Channel Estimation Techniques for Quantized Distributed Reception in MIMO Systems

Transcription

1 Channel Esimaion Techniques for Quanized Disribued Recepion in MIMO Sysems Junil Choi David J Love and D Richard Brown III School of Elecrical and Compuer Engineering Purdue Universiy Wes Lafayee IN Elecrical and Compuer Engineering Deparmen Worceser Polyechnic Insiue Worceser MA Absrac The Inerne of Things (IoT) could enable he developmen of cloud muliple-inpu muliple-oupu (MIMO) sysems inerne-enabled devices can work as disribued ransmission/recepion eniies We expec ha spaial muliplexing wih disribued recepion using cloud MIMO would be a key facor of fuure wireless communicaion sysems In his paper we firs review pracical receivers for disribued recepion of spaially muliplexed ransmi daa he fusion cener relies on quanized received signals conveyed from geographically separaed receive nodes Using he srucures of hese receivers we propose pracical channel esimaion echniques for he blockfading scenario The proposed channel esimaion echniques rely on very simple operaions a he received nodes while achieving near-opimal channel esimaion performance as he raining lengh becomes large I INTRODUCTION The Inerne of Things (IoT) could fundamenally change he wireless communicaion indusry as more and more devices (eg labops smarphones ables and home appliances) are conneced hrough wired/wireless neworks [1] Geographically separaed bu closely locaed inerne-enabled devices could form clusers hrough a local area nework (LAN) and work as massive disribued muliple-inpu muliple-oupu (MIMO) sysems We dub such sysems Cloud MIMO in his paper I is imporan o poin ou ha cloud MIMO is differen from wireless sensor neworks (WSNs) ie he former is focused on daa ransmission and recepion while he laer is aimed o esimae he behavior of local environmens [2] [4] Sill here are many similariies beween he wo eg geographically disribued nodes may cooperae wih each oher o perform disribued ransmission/recepion and i is desirable for each disribued node o perform only simple operaions considering processing power or baery life These similariies allow us o uilize many echniques developed for WSNs o design cloud MIMO sysems For example coded disribued diversiy echniques have been proposed o increase he diversiy order of disribued recepion when he ransmier is equipped wih a single anenna [5] [6] inspired by exploiing channel coding heory for disribued faul-oleran classificaion in WSNs [7] [8] Cloud MIMO will be paricularly imporan a he mobile side A base saions we can deploy a large number of anennas wihou having sric resricion in space which is known as massive MIMO [9] [10] However i may be difficul o deploy many anennas a a mobile such as a smarphone or a lapop due o is limied space The limiaion can be overcome by cloud MIMO which explois he IoT environmen Recenly he scenario ha combines cloud MIMO a he receiver side and spaial muliplexing wih muliple ransmi anennas is sudied in [11] By having only a few quanizaion bis for he received signal a each receive node an opimal maximum likelihood (ML) receiver and a subopimal lowcomplexiy zero-forcing(zf)-ype receiver a he fusion cener are proposed I is shown analyically and numerically ha symbol error raes (SERs) of boh receivers can become arbirary small by increasing he number of disribued receive nodes However he resuls in [11] are based on he ideal assumpion of perfec global channel knowledge a he fusion cener In his paper we exend he work in [11] and propose pracical channel esimaion echniques Using analyical ools developed in[11] we are able o show ha channel esimaion error can be made arbirary small by increasing he lengh of raining phase even wih small quanizaion bis a he receive nodes Numerical resuls also show he effeciveness of he proposed channel esimaion echniques Noaion: Lower and upper boldface symbols represen column vecors and marices respecively a denoes he wo-norm of a vecor a and A T A H A are used o denoe he ranspose Hermiian ranspose and pseudo inverse of he marix A respecively Re(b) and Im(b) denoe he real and complex par of a complex vecor b respecively 0 m represens he m 1 all zero vecor and I m is used for he m mideniymarixc m (R m )andc m n (R m n )represen he se of all m 1 complex (real) vecors and he se of all m n complex (real) marices respecively II SYSTEM MODEL We consider a nework consising of a ransmier fusion cener and K geographically separaed receive nodes We assume he ransmier is equipped wih N anennas while all oher eniies in he nework have a single anenna The ransmiersimulaneouslyransmisn independendaasymbols by spaial muliplexing and each receive node conveys is processed (or quanized) received signal o he fusion cener hrough some sor of local area nework The fusion cener decodes he ransmied daa symbols using quanized received signals and is(esimaed) channel knowledge The concepual figure of his scenario is depiced in Fig 1

2 H = [ h 1 h 2 h K ] H ŷ 2 ŷ 1 We assume ŷ k can be forwarded o he fusion cener wihou any error The collecion of he quanized received signal a he fusion cener is given as ŷ = [ ŷ 1 ŷ 2 ŷ K ŷ K Fig 1: The concepual figure of spaial muliplexing wih a cloud MIMO receiver Wih his seup he inpu-oupu relaion is given as 1 y = H H x+n N y = [ y 1 y 2 y K H = [ h 1 h 2 h K ] x = [ x 1 x 2 x N n = [ n 1 n 2 n K Noe ha y k is he received signal a he k-h received node ρ is he ransmi signal-o-noise raio (SNR) h k CN(0 N I N ) is he independen and idenically disribued Rayleigh fading channel vecor beween he ransmier and he k-h receive node n k CN(01) is complex addiive whie Gaussian noise (AWGN) a he k-h node and x i is he ransmied signal from a sandard M-ary consellaion S = {s 1 s M } C a he i-h ransmi anenna We assume S is a phase shif keying (PSK) consellaion meaning s m 2 = 1 for all m and x 2 = N We assume x i is drawn from S wih all symbols equally likely Followinghesameseupasin[11]weassumehereceived signal y k is quanized wih wo bis using one bi for each of herealandimaginaryparsofy k Thenhequanizedreceived signal ŷ k is given as ŷ k = sgn(re(y k ))+jsgn(im(y k )) sgn( ) is he sign funcion defined as { 1 if x 0 sgn(x) = 1 if x < 0 1 Weconsider heblock-fading channel modelodevelop channel esimaion echniques in Secion IV III REVIEW OF ML AND ZF-TYPE RECEIVERS For he scenario of ineres he opimal ML receiver and he low-complexiy ZF-ype receiver based on he assumpion of perfec global channel knowledge a he fusion cener are developed in [11] We briefly review hese wo receivers in his secion A ML receiver To simplify noaion we firs conver all expressions ino he real domain as Re(hk ) Im(h H Rk = k ) = h Im(h k ) Re(h k ) Rk1 h Rk2 Re(x) x R = Im(x) Re(nk ) n Rk = Im(n k ) Re(yk ) yrk1 y Rk = = Im(y k ) h Rk1 = y Rk2 Re(hk ) h Im(h k ) Rk2 = Im(hk ) Re(h k ) Then he inpu-oupu relaion can be rewrien as y Rk = H T Rk N x R +n Rk (1) The vecorized version of he quanized ŷ k in he real domain is given as ] sgn(re(yk )) [ŷrk1 ŷ Rk = = sgn(im(y k )) ŷ Rk2 We also le S R be { } Re(s1 ) Re(sM ) S R = Im(s 1 ) Im(s M ) Based on ŷ Rk he fusion cener generaes he sign-refined channel marix H Rk = h Rki is defined as [ hrk1 hrk2 ] h Rki = ŷ Rki h Rki Wih hese definiionshe ML receiveris definedin [11]as ˆx RML = argmax x R SN R 2 K i=1 k=1 ( ) 2ρ Φ N Rki x R ht (2)

3 Φ() = 1 2π e τ2 2 dτ and S N R is he N -ary Caresian produc se of S R We can also define he ML esimaor by relaxing he consrain x R SN R in (2) as 2 K ˇx RML = argmax x R R2N x R 2 =N i=1k=1 ( 2ρ Φ N Rki x R ht ) (3) Noe ha he opimizaion problem in (3) is no convex because of he norm consrain on x R The following lemma which is derived in [11] shows he performance of he ML esimaor Lemma1 For arbirary ρ > 0 ˇx RML converges o he rue ransmied vecor x in probabiliy ie as K ˇx RML p x R The lemma can be proved by using he weak law of large numbers and he sochasic dominance heorem Please see Lemma 2 in [11] for deails B ZF-ype receiver The low-complexiy ZF-ype receiver is developed in [11] which is given as ˇx ZF = ( H H) ŷ Based on ˇx ZF he fusion cener can perform symbol-bysymbol deecion as ˆx ZFi = argmin ˇx ZFi x 2 x S ˇx ZFi is he i-h elemen of ˇx ZF To show he performance of he ZF-ype esimaor we firs define he mean-squared error (MSE) beween x R and ˇx RZF as MSE ZF = 1 N E [ x ˇx ZF 2] he expecaion is aken over he realizaions of channel and noise Wih reasonable assumpions he MSE of he ZFype esimaor is derived in [11] which is rewrien in he following lemma Lemma 2 If we approximae he quanizaion error using an addiional Gaussian noise w as ŷ = H H x+n+w N wih w CN(0 K σ 2 q ρ N I K ) and assume 1 K HHH = I N he MSE of he ZF-ype esimaor is given as MSE ZF = N ρ 1 +σq 2 K The lemma can be shown using he analyical ools developedin frameexpansion[12]please see Lemma4in [11]for deails Noe ha he assumpion 1 K HHH = I N holds when K Moreover Lemma 2 shows ha he MSE of he ZF-ype esimaor can be made arbirary small by increasing he number of receive nodes IV CHANNEL ESTIMATION TECHNIQUES Noe ha he analyical resuls in he previous secion are based on he assumpion of perfec channel knowledge a he fusion cener In pracice global channel knowledge should be esimaed using raining signals Moreover channel esimaion echniques should be based on simple operaions a he receive nodes as for disribued recepion Because he channel beween he ransmier and each receive node can be esimaed separaely we focus on he channel vecor of k-h receive node h k To develop channel esimaion echniques we consider a block-fading channel ie he channel is saic for he coherence block lengh of L channel uses and changes independenly from block-o-block Then he inpu-oupu relaion a he k-h receive node can be rewrien as y km [l] = h H km N x m[l]+n km [l] for he l-h channel use in he m-h fading block We assume ha he firs T < L channel uses are used for a raining phase and he remaining L T channel uses are dedicaed o a daa communicaion phase We can wrie he firs T received signals ino a vecor form as y kmrain = X H N mrainh km +n kmrain y kmrain = [ y km [0] y km [1] y km [T 1] ] H X mrain = [ x m [0] x m [1] x m [T 1] ] n kmrain = [ n km [0] n km [1] n km [T 1] ] H Noe ha y kmrain C T X mrain C N T and n kmrain C T In he raining phase X mrain is known o bohhe ransmierandhefusioncenerwhile h km needso be esimaed a he fusion cener We focus on uniary raining and assume X mrain saisfies [13] X H mrainx mrain = I T if N T X mrain X H mrain = T N I N if N < T The normalizaion erm T/N in he case of N < T ensures he average ransmi SNR is equal o ρ in each channel use Similar o Secion III-A we can reformulae hese expressions ino he real domain as y Rkmrain = X T N Rmrainh Rkm +n Rkmrain (4)

4 Re(ykmrain ) y Rkmrain = Im(y kmrain ) Re(Xmrain ) Im(X X Rmrain = mrain ) Im(X mrain ) Re(X mrain ) Re(hkm ) h Rkm = Im(h km ) Re(nkmrain ) n Rkmrain = Im(n kmrain ) I is easy o show ha y Rkmrain R 2T X Rmrain R 2N 2T h Rkm R 2N and n Rkmrain R 2T I is imporan o poin ou ha (4) has he same form as (1) while he roles of he channel and he raining signal are reversed Thus using he same echniques in Secion III we can develop channel esimaors based on he knowledge of he quanized signal ŷ Rkmrain and X Rmrain We define he i-h column of X Rmrain as x Rmraini and ŷ Rkmraini = sgn(y Rkmraini ) y Rkmraini is he i-h elemen of y Rkmrain Then he sign-refinemen based on ŷ Rkmraini is performed as x Rmraini = ŷ Rkmraini x Rmraini and he ML channel esimaor is given as 2T ( ) 2ρ ȟ RkmML = argmax Φ h R R2N N i=1 x T Rmrainih R 2T ( log Φ = argmax h R R2N i=1 ( 2ρ N x T Rmraini h R )) Because Φ( ) is a log-concave funcion we can efficienly solve (5) using sandard convex opimizaion mehods [14] However if T is no large enough he ML channel esimaor reurns an inaccurae channel esimae because here are no enough samples o esimae he rue channel For example if we only consider i = 1 hen here are many possible choices for h R ha give Φ( ) equals o one This rend is shown by numerical sudies in Secion V We can also define he ZF-ype channel esimaor as (5) ȟ RkmZF = ( X T Rmrain) ŷrkmrain (6) If N < T hen (6) can be also wrien as ȟ RkmZF = N T X Rmrainŷ Rkmrain because X mrain X H mrain = T N I N Noe ha he norm of ȟrkmzf highly depends on he norm of ŷ Rkmrain However ŷ Rkmrain is based on he sign funcion and does no have any norm informaion of y Rkmrain Thus we consider he MSE of he normalized channel esimae which is defined as MSE xh = 1 h Rkm E[ N h Rkm ȟrkmx 2] ȟrkmx MSE of normalized channel esimaion T ML esimaor SNR=10dB ZF esimaor SNR=10dB ML esimaor SNR=20dB ZF esimaor SNR=20dB Fig 2: The MSEs of he normalized channel esimaes wih N = 4 and differen values of T (raining channel uses) and ρ (ransmi SNR) for he performance meric of a receiver x in Secion V Alhough similar we are no able o apply Lemma 1 o analyze he performance of he ML channel esimaor because Lemma1isbasedonhenormconsrainon x R whileheml channel esimaor does no have such a consrain However we can sill analyze he performance of he ZF-ype channel esimaor wih he same assumpion of quanizaion error as in Lemma 2 Corollary1 If N < T and we approximae he quanizaion error of he firs T received raining signals in he m-h fading channel a he k-h receive node using an addiional Gaussian noise w kmrain as ŷ kmrain = X T mrain N h km +n kmrain +w kmrain wih w kmrain CN(0 T σqrain 2 N I T ) he MSE of he ZF-ype channel esimaor is given as MSE ZFrain = N3 ρ 1 +N 2σ2 qrain T The resul is a direc consequenceof Lemma 2 The lemma shows ha we can make he MSE of he ZF-ype channel esimaor arbirary small by increasing he lengh of he raining phase T Numerical sudies in Secion V shows he same resul holds for he ML channel esimaor as well V SIMULATION RESULTS We perform Mone-Carlo simulaions o evaluae he proposed channel esimaion echniques In Fig 2 we firs comparehe MSEs ofhenormalizedchannelesimaes ofheml andzf-ypechannelesimaorsiemse MLh andmse ZFh defined in he previous secion The resuls are averaged over channel realizaions of a single receive node wih N = 4 As expeced he MSEs of boh channel esimaors ρ

5 SER T=50 T=100 Perfec channel knowledge K=50 K= SNR (db) Fig 3: Symbol error rae (SER) vs SNR in db scale for he ZF-ype receiver wih differen levels of channel esimaion qualiy We se N = 4 and 8PSK for S go o zero as T increases The convergence rae of he ZFype channel esimaor(and he ML channel esimaor as well) is proporional o 1 T as derived in Corollary 1 Noe ha he ML channel esimaor is inferior o he ZF-ype channel esimaor when T is small which is explained in Secion IV However he ML channel esimaor ouperforms he ZF-ype channel esimaor as T becomes large The gap beween he wo channel esimaors is no significan wih 10dB SNR bu here is a noable gap beween he wo wih 20dB SNR In Fig 3 we plo he SER of he ZF-ype receiver 2 based on he ZF-ype channel esimaor The SER is defined as SER = 1 N E[Pr(ˆx n x n x senhnρn KS)] N n=1 he expecaionis akenover x H and n We againfix N = 4 and adop 8PSK consellaion for S As T increases he SER performance approaches o he case of perfec channel knowledge Alhough he ZF-ype receiver suffers from an error rae floor in high SNR regime he error floor can be miigaed by having more receive nodes for boh cases of perfec and esimaed channel knowledge we can exploi long-erm channel saisics ie spaial and/or emporal correlaion of channels as in [15] which would be an ineresing fuure research opic REFERENCES [1] L Azori A Iera and G Morabio The inerne of hings: A survey Elsevier Compuer Neworks vol 54 no 15 pp Oc 2010 [2] I F Akyildiz W Su Y Sankarasubramaniam and E Cayirci Wireless sensor neworks: A survey Compuer Neworks vol 38 no 4 pp Mar 2002 [3] V Mhare and C Rosenberg Design guidelines for wireless sensor neworks: Communicaion clusering and aggregaion Ad Hoc Neworks vol 2 no 1 pp Jan 2004 [4] R Viswanahan and P K Varshney Disribued deecion wih muliple sensors: Par I-fundamenals Proceedings of he IEEE vol 85 no 1 pp Jan 1997 [5] D J Love J Choi and P Bidigare Receive spaial coding for disribued diversiy Proceedings of IEEE Asilomar Conference on Signals Sysems and Compuers Nov 2013 [6] J Choi D J Love and P Bidigare Coded disribued diversiy: A novel disribued recepion echnique for wireless communicaion sysems IEEE Transacion on Signal Processing submied for publicaion [Online] Available: hp://arxivorg/abs/ [7-Y Wang Y S Han P K Varshney and P-N Chen Disribued faul-oleran classificaion in wireless sensor neworks IEEE Journal on Seleced Areas in Communicaions vol 23 no 4 pp Apr 2005 [8-Y Wang Y S Han B Chen and P K Varshney A combined decision fusion and channel coding scheme for disribued faul-oleran classificaion in wireless sensor neworks IEEE Transacions on Wireless Communicaions vol 5 no 7 pp Jul 2006 [9 L Marzea Noncooperaive cellular wireless wih unlimied numbers of base saion anennas IEEE Transacions on Wireless Communicaions vol 9 no 11 pp Nov 2010 [10] F Rusek D Persson B K Lau E G Larsson T L Marzea E O and F Tufvesson Scaling up MIMO: Opporuniies and challenges wih very large arrays IEEE Signal Processing Magazine vol 30 no 1 pp Jan 2013 [11] J Choi D J Love D R Brown III and M Bouin Disribued recepion wih spaial muliplexing: MIMO sysems for he Inerne of Things IEEE Transacion on Signal Processing submied for publicaion [Online] Available: hp://arxivorg/abs/ [12] V K Goyal M Veerli and N T Thao Quanized overcomplee expansionsin R n :Analysissynhesisandalgorihms IEEETransacions on Informaion Theory vol 44 no 1 pp Jan 1998 [13] W Sanipach and M L Honig Opimizaion of raining and feedback overhead for beamforming over block fading channels IEEE Transacions on Informaion Theory vol 56 no 12 pp Dec 2010 [14] S Boyd and L Vandenberghe Convex Opimizaion Cambridge Universiy Press 2009 [15] J Choi D J Love and P Bidigare Downlink raining echniques for FDD massive MIMO sysems: Open-loop and closed-loop raining wih memory IEEE Journal of Seleced Topics in Signal Processing vol 8 no 5 pp Oc 2014 VI CONCLUSION We sudied he scenario ha combines spaial muliplexing and cloud MIMO for disribued recepion in his paper To relax he ideal assumpion of perfec global channel knowledge considered in [11] we proposed pracical channel esimaion echniques ha rely on very simple operaions a he receive nodes We showed ha even wih very coarse quanizaion a he receive nodes he fusion cener can esimae he channel wih high accuracy if he lengh of he raining phase is sufficienly large To reduce he overhead of he raining phase 2 Because weconsider henormalized channel esimaeswedonocompare he SER of he ML receiver