw +(+σ w )σ v, which is independent of N Thus the rtio of verge signl to interference nd noise powers scles linerly with N This seems good considering tht the SNR in the no-interferer cse given by (7) lso scles linerly with N Note however tht the verge interference power given by the denomintor of (8) does not decrese with N even though the degrees of freedom for interference 63

4 cncelltion increses This is strkly unlike the sitution without weight errors, where lrger number of ntenns permits the complete nulling of lrger number of interferers This indictes tht there might be some suboptimlity in the scling behvior in (8) We show next tht, indeed, much better scling behvior cn be obtined by dopting more optiml pproch III IMPROVING INTERFERENCE SUPPRESSION We now consider the problem of computing the optiml weights in the presence of quntiztion errors Let us represent the i th component of vector x by x[i], eg the weight pplied to ntenn element i is w[i] Let B be the number of quntiztion bits, ie the I nd Q components of ech weight element is quntized to one of B levels There re then N w = BN totl weights vilble from which to choose 4 Let W be the set of these weight vectors We cn then formulte the problem of computing the optiml weight s SINR mximiztion problem: ŵ H h 0 w opt = rg mx ŵ W ŵ H h + ŵ σv Remrk It is not trivil tsk to compre two lgorithms bsed on their SINR performnce If one lgorithm outperforms nother for every possible reliztion of the desired nd interference chnnels, then it is clerly better However, such definition is too stringent to be useful, nd in prctice, we my hve to settle for compring some verges At first glnce, it seems s if the nturl choice would be to compre the verge SINR, ie E [ SINR ] ; however this mesure suffers from some fundmentl disdvntges First of ll, it is rther unwieldy nlyticlly Secondly, the verge cn be dominted by smll number of chnnel reliztions where the SINR becomes very lrge We therefore choose different mesure: the rtio of verge signl power to verge interference power (we neglect noise): SINR = E[ ŵh h 0 ] E[ ŵ H h ] A An pproximte lower bound on chievble SIR The problem of choosing the optiml weight w cn lso be considered s problem of choosing the optiml w We hve seen from (8) tht the verge signl power scles linerly with N in the presence of the errors w However, s we hve seen, the verge interference power is independent of N if the components of w re chosen independently We show next tht if the components of w re chosen dependently of ech other, the interference power cn be mde to decrese significntly with N Given n interference vector h, our gol is to understnd the term w H h Let us first consider simpler problem For n interference vector h, suppose we simply wish to minimize u H h, where u tkes ll possible vlues in U = 4 Note tht some of these weight vectors my be scled versions of others, nd therefore N w is ctully n upper-bound on the number of distinct weight vectors vilble to choose from N {, +} N (ie the elements of U re binry vectors normlized to unit norm) Let Z(u) = u H h Note tht Z(u) CN(0, ) for ny u U since the fding coefficients of h re ssumed to be CN(0, ) Note lso tht Z(u ), Z(u ) re jointly proper complex Gussin with covrince (nd normlized correltion) equl to u H u Let us now choose subset U o of U which forms n orthonorml bsis in N dimensions: the Wlsh-Hdmrd codes Then {Z(u) : u U o } re N uncorrelted, nd hence independent, CN(0, ) rndom vribles The corresponding powers { Z(u),u U o } re therefore exponentil rndom vribles with men one The minimum interference power cn now be upper bounded s P min = min u U Z(u) min u U o Z(u) (9) It is esy to show tht the minimum of N iid exponentil rndom vribles of men one is n exponentil rndom vrible with men N Thus, for our simplified model, the interference power scles down t lest s fst s N with N Let us now return to our originl problem Consider Z( w) w H h For ech coefficient of the ZF weight vector, suppose tht we restrict ourselves to two options: round both I nd Q coefficients up or round them both down Thus, if the quntiztion intervl is, the corresponding coefficient of w is being set to ( X) or X, respectively, where X is the distnce of the unquntized coefficient from the bottomleft edge of the quntiztion bin in which it flls Thus, if we let + indicte rounding down coefficient, nd - indicte rounding up, there re N possible choices of w with this strtegy, which mp to the vectors in U = N {, +} N considered in our prior simplified exmple While the Wlsh- Hdmrd vectors u i no longer yield uncorrelted Gussin rndom vribles Z(b i ) when the distribution of the X s re tken into ccount, since U is n N-dimensionl vector spce, it is lwys possible to find set of bsis vectors v i which yield uncorrelted rndom vribles Z(v i ) nd for which (9) holds This leds us to expect tht the minimum interference power here will lso fll off t lest s fst s N B A geometric upper-bound on chievble SIR Consider the N-dimensionl hypersphere generted by the I nd Q coefficients of the unit vector x corresponding to the weight errors, ie x = w [ R( w[]), I( w[]), R( w[]), I( w[]), ] We wnt to choose x so s to minimize the interference power given by w H h w x H h, from the set X of vilble vectors x: x opt = rg min x X xh h (0) We cn lso obtin n upper-bound on the chievble SIR by ssuming tht the vector x tht minimizes the interference power lso simultneously chieves the mximum possible signl power Intuitively, when the number of ntenns increses, we expect the minimum interference power s given in (0) to 64

5 decrese for two resons First, s the dimensionlity of the vector spce of the weight vectors increses, rndomly chosen vector is more orthogonl to the interference vector (this is explined more precisely in Appendix I) Second, the number of quntized weight vectors to choose from increses exponentilly with N We now quntify these fctors nd obtin n upper bound on the interference suppression cpbility of n rry Without loss of generlity, we ssume tht the interference vector h is ligned with one of the coordinte xes Let θ x = θ x Our be the ngle between h nd x nd let φ x gol is to find x such tht φ x is s smll s possible If x is chosen rndomly nd uniformly on the unit hypersphere, the probbility density function of φ x is given by (see Appendix I): f φ (φ x ) = cos N φ ( x F, 3 N ; 3 () ; ) where F (, ; ; ) is the Guss hypergeometric function [6] Consider the cumultive distribution function corresponding to () given by F φ (φ) = Pr( φ x φ) φ φ f φ (φ x )dφ x From (6) we hve tht F φ (φ) is n incresing function of N for ny φ [0, ] Thus we hve F φ (φ) F 0 (φ) φ () Finlly if ll N w BN vilble weight vectors re ssumed to be distributed uniformly over the surfce of the unit hypersphere, there exists t lest one vector x such tht F φ (φ x ) N w Therefore using () we conclude tht t lest one weight vector x exists such tht φ x N w +BN (3) nd the verge interference power corresponding to this vector x is upper-bounded by E [ h w sin φ x ] σ w 4BN which decreses exponentilly with N The key ssumption in the bove derivtion is tht the vilble weight vectors x re distributed uniformly over the hypersphere with respect to ny rbitrry interference vector h This ssumption is too optimistic; more relistic ssumption is tht the weight vectors re uniformly distributed over hypercube Intuitively there re very few sprse weight vectors in X, wheres most vilble vectors hve significnt (non-zero) coefficients over lrge proportion of ntenn elements Thus for instnce n interference vector tht is highly sprse is difficult to suppress To tke n extreme exmple, n interference vector tht is zero everywhere except single ntenn element will be difficult to suppress 5 5 This is dependent on the ssumption tht none of the weight vectors in the codebook contin coefficients equl to zero Therefore it is not cler how tight the bound in (3) is Also, the bove resoning leds us to expect tht the verge interference suppression performnce will depend strongly on the fding distribution, eg the Ryleigh distribution is more likely to give unblnced or sprse vectors thn line-ofsight distribution We leve more detiled explortion of these ides to future work IV CONSTRUCTIVE ALGORITHMS FOR INTERFERENCE SUPPRESSION We hve seen tht the nive pproch to quntizing the ntenn weight vector, bsed on pplying sclr quntizer to ech coefficient independently, does not chieve the optimum interference suppression On the other hnd, becuse there is totl of BN possible quntized vectors, implementing n optiml vector quntizer by exhustively serching through the set of reconstruction levels hs computtionl cost tht is exponentil in the number of ntenns nd the number of quntiztion bits Thus, vector quntiztion by exhustive serch is infesible in prcticl scenrios We propose insted simple, suboptiml vector quntiztion scheme which is bsed on coordinte descent optimiztion This scheme substntilly improves over the nive method, yet hs computtionl cost tht is liner in the number of ntenns The suboptiml scheme greedily quntizes the weight vector by serching through the set of reconstruction levels for ech element individully, insted of jointly s in the exhustive serch lgorithm We begin by computing n initil weight vector This cn be done in severl wys, such s pplying sclr quntizer to the optiml MVDR weights given by (5), pplying sclr quntizer to the desired chnnel response h 0 (mtched filter), or even rndomly drwing vector from the set of vlid quntized weights Our simultions show tht initilizing with quntized version of the optiml weights or the mtched filter yield fr superior results to using rndom initiliztion In most of our simultions, we used the mtched filter Next, we serch through ll B vlid reconstruction levels of the first coefficient, while keeping the other N coefficients fixed The vlue which mximizes the output SINR is chosen s the quntized vlue of the first coefficient We then proceed to quntize the second coefficient, keeping the first coefficient fixed t its quntized vlue nd coefficients 3 through N fixed t their initil vlues, by similr serch This method is pplied to ech of the N coefficients of w, yielding suboptiml, vector quntized weight vector The coordinte descent quntizer hs computtionl cost of N B We note tht the output of this quntizer depends on both the method of selecting the initil vector nd the order in which the coefficients re quntized V SIMULATION RESULTS In this section, we provide numericl results tht verify the nlysis of this pper, nd compre the performnce of the vrious quntiztion schemes 65

6 A Generl errors in ntenn weights Figure () demonstrtes the reltionship between interference power fter bem-nulling nd the men squre error in the ntenn weights The plot compres the cse where independent Gussin noise is dded to the rel nd imginry components of the optiml weight vector to the cse where the noise is dded to the mgnitude nd phse, s well s compring the Ryleigh nd LOS chnnels In this simultion there were N = 00 ntenns nd K = 0 interferers The results show tht the interference lekge is directly proportionl to the totl error power, nd does not depend on whether the error is modeled s dditive in the Crtesin or polr coordintes The chnnel hs very little effect on the performnce B Sclr quntiztion of ntenn weights In this section, we present numericl results of the nive, sclr quntiztion scheme With sclr quntiztion, the vrince of the quntiztion noise grows s σ w B nd therefore the SIR is proportionl to N K B Figure (b) shows the SIR (normlized by the number of ntenns, N) s function of N, with the number of interferers fixed t K = 0 nd the quntizer size fixed t B = 4 bits Figure (c) shows the SIR (multiplied by K) s function of K, with N = 000 nd B = 4 These plots demonstrte tht the SIR is linerly proportionl to N/K for the sclr quntiztion method The slight downwrd slope in (c) is due to the fct tht for fixed number of ntenns, s the dimension of the interference subspce increses more of the desired signl lies in tht subspce Hence, the signl power tht is orthogonl to the interference begins to decrese Figure (d) shows the SIR s function of B, with N = 000 nd K = 0 The slope of the curve closely mtches the predicted gin of 6dB for ech quntiztion bit C Vector quntiztion of ntenn weights We lso simulted the coordinte descent vector quntizer, to quntify the performnce nd demonstrte the gins over sclr quntiztion While it is difficult to extrpolte the exct dependence of the SIR on the vribles, N, K, nd B (or if they re even seprble); the generl trend seems to show tht the SIR grows polynomilly in N, polynomilly in K, nd exponentilly in B Figure 3() shows the SIR s function of the number of ntenns, for vrious chnnels nd quntizer bit rtes,with K = 5 interferers The quntizer ws initilized by pplying sclr quntizer to the mtched filter In contrst to the simulted performnce of the sclr quntizer in Figure (b) where SIR/N ws constnt, we see tht for the coordinte descent vector quntizer SIR/N grows super-linerly Thus, the SIR grows fster thn qudrticlly with the number of ntenns Figure 3(b) shows tht the verge interference power (scled by the number of ntenns) is slightly decresing in the sme simultion Thus, the verge interference power with vector quntiztion decreses slightly fster thn linerly with N This is in contrst to the sclr quntizer, where the interference power ws constnt Therefore, the signl strength continues to grow linerly Thus, the coordinte descent method does not cuse significnt reduction in SNR Figure 3(c) shows the quntity K SIR s function of the number of interferers K, with N = 00 ntenns nd B = bit quntizer The plot shows tht the SIR is decresing slightly fster thn by fctor of /K Tken together, Figures 3() nd (c) imply tht for the coordinte descent vector quntizer, the SIR is proportionl to (N/K) α, with α slightly greter thn Recll tht with sclr quntiztion the SIR ws proportionl to N/K The functionl dependence of SIR on N nd K through the rtio N/K is corroborted by Figure 3(d), which shows tht the SIR is constnt when N is vried nd K is set to N/4 The dependence of the coordinte descent method on the number of quntiztion bits is presented in Figure 3(e) Just s in sclr quntiztion, the SIR grows exponentilly in the bit rte, with gin of pproximtely 6 db per bit Finlly, we lso simulted the performnce of the exhustive serch vector quntizer Figure 3(g) shows the SIR in db s function of N, with K = nd one bit quntiztion Figure 3(h) shows the SIR in db s function of K, with N = 0 nd B = bit quntiztion While the computtionl complexity of the exhustive serch method presents n obstcle to providing more extensive simultions, it ppers tht the SIR (in liner units) grows exponentilly in N nd decreses exponentilly in K Thus, there my still be significnt gp between this optiml vector quntizer nd the coordinte descent lgorithm, where SIR grew polynomilly in N/K This potentil gin motivtes future work on computtionlly efficient vector quntizers tht outperform coordinte descent D Bemforming gin While we hve focused in this work on the effect of vrious quntiztion schemes on SIR, the bemforming gin cn lso be improved Figure (e) shows the normlized rry gin of the sclr quntizer, s well s the exhustive serch nd coordinte descent vector quntizers, when ll phses were quntized to two levels The gp between vector nd sclr quntiztion is smller becuse bemforming is more robust to errors thn interference rejection, nd thus the nive method does not incur lrge penlty E Effect of noise Up to this point, we hve neglected ll sources of error other thn quntiztion However, in prctice there my be smll dditionl noise due to vrious hrdwre imperfections or therml vritions Figure 3(f) shows the SIR s function of the vrince of uniform rndom vrible tht is dded to ech component of the ntenn weight vector fter quntiztion The plot shows tht the performnce of the coordinte descent method is still superior to the nive sclr quntiztion scheme VI CONCLUSION We hve shown tht coefficient quntiztion in digitl receiver implementtions cn hve profound effect on the 66

7 verge interferer power (db) IQ error/ryleigh Phse-Amp error/ryleigh IQ error/los Phse-Amp error/los SIR/N (db) K* σ w (db) number of interferers (K) () (b) (c) normlized rry gin sclr quntizer exhustive serch vector quntizer coordinte descent vector quntizer number of bits (B) (d) (e) Fig () Simulted interference power s function of errors in the weight vector (b)-(d) Simulted effect of sclr quntiztion on the performnce of the MVDR bemnulling scheme s function of the number of ntenns N, the number of interferers K, nd the number of quntiztion bits B (e) Comprison of the bemforming gin with sclr, exhustive serch vector, nd coordinte descent vector quntizers performnce of interference-limited multi-ntenn systems In prticulr, quntiztion of bemforming weights criticlly ffects interference suppression performnce, nd must be performed crefully in order to exploit the degrees of freedom gins from using n incresed number of ntenns While our suboptiml greedy sequentil strtegy provides lrge gins over the nive sclr quntiztion strtegy, our nlyticl estimtes indicte tht it might be possible to do much better Importnt open problems include refinement of the nlysis to provide tight upper nd lower bounds on SIR scling, nd devising efficient dptive nd non-dptive lgorithms for finding the quntized weights tht pproch the performnce of the optiml quntized weights APPENDIX I DISTRIBUTION OF THE ANGLE WITH RESPECT TO AN INTERFERENCE VECTOR We strt with the well-known formul for the re of surfce element of d-dimensionl unit hypersphere [7], [8]: ds = sin d θ d dθ d sin d 3 θ d 3 dθ d 3 sinθ dθ dθ 0 where θ d, θ d 3,, θ, θ 0 re the d ngulr coordintes of the hyper-sphericl coordinte system If we ssume (without loss of generlity) tht the interference vector is ligned with one of the coordinte xes, then θ x θ d is the ngle mde by the unit vector x on the surfce element bove with the interference vector If the unit vector is ssumed to be distributed uniformly on the surfce of the hypersphere, then the probbility density of the ngle is proportionl to the re element ie f θ (θ x ) sin d θ d It is more convenient for us to work with the trnsformed ngle φ x = θ x, nd we then hve for the probbility density of φ x : f φ (φ x ) = sin d ( φ ) x sin d ( φ ) x dφx cos d φ x 0 sind φ x dφ x Define I n (t) = t 0 sinn x dx We hve the identity ( I n (t) cost F, n ; 3 ; cos t ) where F (, ; ; ) is the Guss hypergeometric function However, it is more convenient to use the recursive formul: I n (t) sinn t cost n We then hve Pr( φ x φ) = Pr ( φ < φ x From (4), we hve I n ( ) n (4) in (5) we hve + n n I n (t) (4) n I n ( ) = I n (φ) I n ( ) (5) ), nd further using Pr( φ x φ) = sinn φcosφ ( (n )I n ( ) + I ) n (φ) I n ( ) (6) We see from (6) tht the cumultive probbility distribution (CDF) of φ x, F (n) (φ) = Pr( φ x φ) is n incresing function of n From this, we hve Pr( φ x φ) > φ x Finlly we hve the following lemm: Lemm Let F (x), F (x) be two probbility distributions 67

8 SIR/N bit coordinte descent/ryleigh -bit coordinte descent/ryleigh -bit coordinte descent/los 0log(N*(verge interferer power)) bit coordinte descent/ryleigh -bit coordinte descent/los SIR*K (db) bit coordinte-descent/ryleigh -bit coordinte-descent/los bit coordinte descent/los -bit coordinte descent/ryleigh number of interferers (K) () (b) (c) 45 coordinte descent/ryleigh 8 -bit coordinte descent -bit sclr quntizer number of ntenns (N=4K) number of bits (B) noise (db) (d) (e) (f) 90 -bit exhustive serch/los -bit exhustive serch/los (g) number of interferers (K) (h) Fig 3 ()-(f) Simulted performnce of the coordinte descent vector quntizer (g)-(h) Simulted performnce of the exhustive serch vector quntizer (CDFs) in n intervl [, b] such tht F (x) F (x), x [, b] nd let g(x) be non-incresing, differentible function in [, b] Then E F [g(x)] E F [g(x)] Proof Since g(x) is non-incresing, differentible function, we hve g (x) 0, x [, b] Consider E F [g(x)] = = b g(x)df (x) [ g(x)f (x) g(b) g(b) ] b b b b g (x)f (x) dx g (x)f (x) dx (7) g (x)f (x) dx E F [g(x)] where we used integrtion by prts in (7) From the bove lemm with g(x) = cos x in [0, ] nd (6) then n < n we get E F (n )[sin θ x ] E F (n )[cos φ x ] < E F (n )[sin θ x ] (8) REFERENCES [] D J Love, R W Heth, W Sntipch, nd M L Honig, Wht is the vlue of limited feedbck for MIMO chnnels? IEEE Communictions Mgzine, vol 4, no 0, pp 54 59, October 004 [] K Mukkvilli, A Sbhrwl, B Azhng, nd E Erkip, Performnce limits on bemforming with finite rte feedbck for multiple ntenn systems, in Thirty-Sixth Asilomr Conference on Signls, Systems nd Computers, vol, November 00, pp [3] L Zheng nd D N C Tse, Sphere pcking in the Grssmnn mnifold: geometric pproch to the noncoherent multi-ntenn chnnel, in IEEE Interntionl Symposium on Informtion Theory, June 000, p 364 [4] D J Love nd R W Heth, Multi-mode precoding using liner receivers for limited feedbck MIMO systems, in IEEE Interntionl Conference on Communictions, vol, June 004, pp [5] D Mnolkis, V Ingle, nd S Kogon, Sttisticl nd Adptive Signl Processing: Spectrl Estimtion, Signl Modeling, Adptive Filtering nd Arry Processing McGrw-Hill, 000 [6] E W Weisstein, Hypergeometric function, Wolfrm MthWorld [Online] Avilble: html [7] J S Hicks nd R F Wheeling, An efficient method for generting uniformly distributed points on the surfce of n n-dimensionl sphere, Communictions of the ACM, vol, no 4, pp 7 9, 959 [8] Hypergeometric function, Mthemtics Wiki [Online] Avilble: http: //wwwmthemticsthetngentbundlenet/wiki/sphericl\ coordintes If we set n = 0, this gives E F (n)[sin θ x ] > 68