WIRELESS networks are growing rapidly as demand for

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL XX, NO XX, MONTH 009 Amplify-and-Forward Relay Neworks Under Received Power Consrain Alireza Shahan Behbahani, Suden Member, IEEE and A M Elawil, Member, IEEE Absrac Relay neworks have received considerable aenion recenly, especially when limied size and power resources impose consrains on he number of anennas a each node While fixed and mobile relays can co-operae o improve recepion a he desired desinaion, hey also conribue o un-inended inerference for neighboring cells reusing he same frequency In his paper, we propose and analyze a relay scheme o simulaneously imize SNR and minimize MSE, for an amplify-and-forward (AF) relay nework operaing under a receive power consrain guaraneeing ha he received signal power is bounded o conrol inerference o neighboring cells If he inended desinaion lies a he periphery of he cell, hen he proposed scheme guaranees ha he oal power leaking ino neighboring cells is bounded The opimal relay facors are provided for boh correlaed and uncorrelaed noise a he relays Simulaion resuls are presened o verify he analysis Index Terms Minimum-mean-square-error (MMSE), signalo-noise raio (SNR), relay neworks, relay opimizaion, inerference, noise correlaion I INTRODUCTION WIRELESS neworks are growing rapidly as demand for reliable, high daa rae, and efficien communicaion echnologies has inensified during recen years To address he requesed increase in capaciy and o improve wireless link performance, co-operaive wireless relay based neworks have been adoped Various relay sraegies have been sudied in lieraure [] Among hese sraegies, amplify-and-forward has been more widely adoped due o is inheren simpliciy, where he relays amplify he received signal and forward he scaled signal o he desinaion The problem of finding opimal relay facors for differen cos funcions and power consrains has been sudied in he lieraure for single and muliple anennas relays [], [3], [4] However, in mos preceding work, power consrains are placed a he ransmiing nodes, where he assumpion is ha hese nodes are fixed in space and hus heir effec in erms of inerference is limied and should no degrade service in neighboring cells By referring o he 806j working group documens [5], relay saions are no necessarily fixed in locaion bu raher he sandard defines hree classes of relay saions a) fixed, b) nomadic and c) mobile This creaes an ineresing scenario where raher han assuming ha ransmi power conrol can be cenralized a one common "base saion", i could be considered raher as a "nework" Manuscrip received November 4, 008; revised June 5, 009; acceped Augus 3, 009 The associae edior coordinaing he review of his leer and approving i for publicaion was J M Shea The auhors are wih he Dep of Elecrical Engineering and Compuer Science of Universiy of California, Irvine, CA 9697, USA ( {sshahanb, aelawil}@uciedu) This work was suppored in par under gran number NIJ/DOJ 006-IJ-CX- K044 Digial Objec Idenifier 009/TWC /08$500 c 009 IEEE parameer ha is a funcion of he number and geographical locaions of relays wihin a cell, where he relays co-operae o achieve a desired receive power Recenly here has been some research conribuions in [6], [7], where he power consrain is considered a he receiver sie (see Sec II-B for definiion and moivaion of power consrain) In [3] opimal resuls are provided for a MIMO amplify-and-forward relay nework under power consrain a he desinaion In his paper, we furher invesigae his concep and consider a SISO AF relay nework where each node is equipped wih anenna The power consrain is defined such ha he received signal power is bounded beween wo values The lower bound is defined in order o be able o decode he signal a he desinaion reliably, and he upper bound is defined o limi he inerference on oher neighboring neworks which are using he same specrum We minimize he MSE and show ha i is equivalen o imizing he receive SNR Simulaion resuls are presened ha quanify he performance of he proposed scheme A Noaion We shall use bold lower case for vecors, and bold capial leers for marices Furher ( ) and ( ) T sand for complex ransposiion and ransposiion respecively Also [A] ij denoes he elemen in row i and column j of marix A λ (A) and v (A) represen principal eigenvalue and eigenvecor of marix A respecively We denoe by (a a ) a diagonal marix wih diagonal elemens given by a i AlsoE sands for expecaion operaor II PROBLEM FORMULATION A Sysem Model We consider an amplify-and-forward (AF) relay nework consising of ransmi anenna a he source, receive anenna a he desinaion, and K relays, each equipped wih anenna which can be used for boh ransmission and recepion - see Fig We denoe by h s =[h s,h s h sk ] T he K channel vecor beween he source and relay nodes, while h =[h,h h K ] is he K channel vecor beween he relay nodes and he desinaion The ih elemen of he vecor h s, h si, is he channel beween he source and ih relay, which is called he backward channel In he same way h i, he ih elemen of he vecor h, is he channel beween he ih relay and he desinaion which is called he forward channel The channel vecors are memoryless and a quasi-saic fading condiion is assumed, Furhermore, heir elemens are assumed o be independen idenically disribued (iid) zero mean complex Gaussian wih variances σh s and σh which are h si CN(0,σh s ) and h i CN(0,σh )We

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL XX, NO XX, MONTH 009 hs hs h s hs K s Fig source v s A wireless relay nework relay relay relay K h h v h h K desinaion also assume ha he source has no channel sae informaion (CSI), and he desinaion has complee knowledge of all channels Each relay could have local CSI (local backward and forward channels informaion), or no channel informaion a all (see SecV) The need for more or less CSI was found o be dependen on he assumpion made regarding he correlaion of noise a he relays I is imporan o noe ha while in mos prior work in lieraure, noise a relays are considered uncorrelaed, here are pracical siuaions where noise a relays could be correlaed A direc example of his is he case of a wireless nework where each relay is exposed no only o is local noise, bu also o common inerference from oher nodes in neighboring neworks This resuls in correlaed noise a each relay node, even hough nodes are spaially separaed [8] For his reason we consider boh correlaed and uncorrelaed noise a relays The relay marix is represened by a K K diagonal marix F,whereih diagonal elemen, f i,isherelayih gain facor The received signal is modeled as y = h Fh s s + h Fv s + v, () where v s and v are zero mean addiive Gaussian noise (AGN) wih covariance marix R vs and power σv respecively and s is he ransmied signal wih power σs = E s I is assumed ha he ransmier always uses is imum power which is σs Our goal is o find he relay marix F o minimize MSE beween he source and desinaion under a power consrain (see secion II-B) which guaranees ha he received signal power is always less han a imum value (inerference level) We will show ha MSE minimizaion is equivalen o SNR imizaion B Power Consrain Tradiionally, consrains on power were placed separaely a he ransmission devices due o heir limied power capabiliy (source or relay) or due o regulaions specifying he imum power per ransmier This approach ypically assumes a cenralized "base saion" wih a specific ransmission mask y such ha inerference o neighboring cells is minimized However, due o significan aciviy in relay based neworks, here has been a recen rend o evaluae "nework" level power consrains, where he limiaion is no longer on he abiliy of a specific ransmier o emi power, bu raher on he abiliy of a se of ransmiers (fixed or mobile), o mee a power or inerference consrain a he receiver sie [3], [6], [7] By incorporaing knowledge of he locaion of he relays wihin a cell, and he imum power allowed a he periphery, one can limi inerference o neighboring cells by bounding he power received by a desinaion a he periphery from all allowable ransmiers (relays) wihin he cell Furhermore, he received signal power should exceed he minimum power required o be able o decode he desired signal correcly In order o saisfy he aforemenioned requiremens, we define he power consrain as he summaion of powers a he oupu of he forward channel of each relay (noe ha his is no he received signal power a he desinaion) In order o do ha we define he kh received signal a he desinaion as y k = h k f k h sk s + h k f k v sk, k =,,,K () which conribues he power p k = f k h k (σs h sk +[R vs ] kk ), (3) where [R vs ] kk denoes he kh diagonal elemen of R vs,and is he received signal power a relay k Now,wedefine he power consrain as f k h k (σs h sk +[R vs ] kk ) =, (4) which guaranees ha he received signal power is bounded by and K as given below h Fh s σ s + h FR vs F h K (5) The equaliy holds when here is only one relay in he nework which is K = In his case, we achieve he exac power of a he desinaion I should be menioned ha for a given and by increasing he number of relays, K, he received signal power a he desinaion will increase The lower and upper bound can be conrolled by he wo parameer and K, where subsequenly hese wo parameers can be conrolled by he nework In oher words, he number of relays employed and also could be decided by he nework o achieve he desired lower and upper bound power III MSE MINIMIZATION, (SNRMAXIMIZATION) In his secion, we find he relay marix F such ha we minimize MSE under he power consrain and show ha i is equivalen o SNR imizaion The MSE minimizaion is defined as min k 0,F ξ(f ) min k 0,F E s k 0y, (6) where he opimal k 0 is in fac he Wiener filer which is given by [9],

3 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL XX, NO XX, MONTH k 0 =(σs +h s F h (h FR vs F h + σ v ) h Fh s ) h sf h (h FR vs F h + σv ) (7) Afer subsiuing (7) ino he MSE cos funcion in (6), he MSE can be expressed as ξ(f )=(σs + h sf h (h FR vs F h + σv ) h Fh s ) (8) Define he K vecor f =[f f K ] such ha i includes he diagonal elemens of F, hen he power consrain in (4) can be expressed as f k h k (σs h sk +[R vs ] kk ) = fwf =, (9) where W is a diagonal marix, and can be expressed as W = [ h σr h K σr K ] The problem is now o solve min ξ(f ) s F fwf = (0) However, he MSE cos funcion, ξ(f ), can be furher modified as ξ(f )= [ σs + h s F h (h FR vs F h + σ v ) ] h Fh s [ = σs + h Fh s h s F h ] h FR vs F h + () σ v The opimizaion problem can now be expressed as [ h Fh s h sf h ] F h FR vs F h + σv s fwf = () I should be menioned ha he above cos funcion is he SNR a he desinaion excep ha he ransmier power, σs, does no appear a he numeraor which has no effec on he opimizaion problem since he ransmier always operaes a is imum power Therefore, from now on, we consider he opimizaion problem as an SNR imizaion Δ By defining D h = [ h h K ], such ha h F = fd h, he imizaion problem can be expressed in he form of a Rayleigh-Riz raio f fd h h s h s D h f fd h R vs D h f + σ v s fwf = (3) By defining f Δ = fw /, he problem can be equivalenly expressed as f fa f fb f s f =, (4) wherewehavedefined,a= Δ W / D h h s h sd h W /, B =(W Δ / D h R vs D h W / + βi), andβ = Δ σv / Noice ha he marix B is Hermiian and posiive definie, herefore we can decompose i ino Cholesky facors as B = L L, and he soluion o (4) is given by f v ( L A L ) (5) ( ) = α v L W / D h h s h s D / h W L (6) ( ) = α L W / D h h s (7) where (7) follows from he fac ha marix A is rank marix and α is used o adjus he norm of f so ha f = [see (4)], which implies ha α = pc h sd h W / B W / D h h s (8) Finally, by subsiuing (8) ino (7), he relay coefficiens can be expressed as f = pc h sd h W / B W / D h h s ( L W / D h h s ) W / (9) For uncorrelaed noise a he relays where R vs = σv s I,he relay coefficiens can be expressed as pc h s f k = k h k (0) h k σr σv k + σ K h vs si σr i= k σ v pc σ r i +σ vs As shown in (0), if he forward channel is weak, he opimal relay ransmi power could be high We consider a simple scheme where each relay is capped o a imum possible value p I will be shown in he simulaion secion (Sec VI) ha he impac of his scheme is a funcion of he number of relays employed and he desired power consrain IV SNR BEHAVIOR In his secion, we derive and examine how he oal SNR a he desinaion behaves From (4) and by uilizing Rayleigh Quoien we have σs ( λ B A ) () ) = σs λ (h s D W / B W / D h h h s () ) = σs (h s D W / B W / D h h h s (3) Now, considering uncorrelaed noise a he relays, R vs = I, he receive SNR can be expressed as σ s h s k ( σ v ) (4) I can be observed ha he opimal receive SNR, SNR,for uncorrelaed noise, is a funcion of he backward channels and no he forward channels However, i is imporan o undersand ha he effec of he forward channels is implicily accouned for by conrolling he relay ransmi power If is large such ha σ v σ vs,hen σ s h sk = σ s h s (5)

4 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL XX, NO XX, MONTH 009 The above resul show ha, in his limiing case, he sronger he backward channels he higher he SNR On he oher hand, if is small such ha σ v σ v, and also assuming ha σs h sk + σv s σs h sk (which means high SNR in he firs hop) hen σ v = K σ v (6) In his case, he sysem behaves like an AWGN channel wih ransmi power of K and noise a he desinaion wih power of σ v BER K= K= K=0 V DISCUSSION In his paper, we assume ha he source has no CSI and he desinaion has complee knowledge of all channels Acquiring complee knowledge of channels a he desinaion can be performed hrough raining However depending on noise correlaions a relays, he relays coefficiens can be calculaed disribuively or a he desinaion We address hese wo differen cases in he following A Uncorrelaed noise a he relays In his case, noise a relays are uncorrelaed such ha R vs = σv s I and relays coefficiens can be found disribuively using (0) Here, each relay only needs o have knowledge of is local backward and forward channels, and also each relay K h needs o know an exra coefficien c = si i= σv pc σ r +σ i vs which is he same for all relays and can be broadcased o he relays by he receiver hrough feedback Each relay can acquire is local backward channel hrough sandard raining mehods Obaining each relay s forward channel is equivalen o obaining ransmi CSI in poin-o-poin wireless sysems B Correlaed noise a he relays In his case, finding relays coefficiens disribuively requires ha each relay knows no only is local backward and forward channels bu also all oher relays channels (see (9)), which require significan overhead However as a alernaive, he opimizaion process can be performed a he desinaion and sen back o he relays by using feedback In his case, relays do no have CSI and requires less overhead Simulaion resuls verify ha having correlaed noise a he relays acually improves performance However, he receiver should be aware of he correlaion in order o ake advanage of i by using (9) If he receiver is no aware of noise correlaion a relays, he relays coefficiens are he same as in he case of uncorrelaed noise which is expressed in (0) VI SIMULATION RESULTS In his secion we provide numerical resuls o verify he performance of he proposed scheme and our analyical calculaions We assume ha all relays are a equal disance from he source and desinaion such ha he forward and backward channels have he same saisics, which are generaed as zeromean and uni-variance independen and idenically disribued Fig BER performance of SNR imizaion subjec o power consrain of 0 db for differen number of relays Received signal power (linear scale) K= K= K= Fig 3 The received signal power a he desinaion in linear scale for power consrain of 0 db, =, and for differen number of relays (iid) complex Gaussian random variables or equivalenly h sk,h i CN(0, ) The source ransmis independen daa wih ransmi power of 0 db All simulaions are conduced using a QPSK consellaion, and noise a he relays and desinaion are assumed o be iid wih he same variance We plo bi error rae (BER) curves versus SNR, which is defined as SNR = σ s σ hs σ = σ s σ h vs σ where as menioned above σ v h s and σh are equal o one The BER provided here is averaged over differen channel realizaions, and he noise a he relays and desinaion is considered iid unless oherwise menioned Figure shows BER of he proposed scheme (0) versus SNR for differen number of relays K, where he power consrain is se o 0 db ( = ) Increasing he number of relays improves he sysem performance which is mainly due o he disribued diversiy order obained via adding more relays Noe ha for a given K, he diversiy order exhibied is larger han K This is aribued o he fac ha he compued relay marix F incorporaes knowledge of boh forward and backward channels o conrol power a he relays, resuling in higher diversiy order [0]

5 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL XX, NO XX, MONTH BER K=0, power clipped, power clipped K=0, power clipped Fig 4 BER performance comparison for he case ha here is no limi on he relay ransmi power and where each relay limis is oupu power o p =0dB if i is higher han p Here he power consrain is se o 0dB Rae bps/hz K=0, no corr K=0, corr=05 K=0, corr=09 K=40, no corr K=40, corr=05 K=40, corr= Fig 5 Comparison of imum achievable rae for differen noise correlaion a he relays Here he power consrain is se o 0 db and K=0, 40 Figure 3 is provided o show he received signal power (in linear scale) a he desinaion versus SNR wih he same seing as Fig Figure 3 shows ha he received signal power increases as he number of relays increases and also i verifies ha for a given power consrain, here =or 0 db, he received signal power is always greaer han or equal (equaliy holds for K =) and less han K which is consisen wih (5) where i guaranees ha he received signal power is always beween and K As shown in (0), he oupu power of some relays could be high if he forward channels are weak Figure 4 depics he sysem performance for he scheme where if he relay ransmi power is higher han p, which is 0 db here, he relay clips is oupu power o p In his simulaion, he power consrain is se o 0 db and we consider 3, 5, and 0 relays The BER performance degrades as we clip he power, however as we increase he number of relays his degradaion becomes significanly smaller The reason is ha by increasing he number of relays less power is needed a each relay o achieve he power consrain Finally, Figure 5 compares imum achievable rae for a relay nework wih uncorrelaed noise versus correlaed noise wih correlaion coefficien of 05 and 09 and [R vs ] ii = for all i The power consrain is se o 0 db and K is se o 0 and 40 As shown in he figure, increasing noise correlaion improves he imum achievable rae VII CONCLUSIONS In his paper we proposed and analyzed an AF relay scheme such ha he received signal power a he desinaion is always bounded in order o limi he inerference o neighboring cells using he same specrum We derived opimal relay facors o minimize MSE, under power consrain a he desinaion and showed ha i is equivalen o imizing SNR a he desinaion subjec o he same power consrain The opimal relay facors are provided for boh correlaed and uncorrelaed noise a he relays Simulaions are provided ha presen he performance of he proposed scheme REFERENCES [] A Scaglione, D L Goeckel, and J N Laneman, Cooperaive communicaions in mobile ad hoc neworks," IEEE Signal Processing Mag, pp 8-9, Sep 006 [] N Khajehnouri, and A H Sayed Disribued MMSE relay sraegies for wireless sensor neworks," IEEE Trans Signal Processing, vol 55, no 7, pp , July 007 [3] A S Behbahani, R Merched, and A M Elawil Opimizaions of a MIMO relay nework," IEEE Trans Signal Processing, vol 56, no 0, pp , Oc 008 [4] Y Jing and H Jafarkhani, Nework beamforming using relays wih perfec channel informaion," [Online] Available: hps://webfilesuciedu/yjing/www/papers/neworkbfpdf [5] Air inerface for fixed and mobile broadband wireless access sysems, mulihop relay specificaion," IEEE 806 s Relay Task Group, [Online] Available: hp://ieee80org/6/relay/docs/806j-06 06r4zip [6] M Gaspar, On Capaciy under received-signal consrains," in Proc Alleron Conf Commun, Conrol Compuing, Monicello, IL, pp 3-33, Oc 004 [7] M Gaspar, On capaciy under receive and spaial specrum-sharing consrains," IEEE Trans Inform Theory, vol 53, no, pp , Feb 007 [8] K S Gomadam, and S A Jafar, The effec of noise correlaion in amplify-and-forward relay neworks," IEEE Trans Inform Theory, vol 55, no, pp , Feb 009 [9] A Scaglione, P Soica, and S Barbarossa, G B Giannakis, and H Sampah, Opimal designs for space-ime linear precoders and decoders," IEEE Trans Signal Processing, vol 50, no 5, pp , May 00 [0] N Ahmed, M A Khojasepour, A Sabharwal, and B Aazhang, Ouage Minimizaion wih Limied Feedback for he Fading Relay Channel," IEEE Trans Commun, vol 54, no 4, pp , Apr 006