Envelope and Phase Distribution of Two Correlated Gaussian Variables

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1 IEEE RANSACIONS ON COMMUNICAIONS, VOL. 57, NO. 4, APRIL 9 95 Envelope and Phase Disibuion of wo Coelaed Gaussian Vaiables Pahapasinghe Dhamawansa, Membe, IEEE, Nandana Rajaheva, Senio Membe, IEEE, and Chinhananda ellambua, Senio Membe, IEEE Absac Pobabiliy densiy funcions pdf s ae deived fo he phase and ampliude envelope of he complex gain X + jy j =, wheex and Y ae wo coelaed non zeo-mean Gaussian andom vaiables. he pdf of he ampliude is deived as an infinie seies, bu educes o a closed-fom expession when he means ae zeo. he classical Rayleigh and Rician pdf s un ou o be special cases of he deived pdf. his pdf is used o analyze he eo pefomance of non-coheen binay fequency shif keying BFSK wih in-phase/quadauei/q imbalance ove an addiive whie Gaussian noise AWGN channel. he esuling bi eo ae BER expession is deived as an infinie seies. he analyical expessions ae validaed by simulaion, and he I/Q imbalance elaed pefomance degadaion is quanified. Convegence of he PDF seies and he BER seies is esablished. Index ems Chaaceisic funcion chf, coelaed Gaussian, fequency shif keying, in-phase/quadaue imbalance, pobabiliy densiy funcion pdf, Rayleigh densiy. I. INRODUCION WHA is he pobabiliy densiy funcion pdf of he ampliude envelope of he complex gain X + jy j =, wheex and Y ae wo coelaed non zeo mean Gaussian andom vaiables? Mos wieless enginees ae familia wih he answe o a special case of his quesion; i.e, when X and Y have zeo coelaion, one obains he classical Rayleigh pdf o he Rician pdf, which ae of fundamenal impoance no only in wieless communicaions, bu also in many ohe fields of eseach. he pdf of he phase of he complex gain is of inees oo. Moeove, he pdf of X + Y is of inees as a genealizaion of he classical Rayleigh and Rician pdfs. I has applicaion in in-phase/quadauei/q imbalance poblems, which commonly occu in adio fequency eceives ha employ analog quadaue down mixing, when hee is a loss of ohogonaliy beween he inphase and quadaue phase signals geneaed by he local oscillao. he effecs of I/Q imbalance and he compensaion mehods ae addessed in ] 3]. Pape appoved by V. A. Aalo, he Edio fo Divesiy and Fading Channel heoy of he IEEE Communicaions Sociey. Manuscip eceived May 3, 7; evised Sepembe 9, 7 and Decembe 7, 7. P. Dhamawansa was wih he elecommunicaions Field of Sudy, School of Engineeing and echnology, Asian Insiue of echnology, P. O. Box 4, Klong Luang, Pahumhani, hailand. He is now wih he Depamen of Eleconic and Compue Engineeing, HKUS, Clea Wae Bay, Kowloon, Hong Kong eesinghe@us.hk. N. Rajaheva is wih he elecommunicaions Field of Sudy, School of Engineeing and echnology, Asian Insiue of echnology, P. O. Box 4, Klong Luang, Pahumhani, hailand ajah@ai.ac.h. C. ellambua is wih he Depamen of Elecical and Compue Engineeing, Univesiy of Albea, Edmonon, AB 6G V4, Canada chinha@ece.ualbea.ca. Digial Objec Idenifie.9/COMM /9$5. c 9 IEEE Beckmann 4] has addessed he deivaion of his pdf. He noes ha a pai of coelaed vaiables can be ansfomed ino a pai of independen ones by oaing hough an angle. Based on his fac, he deives he pdf fo he case of an independen bu no idenically disibued X and Y. As a esul, he pdf 4, 4.6-8] does no include he coelaion coefficien explicily. In 5] Pawla e al. inoduced he disibuion of phase angle beween wo vecos peubed by coelaed Gaussian noise. he Pawla F funcion has been used o analyze he eo pefomance of coheen M-ay phase shif keying MPSK. Refeence 6] deived a finie ange inegal expession fo he symbol eo ae SER of MPSK impaied wih I/Q imbalance. he pdf expessions of he diffeence of phase angles in 5] wee fuhe modified o give new foms of expessions in 7]. A moe compac se of equaions involving Caig fom of finie ange inegals wee deived in 8] fo he SER pefomance of coheen MPSK modulaion wih I/Q imbalance. Vihaladevuni and Alouini 9] inoduced exac analyical bi eo ae expessions of genealized hieachical PSK consellaions wih impefec phase o iming synchonizaion ove an AWGN channel using Pawla F funcions. Moe geneal Nakagami-m disibuion esuls ae given in ], ]. he eo pefomance of non-coheen deecion of binay ohogonal, nonohogonal signals is analyzed in ], paially coheen deecion in 3] and M-ay FSK in 4]. Howeve, efeences ] 4] assume ideal I/Q balance. he convenional analysis of envelope deecion of M-ay ohogonal signaling also assume an ideal I/Q balance. Moe ealisic analysis of such sysems equies ha I/Q imbalance is fully accouned fo. he pdf expessions deived in his pape may be useful in hose poblems. heefoe, his lee deives he pdf s of he phase and ampliude envelope of he complex gain X + jy,wheex and Y ae a pai of coelaed Gaussian andom vaiables. he pdf of he ampliude is deived as an infinie seies, bu educes o a closed-fom expession when he means ae zeo. he classical Rayleigh and Rician pdf s un ou o be special cases of he deived pdf. he esuls ae applied o analyze he eo pefomance of binay fequency shif keying BFSK wih in-phase/quadauei/q imbalance ove an addiive whie Gaussian noise AWGN channel. he esuling bi eo ae BER expession is deived as an infinie seies. he analyical expessions ae validaed by simulaion, and he pefomance degadaion due o he I/Q imbalance is quanified. Convegence of he PDF seies and he BER seies is esablished. he pape is oganized as follows. Secion II deives he infinie seies epesenaions fo he phase and ampliude

2 96 IEEE RANSACIONS ON COMMUNICAIONS, VOL. 57, NO. 4, APRIL 9 p R = σ X σ exp X σy + Y σ X ρ X Y σ X σ Y Y ρ σx σ Y ρ n= p Θ θ = σ3 X σ3 Y πδ ψ θ + ε n cos n β ψ ρ 3 πδ X,Y Δ ψ θ I n σ X σ Y +4ρ σ X σ Y 4σ X σ Y ρ exp X σy + Y σ X ρ } X Y σ X σ Y σx σ Y ρ Δ X,Y cos θ βefc cos θ β Δ ψ θ σ X + σ Y } 4σX σ Y ρ ] I n σx σ Y ρ, } exp Δ ψ θ cos θ β ρ f Θ θ = π ρ sin θ exp X + Y X Y ρ σ ρ } exp Ωθ cos θ φ + } πω X,Y Ωθ cos θ φefc Ω X,Y Ωθ cos θ φ 4 envelope of he complex gain X + jy,wheex and Y ae wo coelaed non zeo-mean Gaussian andom vaiables. Special cases of he geneal esuls ae idenified. In Secion III, he BER pefomance of non-coheen BFSK in AWGN is analyzed. Secion IV summaizes he main esuls of he pape. he deivaion of he infinie seies and convegence esuls ae pesened in he appendices. II. PDF S OF HE AMPLIUDE AND PHASE OF A PAIR OF CORRELAED GAUSSIAN RANDOM VARIABLES he key esuls on he disibuions of he envelope and phase ae now pesened. heoem : Le X and Y be wo eal Gaussian andom vaiables wih VARX] =σx, VARY ]=σ Y and mean values EX] = X and EY ] = Y. he coelaion coefficien beween X and Y is ρ. he pdf of R = X + Y is given by and he disibuion of Θ=an Y X is given by, whee Δ X,Y = X σ4 Y + Y σ4 X + ρ σx σ Y X + Y ρ X Y σ X σ Y σ X + σy Δ ψ θ =σxσ Y. σx cos ψ = σ Y σx σ Y +4ρ σx σ Y ρσ x σ Y sin ψ = σx σ Y +4ρ σx σ Y cos β = σ Y X σ Y ρ Y σ X sin β = σ X Y σ X ρ X σ Y ρ σ X+σ Y ], σx σ Y +4ρ σx σ Y cos θ ψ efcz = exp d is he complemenay eo z funcion, ε n is he Neumann faco 5] i.e., ε = and π ε n = fo all n =,,... andi n z is he n-h ode modified Bessel funcion of he fis kind. Poof: See Appendix A. he above envelope densiy funcion can easily be educed o 4, eq.4.6-8] when ρ =. Fuhemoe, he disibuion of phase genealizes mos of he esuls given in he lieaue including 6, eq. D.5] which can easily be deived assuming ρ =and Y =. Coollay : Le X and Y be wo eal Gaussian andom vaiables wih VARX] =VARY ]=σ and mean values EX] = X and EY ] = Y. he coelaion coefficien beween X and Y is ρ. he pdf of R = X + Y is given by f R = σ ρ exp X + Y X Y ρ } + σ ρ ε n cos n φ π ρ ] I n 4 σ ρ n= ] I n σ ρ and he disibuion of Θ=an Y X is given by 4 above, whee Ω X,Y = X + Y +ρ 4 X Y ρ, Ωθ = σ ρ X Y ρ ρ sin θ, cos φ = and sin φ = Y Xρ. Poof: Subsiuing σ X = σ Y = σ in and gives ψ = π/, and 3 and 4 follow. Coollay : Le X and Y be wo eal idenical Gaussian andom vaiables wih VARX] =VARY ]=σ and mean values EX] =EY ]=. he coelaioncoefficienbeween 3

3 DHARMAWANSA e al.: ENVELOPE AND PHASE DISRIBUION OF WO CORRELAED GAUSSIAN VARIABLES 97 P R = σ Xσ Y ρ σx + exp X σy + Y σ X ρ } X Y σ X σ Y σ Y σx σ Y ρ σ X σy p+ n +4ρ σx σ Y 4σ X + σ Y Ω X,Y σ X σ Y ρ σ X + σ Y n,p,q= ε n cos n β ψ p!q!n + p!n + q! q+n γ p +n + q +, σx + σ Y 4σX σ Y ρ X and Y is ρ. he pdf of R = X + Y is given by g R = σ ρ exp ρ+ } σ ρ ρ ] ] ε n I n σ ρ I n σ + ρ. n= Poof: Accoding o 3 φ = π 4 when X = Y =. Hence by subsiuing hem in 3 gives 5. Coollay 3: Le X and Y be wo eal idenical Gaussian andom vaiables wih VARX] =VARY ]=σ and mean values EX] =EY ]=. he coelaion coefficien beween X and Y is ρ. he pdf of R = X + Y is given by exp h R = } σ ρ σ I ρ 5 ρ ] σ ρ. 6 Poof: By subsiuing zeo fo in 5 and using I =, he esul follows. he cumulaive disibuion funcion cdf of R fo he case descibed in heoem can easily be deived wih he seies expansion of each of he Bessel funcion ems in followed by em by em inegaion. hus, he cdf of R is obained as given a he op of he page, whee γa, z = z a exp d is he incomplee gamma funcion. Noe ha he classical esuls fo Rayleigh i.e., when X = Y =,σ X = σ Y = σ and ρ = and Rician densiies i.e., when ρ =and σ X = σ Y = σ wih he Rician faco K = X + Y σ ae some special cases of. Figue depics he heoeical pdf, g R calculaed wih 5 and he simulaed values fo diffeen values of α hee, wihou loss of genealiy, ρ =sinα wih =,σ =. III. HE BER PERFORMANCE OF NON-COHEREN BFSK WIH I/Q IMBALANCE his secion pesens an illusaive applicaion of he peceding esuls. Alhough only one applicaion is consideed fo beviy, poenial ohes include pefomance analysis of M-ay FSK unde I/Q imbalance. Conside a noncoheen BFSK sysem which employs wo caie fequencies f,f wih f f =, whee is he bi duaion. his minimum fequency diffeence cieion ensues ha he caies ae ohogonal 7]. Le us conside he simplified block diagam of he BFSK eceive shown in Fig., whee α epesens he I/Q imbalance phase. Fo simpliciy, he I/Q imbalance phase angles ae assumed o be idenical fo boh he caie fequencies. he eceived signal can be wien as Eb = cos πf i + ϕ+n, i =, 7 g R α = π/, π/4, π/3 Analyical Simulaion Fig.. he pdf of g R fo diffeen values of ρ =sinα when = and σ =. Received signal Fig.. cos πf sinπf α cos πf sinπ f α d d d d Sample = Block diagam of he non-coheen BFSK eceive. Envelope Deeco Oupu whee ϕ is a andom phase angle unifomly disibued ove, π, E b denoes he enegy pe BFSK symbol and n denoes a whie Gaussian noise pocess having he auocoelaion funcion R n τ = N δτ. Wihou loss of genealiy, we assume ha f and f ae equipobable and ha f is ansmied. he sample values a he inpu o he envelope deeco as = E b cos ϕ+n, = E b sin ϕ + α+n and = n, = n whee n,n and n,n ae independen Gaussian andom pais while he andom vaiables in each pai ae joinly Gaussian wih vaiance N and coelaion coefficien ρ = sinα. An eo occus if + < + o he nomalized vaiables saisfy + N/ = R < + N/ = R 7]. Following 7],

4 98 IEEE RANSACIONS ON COMMUNICAIONS, VOL. 57, NO. 4, APRIL 9 he condiional BER of BFSK can be expessed as P ɛ ϕ = f R d f R d 8 whee f R can be wien using 3 as hee X = Eb cos ϕ, Y = E b sinϕ + α and σ = N f R = sec α exp γ b sec α ε n cos n β π sin α sec α I n 4 n= I n sec α γ b sin ϕ +cos α + ϕ, 9 f R can be expessed using 6 as f R = sec α exp sec α sin α sec α I cos α + ϕ cos β = sin ϕ +cos α + ϕ sin ϕ sin β = sin ϕ +cos α + ϕ whee γ b = E b N. Since a closed-fom soluion does no exis fo he inne inegal in 8, an infinie seies soluion is sough. Using he equivalen infinie seies epesenaion of modified Bessel funcions and 8, eq.3.35.], he inne inegal can be wien as f R d = k k= l= k! sec l α sin k α exp l!k! k+l sec α l. By subsiuing and 9 in 8 followed by he expansion of Bessel funcion ems wih equivalen infinie seies and em-by-em inegaion, he condiional BER is obained as k ε n k!η!sin η α cos αγ q+n P ɛ ϕ = k! l!p!q!p+n!q+n! k= l= n,p,q= exp γcosn β π 4 sin η+η+ ϕ +cos α + ϕ ] q+n whee η =k +p + n, η =p +n + l + q. he fom of condiional BER given in is impoan in he sense of obaining he eo ae expession in he absence of I/Q imbalance. When he inphase and quadaue componens ae phase shifed by π/ i.e., α = he nesed summaions in degeneae ino he following simple fom P ɛ ϕ = exp γ q= γ q q q! 3 which can easily be inegaed wih espec o ϕ o obain he classical expession fo BER of he noncoheen BFSK as exp γ 7]. Wih he unifomly disibued andom phase, yields he BER as P ɛ ϕ = k k= l= n,p,q= π exp γ η+η+ π ε n k!η!sin η α cos αγ q+n k! l!p!q!p+n!q+n! cos n β π 4 sin ϕ +cos α + ϕ ] q+n dϕ. 4 Now le us conside he inegal in 4 which can be expessed using as π Iα = cos n β π sin ϕ +cos α + ϕ ] q+n dϕ 4 =cos nπ n n s Js, n s s s= +sin nπ n n s Js +, n s s + s= 5 whee π Jα, u, v = sin u ϕ cos v α + ϕ sin ϕ +cos α + ϕ ] q dϕ 6 wih u, v =,,,..and m n denoes he binomial coefficien. Hee, he cos nβ and sin nβ ems have been expanded using cosine and sine ems. he inegal given in 6 can fuhe be decomposed using he binomial heoem ino simple inegals involving sine and cosine ems. Howeve, his would incease he numbe of summaions and hence affec he compuaional complexiy. aking his difficuly ino accoun, 5 and 6 in 4 and numeical inegaion echniques can be used o find he exac BER values. his allows a adeoff beween analyical acabiliy and numeical efficiency. Nex, ou heoeical esuls ae validaed by using he simulaion esuls Fig. 3. A noncoheen BFSK in AWGN is simulaed, and he numeical esuls based on 4 ae compaed wih he simulaion esuls. A span of db is seleced along SNR axis and α =,π/,π/8 fo calculaing he heoeical esuls wih 4. he degee of accuacy depends on he numbe of ems used o uncae he infinie seies. Since he incease in eihe of he quaniies γ o α would incease he ems o be summed wih 4, we have seleced SNR and α values menioned above. We use a minimum of 7, 3, 7, 7 and a maximum of 3, 3, 3, 3 ems in each index of k, n, p, q especively. Mahemaica is used o pefom he numeical inegaions given in 5. Fig. 3 shows ha ou analyical esuls agee well wih he simulaion esuls. he pefomance degadaion due o he I/Q imbalance inceases wih he incease of mismach, as o be expeced. IV. CONCLUSION his lee has deived pdf expessions fo he phase and ampliude envelope of he complex gain X + jy,wheex and Y ae wo coelaed non zeo mean Gaussian andom vaiables. he classical Rayleigh and Rician pdf s ae special

5 DHARMAWANSA e al.: ENVELOPE AND PHASE DISRIBUION OF WO CORRELAED GAUSSIAN VARIABLES 99 BER.E+.E-.E- α =, π/, π/8 ρ =,.3,.38 Analyical Simulaion.E SNR db Fig. 3. BER pefomance of he non-coheen BFSK eceive wih I/Q imbalance. cases of he deived pdf. Convegence esuls have been esablished. he pdf has been used o analyze he eo pefomance of BFSK wih I/Q imbalance ove an AWGN channel. his analysis can also be exended o M-ay FSK M >. he analyical expessions wee validaed by simulaion. he pdf expessions may also be useful fo ohe I/Q imbalance poblems and/o coelaed Gaussian noise poblems. APPENDIX A PROOF OF HEOREM Using he classical ecangula-o-pola coodinae ansfomaion i.e., x = cos θ, y = sin θ he join pdf of wo coelaed Gaussian andom vaiables X, Y having paamees X, Y,σX,σ Y,ρ wih ρ < can be wien as p R,Θ, θ = e e e } X σ Y + Y σ X ρ X Y σ X σ Y σ X σ Y ρ πσ X σ Y ρ σ X +σ Y 4σ X σ Y ρ } C +D cosθ β] A e +B cosθ ψ] ρ whee A = σ X σ Y ρ,b = σ X σ Y X σ Y ρ Y σ X Y σx ρx σy,d = σx σy ρ σy σx ρ 7 4σX σ Y ρ,c = and ψ, β ae definedinsecionii.ofind he maginal pdf of R, 7 mus be inegaed wih espec o θ, π. By obseving he peiodiciy of he composie igonomeic expession wihin he exponenial em in 7 and using he following ideniy 9] exp z cos θ = ε n I n zcosnθ, 8 n= R can be wien as p R = πσ X σ e Y ρ e σ X +σ Y 4σ X σ Y ρ } } X σ Y + Y σ X ρ X Y σ X σ Y σ X σ Y ρ n= ] ε n I n A +B π cos n θ + β ψ C +D cosθ] e dθ. Nex, he cos n θ + β ψ em is expanded as a sum of poduc of cosine and sine ems and obseving he fac ha he sine is an odd funcion followed by he inegaion using 9, eq.7.34] gives he desied esul. Nex we pove saing wih 7. By eaanging he ems in 7, he join pdf becomes p R,Θ, θ = e } X σ Y + Y σ X ρ X Y σ X σ Y σ X σ Y ρ Δ ψ θ 4σ X σ Y ρ πσ X σ Y ρ e +Sθ Rθ 9 σx σ Y ρ cosθ whee Rθ = and Sθ = β. Now i is obvious ha 9 has o be inegaed wih espec o o obain he maginal of Θ. Compleing he squae of he quadaic expession in he exponen and subsequen inegaion wih espec o yields he maginal of Θ as. APPENDIX B CONVERGENCE OF INFINIE SERIES A. Convegence of Befoe poving he convegence, we esablish he following inequaliy. I n z <I z, n =,,... Fom he definiion of I n z, i follows ha π I n z = exp z cos θcosnθdθ π < π exp z cos θ cos nθ dθ. π Since cos nθ <, he inequaliy follows immediaely. Nex we show ha he infinie seies conveges absoluely. o his end, is ansfomed as e p R < X σ Y + Y σ X ρ X Y σ X σ Y σ X σ Y ρ σ X +σ Y 4σ X σ Y ρ σ X σ Y ρ σx ε n I n σ Y +4ρ σx σ Y 4σ n= X σ Y ρ ] I n σx σ Y ρ,

6 9 IEEE RANSACIONS ON COMMUNICAIONS, VOL. 57, NO. 4, APRIL 9 and applying i becomes X σ Y + Y σ X ρ X Y σ X σ Y σ X σ Y ρ p R < e σ X σ Y ρ ] I σx σ Y ρ ε n I n n= σ X +σ Y 4σ X σ Y ρ σ X σ Y +4ρ σ X σ Y 4σ X σ Y ρ. Now using he elaion i σ is easy o see ha he infinie } X summaion conveges o exp σ Y +4ρ σx σ Y and 4σX σ Y ρ hence we have e p R < X σ Y + Y σ X ρ X Y σ X σ Y σ X σ Y ρ σ X σ Y ρ e Δ 4σ 4 X σ4 Y ρ I σ X +σ Y 4σ X σ Y ρ σ X σ Y ρ ] which complees ou poof. Since we poved ha he absolue sum is uppe bounded wih a coninuous funcion, he unifom convegence of infinie seies can be assumed wihou loss of genealiy. B. Convegence of 4 he following esul is woh of menioning befoe a deailed discussion on he convegence of 4. Fo L =,,... we can wie n= n + L! n! n = L! F L +; ; = L! L+ 3 whee p F q a,a,,a p ; c,c,,c q ; x is he genealized hypegeomeic funcion 9]. In wha follows we show ha he seies 4 conveges absoluely. I is easy o see ha 4 can easily be wien as P ɛ ϕ < k k= l= n,p,q= ε n k!η! sin η α cos α k! l!p!q!p+n!q+n! γ q+n exp γ η+η+ sin ϕ +cos α + ϕ ] q+n. he igh side of above inequaliy can be uppe bounded as P ɛ ϕ < k= l= n,p,q= ε n k!η! sin η α cos α k! l!p!q!p+n!q+n! γ q+n exp γ η+η+ sin ϕ +cos α + ϕ ] q+n and he eaangemen of ems yields P ɛ ϕ < cos Γ k + α exp γ sin k α πk! n,p,q= k= ε n η! sin p+n α γ q+n p!q!p+n!q +n! 4p+3n+q+ sin ϕ +cos α + ϕ ] q+n p +n + q + l! l! l l= 4 whee we have used he ideniy n! = n n! π Γ n +,n=,,... A caeful inspecion yields he summaion of fis infinie seies as F ; ;sin α = sec α and an applicaion of 3 gives he las infinie summaion as p + n+q! p+n+q+. Upon subsiuion of hose values in 4 esuls in P ɛ ϕ < cos α exp γ n,p,q= ε n p +n + q! p!q!p+n!q +n! sin p+n α γ q+n 4p+3n+q+ sin ϕ +cos α + ϕ ] q+n 5 which is he esul we ge if em by em inegaion is pefomed on f u R ove he span,. One should noe ha f u R sands fo he igh side of coesponding o ou case of inees. Following we can uppe bound f u R as fr u < sec α exp γ exp sin } α sec α I sec α γ sin ϕ +cos ϕ + α ] and now 5 becomes P ɛ ϕ < 6 f u R d. 7 he em by em inegaion is jusified hee, since ha seies also conveges unifomly. heeafe wih he help of 6 and, Appendix, eq..4.3] z exp λz I αzdz = α λ exp 4λ we can uppe bound he inequaliy 7 as P ɛ ϕ < which complees ou poof. cos α exp γ e sin ϕ+cosϕ+α] γ sin α sin α ACKNOWLEDGMEN he auhos would like o hank he associae edio and he anonymous eviewes fo hei ciical commens ha gealy impoved his pape and fo binging 4] o ou aenion. he fis auho would like o hank he govenmen of Finland and he fome AI-Finnish pojec dieco lae Pof. A. B. Shama fo he docoal scholaship povided o him.

7 DHARMAWANSA e al.: ENVELOPE AND PHASE DISRIBUION OF WO CORRELAED GAUSSIAN VARIABLES 9 REFERENCES ] M. Valkama, M. Renfos, and V. Koivunen, Advanced mehods fo I/Q imbalance compensaion in communicaion eceives, IEEE ans. Signal Pocessing, vol. 49, no., pp , Oc.. ] X. Huang, On ansmie gain/phase imbalance compensaion a eceive, IEEE Commun. Le., vol. 4, no., pp , Nov.. 3] J. K. Caves and M. W. Liao, Adapive compensaion fo imbalance and offse losses in diec convesion ansceives, IEEE ans. Veh. echnol., vol. 4, no. 4, pp , Nov ] P. Beckmann, Pobabiliy in Communicaion Engineeing. Hacou, Bace & Wold, Inc., ] R. Pawula, S. Rice, and J. Robes, Disibuion of he phase angle beween wo vecos peubed by gaussian noise, IEEE ans. Commun., vol. 3, no. 8, pp , Aug ] M. K. Simon and D. Divsala, Some new wiss o poblems involving he gaussian pobabiliy inegal, IEEE ans. Commun., vol. 46, no., pp., Feb ] R. F. Pawula, Disibuion of he phase angle beween wo vecos peubed by gaussian noise II, IEEE ans. Veh. echnol., vol. 5, no., pp , Ma.. 8] S. Pak and S. H. Cho, SEP pefomance of coheen MPSK ove fading channels in he pesence of phase/quadaue eo and i-q gain mismach, IEEE ans. Commun., vol. 53, no. 7, pp. 88 9, July 5. 9] P. K. Vihaladevuni and M.-S. Alouini, Effecs of impefec phase and iming synchonizaion on he bi-eo ae pefomance of PSK modulaions, IEEE ans. Commun., vol. 53, no. 7, pp , July 5. ] G. K. Kaagiannidis, D. A. Zogas, and S. A. Kosopoulos, On he mulivaiae Nakagami-m disibuion wih exponenial coelaion, IEEE ans. Commun., vol. 5, no. 8, pp. 4-44, Aug. 3. ] G. K. Kaagiannidis, D. A. Zogas, and S. A. Kosopoulos, An efficien appoach o mulivaiae Nakagami-m disibuion using Geen s maix appoximaion, IEEE ans. Wieless Commun., vol., no. 5, pp , Sep. 3. ] C. W. Helsom, he esoluion of signals in whie, Gaussian noise, in Poc. IRE, Sep. 955, pp. 8. 3] A. Viebi, Opimum deecion and signal selecion fo paially coheen binay communicaion, IEEE ans. Infom. heoy, vol., pp , Ap ] M. Nesenbegs, Opimum ecepion of coded muliple fequency keying in pesence of andom vaiable phases, IEEE ans. Commun., vol. 9, pp. 77 7, Oc ] G. N. Wason, A eaise on he heoy of Bessel Funcions. Cambidge, U.K.: Cambidge Univ. Pess, ] S Haykin, Digial Communicaions. John Wiley and Sons, ] J. G. Poakis, Digial Communicaions, 3d ed. McGaw-Hill, ] I. S. Gadsheyn and I. M. Ryzhik, ables of Inegals, Seies and Poducs, 4h ed. New Yok: Academic Pess, 98. 9] L. C. Andews, Special Funcions of Mahemaics fo Enginees, nd ed. Bellignham, MA: SPIE Pess, 998. ] K. S. Mille, Mulidimensional Gaussian Disibuions. John Wiley, 964.

STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION

Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE