Spectral and power efficiency trade-offs for multiple-antenna reception techniques under co-channel interference

Transcription

1 EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONS Eur. Tras. Telecomms. ; :5 7 Published olie i Wiley Olie Library (wileyolielibrary.com). DOI:./ett.535 RESEARCH ARTICLE Spectral ad power efficiecy trade-offs for multiple-atea receptio techiques uder co-chael iterferece Özgür Ertuğ* Electrical ad Electroics Egieerig Departmet, Gazi Uiversity, Telecommuicatios ad Sigal Processig Laboratory, 657 Akara, Turkey ABSTRACT Spatial receive diversity ad beamformig are two competig multiple-atea receptio strategies for ehacemet of spectral ad power efficiecy of wireless commuicatio systems because they both use the same resource, that is the spatial degrees of freedom. We aalyse i this paper ergodic, block-fadig outage ad delay-limited outage spectral efficiecies as well as average bit error rates ad service outage probabilities of a hybrid diversity-beamformig receiver with atea partitioig over flat Rayleigh-fadig uplik chaels uder co-chael iterferece. The results validated by Mote Carlo simulatios decipher the trade-offs i terms of spectral ad power efficiecies i betwee all-diversity, hybrid diversity-beamformig ad all-beamformig receptio for differet scearios represeted by average sigal-tooise ratio per path ad desired sigal-to-iterferece ratio for usig these multiple-atea receptio techiques i iterferece-limited wireless commuicatio systems. Copyright Joh Wiley & Sos, Ltd. *Correspodece Özgür Ertuğ, Electrical ad Electroics Egieerig Departmet, Gazi Uiversity, Telecommuicatios ad Sigal Processig Laboratory, 657 Akara, Turkey. ertug@gazi.edu.tr Received 3 September ; Revised 6 October ; Accepted 6 October. INTRODUCTION Cocurret ad ext-geeratio wireless cellular commuicatio systems are expected to provide multimedia services such as image ad video commuicatios with wireless access to Iteret besides voice ad data that demad high data rates ad lik reliabilities. Sigal fadig as well as co-chael iterferece due to frequecy reuse are the key detrimetal factors o the performace of these services that have to be combatted agaist [ 3]. Multiple ateas at the trasmitters ad receivers of wireless commuicatio systems ca be used for two distict purposes i fuctioally differig form to combat agaist these detrimetal factors over wireless chaels. First use may be to reduce sigal fadig by providig spatial diversity gai via trasmittig/receivig ucorrelated sigals [, 5]. Secod use may be to suppress co-chael iterferece directioally by providig array gai via trasmit/receive beamformig [6]. Both approaches are promisig for these commuicatio services to achieve the required spectral ad power efficiecies ad are uder itese research ad developmet leadig to various system desig cocepts. Because both approaches use the same resource that is the spatial degrees of freedom, spatial diversity ad beamformig are two competig techiques for ehacemet of detectio performaces, ad the workig priciples behid spatial diversity ad beamformig are quite differet. Cosiderig the receive side first, i spatial receive diversity, the mai idea is to use ateas separated by several wavelegths to achieve ucorrelated fadig at the receive ateas so that these compoets ca be combied coheretly i a optimum fashio via maximum ratio combiig (MRC) to ehace received average sigal-to-oise ratios (SNRs). O the other had, i provisio of array gai via receive beamformig, the ateas are separated by a small distace such as about half the carrier wavelegth just aroud spatial Nyquist samplig rate; thus, the received sigals at the ateas oly differ by a phase shift. Combiig these sigals with appropriate complex weights, array respose ca be steered towards desired sigals while leavig uwated iterferig sigals icludig self-multipath iterferece suppressed. O the trasmit side i the uplik chaels, the idea of trasmit space diversity sigallig is almost the same as Copyright Joh Wiley & Sos, Ltd. 5

2 Multiple-atea receptio techiques uder co-chael iterferece Ö. Ertuğ that of receive space diversity; however, because of the space limitatio at mobile statios, i geeral the trasmit atea elemets for space diversity have to be placed closely resultig i the correlatio of the fadig processes o the trasmit space diversity braches ad hece reducig the diversity gai achievable. Similarly, i trasmit beamformig, the mai aim is to appropriately weight the trasmit atea sigals to direct the trasmitted sigal eergy towards base statio i the uplik i azimuthal agular space ad to miimise the spatial iterferece to other directios. However, i both approaches of trasmit diversity ad trasmit beamformig, the space cosideratios that limit the iclusio of higher umber of atea elemets to achieve higher diversity or array gais as well as cosequet brach correlatio iduced due to low agular spreads reducig gais i both cases ad the eed to shift the sigal processig burde from the power/area limited mobile statios to the base statio dictate the implemetatio of diversity or beamformig approaches o the base statio side rather tha the mobile statio s especially i the uplik. Spatial diversity gai versus array gai via beamformig i multiple-atea wireless commuicatios are previously aalysed comparatively i some cotributios i the literature. Friedlader ad Scherzer aalysed the trasmit beamformig versus trasmit diversity uder flat fadig chael model i the forward lik of a cellular commuicatio system i [7, ] i terms of the ergodic capacity, outage capacity ad trasmit power required. For all these metrics, they preseted that uder ideal coditios such as ucorrelated fadig ad o had-offs, trasmit diversity is superior to trasmit beamformig i terms of spectral ad power efficiecy. O the other had, whe there is correlatio fadig due to low agular spreads ad there are soft had-offs of mobiles betwee cells (hece base-statio diversity exists), it is show that trasmit beamformig is superior to trasmit diversity. Similarly, Che et al. ivestigated the effects of icreased vehicular speed o the comparative performace of trasmit diversity ad trasmit beamformig for Uiversal Mobile Telecommuicatios System High-Speed Dowlik Packet Access compoet i [9] ad showed that for low vehicular speeds resultig i low Doppler spreads ad low fadig correlatio betwee diversity braches, trasmit diversity has better power efficiecy with respect to covetioal trasmit beamformig whereas for high vehicular speeds resultig i higher brach correlatio, trasmit beamformig turs out to be superior to trasmit diversity. Because both spatial diversity receptio ad beamformig use the same resource of spatial degrees of freedom, there also exists a trade-off i the partitioig of spatial degrees of freedom betwee the two techiques, ad some cotributios i the literature also proposed to combie trasmit diversity ad beamformig techiques as a hybrid withi the cotext to aalyse the achievable spectral ad power efficiecy gais. As examples, Nezafat ad Kaveh i [] aalysed the ergodic ad % outage capacity of a multiple-atea system over flat Rayleigh fadig both for reverse ad forward liks where ateas are divided ito beamformig groups while beamformig groups are combied for spatial diversity as well. Their aalysis presets that the reverse lik capacity i the ergodic regime of a diversity system is higher tha that of a beamformig system, the beamformig system ca provide higher capacity tha the diversity system i the forward lik ad a hybrid diversity-beamformig system provide comparable capacity to the diversity system i the reverse lik ad to the beamformig system i the forward lik. They also compare outage capacities of these three systems showig that the diversity system provides higher outage capacity both i reverse ad forward liks. I aother work, Pederse ad Morgese preseted a hybrid dowlik atea array algorithm i which both Alamouti-based space time trasmit diversity o slated dual-orthogoal polarised atea arrays are employed followed by a covetioal trasmit beamformer ad aalysed the algorithm for widebad code divisio multiple access Uiversal Mobile Telecommuicatios System dowlik [] maily emphasisig the power efficiecy gai of usig dual-polarised atea arrays i cojuctio with covetioal beamformig over the trasmit diversity oly with sigle-polarised ateas. Furthermore, Soi et al. proposed i [] joit trasmit diversity ecodig via space time spreadig (STS) combied with trasmit beamformig that they call steered STS where the total umber of ateas are divided ito two beamformig groups that are spaced at least c apart to achieve ucorrelated fadig betwee groups for trasmit diversity. Their aalysis of steered STS for CDMA dowlik over flat ad two-path fadig chaels versus mobile speed shows the sigificat power efficiecy gais of steered STS versus sigle-atea trasmissio as well as STS oly. May of these previous works o the comparative aalysis of diversity, beamformig ad hybrid diversitybeamformig systems i terms of spectral ad power efficiecy gais maily cocetrate o the dowlik chaels except the work of Nezafat ad Kaveh i []. Furthermore, i [], the trasmit or receive diversity, beamformig ad hybrid diversity-beamformig are compared with o co-chael iterferece existig due to frequecy reuse that directly has sigificat impact o the spectral ad power efficiecy gais of diversity, beamformig ad hybrid diversity-beamformig receivers. Similar system model assumptios also hold i [7, ]. We aalyse i this paper spectral ad power efficiecy trade-offs betwee spatial diversity receptio ad covetioal receive beamformig usig hybrid diversitybeamformig receivers with atea partitioig, a example of which is preseted i the work of Nezafat ad Kaveh [] for asychroous uplik of a multicell wireless commuicatio system with co-chael iterferece due to frequecy reuse over flat Rayleigh fadig chaels. I Sectio, we preset the system/chael model, the mathematical prelimiaries ad the hybrid diversitybeamformig receiver structure over which our aalyses are performed. I Sectio 3, we statistically characterise 6 Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett

3 Ö. Ertuğ Multiple-atea receptio techiques uder co-chael iterferece the desired user sigal-to-iterferece-ad-oise ratio (SINR). I Sectio, iformatio-theoretic ergodic, blockfadig outage ad delay-limited outage spectral efficiecies are formulated for evaluatio. The formulatio of average bit error rates (BERs) ad service outage probabilities for power efficiecy aalysis are pursued i Sectio 5. Numerical results ad discussios out of the formulatios i Sectio, ad Sectio 5 are preseted i Sectio 6 followed by our coclusios i Sectio 7.. SYSTEM MODEL AND PRELIMINARIES Cosiderig the detectio of a desired user i a asychroous uplik multicell wireless commuicatio system i the frequecy bad W s with cetre frequecy f c over time-varyig flat Rayleigh fadig chaels, a sigle domiat co-chael iterferig user from the eighbourig cells are assumed to iterfere with the desired user withi the same frequecy bad due to frequecy reuse. A sigle domiat iterferer assumptio well models i aalytically more tractable form the case where more tha oe iterferer exists ad the time-varyig flat Rayleigh fadig chael correspods to a o-lie-of-sight propagatio situatio with large coherece badwidth. Both desired ad iterferig users are assumed to be equipped with a sigle trasmit atea, whereas M omidirectioal vertically orieted dipole atea elemets are cosidered to be available at the base statio of the cell i which the desired user reside. The block diagram of the hybrid spatial receive diversity ad covetioal beamformig receiver is depicted i Figure where M available ateas are divided ito M d spatial diversity groups of M b beamformig ateas for spatial iterferece suppressio filterig; that is, M D M d M b. The diversity groups are spatially separated by c to assure idepedece of fadig betwee diversity groups, whereas the beamformig ateas i each diversity group are separated by a distace d m D c =, where c D c=f c is the carrier wavelegth for carrier frequecy f c ad c D 3 m/s is the speed of light. Assumig the directio of arrival (DOA) of the desired sigal source with respect to the atea array ormal is kow via a directio-fidig algorithm at the receiver, the spatial array respose of uiform liear array uder covetioal beamformig is give by MX b F./D g i exp.j r s si.// D g H. /a./ id () where g. / is the vector of beamformig coefficiets of the form g i D C i exp.j i' i /, a./ D Œ exp.j / exp.j / :::exp.j.m // T is the array steerig vector associated with azimuth agle ad r s D d m = is the spatial samplig rate; i.e. <r s =; where the array gai is maximum at r s D =. If the parameters ' i ad C i are set to ' i D.= c /d m si. / ad C i D =M i, the db maximum ormalised array respose occurs at DOA, ad all other sigals arrivig from differet directios will be suppressed accordig to the magitude of the array respose at the correspodig DOAs. The propagatio sceario cosidered i this paper is composed of a desired user with DOA with respect to the partitioed atea array ormal, whereas the DOA of the co-chael iterferig user is represeted by with respect to the partitioed atea array ormal ad the beamformers of the hybrid receiver are matched to the DOA of the desired user. For the iformatio-theoretic spectral efficiecy aalysis i Sectio, we assume users employ.n ; NR/ symbol trasmit power limited complex-gaussia radom block ecodig as block-legth N!o the trasmitter side such that Efjs Œj gp ad Efjs Œj gp,where R is the code rate i bits/symbol, P is the desired user s sigal power, P is the iterferig user s sigal power ad is the average oise power ad the use of maximumlikelihood decodig o the receiver side. O the other had, for the power efficiecy aalysis through average BERs ad service outage probabilities i Sectio 5, we assume users employ atipodal biary phase-shift keyig (BPSK) sigallig, that is s k Œ D A k g.t/ cos.f c t C i/;k f; g; i f; g with hard-decisio detectio o the receiver side. Figure. System model of hybrid spatial receive diversity ad beamformig receiver structure. Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett 7

4 Multiple-atea receptio techiques uder co-chael iterferece Ö. Ertuğ Via the large spacig betwee spatial diversity groups, the fadig processes of each user at each diversity group are idepedet over spatial diversity groups, whereas each of these fadig processes differ by a phase shift over each beamformig atea i a spatial diversity group. Therefore, there exists M d mutually idepedet statioary ad ergodic Rayleigh fadig processes h km.t/ i the system for each user assumed to have uity path power. This results i zero-mea uity-variace complex-gaussia distributed fadig chael coefficiets after samplig of the matched filter outputs i each chael symbol sigallig iterval, that is h km Œ; m f;;:::;m d g. At the iput to the hybrid diversity-beamformig receiver after radio frequecy dowcoversio ad matched filterig with rectagular pulse-shapig filter g.t/, the received discrete-time basebad vectorial sigal at M receive ateas for th symbol is the give by YŒ D h ;d Œs Œ C h ;d Œs Œ C Œ () where s Œ ad s Œ are the radomly coded or BPSKmodulated data streams of the desired user ad iterferig users, respectively, h ;d Œ ad h ;d Œ are the desired user s ad iterferig users directioal chael impulse respose vectors give after droppig the time-idex: h ;d D 6 h ;d D 6 h a. / h a. / : h Md a. / h a. / h a. / : h Md a. / D h a. / (3) D h a. / () where the operator stads for matrix Kroeecker product ad is the zero-mea spatio-temporally white complex- Gaussia backgroud oise vector over M o receive ateas with autocovariace matrix E H D I,thatis D N W s with N = beig the basebad oise power spectral desity. The average received SNR per path of the desired ad iterferig users are the defied as N D Efjh i j gp N D Efjh i j gp D P D P as a result of uity path power chael coefficiets ad the average received desired sigal-to-iterferece ratio (SIR) per diversity group is give by (5) (6) N D Efjh i j gp Efjh i j gp D P P D N N (7) Furthermore, assumig the statistical idepedece of the desired sigal, the iterferig sigals ad the backgroud oise, the covariace matrix of the received sigal is give by R YY D R SS CR NN D h ;d h H ;d P Ch ;d h H ;d P C I () where R SS D h ;d h H ;d P (9) is the desired sigal covariace matrix ad R NN D h ;d h H ;d P C I () is the covariace matrix of the co-chael iterferig user plus backgroud oise. 3. STATISTICAL CHARACTERISATION OF DESIRED USER S SIGNAL-TO-INTERFERENCE- AND-NOISE RATIO The derivatio of spectral ad power efficiecies of the desired user with hybrid diversity-beamformig receiver uder co-chael iterferece requires accurate statistical characterisatio of the output SINR of the desired user. Via the structure of the hybrid MRC diversity combiig ad covetioal beamformig receiver i Figure, the spatio-temporal diversity combiig ad covetioal beamformig filter vector is give by w D M b h a. / () It is worthwhile otig here that MRC combiig is optimum i the sese of maximisig output SNR i additive white Gaussia oise but ot ecessarily i the presece of co-chael iterferers with the optimum combiig [3] beig the optimum SINR-maximisig diversity combiig scheme. Furthermore, the optimum SINR-maximiser combier scheme uder co-chael iterferece is also the miimum-variace beamformer or Capo s beamformer []. However, i terms of diversity combiig, optimum combiig is quite complex ad very sesitive to chael estimatio accuracies i practice with respect to MRC. Moreover, i terms of beamformig, although Capo s beamformer maximises SINR, its performace is drastically reduced i practice uder DOA mismatches. Therefore, throughout the sequel, we maily costrai our attetio o the use of covetioal beamformig ad MRC i hybrid diversity-beamformig receiver structure as preseted i Figure. Assumig the oise has same power o all atea braches, the output SINR of the desired user with hybrid receiver is give by Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett

5 Ö. Ertuğ Multiple-atea receptio techiques uder co-chael iterferece D wh h ;d h H ;d w w H () R NN w Our mai goal is to derive a geeral expressio for the probability desity fuctio (pdf) of the output SINR of the desired user for further derivatios. Let x k D h H k;d h k;d;kd ; (3) ad ˇ ˇh H ;d h ˇ ;d ˇ D () x x be two radom variables. The the SINR of the desired user is defied as x D ˇˇwH x C E ˇˇo D x x C (5) The beamformig power loss factor or iverse array gai ˇ o the radom variable x k is S b D ˇg H ˇ. k /a. k / ˇ D such that x k D h H k h k;k D ;, ad the radom variable x k is thus chi-squared distributed with M d degrees of freedom havig pdf f.x k /D.M k d / xm d exp.x k k =P k /; x k (6) havig mea power P k where.x/ D R t x e x dx is the complete Gamma or factorial-iterpolatig fuctio. O the other had, the radom variable ca be expaded as ˇ ˇh H h ˇ ˇ D (7) x x havig mea power S b P where the beamformig power loss factor S b o the iterferig sigal arrivig from with a covetioal beamformer matched to DOA of the desired user at is expressed as ˇ S b D ˇg H ˇ. /a. / ˇ D M X b M exp.j ir s.si. / si. ///j b ˇ id () that is preseted i Figure versus ad M b for D 9 ı ad r s D = where array gai is maximal for fixed ad M b. From the aalysis of the beamformig power loss surface, which ca be exteded to ay value, the array gai icreases as the separatio betwee desired ad iterferig users DOAs icrease as well as with the icreased umber of beamformig ateas. Through the use of Melli covolutio [5], it ca be show that the radom variable z D x is expoetially distributed with mea S b P : f.z / D exp z ; z (9) S b P S b P The pdf of the radom variable t D z C is also give by [6]! f t.t/ D exp t ; t () S b P S b P Represetig the SINR radom variable by variable D x =t, the radom variables x ad t are statistically idepedet; thus, their joit pdf ca be writte as S b M b φ 6 Figure. Beamformig power loss factor or iverse array gai versus ad M b for D 9 ı ad r s D =. Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett 9

6 Multiple-atea receptio techiques uder co-chael iterferece Ö. Ertuğ the product of the margial pdfs of x ad t: f x ;t.x ;t/d f x.x /f t.t/ () Through the result of Papoulis [6], the pdf of a radom variable i divisio form as D x =t ca be derived as Z f. / D tf x M d ;t. t;t/dt () which ca be brought ito the form usig the statistical idepedece priciple i Equatio () as Z f. / D tf x. t/f t.t/dt (3) ad the desity fuctio of the SINR radom variable is arrived after some algebra as MX d Md f. / D i id M d Md i P C S b P ic e S b P c.i C / c.m d / P ; () A Mote Carlo simulatio validatio of the pdf for obtaied i Equatio () is preseted i Figure 3, ad the histogram obtaied via Mote Carlo simulatio of Equatio (5) with 6 rus exactly matches the derived aalytic pdf i Equatio ().. SPECTRAL EFFICIENCIES.. Fast-fadig ergodic spectral efficiecy Ergodic mea/average spectral efficiecy e uder chael state iformatio receiver (CSIR) oly i bits/ secod/hertz is the operatioal chael capacity per uit badwidth otio over sufficietly fast-fadig chaels ad is equivalet to ifiite-horizo time-series averagig of idepedet ad idetically distributed maximum mutual iformatio radom variables through strog law of large umbers. Although the exact sum-rates capacity regio of twouser iterferece chaels are ot yet kow i the literature eve for Gaussia chaels ad is approximated by upper ad lower bouds, the sigle-user capacity per uit badwidth for two-user iterferece chael uder sigleuser maximum-likelihood decodig i bits/secods/hertz is defied i terms of SINR as [7] e D C e D lim W s N! N NX I max Œ D NX D lim max H.Y I X jx / N! N f D X.X / NX D lim log N! N. C Œ/ D D E flog. C /g (5) The averagig over the distributio of SINR i () yields Md i id Z log. C / MX d e D Md i S b P c.i C / c.m d / M d e P ic d (6) P C S b P f γ (γ ) Pdf of theoretical SINR Histogram of SINR Ufortuately, the ier itegral i the form of the oe i Equatio (6) caot be take i closed-form, ad similar fiite-limit or ifiite-limit itegrals appear throughout the sequel for the rest of the performace metrics derived i the followig sectios. Hece, i these cases, we provide umerical results through high-resolutio umerical trapezoidal itegratio of these formulas as well as their Mote-Carlo simulatio validatio i Sectio γ Figure 3. Mote Carlo simulatio based validatio of the theoretical probability desity fuctio of output sigal-toiterferece-ad-oise ratio at M d D, M b D, r s D :5, P D db, DIR D db, D 3 ı ad D 6 ı... Block-fadig outage spectral efficiecy I block-fadig chael model with CSIR, the fadig is assumed to be sufficietly slow such that a whole codeword trasmissio experieces the same fade level, ad the probability of outage for a outage spectral efficiecy o is the probability of average maximum mutual iformatio over the whole block legth fallig below the outage spectral Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett

7 Ö. Ertuğ Multiple-atea receptio techiques uder co-chael iterferece efficiecy defied i terms of as 9 < P out.c/ NX = D Pr I : max Œ o N ; D 9 < NX = D Pr log : N. C Œ/ o ; D D Pr flog. C / o g Z o D log. C /f. /d (7) Takig the probability of outage itegral over the distributio of SINR i Equatio () yields P out.c/ D X M d Md id i Md i S b P c.i C / c.m d / Z o M d e P log. C / ic d () P C S b P.3. Delay-limited outage spectral efficiecy Delay-limited outage spectral efficiecy d uder both chael state iformatio trasmitter (CSIT) ad CSIR with optimal water-fillig power allocatio over all fadig states [] is defied as the maximisatio of outage spectral efficiecy as the limit of outage probability teds to zero with probability oe for all fadig states subject to log-term average power costraits ad is also frequetly deoted as zero-outage spectral efficiecy. I our sceario cosidered with co-chael iterferece ad with CSIT ad CSIR for both desired ad iterferig users, respectively, the delay-limited spectral efficiecy is formulated as d D lim max Pr flog!. C / g s:t:e fp.h /g P ( ) D lim max Pr log!. C kh k P x C M d / (9) For miimal power usage to be achieved, the optimal power allocatio must satisfy P.h / D. / x C M d kh k (3) where the outage spectral efficiecy should be chose such that E P.h / D P ad the average of optimal power allocatio is E P.h / D. /E ( ) x C M d kh k (3) For the log-term average power costrait to be met, the optimal power allocatio is the give by P.h / D E P x CM d kh k x C M d kh k D P (3) ad the delay-limited outage spectral efficiecy equals to where E d D log C P C E x CM d A kh k D C o A (33) E MX d D id Md i Md i S b P c.i C / c.m d / Z M d e P ic d (3) P C S b P 5. POWER EFFICIENCIES 5.. Average bit error rate For BPSK modulatio over slowly fadig chaels, the coditioal BER of the desired user is give by P b.j / D Q.p / (35) where Q.x/ D = p R x ex dx is the complemetary Gaussia error fuctio ad the average BER of the desired user is equal to the averagig of coditioal error rate over the distributio of SINR : N P b./ D Z Q. p /f. /d (36) However, averagig of Q./ fuctio directly over a desity fuctio is hard to evaluate because the itegrad is i fact i the lower limit of the itegral i the defiitio of Q./ fuctio. Istead, by usig the exact Craig s represetatio of the Q./ fuctio [9] as Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett

8 Multiple-atea receptio techiques uder co-chael iterferece Ö. Ertuğ Z Q.x/ D! exp x si d (37)./ which is valid for all x, the coditioal error rate of the desired user equals to P b.j / D Z exp si d (3)./ ad the average probability of bit error is give by PN b./ D Z Z exp si f. /d d./ (39) Performig the averagig over the desity fuctio of SINR i Equatio (), PN b./ D MX d Md i id Z Z M d e Md i c.i C / S b P c.m d / P C si./ P C S b P ic 5.. Service outage probability d d () Service outage probability i iterferece-limited wireless commuicatios is defied as the output SINR of the desired user fallig below a certai threshold deoted as power protectio threshold ˇ [5]: P.s/ out D Pr f ˇg () Via the desity fuctio of SINR i Equatio (), the probability of service outage of the desired user is the give by M P out.s/ X d Md D i id Md i S b P c.i C / c.m d / Zˇ M d e P ic d () P C S b P 6. NUMERICAL RESULTS AND DISCUSSION The Mote Carlo simulatio based validatio of the ergodic, outage ad delay-limited spectral efficiecies umerically evaluated with high-resolutio trapezoidal itegratio i Equatios (6), () ad (33) (3) as well as average BER ad service outage probabilities i Equatios () ad () by usig 6 averagig rus over the SINR defiitio i Equatio (5) are preseted i Figures ad 5, respectively, ad the Mote Carlo based estimatios are quite close matches for all five performace metrics cosidered to the aalytic curves obtaied with the aid of umerical itegratios. For the variatio of achievable spectral ad power efficiecies to be aalysed, the ergodic, block-fadig outage ad delay-limited outage spectral efficiecies as well as average BER ad service outage probabilities are preseted from Figure 6 versus average SNR per path i decibels at r s D :5 ad at differet M d =M b combiatios of a total umber of M D ateas for two differet scearios. The first sceario is for SIR D db ad DOA pair of D 3 ı / D ı correspodig to a high iterferece/low array gai situatio for fixed M b. Secod sceario is for SIR D db ad DOA pair of D Spectral efficiecy (bits/sec/hz) Ergodic SE Aalytic Ergodic SE Simulatio % BF Outage SE Aalytic % BF Outage SE Simulatio DL Outage SE Aalytic DL Outage SE Simulatio Average SNR per path Figure. Mote Carlo simulatio based validatio of spectral efficiecies at M d D, M b D, r s D :5, SIR D db, D 3 ı ad D 6 ı. Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett

9 Ö. Ertuğ Multiple-atea receptio techiques uder co-chael iterferece Log(P) 3 Average BER Aalytic Average BER Simulatio db Outage Probability Aalytic db Outage Probability Simulatio Average SNR per path Figure 5. Mote Carlo based validatio of average bit error rate ad db service outage probability at M d D, M b D, r s D :5, SIR D db, D 3 ı ad D 6 ı. Spectral efficiecy (bits/sec/hz) Average SNR per path (db) Figure 6. Ergodic spectral efficiecies of hybrid diversitybeamformig receiver for various partitioigs of a total of M D ateas at r s D :5, for SIR D db ad directio of arrival pair of D 3 ı / D ı correspodig to a high iterferece/low array gai sceario for fixed M b or for SIR D db ad directio of arrival pair of D 3 ı / D 6 ı correspodig to a low iterferece/high array gai sceario, for fixed M b ; -: M d D =M b D, SIR D db ad D 3 ı = D ı, o-:m d D =M b D, SIR D db ad D 3 ı = D ı, x-:m d D =M b D, SIR D db ad D 3 ı = D ı, C-:M d D =M b D, SIR D db ad D 3 ı = D ı, :M d D =M b D, SIR D db ad D 3 ı = D 6 ı, o :M d D =M b D, SIR D db ad D 3 ı = D 6 ı, x :M d D =M b D, SIR D db ad D 3 ı = D 6 ı, C :M d D =M b D, SIR D db ad D 3 ı = D 6 ı. 3 ı / D 6 ı correspodig to a low iterferece/high array gai situatio for fixed M b. Cosiderig first all five graphs together, both spectral efficiecy curves as well as the average BER ad service outage probability curves for the same umber of diversity/beamformig ateas for two differet scearios start closely for low average SNRs per path ad the differece i-betwee icreases for icreased average SNR per path. All the curves also coverge to certai floor values as a result of co-chael iterferece idetified by SIR level. Furthermore, for a certai sceario, diversity/beamformig atea partitioig ad average SNR per path, the ergodic spectral efficiecy presets the highest data rate per uit badwidth achievable followed by delay-limited outage spectral efficiecy due to availability of CSIT, ad the lowest data rate per uit badwidth is represeted by the block-fadig outage spectral efficiecy. Aalysig the spectral ad power efficiecy curves idividually i terms of the variatio o the partitioig of a total umber of eight ateas i-betwee spatial diversity receptio ad beamformig, the ergodic spectral efficiecy curves i Figure 6 for high iterferece/low array gai sceario, the highest ergodic spectral efficiecy is obtaied with a all-diversity receiver usig M d D =M b D combiatio due to higher diversity gai offered followed by hybrid diversity-beamformig M d D =M b D combiatio. For hybrid diversity-beamformig M d D =M b D ad all-beamformig M d D =M b D combiatios, at low SNR, M d D =M b D combiatio with higher diversity order provides higher ergodic spectral efficiecy tha all-beamformig M d D =M b D combiatio, whereas after a average SNR per path of 5 db, all-beamformig M d D =M b D combiatio with higher array gai at M b D provides higher ergodic spectral efficiecy tha M d D =M b D combiatio. O the other had, for low iterferece/high array gai sceario, at low average SNR values, a alldiversity M d D =M b D combiatio provides the highest ergodic spectral efficiecy due to diversity gai, ad a all-beamformig M d D =M b D combiatio provides the lowest array gai. However, at high average SNR regio, hybrid diversity-beamformig M b D /M d D combiatio provides the highest ergodic spectral efficiecy followed by a all-beamformig M b D /M d D combiatio due to hybrid diversity ad array gai. Cosiderig the aalysis of % block-fadig outage spectral efficiecies versus average SNR per path i decibels preseted i Figure 7, for high iterferece/low array gai sceario, the highest ergodic spectral efficiecy for fixed average SNR per path is provided by all-diversity M d D =M b D combiatio, ad the lowest ergodic spectral efficiecy is provided by all-beamformig M d D =M b D combiatio favourig the preferece of allocatio of all receive ateas to diversity. O the other had, for low iterferece/high array gai sceario, at low average SNR per path values, the highest erogdic spectral efficiecy is agai provided by all-diversity M d D =M b D combiatio, ad the lowest ergodic spectral efficiecy is provided by all-beamformig M d D =M b D combiatio favourig the preferece of allocatio of all receive ateas to diversity, whereas at sufficietly high average received SNR per path values, hybrid diversitybeamformig M d D =M b D combiatio provides Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett 3

10 Multiple-atea receptio techiques uder co-chael iterferece Ö. Ertuğ Spectral efficiecy (bits/sec/hz) Average SNR per path (db) Figure 7. % block-fadig outage spectral efficiecies of hybrid diversity-beamformig receiver for various partitioig of a total of M D ateas at r s D :5, for SIR D db ad directio of arrival pair of D 3 ı = D ı correspodig to a high iterferece/low array gai sceario for fixed M b or for SIR D db ad directio of arrival pair of D 3 ı = D 6 ı correspodig to a low iterferece/high array gai sceario for fixed M b ; -: M d D =M b D, SIR D db ad D 3 ı = D ı, o-:m d D =M b D, SIR D db ad D 3 ı = D ı, x-:m d D =M b D, SIR D db ad D 3 ı = D ı, C-:M d D =M b D, SIR D db ad D 3 ı = D ı, :M d D =M b D, SIR D db ad D 3 ı = D 6 ı, o :M d D =M b D, SIR D db ad D 3 ı = D 6 ı, x :M d D =M b D, SIR D db ad D 3 ı = D 6 ı, C :M d D =M b D, SIR D db ad D 3 ı = D 6 ı. the highest spectral efficiecy, ad all-beamformig M b D =M d D combiatio provides higher spectral efficiecy tha hybrid diversity-beamformig M b D =M d D combiatio. Cosiderig the delay-limited outage spectral efficiecies with CSIR ad CSIT at Figure, first the delay-limited ergodic spectral efficiecies provided by all-beamformig M d D =M b D combiatios i both scearios are zero as expected because the delay-limited outage spectral efficiecy is zero for a scalar sigle-atea lik, ad a all-beamformig receiver forms actually a sige-atea lik ehaced with array gai. For high iterferece/low array gai sceario, the highest ergodic spectral efficiecy for fixed average SNR per path is agai provided by alldiversity M d D =M b D combiatio, ad the lowest ergodic spectral efficiecy is provided by hybrid diversitybeamformig M d D =M b D combiatio favourig the preferece of allocatio of all receive ateas to diversity. O the other had, for low iterferece/high array gai sceario, at low average SNR per path values, the highest erogdic spectral efficiecy is agai provided by alldiversity M d D =M b D combiatio, ad the lowest ergodic spectral efficiecy is provided by hybrid diversitybeamformig M d D =M b D combiatio favourig the preferece of allocatio of all receive ateas to diversity, whereas at sufficietly high average received SNR per path values, hybrid diversity-beamformig M d D =M b D combiatio provides the highest spectral efficiecy. By aalysig the average BER ad service outage probability curves for the hybrid receiver for power efficiecy i Figures 9 ad, it turs out that for both high iterferece/low array gai ad low iterferece/high array gai scearios, the lowest average BER ad service outage probability are provided by all-diversity M d D =M b D combiatio ad the highest average BER ad service Spectral efficiecy (bits/sec/hz) Average SNR per path (db) Figure. Delay-limited outage spectral efficiecies of hybrid diversity-beamformig receiver for various partitioig of a total of M D ateas at r s D :5, for SIR D db ad directio of arrival pair of D 3 ı = D ı correspodig to a high iterferece/low array gai sceario for fixed M b or for SIR D db ad directio of arrival pair of D 3 ı = D 6 ı correspodig to a low iterferece/high array gai sceario for fixed M b ; -: M d D =M b D, SIR D db ad D 3 ı = D ı,o-:m d D =M b D, SIR D db ad D 3 ı = D ı,x-:m d D =M b D, SIR D db ad D 3 ı = D ı, C-:M d D =M b D, SIR D db ad D 3 ı = D ı, :M d D =M b D, SIR D db ad D 3 ı = D 6 ı,o :M d D =M b D, SIR D db ad D 3 ı = D 6 ı,x :M d D =M b D, SIR D db ad D 3 ı = D 6 ı, C :M d D =M b D, SIR D db ad D 3 ı = D 6 ı. Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett

11 Ö. Ertuğ Multiple-atea receptio techiques uder co-chael iterferece Log(P e ) Average SNR per path i db Figure 9. Average bit error rate of hybrid diversity-beamformig receiver for various partitioig of a total of M D ateas at r s D :5, for SIR D db ad directio of arrival pair of D 3 ı = D ı correspodig to a high iterferece/low array gai sceario for fixed M b or for SIR D db ad directio of arrival pair of D 3 ı = D 6 ı correspodig to a low iterferece/high array gai sceario for fixed M b ; -: M d D =M b D, SIR D db ad D 3 ı = D ı,o-:m d D =M b D, SIR D db ad D 3 ı / D ı,x-:m d D =M b D, SIR D db ad D 3 ı / D ı, C-:M d D =M b D, SIR D db ad D 3 ı / D ı, :M d D =M b D, SIR D db ad D 3 ı / D 6 ı,o :M d D =M b D, SIR D db ad D 3 ı / D 6 ı,x :M d D =M b D, SIR D db ad D 3 ı / D 6 ı, C :M d D =M b D, SIR D db ad D 3 ı / D 6 ı. outage probability is preset with all-beamformig M d D =M b D combiatio for all average SNR per path values dictatig the allocatio of all receive ateas to diversity receptio for power efficiecy maximisatio. Through these results, the followig statemets through observatios are made for the spectral ad power efficiecy trade-offs betwee multiple-atea receptio strategies of spatial diversity receptio ad beamformig receptio uder the realistic sceario of the existece of co-chael iterferece: I terms of ergodic spectral efficiecies, at high iterferece/low array gai scearios, a all-diversity system provides the highest spectral efficiecy, whereas at low iterferece/high array gai scearios, although the ergodic spectral efficiecy of a all-diversity system may still be higher at low average SNR per paths, a hybrid diversity-beamformig system ad a all-beamformig system may provide higher ergodic spectral efficiecy tha a all-diversity system at high average SNRs per path. I terms of block-fadig outage spectral efficiecies, at high iterferece/low array gai scearios, a all-diversity system is uiformly superior to a hybrid diversity-beamformig system ad a all-beamformig system, whereas at low iterferece/high array gai scearios, although a alldiversity system may provide the highest block-fadig (s) Log(P out ) Average SNR per path i db Figure. Service outage probabilities of hybrid diversitybeamformig receiver for various partitioig of a total of M D ateas at r s D :5, for SIR D db ad directio of arrival pair of D 3 ı = D ı correspodig to a high iterferece/low array gai sceario for fixed M b or for SIR D db ad directio of arrival pair of D 3 ı = D 6 ı correspodig to a low iterferece/high array gai sceario for fixed M b ; -: M d D =M b D, SIR D db ad D 3 ı = D ı,o-:m d D =M b D, SIR D db ad D 3 ı = D ı,x-:m d D =M b D, SIR D db ad D 3 ı = D ı, C-:M d D =M b D, SIR D db ad D 3 ı = D ı, :M d D =M b D, SIR D db ad D 3 ı = D 6 ı,o :M d D =M b D, SIR D db ad D 3 ı = D 6 ı,x :M d D =M b D, SIR D db ad D 3 ı = D 6 ı, C :M d D =M b D, SIR D db ad D 3 ı = D 6 ı. outage spectral efficiecy at low average SNRs per path, a hybrid diversity-beamformig system may provide higher block-fadig outage spectral efficiecy tha a all-diversity system. I terms of delay-limited outage spectral efficiecies, at high iterferece/low array gai scearios, the highest delay-limited outage spectral efficiecy is provided by a all-diversity system, whereas at low iterferece/high array gai scearios, although a all-diversity system may provide the highest delaylimited outage spectral efficiecy at low average SNRs per path, the delay-limited outage spectral efficiecy provided by a hybrid diversity-beamformig system may be higher tha that of a all-diversity system. I terms of power efficiecies through the aalysis of average BERs ad service outage probabilities, a all-diversity system is uiformly superior i terms of power efficiecy to achieve a fixed average BER or service outage probability to a hybrid diversity-beamformig system or a all-beamformig system. 7. CONCLUSIONS We aalysed i this paper ergodic, block-fadig outage ad delay-limited outage spectral efficiecies as well as Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett 5

12 Multiple-atea receptio techiques uder co-chael iterferece Ö. Ertuğ average BERs ad service outage probabilities of hybrid diversity-beamformig receivers with atea partitioig over flat Rayleigh-fadig uplik chaels with co-chael iterferece to elucidate the spectral ad power efficiecy trade-offs betwee all-diversity, hybrid diversitybeamformig ad all-beamformig receptio. The derived results evaluated by the aid of umerical itegratios ad validated by Mote Carlo simulatios preset as a complete picture uder the realistic model with co-chael iterferece that although a all-diversity system is uiformly superior to a hybrid diversity-beamformig system or a all-beamformig system i the high iterferece/low array gai scearios for all average SNRs per path, the ergodic, block-fadig outage or delay-limited outage spectral efficiecy provided by a hybrid diversity-beamformig ad/or a all-beamformig system may be higher tha that of a all-diversity system at high average received SNRs per path i low iterferece/high array gai scearios. O the other had, i terms of power efficiecies evaluated through average BERs ad service outage probabilities, a all-diversity system is uiformly superior i the uplik to a hybrid diversity-beamformig system ad a all-beamformig system dictatig the allocatio of all receive ateas to spatial diversity receptio for power efficiecy maximisatio. These theoretical ad simulatio-based fidigs o the spectral ad power efficiecy trade-offs betwee all-diversity, hybrid diversity-beamformig ad allbeamformig receptio ca also be further used to improve the achievable data rates ad power efficiecies o the average by adaptively switchig i-betwee these modes o the basis of the trasmit SIR ad average received SNR i software-defied base statios. Aother iterestig ope research questio is also to cosider multipleateas at the trasmitter ad to aalyse the spectral ad power efficiecies whe there is atea partitioig ibetwee trasmit diversity ad trasmit beamformig as at the receiver. REFERENCES. Lee WCY. Essetials of Wireless Commuicatios. McGraw Hill,.. Simo MK, Alouii MS. Digital Commuicatio Over Fadig Chaels: A Uified Approach to Performace Aalysis. Joh Wiley ad Sos: Bosto, MA, USA,. 3. Rappaport TS. Wireless Commuicatios: Priciples ad Practice. Pretice Hall: Hoboke, NJ, USA, Milovic DM, Stefaovic M, Pokrajac DM. Stochastic approach for output SINR computatio at SC diversity systems with correlated Nakagami-m fadig. Europea Trasactios o Telecommuicatios August 9; (5): da Silva MM, Correira AMC, Diis R. O tramissio techiques for multi-atea CDMA systems. Europea Trasactios o Telecommuicatios Jauary 9; (): Zha D. Mobile commuicatio array processig based o radial-basis fuctio eural etwork ad fractioal lower order statistics. Europea Trasactios o Telecommuicatios April 9; (3): Friedlader B, Scherzer S. Beamformig vs. trasmit diversity i the dowlik of a cellular commuicatios system, I Coferece Record of 35 Asilomar Coferece o Sigals Systems ad Computers, Asilomar, CA, November ;.. Friedlader B, Scherzer S. Spatial diversity vs. array gai i cellular commuicatio systems, I Coferece Record of 36 Asilomar Coferece o Sigals Systems ad Computers, Asilomar, CA, November ; Che Y, Speidel J, Hoek C, Russ S. Ivestigatio of diversity ad beamformig techiques for UMTS high speed dowlik packet access, I Proceedigs of Iteratioal Symposium o Persoal Idoor ad Mobile Radio Commuicatios 5, September 5; Nezafat M, Kaveh M. Aalysis of diversity, beamformig ad hybrid-diversity beamformig systems, I Proceedigs of 3 IEEE Iteratioal Coferece o Acoustics Speech ad Sigal Processig, April 3; Pederse KI, Morgese PE. A simple dowlik atea array algorithm based o a hybrid scheme of trasmit diversity ad covetioal beamformig, I Proceedigs of Vehicular Techology Coferece Sprig, May ; Soi RA, Buehrer RM, Beig RD. Trasmit beamformig combied with diversity techiques for CDMA systems, I Proceedigs of IEEE Iteratioal Coferece o Acoustics Speech ad Sigal Processig, May ; Li S, Fag H. Accurate performace aalysis of optimum diversity combiig ad equalizatio over mobile radio fadig chael with CCI, I Proceedigs of ICICS, 5; 5.. Lorez RG, Boyd S. Robust miimum variace beamformig. IEEE Trasactios o Sigal Processig 5; 53(5): Cui J, Sheikh AUH. Outage probability of cellular radio systems usig maximal ratio combiig i the presece of multiple iterferers. IEEE Trasactios o Commuicatios August 999; 7():. 6. Papoulis A. Probability, Radom Variables ad Stochastic Processes. McGraw-Hill: New York, NY, USA, Hamdi KA. A useful lemma for capacity aalysis of fadig iterferece chaels. IEEE Trasactios o Commuicatios February ; 5(): 6. 6 Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett

13 Ö. Ertuğ Multiple-atea receptio techiques uder co-chael iterferece. Caire G, Taricco G, Biglieri E. Optimum power cotrol over fadig chaels. IEEE Trasactios o Iformatio Theory July 999; 5(5): Craig JW. A ew, simple ad exact result for calculatig the probability of error for two-dimesioal sigal costellatios, I Proceedigs of IEEE Military Commuicatios Coferece (MILCOM 9), McLea, VA, October 99; AUTHOR S BIOGRAPHY Özgür Ertuğ received his BSc degree from the Uiversity of Souther Califoria i 997, his MSc degree from William Marsh Rice Uiversity i 999 ad his PhD degree from Middle East Techical Uiversity i 5, all i Electrical ad Electroics Egieerig specializig o Telecommuicatios ad Sigal Processig. Sice 7, he serves i the faculty of Electrical ad Electroics Egieerig Departmet of Gazi Uiversity as a assistat professor. Dr Ertuğ serves as a techical committee member ad reviewer to major archival jourals ad cofereces. He is a member of IEEE Commuicatios Society ad IEEE Iformatio Theory Society. His mai research iterests lie i wireless commuicatio systems ad etworkig, sigal processig for wireless commuicatios, iformatio theory ad codig for wireless commuicatios, software-defied radio ad ateas, ad propagatio issues. Eur. Tras. Telecomms. :5 7 () Joh Wiley & Sos, Ltd. DOI:./ett 7