RELAY-ASSISTED networks have received renewed significant

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1 134 I TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 4, APRIL 13 OFM AF Relaying Unde I/Q Imbalance: Pefomance Analysis and Baseband Compensaion Mohamed Mokha, Ahmad Gomaa, and Naofal Al-hahi, Fellow, I Absac We analyze he ouage pefomance of half-duplex amplify-and-fowad elaying in an OFM sysem wih MRC deecion in he pesence of I/Q imbalance and compae i wih ha of he diec ansmission mode. Boh analyical and numeical esuls demonsae ha he diec mode can oupefom he amplify-and-fowad mode even unde modeae levels of uncompensaed I/Q imbalance. The coss-ove I/Q imbalance levels ae deemined analyically o be invesely popoional o he cube of he signal consellaion size. In addiion, we pesen a low-complexiy eceive-based digial baseband I/Q imbalance compensaion scheme fo he amplify-and-fowad mode and analyze is VM pefomance. Fuhemoe, we deive accuae analyical appoximaions fo he VM pefomance as a funcion of elay locaion and I/Q imbalance level wih and wihou compensaion. Index Tems I/Q imbalance, amplify-and-fowad elay, OFM, VM. I. INTROUCTION RLAY-ASSIST newoks have eceived enewed significan aenion in he pas few yeas afe he pioneeing wok in 1. Fuhemoe, elaying has been ecenly adoped in he long-em evoluion LT-advanced sandad o mee he inenaional Mobile Telecommunicaions IMT- advanced equiemens. Two of he mos common elaying poocols ae decode-and-fowad F and amplifyand-fowad AF. Ou focus in his pape is on AF elays because hey ae moe cos effecive since he signal is only amplified and no decoded a he elay compaed o F elays. The pefomance of AF-based elaying has been analyzed in seveal papes such as 3, assuming an ideal adio fequency RF fon-end. In pacice, and fo hadwae implemenaion easons, he AF elays ypically down-conve, amplify a baseband, and hen up-conve he signal fo moe deails, see 4. Moeove, and mos impoanly, AF elays, specifically half-duplex elays, mus buffe he eceived signal block in he fis ime slo unil i is ansmied afe amplificaion in he second ime slo which is essenial fo synchonizaion. This buffeing opeaion canno be efficienly done in he analog domain. Insead, i is done digially a baseband 5. Theefoe, due o he down/up convesion opeaions a he AF elay and he cos consains on is RF fon end, AF elays can suffe fom seveal majo RF impaimens such as caie fequency Manuscip eceived Augus 6, 1; evised Ocobe 19 and Novembe 9, 1. The associae edio coodinaing he eview of his pape and appoving i fo publicaion was M. Juni. This pape is suppoed by a NPRP gan fom QNRF. The saemens made heein ae solely he esponsibiliy of he auhos. The auhos ae wih he Univesiy of Texas a allas, USA mmm961@udallas.edu, aag 1@yahoo.com, aldhahi@udallas.edu. igial Objec Idenifie 1.119/TCOMM offse CFO, phase noise PN and I/Q imbalance IQI 6. In 5, he auhos consideed CFO and analyzed is effecs on AF elays, while he impac of PN on he pefomance of AF-based elay newoks was analyzed in 7 fo ohogonal fequency-division muliplexing OFM sysems. IQI effecs in AF elay newoks wee invesigaed in 8. In conas o his pape, he auhos in 8 did no conside OFM sysems and did no conside IQI effecs a he elay iself. We summaize ou main conibuions as follows. Fis, we compae he ouage pefomances of he AF and diec no elay modes and show ha he diec mode can oupefom he elay mode even fo modeae IQI levels. Second, we deive he IQI heshold level above which he diec mode oupefoms he AF elay mode in he ouage pobabiliy sense. Finally, we dive accuae analyical appoximaions fo he eo veco magniude VM of join channel and IQI compensaion scheme based on boh zeo-focing ZF and convenional maximum aio combinings MRC deecion. To he bes of ou knowledge, his is he fis pape o analyze IQI effecs in OFM AF elay sysems and conside IQI a he elay iself. The es of his pape is oganized as follows. The sysem model is descibed in Secion II and he pefomances of he AF and diec modes wihou IQI compensaion ae analyzed in Secion III. We pesen channel esimaion and digial baseband compensaion algoihms in Secion IV. In he same secion, we analyze he VM pefomance of he pesened compensaion algoihm. Simulaion esuls and conclusions ae given in Secions V and VI, especively. Noaions: Unless ohewise saed, lowe and uppe case bold lees denoe vecos and maices, especively. The maices I and F denoe, especively, he ideniy maix and he Fas Fouie Tansfom FFT maix whose middle ow coesponds o he C, and hei subscips denoe heis sizes. Fo maices, A # ĨN A Ĩ N, while fo vecos, a # ĨN a whee ĨN =F N F T N is he evesal image pemuaion maix. Also, H,,and T denoe he maix complex-conjugae anspose, complex-conjugae, and anspose opeaions, especively. The opeaos and denoe he saisical expecaion and he absolue value, especively /13$31. c 13 I II. SYSTM MOL We conside he downlink 1 ansmission scenaio in elayassised OFM sysems wih one anenna a each of he base saion BST, elay saion RS and use equipmen U. The RS opeaes in he AF mode whee he ansmission 1 The pesened modeling appoach can be easily exended o he uplink.

2 MOKHTAR e al.: OFM AF RLAYING UNR I/Q IMBALANC: PRFORMANC ANALYSIS AN BASBAN COMPNSATION 135 occus ove wo ime slos, as shown in Fig. 1. In he fis slo, he BST ansmis o boh he RS and U. In he second slo, he elay amplifies is eceived signal and fowads i o he U while he BST emains silen. The fading channels of he hee links ae assumed independen, fequency-selecive and fixed ove a leas one OFM symbol duaion. The RF fon-end of he BST is assumed IQI-fee while hose of he RS and U ae boh IQI-impaied whee he RS suffes fom boh ansmi and eceive IQI. We conside fequency-independen IQI caused by he gain and phase mismaches beween he I and Q banches pahs 6 and hey ae denoed, especively, by ɛ / x and θx / whee he subscip x is he eminal idenifie BST, RS o U and he supescip / denoes he up/down-convesion pocess, especively. The ime-domain T baseband epesenaion of he IQI-impaied signal is given by g / IQI 6 whee g / IQI =μ/ g+ν / g 1 whee μ / θ = cos /+ iɛ θ sin, he minus and plus signs ae fo up and down convesion, especively, ν / = ɛ θ cos θ i sin,andg is he T baseband IQIfee signal and he g em aises due o IQI effecs. The T eceived signals in he fis ime slo a he RS and U afe emoving he cyclic-pefix CP ae given, especively, by R = μ H R SR s + νr H SR s + μ R n R + ν R n R 1 = μ H S s+ν H S s + μ n1 +ν n1 3 whee S, R and denoe he BST, RS and U, especively, and s is he T N 1 daa veco wih s s H =η o I N.The N N ciculan 3 channel maix H XY models he channel fom node X o Y and is fis column is ht T XY 1 N L, whee h XY is he channel impulse esponse CIR wih L complex Gaussian aps. Moeove, n R and n 1 ae he ciculaly symmeic complex addiive whie Gaussian noise AWGN vecos wih single-sided powe specum densiies of N o. In he second ime slo, he elay amplifies he eceived signal and fowads i o he desinaion. To keep complexiy low a he elay, he channel is no esimaed a he elay. Hence, he amplificaion faco is defined as a = η1 R whee η 1 is he aveage enegy ansmied by he RS. Theefoe, he T baseband equivalen signal eceived a he U in he second ime-slo is given by = μ H 1 +ν H s+ ν H 1 +μ H s + z 4 whee z is he composie noise veco and H 1 = μ R μ R +ν R νr a HSR HR 5 H = μ R ν R + ν R μ R a H SR H R 6 Based on he above model, he fequency-domain F eceived signals a he U in he fis and second ime slos The exension o fequency-dependen IQI is saighfowad following, fo example, he appoach in 9. 3 ue o he CP inseion and channel saionaiy assumpions. Ideal Tx H SR Rx IQI, R R Amp RS H S Tx IQI, R R Rx IQI, BST U Fig. 1. Sysem block diagam whee he solid and hollow aows denoe ansmissions duing he fis and second ime slos, especively. ae obained by aking he FFT of 3 and 4, especively, and can be wien as follows 1 = μ H S s + νh # S s# + μ n 1 + ν n 1# 7 =μ H 1 + ν H s + ν H# 1 + μ H# s # + z 8 whee H 1 and H can be easily obained fom 5 and 6 by eplacing each H XY and H XY by H XY and H # XY, especively, whee H XY is he N N diagonal maix whose diagonal elemens epesen he channel fequency esponse CFR fom node X o Y. Moeove, H XY k, whee H XY k is he k h elemen on he main diagonal of he maix H XY, is an exponenially-disibued andom vaiable wih mean σk XY which is popoional o d XY ξ whee d XY is he disance beween node X and node Y and ξ is he pahloss exponen. In addiion, s, n 1, and z ae he fequencydomain epesenaions of s, n 1, and z, especively. Fo he definiion of #, please efe o he noaion definiions in Secion I whee complex conjugaion in T esuls in evesal and complex conjugaion in F. Hence, IQI esuls in image mio inefeence in F whee he k h subcaie is disoed by is image subcaie whose index is k. III. PRFORMANC ANALYSIS WITHOUT IQI COMPNSATION In his secion, we invesigae he effec of IQI on he pefomance a he U whee he channel is assumed pefecly known a he U and we use MRC, which ignoes IQI, fo deecion. I is known ha MRC achieves he maximum likelihood ML pefomance in he pesence of AWGN only, i.e. no IQI, as shown in 3. We conside wo pefomance meics: in Secion III-A we analyze he ouage pobabiliy while VM is analyzed in Secion III-B. A. Pefomance Analysis: Ouage Pobabiliy In his secion, we deive he ouage pobabiliies of he diec and AF ansmission modes unde uncompensaed IQI 1. The AF mode povides a highe divesiy gain han he diec mode; howeve, i suffes moe fom IQI due o he non-ideal RF fon-end of he elay saion. 1 Insananeous SINR xpessions: The insananeous signal-o-inefeence-plus-noise aio SINR expessions ae equied fo he ouage pobabiliies calculaions in Secions III-A and III-A3. We compue he SINR a he k h subcaie H R

3 136 I TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 4, APRIL 13 in he diec mode fom 7 as follows μ γ i k = H S k 9 ν H S k + No η μ + ν Assuming MRC 11 of 1 and,hesinrahekh subcaie in he AF mode is uppe-bounded by γ AF whee 7 γ AF k =γ i k+min γ SR k,γ R k 1 and γ SR k has he same fom as γ i k wih μ, ν and H S eplaced by μ R, ν R and H SR, especively. Nex, γ R k is compued fom 8 as follows γ R k = 11 μ R μ H Rk+ν R ν H R k ν R μ H Rk+μ R ν H R k + No μ + ν η 1 μ R The iec Mode Ouage Pobabiliy: The diec mode ouage pobabiliy is given by P i R =P log 1 + γ i k <R = P γ i k < R 1 = F γi k R 1 1 whee R is he equied ae and F γi k u is he cumulaive densiy funcion CF of γ i k given by F γi k u = y= u a b y+c x= f X,Y x, ydxdy 13 whee X = H S k, Y = H S k, a = μ, b = ν and c = No η μ + ν. xcep fo cene subcaies, he coelaion beween H S k and H S k is small hanks o hei lage specal sepaaion. Hence, hey ae consideed uncoelaed and independen 4,andsoaeX and Y because hey ae funcions of H S k and H S k, especively. Hence, f X,Y X, Y is appoximaed by f X,Y f X xf Y y and F γi k u is given by F γi k u y= f Y y u a b y+c x= f X xdxdy 14 whee X and Y ae exponenially-disibued andom vaiables wih he densiies f X x = 1 x exp and f σk S σk S Y y = 1 exp y, as defined in Secion II. Caying ou he σ k S σ k S inegaion in 14, we ge c F γi k u 1 exp a σ S k u 1+ b σs Subsiuing R 1 fo u in 15, we ge 1 k u 15 a σk S c R 1 P i R 1 exp 1+ b 1 σs k R 1 16 a σk S a σk S 4 Because hey ae joinly Gaussian. 3 The AF Mode Ouage Pobabiliy: The AF mode ouage pobabiliy is lowe-bounded by 7 P AF R R 1 F γrelayk R 1 uf γikudu 17 whee f γiku = F γi ku/ u is he pobabiliy densiy funcion PF of γ i k. Also,R is muliplied by in 17 o compae he ouage pobabiliies of boh modes a he same ae since he MRC is pefomed ove wo ime slos in he AF mode 7. Fuhemoe, F γrelayku is he CF of γ Relay k min γ SR k,γ R k givenby7 1 F γrelayku =1 1 F γsr ku FγR k u 18 whee F γsr ku and F γr ku ae he CFs of γ SR k and γ R k, especively. F γsr ku has he same fom as F γi k u in 15 wih he U paamees eplaced by hose of he RS. Nex, we compue F γr ku whee, fo noaional convenience, we e-wie γ R k in 11 as follows Z γ R k W + N 1 Z W + N 1 19 whee Z = a 1 H R k +b 1 H R k, W = c1 H R k + d 1 H R k and N 1 = No μ + ν wih a η 1 μ R 1 = μ R μ, b 1 =νr ν, c 1 = νr μ and d 1 =μ R ν. Simila o F γi k u, we appoximae F γr ku by F γr ku w= uw+n1 f W w f Z zdz dw z= }{{} I 1 whee W and Z ae no exponenially-disibued because W and Z ae no pope andom vaiables 1, W = c 1 d 1 ρ k and Z = a 1 b 1 ρ k whee ρ k H R kh R k. ThePFofZ is given by 13 c z j zj 1 exp β z z f Z z = j!β z j+1 1 j= whee β z =λ z 1λ z /λ z 1 + λ z, c z = β/ λ z 1 λz and c z j = 1 j j 1 m= d z j m cz m, j > d z j = 1 β j z λ z + 1 β j z 1 λ z 3 whee λ z 1 and λz ae he eigenvalues of he coelaion maix V z = ZZ T whee Z = T ZR ZI and ZR and Z I denoe he eal and imaginay pas of Z, especively. The inegal I 1 in is given by 14 c z uw+n1 j 1 I 1 = j!β z j+1 z j exp z dz 4 β z = j= j= c z j 1 exp uw+n 1 β z j = u w + N 1!β z c z j 1 q j j=

4 MOKHTAR e al.: OFM AF RLAYING UNR I/Q IMBALANC: PRFORMANC ANALYSIS AN BASBAN COMPNSATION 137 Subsiuing I 1 back ino and wiing f W w in a fom simila o f Z z in 1, we ge c z { j F γr ku cw l 1 l!β j= w l+1 w l exp w dw l= β w } 1 q j w l exp w dw 5 β w whee w l 1 exp β w w dw = l!β w l To simplify he second inegal in 5, we appoximae q j by q j exp uw + N j 1 wu 1 + N1 w β z!β = z 6 whee we used 1+ N1 w 1+ N1 w as a valid appoximaion a high SNR whee N 1 <<1. Usingq j in 6, we ge F γr ku 1 c z j c w l j= l= j α 1 sl = j α sl + =1 whee α 1 = l+ β, w l βz α = l+ 1 β w l βz β z 1+ βwu 7 N1 β w n and sn = u exp un1 β z u exp N1+nβwu β z. We used he appoximaion n 1+ βw β z u exp nβwu β z which is a valid appoximaion β fo pacical IQI levels whee w βz << 1 as shown in 15. Fuhemoe, in 7, we used he elaion j= cz j = l= cw l = 1 which follows diecly fom he elaions f Z zdz = f W wdw =1 and w l 1 exp β w w dw = l!β w l Using F γr ku, we ge F γrelayku fom 18 and subsiue back ino 17 o compue he ouage pobabiliy of he AF mode as follows P AF R 1 exp τ S R F τ S c z j c w l j α 1 hl = j= l= j α hl + =1 whee R F = R 1, τ S = c +b σs k a σk S exp τ SR + τ R nr F 1 exp +RFn R +1 F c R +b R σ SR 8 and hn = +R Fn whee τ SR = k, τ a R σk SR R n = N1+nβw β z and n=τ S τ SR τ R n. To each he expession in 8, we appoximae ems of he fom 1 + x n by exp nx whee x<<1. 4 iec vs AF Tansmission Unde Uncompensaed-IQI: We compae he ouage pobabiliies of he diec and AF modes a high SNR N o whee IQI effecs dominae pefomance. Fuhemoe, we assume he same IQI levels ɛ and θ a he RS and U and ha σ S k =σs k, σsr k =σsr k and σk R =σr k which is he case when he CIR coefficiens ae uncoelaed. enoing by he asympoic equivalence a high SNR, we have 1 P i R 1 1+ br 1 br 1 9 a a whee b a = ν μ << 1 fo pacical IQI levels wih he subscips of b and a omied because he U and RS ae assumed o have he same IQI levels. Alhough P AF R in 8 has infinie ems, he fis em j = l = ishe mos significan fo non-cene subcaies whee H R k and H R k ae almos uncoelaed. Noing ha α and consideing only he fis em in 8, we wie P AF R b a R F c z b b cw R F a a + β w R 3 F β z In 15, we showed ha c z 1, cw βw 1 and β z b a. Hence, b P AF R 3 R 1 31 a Compaing 9 and 31, we calculae he maximum IQI level b i.e. a = ν μ fo which he AF mode ouage pobabiliy is sill lowe han ha of he diec mode as follows P AF R P i R 1 ν μ R 1 3 R R 3 δ 3 Hence, if ν μ exceeds he heshold δ in 3, he AF mode is oupefomed by he diec mode. The heshold δ is invesely popoional o he cube of he signal consellaion size R. Hence, he bigge he signal consellaion size is, he ealie he AF-iec modes cossove poin occus. B. Pefomance Analysis: VM To deive an expession fo he VM a he k h subcaie a medium and high SNR whee IQI dominaes he pefomance, we exploi he CFR independence beween he hee links S, S R, and R. Moeove, he k h subcaie and is image ae assumed uncoelaed wih equal vaiance 5. In addiion, o simplify he VM expession, hencefoh, we make he following assumpions wihou loss of genealiy: 1 we assume he same IQI level a all nodes, we nomalize ξ, d σ S =1,3weseσ SR = σ R = σ = d S whee d = d SR = d R he elay always lies on he pependicula line dividing he S- link ino equal pas, which is jusified by he esul shown in 16 ha he opimal AF elay locaion is a he midpoin beween he BST and he U. Finally, based on 3 and fom 7 and 8, he esimaed signal a he k h subcaie using MRC is ŝk =ψ k sk+λ k s k+n k 33 whee ψ k and λ k epesen he combined effecs of he channel and IQI given by ψ k = μ η o H S k + a ηo H SR kh R k N o N o 1 + a H R k Ck 34 λ k = ν η o H S N kh S k+ a ηo H SR kh R k o N o 1+a H R k k 35 5 The CFR has a unifom powe disibuion ove subcaies if he CIR aps ae uncoelaed which is he case in pacice fo uncoelaed scaeing.

5 138 I TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 4, APRIL 13 Fuhemoe, he noise em in 33 is given by ηo H S n k = k μn 1 N k+ν n 1 k o + a η o H SR kh R k N o 1 + a H R k μn k+ν n k + a η o H SR kh R k N o 1 + a H R k μak+ν Bk n R k +νak+μ Bk n R k 36 whee Ck and k ae given by Ck =μh SR kak+ν H SR kbk 37 k =νh SR kak+μ H SR kbk 38 Ak =a μ H R k+ ν H R k 39 Bk =aμν H R k+h R k 4 enoing he eo em a he k h subcaie by e MRC k = ŝk ψ k sk, he VM of his subcaie is defined as follows e MRC k VM MRC k η o ψ k 41 Fo pacical IQI levels and using 34 and 37, ψ k is appoximaed by ψ k.4 5 μ γo η + μ 4 σγo + μ 6 σ γ o 4 o whee γ o = ηo N o is he inpu SNR, and he poof is povided in Appendix A. Since he noise and ansmied symbols ae assumed independen, e MRC k =η o λ k + n k. Theefoe, fom 35 and 38, we ge λ k ν γo +.84 ν μ 4 σ γo /ηo 43 n k μ γ o +.19 μ σγ o +.11 μ +4 ν μ 4 σγ o 44 wih he poof given in Appendix A. Subsiuing 4-44 ino 41, we ge he VM expession in 45 shown on he nex page as a funcion of he IQI level, inpu SNR, and elay locaion. Fuhemoe, we compue he VM floo by consideing he high SNR scenaio γ o as follows VM MRC k =. ν μ 4 σ μ +.8 μ 4 σ +.4 μ 4 σ 47 Seing σ = 1, i.e. he nodes fom an equilaeal iangle,. and subsiuing back ino 47, we ge VM MRC k = ν μ 4 μ +1. μ, which shows he dependence of VM on he 4 IQI paamees. IV. IGITAL BASBAN COMPNSATION The VM analysis in Secion III-B showed ha MRC deecion fo AF elays esuls in an eo floo in he pesence of IQI. This moivaes us o implemen a moe efficien deecion scheme which compensaes fo IQI effecs. In Secion IV-A, we descibe a low-complexiy appoach fo join channel and IQI paamees esimaion in he T. oing he esimaion in he T equies one OFM symbol, as opposed o wo OFM symbols fo pefoming he esimaion in he F. We descibe a low-complexiy deecion appoach in Secion IV-B and analyze is VM pefomance in Secion IV-C. A. Pilo-Aided Channel and IQ Imbalance simaion Since H S is a ciculan maix and fo he aining phase, quaion 3 can be e-wien in he maix fom as in quaion 46 shown on he nex page whee S 1 is he N L ciculan aining maix whose fis column is he aining sequence ansmied in he fis ime slo. The LS esimae of h S,eq is X 1H X1 1 XH and he esimaion eo, ē = ˆ hs,eq,ls h S,eq, vaiance is popoional o 1H X X Moeove, o minimize he eo vaiance subjec o a powe consain s ove he ansmied aining signal, he condiion X 1H X1 = s I L L needs o be saisfied 17. This condiion anslaes o he aining signal design cieia S 1H S = S1T S1 = IL and S 1H S1 = S1T S = L L. Hence, he maix invesion of X 1H X1 is no longe needed and ˆ hs,eq,ls is simplified o ˆ hs,eq,ls = 1 s 1T S1 1T S T 1. In he second ime slo, since he equivalen channel maices ae ciculan 6, quaion 4 can be wien as follows = S S μ h1 + ν h + z ν h 1 + μ h X hsr,eq + z 48 whee h 1 and h ae he fis columns of H 1 and H, especively, each of lengh L 1. Moeove, S is he N L 1 aining maix whose fis column is ansmied in he second ime slo. Similaly, LS esimao of h SR,eq is given by ˆ h SR,eq,LS = X H X 1 XH and he opimaliy condiions on S ae: S H S = S T S = IL 1 and S H S = ST S = L 1 L 1. Hence, he LS esimao of ˆ hsr,eq,ls is simplified o ˆ hsr,eq,ls = 1 S S T. Seveal opimal designs s T T of S i,i {1, }, wee poposed in 18. Below, we descibe one of hese opimal pilo designs in he F fo N =64 S i = F H PF i, i =1, 1, k {, 4, 8,...,3} p k,k = 1 k 4 8, k {36, 4,...,6}, ohewise 49 whee P is a diagonal maix wih he diagonal enies p k,k and F 1 and F denoe he fis L and L 1 columns of F, especively. B. Zeo-Focing Join qualizaion and IQI Compensaion We pesen a zeo-focing appoach fo join channel equalizaion and IQI compensaion whee each subcaie and is image ae joinly pocessed fo daa deecion 19. The ZF appoach has a low complexiy and does no equie SNR knowledge. Fuhemoe, we show in Secion V ha is pefomance is vey close o he IQI-fee case. Hence, hee 6 The linea combinaion of ciculan maices is also ciculan.

6 MOKHTAR e al.: OFM AF RLAYING UNR I/Q IMBALANC: PRFORMANC ANALYSIS AN BASBAN COMPNSATION 139 VM MRC k ν μ 4 σ γ o + μ ν 1+.19σ +.1 μ μ +4 ν σ μ +.8 μ 4 σ +.4 μ 4 σ 45 γ o 1 = S1 S1 μ hs + μ ν I N ν I n 1 N X 1 hs,eq + n 46 h S 1 k 1 k k }{{ k } y k = μ H Sk ν H S k ν H S k μ H S k μ H 1k+ν H k ν H 1 k+μ H k sk ν H 1 k+μ H k μ H 1 k+ν H k s + ñ 5 k }{{} Γ k n 1 U ñ= U V n 1 k n1 k n k n k n R k n R k T 51 is no need fo he moe complex MMS o ML deecion echniques. enoing he k h eny of he veco x by xk, we sack 1 k, k, 1 k and k ino he veco y k, defined in quaion 5 on he op of his page, whee ñ denoes he composie noise veco given by 51 on he op of his page. Fuhemoe, is he all-zeo maix while U and V ae given by μh1k+νh k νh 1 μ ν k+μh k H U= ν μ V= SRk H SR k ν H 1k+μ H k H SRk μ H 1 k+ν H k H SR k Given he CSI and IQI paamees esimaes a he U, sk and s k ae joinly deeced using he following ZF equalize followed by a slice ŝk ŝ k T =Γ H k Γ k 1 Γ H k y k 5 whee he maix invesion in 5 is easily implemened because Γ H k Γ k is a maix. Moeove, fo pacical IQI levels, Γ H k Γ k 1 exiss wih high likelihood 15. C. VM Pefomance Analysis of ZF Compensaion Since he eceived symbol a he k h subcaie is decoded inefeence-fee, hanks o he ZF equalize, he VM of he k h subcaie is defined as VM ZF k ezf k η o, whee e ZF k =ŝk sk. Fom quaions 5 and 5, i can be easily shown ha see 17, quaion 1.9 e ZF k = e T 1 Σ 1 e 1 whee e 1 = 1 T and Σ=Γ H k R 1 ññ Γ k. Moeove, fom he definiion of ñ in 51, Rññ he covaiance maix of he composie noise veco has he following block diagonal fom Rññ = N o ññ H =N o UU H UU H +VV H 53 Fo pacical IQI levels, i can be shown ha Rññ is a diagonally-dominan maix 7, hen Rññ can be appoximaed by a diagonal maix whose main diagonal is N o σñ1 σñ1 σñ σñ, whee σñ1 = μ and 7 A maix is said o be diagonally dominan if fo evey ow of he maix, he magniude of he diagonal eny in a ow is lage han o equal o he sum of he magniudes of all he ohe non-diagonal enies in ha ow, fo moe deails please see. σñ = μ +a μ 4 μ +4 ν σ. Now, o simplify compuing Σ 1, we used he following linea appoximaion of he maix invese 1 Σ 1 4 Σ I Σ Σ 54 Moeove, by exploiing he subcaie CFR independence beween he hee links and beween he k h subcaie and is image of he same link, VM ZF k is appoximaed as follows he accuacy of his appoximaion will be veified in he simulaions secion e T 1 VM ZF k = Σ 1 e 1 η o 1 Σ 1 et 1 Σe1 Σ 55 η o Because of he independence assumpion beween he CFRs a he k h subcaie and is image, i can be easily shown ha e T 1 Σ e 1 = 1 Σ. Afe saighfowad algeba, we ge Σ = 1+ μ 4 a σ N o 1+ μ 4 a 56 σ Fo medium-o-high SNR whee a 1 σ, and subsiuing 56 ino 55, we ge 1 VM ZF k γ o 1+ μ 4 1+ μ 4 + μ 4 σ 57 which shows ha ou pesened ZF compensaion scheme eliminaes he eo floo due o IQI. V. NUMRICAL RSULTS We sa by veifying he ouage pobabiliy expessions in Secion III-A numeically fo he diec and AF modes. We assume he same IQI levels a he RF fon-ends of he U and RS, i.e. μ = μ R =μ R = μ and ν = ν R = ν R = ν. Howeve, he deived expessions in Secion III-A ae applicable fo any IQI levels. We use a muli-pah channel model wih 8 uncoelaed pahs and a unifom powe-delay pofile. The FFT size is N =64, and he subcaie index of inees

7 131 I TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 4, APRIL σ=1 σ=4 σ=6 Ouage Pobabiliy iec mode AF mode VM % ν/μ = no IQI ν/μ =.3 ν/μ =.15 No IQI SNR db Fig.. Analyical hick lines and simulaed hin lines ouage pobabiliies of he AF solid lines and diec dashed lines modes fo R =bis/sec/hz. Ouage Pobabiliy R = 4 R = 3 R = AF, R= AF, R=3 AF, R=4 iec, R= iec, R=3 iec, R= ν / μ Fig. 3. Ouage pobabiliies of he AF solid lines and diec dashed lines modes vesus he IQI level fo diffeen aes R asnr = 4dB. is k =15.Weseη o = η 1 =1and SNR=1/N o. To compue P AF R in 8, we uncaed each infinie summaion o he fis 15 non-zeo ems since c z j = cw l =fo odd j and l. Fig. shows he numeical and analyical ouage pobabiliies fo boh he diec and AF modes unde vaious IQI levels fo R =bis/sec/hz. We noe ha he numeical esuls and analyical expessions ae vey close o each ohe especially in he diec mode whee hey coincide unde diffeen IQI levels. Fuhemoe, we obseve ha he AF mode becomes wose han he diec mode even unde modeae IQI levels such as ν μ =.15 which coesponds o 1 log ɛ=.6 db gain imbalance and θ= phase imbalance. In Fig. 3, we compae P AF R and P i R vesus ν μ fo diffeen aes R =, 3 and 4 bis/sec/hz whee he maked AF-iec cossove poin occus a smalle ν μ values fo highe aes as pediced by 3. In Figs. 4 and 5, we invesigae he accuacy of he VM expessions fo he MRC deeco fo diffeen elay posiions and IQI levels, especively. Alhough he VM expession in 45 was deived fo medium-o-high SNR, Figs. 4 and 5 demonsae is accuacy even fo low SNR. As expeced, a lowe sigma when he elay is fahe away fom he BST and U is wose a low SNR because he elay is essenially amplifying and e-sending noise which degades he pefomance. Howeve, a highe sigma when he elay is close o he BST and U is wose a high SNR because he link S R has moe IQI, due o boh ansmi and eceive IQI a he elay, han he diec link; hence, he SNR db Fig. 4. Analyical solid lines and simulaed dashed lines VM of MRC vesus SNR fo diffeen elay posiions σ = d ξ d assuming 1 db S gain imbalance and 5 phase imbalance. VM % IQI:1log 1 1+ε=1dB IQI: 1log 1 1+ε=.5dB IQI: 1log 1 1+ε=.5dB SNR db Fig. 5. Analyical solid lines and simulaed dashed lines VM of MRC fo diffeen gain mismach levels wih σ =6and θ =5. elay conibues moe inefeence. We found ha fo pacical IQI levels, whee he phase imbalance is less han 5,VM depends mainly on gain imbalance while is vaiaion wih phase imbalance is negligible 8. Theefoe, in Fig. 5, we fixed he phase imbalance a is wos-case level of 5 and vaied he gain imbalance o veify he accuacy of ou analyical expession fo diffeen IQI levels. Fig. 6 demonsaes he accuacy of ou VM appoximaion fo he ZF equalize in quaion 57 a high SNR fo diffeen elay posiions a 1 db ampliude imbalance. Specifically, he simulaion esul suppos he main conclusion fom 57 ha he ZF equalize eliminaes he VM floo due o IQI. Finally, in Fig. 7, we compae he BR pefomance of he convenional MRC deeco whee he IQI is ignoed wih ha of ou ZF-based IQI compensaion scheme. We obseve ha he pefomance of he ZF deeco is vey close o he IQI-fee case while he MRC echnique suffes fom significan pefomance loss. I is ineesing o obseve ha, wih pefec CSI, he ZF deeco slighly oupefoms he IQI-fee sysem a high SNR hanks o he divesiy gain povided by ansmi IQI a he RS see. In he same figue, we show he impac of channel esimaion inaccuacies using he appoach descibed in Secion IV-A. Alhough he ZF deeco expeiences abou 3dB SNR loss due o channel esimaion inaccuacies, i sill significanly oupefoms he MRC deeco. 8 This follows fom definiions of μ and ν whee μ 1 and ν ɛ fo small θ.

8 MOKHTAR e al.: OFM AF RLAYING UNR I/Q IMBALANC: PRFORMANC ANALYSIS AN BASBAN COMPNSATION 1311 VM % σ=1 σ=4 σ= SNR db Fig. 6. Analyical solid lines and simulaed dashed lines VM of ZF fo diffeen values of σ and IQI level of 1 log ɛ=1 db and θ=5. Uncoded BR MRC w/ IQI & pefec CSI MRC w/ IQI & channel esiamion ZF w/ IQI & pefec CSI ZF w/ IQI & channel esimaion MRC wihou IQI SNR pe bi db Fig. 7. Uncoded BR of ZF and MRC unde IQI 1 log ɛ=1 db and θ =5 wih pefec solid lines and esimaed dashed lines CSI as compaed o he IQI-fee scenaio wih pefec CSI. VI. CONCLUSION We analyzed he VM and ouage pobabiliy of elayassised AF ansmission and MRC deecion in he pesence of IQI and veified he accuacy of he deived expessions numeically. Compaing he ouage pefomances of he AF and diec modes, we found ha he fome loses is pefomance advanage o he lae when he IQI level exceeds a ceain heshold. This heshold was deived analyically and found o be invesely popoional o he cube of he signal consellaion size. Hence, bigge signal consellaions ae much moe sensiive o IQI in he AF mode compaed o he diec mode. Fuhemoe, we pesened a low-complexiy pilo-aided digial baseband IQI esimaion and compensaion scheme fo AF elay sysems and analyzed is VM pefomance a high SNR. Simulaion esuls showed ha he BR pefomance of ou poposed pilo-aided compensaion scheme is significanly bee han convenional MRC deecion schemes which do no ake IQI effecs ino accoun. APPNIX A To calculae ψ k, λ k and n k as defined v in quaions 34-36, we need o evaluae 1+a v and v 1+a v whee v is an exponenial RV wih mean σ v,i.e. v 1 v exp. Thus, we have v 1 + a v = 1σv v v 1 + a v exp dv 58 Making he change of vaiables u =1+a v,wege v 1 + a v = 1 u 1 u 1 a 6 1 u exp a du 59 = 1 1 u u +1 u 1 a 6 exp a 1 u exp a du = 1 1+a a a exp a i a 1 a 6 X whee i pw exp px w x dx is he exponenial inegal funcion 14. Similaly, v 1 + a v = 6 1 a a exp a i a 1 a 4 Y Moeove, using he definiions of Ak and Bk in quaions 39 and 4, Ak, Bk and Re μνakbk ae expanded as in quaions 61 o 63, on he nex page. Obseving ha H R k and H R k ae independen Gaussian andom vaiables wih zeo mean and using quaions 59 o 63, we define he following expecaions which will be used lae in he deivaion HR k Ak 1 + a H R k = μ 4 a 4 σ X ν 4 a Y 64 HR k Bk 1 + a H R k = μ ν a 4 σ X μ ν a Y 65 HR k Re μνakbk 1 + a H R k = μ 4 ν a 4 σ X μ ν 4 a Y 66 Saing wih he evaluaion of ψ k, fom 34 and 37, we have ψ k = μ η o No H S k 4 + η oa HS k Re μh SR kh R kck No 1 + a H R k + a η o HSR k 4 H R k No 1 + a H R k 67 μ Ak +Re μνakbk + ν B In he sequel, we compue he hee saisical expecaions in he igh hand side RHS of 67. Given ha σ S =1,he H S k 4 is compued as follows H S k 4 = v exp v dv = 68 Nex, exploiing independence beween he diffeen links and beween he k h subcaie and is image, we wie HS k Re μh SR kh R kck 1 + a H R k aσ μ 4 H R k 1 + a H R k 69

9 131 I TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 4, APRIL 13 Ak =a μ 4 H R k + μ ν Re H R kh R k + ν 4 H R k 61 Bk =a μ ν H R k +Re H R kh R k + H R k 6 Re μνakbk =a μ ν μ H R k + a ν H R k H whee Rk 1+a H Rk is calculaed fom 14, P341 as H R k 1 + a H R k = 1 v v σ 1 + a v exp dv σ = a σ a exp a i σ a + σ 7 σ Subsiuing 7 ino 69, we ge HS k Re μh SR kh R k Ck = μ 4 a 1 + a H R k 1 1 a exp a i σ + μ Re H R kh R k + ν Re H R kh R k 63 1 a σ + σ 71 The hid saisical expecaions in he RHS of 67 is compued as follows HSR k 4 H R k 1 + a H R k μ Ak +Re μνakbk + ν B = μ σ a μ 4 X ν 4 Y + ν 4 μ σ X Y a + 4 μ ν σ a μ X ν Y μ +4 ν μ 4 σ X 7 a whee X = 1 a σ X and he appoximaion in 7 is valid fo pacical IQI levels unde he assumpions in Secion IV-B whee X >Y. Thus, subsiuing quaions 68, 71, and 7 ino 67, we ge η o ψ k μ γo + μ 6 σ γo X μ 4 γo a exp a i σ a + σ 73 σ Fo medium-o-high SNR whee IQI dominaes noise, a = 1 σ, hus X = +3exp1i 1 =.1 and η o ψ k can be simplified as shown in 4. Nex, fom 35 and 38, λ k can be wien as follows λ k = ν η o No H S k H S k + a η o HSR k H SR k H R k No 1 + a H R k 74 ν Ak +Re νμakbk + μ B The fis saisical expecaion in he RHS of 74 can be easily compued because of he independence assumpion beween he k h subcaie and is image; hence, H S k H S k = 1. Using quaions 64 o 66, he second saisical expecaion em in he RHS of quaion 74 is as follows HSR k H SR k H R k 1 + a H R k ν Ak +Re νμakbk + μ B = ν σ a μ 4 X ν 4 σ σ Y + μ 4 ν σ X a σ σ Y + μ ν σ μ X ν σσ a Y 4 μ 4 ν σ X 75 a Subsiuing H S k H S k =1and 75 ino 74, and consideing medium-o-high SNR levels, quaion 74 can be wien as in 43. Then, using 36, n k is n k = η o μ + ν H S k N o + η oa μ + ν HSR k H R k N o 1 + a H R k + η oa HSR k H R k N o 1 + a H R k 76 μak+ν Bk + νak+μ Bk whee H S =1and HSRk H Rk 1+a H Rk = 1 a Y. 4 Fo pacical IQI levels whee μ ν, he following appoximaion can be used μak+ν Bk + νak+μ Bk 77 μ Ak +4Re μνakbk + μ Bk Hence, he hid expecaion em in he RHS of quaion 76 is compued as follows HSR k H R k 1 + a H R k μ Ak +4Re μνakbk + μ Bk μ σ a μ 4 X ν 4 Y + μ 4 ν σ X Y + 4 μ ν σ a a μ X ν Y μ +4 ν μ 4 σ X 78 a Using 78 and he elaionships HSRk H Rk 1+a H Rk in 76 as follows H S = 1 and = 1 a 4 Y, we appoximae n k n k μ γ o μ a γ oy + μ +4 ν μ 4 σγ o X 79

10 MOKHTAR e al.: OFM AF RLAYING UNR I/Q IMBALANC: PRFORMANC ANALYSIS AN BASBAN COMPNSATION 1313 Fo medium-o-high SNR levels, subsiuing Y = 1 + exp1i 1 =.19 and X =.1 ino 79, we aive a he final esul in 44. RFRNCS 1 J. Laneman,. Tse, and G. Wonell, Coopeaive divesiy in wieless newoks: efficien poocols and ouage behavio, I Tans. Inf. Theoy, vol. 5, no. 1, pp , 4. C. Hoymann, W. Chen, J. Monojo, A. Golischek, C. Kousimanis, and X. Shen, Relaying opeaion in 3GPP LT: challenges and soluions, I Commun. Mag., vol. 5, no., pp , 1. 3 P. Anghel and M. Kaveh, xac symbol eo pobabiliy of a coopeaive newok in a Rayleigh-fading envionmen, I Tans. Wieless Commun., vol. 3, no. 5, pp , 4. 4 P. Muphy, A. Sabhawal, and B. Aazhang, On building a coopeaive communicaion sysem: esbed implemenaion and fis esuls, URASIP J. Wieless Commun. New., 9. 5 Y. Yao and X. ong, Muliple CFO miigaion in amplify-and-fowad coopeaive OFM ansmission, I Tans. Commun., 1. 6 F. Holin and A. Boudoux, igial Compensaion fo Analog Fon- nds. Wiley, 8. 7 P. Rabiei, W. Namgoong, and N. Al-hahi, On he pefomance of OFM-based amplify-and-fowad elay newoks in he pesence of phase noise, I Tans. Commun., vol. 59, no. 5, pp , J. Qi, S. Aissa, and M. Alouini, Analysis and compensaion of I/Q imbalance in amplify-and-fowad coopeaive sysems, in 1 I WCNC. 9 B. Naasimhan,. Wang, S. Naayanan, H. Minn, and N. Al-hahi, igial compensaion of fequency-dependen join Tx/Rx I/Q imbalance in OFM sysems unde high mobiliy, I J. Sel. Topics Signal Pocess., vol. 3, no. 3, pp , 9. 1 B. Maham, O. Tikkonen, and A. Hjoungnes, Impac of ansceive I/Q imbalance on ansmi divesiy of beamfoming OFM sysems, I Tans. Commun., vol. 6, no. 3, pp , J. G. Poakis, igial Communicaion, 4h ediion. McGaw-Hill, 1. 1 P. Scheie and L. Schaf, Second-ode analysis of impope complex andom vecos and pocesses, I Tans. Signal Pocess., vol. 51, no. 3, pp , G. Ropokis, A. Ronogiannis, and P. Mahiopoulos, Quadaic foms in nomal RVs: heoy and applicaions o OSTBC ove Hoy fading channels, I Tans. Wieless Commun., vol. 7, no. 1, pp , I. Gadsheyn and I. Ryzhik, Table of Inegals, Seies, and Poducs. Academic Pess, A. Gomaa, M. Mokha, and N. Al-hahi, Amplify-and-fowad elaying unde I/Q imbalance, in 1 I GLOBCOM. 16. Michalopoulos and G. Kaagiannidis, Pefomance analysis of single elay selecion in Rayleigh fading, I Tans. Wieless Commun., vol. 7, no. 1, pp , S. M. Kay, Fundamenals of Saisical Signal Pocessing: simaion Theoy. Penice Hall, H. Minn and. Munoz, Pilo designs fo channel esimaion of MIMO OFM sysems wih fequency-dependen I/Q imbalances, I Tans. Commun., vol. 58, no. 8, pp. 5 64, L. Anila, M. Valkama, and M. Renfos, fficien miigaion of fequency-selecive I/Q imbalance in OFM eceives, in Poc. 8 I VTC Fall. G. Golub and C. V. Loan, Maix Compuaions. Johns Hopkins Univesiy Pess, N. Le Josse, C. Lao, and K. Amis, fficien seies expansion fo maix invesion wih applicaion o MMS equalizaion, I Commun. Le., vol. 1, no. 1, pp , 8.. Au, Z. Lei, and F. Chin, xploiing he divesiy gain of ansmie I/Q imbalance in single-anenna OFM sysems, in 9 I GLOBCOM. Mohamed Mokha eceived BSc degee in eleconics and communicaions engineeing fom Alexandia Univesiy, gyp, in 8 and MSc fom Nile Univesiy, gyp, in 1. He is cuenly woking owads he Ph.. degee a he Univesiy of Texas a allas, USA. His eseach ineess include coopeaive communicaion and RF impaimens equalizaion a he baseband. Ahmad Gomaa S 9 eceived his B.Sc. and M.Sc. degees in eleconics and communicaions engineeing fom Caio Univesiy, gyp, in 5 and 8, especively. He obained his Ph.. degee fom he Univesiy of Texas a allas, USA, in 1. His eseach ineess include spase FIR files design, RF impaimens compensaion a he baseband, cochannel inefeence cancellaion, and compessive sensing applicaions o digial communicaions. Naofal Al-hahi F 8 eaned his Ph.. degee in lecical ngineeing fom Sanfod Univesiy in Fom 1994 o 3 he was a pincipal membe of he echnical saff a G Reseach and AT&T Shannon Lab. In 3, he joined UT-allas as an Associae Pofesso, became a full Pofesso in 7, and a Jonsson isinguished Pofesso in 1. His cuen eseach ineess include boadband wieless and wieline ansmission, MIMO- OFM ansceives, baseband pocessing o miigae RF/analog impaimens and compessive sensing. He has auhoed ove 3 papes and holds 33 issued US paens. He is coecipien of he I VTC Fall 5 bes pape awad, he 5 I signal pocessing sociey young auho bes pape awad, he 6 I onald G. Fink bes pape awad and is an I fellow.