THE IMPACT OF A PRIORI INFORMATION ON THE MAP EQUALIZER PERFORMANCE WITH M-PSK MODULATION

Transcription

1 5th Europan Signal Procssing Confrnc (EUSIPCO 007), Poznan, Poland, Sptmbr 3-7, 007, copyright by EURASIP THE IMPACT OF A PRIORI INFORMATION ON THE MAP EQUALIZER PERFORMANCE WITH M-PSK MODULATION Chaabouni Sihm ISECS Rout Mnzl Chakr km 0.5 BP 868, 308 Sfax, Tunisia chaabouni sihm@yahoo.fr Noura Sllami ISECS Rout Mnzl Chakr km 0.5 BP 868, 308 Sfax, Tunisia noura.sllami@iscs.rnu.tn Alin Roumy IRISA-INRIA Campus d Bauliu 3504 Rnns Cdx, Franc alin.roumy@irisa.fr ABSTRACT In this papr, w considr a transmission of M-Phas Shift Kying (M-PSK) symbols with Gray mapping ovr a frquncy slctiv channl. W propos to study analytically th impact of a priori information (providd for instanc by a dcodr in a turbo qualizr) on th maximum a postriori (MAP) qualizr prformanc. W driv analytical xprssions of th bit rror probability at th output of th qualizr. Simulations show that th analytical xprssions approximat wll th bit rror rat () at th output of th MAP qualizr at high signal to nois ratio ().. INTRODUCTION Th high data rat communication systms ar impaird by intrsymbol intrfrnc (ISI). To combat th ffcts of ISI, an qualizr has to b usd. Th optimal soft-input soft-output qualizr, that achivs minimum bit rror rat (), is basd on th maximum a postriori (MAP) critrion. In this papr, w considr th cas whr th M-Phas Shift Kying (M-PSK) modulation (with M = q, for q ) is usd and th MAP qualizr has a priori information on th transmittd data. Th a priori information ar providd by anothr modul in th rcivr, for instanc a dcodr in a turbo-qualizr []. In a turbo-qualizr, th qualizr and th dcodr xchang xtrinsic information and us thm as a priori in ordr to improv thir prformanc. In [], th authors studid analytically th impact of a priori information on th MAP qualizr prformanc in th cas of Binary Phas Shift Kying (BPSK) modulation (M = ). Th aim of our papr is to gnraliz th study of [] to th cas of M-PSK modulation. To do this, w driv analytical approximat xprssions of th at th output of th MAP qualizr. Simulations show that ths xprssions approximat wll th at th output of th MAP qualizr at high. This papr is organizd as follows. In sction, w giv th systm modl. In sction 3, w driv analytical xprssions of th at th output of th MAP qualizr. In sction 4, w giv simulation rsults for 8-PSK and 6-PSK modulations. Throughout this papr matrics ar uppr cas and vctors ar undrlind lowr cas. Th oprator (.) T dnots transposition, and R(.) rprsnts th ral valu.. SYSTEM MODEL W considr a data transmission systm ovr a frquncy slctiv channl. W assum that transmissions ar organizd into bursts of T symbols and th channl is invariant during on burst. Th channl is sprad ovr L symbols. Th input information bit squnc b = (b q( L),...,b (qt ) ) T, is mappd to a squnc of M-PSK symbols with Gray mapping whr M = q, and q. Th basband signal rcivd and sampld at th symbol rat at tim k is: L y k = h i x k i + n k, () i=0 whr h i, rprsnts th i th complx channl tap gain, x k, for L k T, ar th transmittd symbols, n k ar modld as indpndnt random variabls of a complx whit Gaussian nois with zro man and varianc σ, with normal probability dnsity function (pdf) N C ( 0,σ ) whr N C ( α,σ ) dnots a complx Gaussian distribution with man α and varianc σ. Equation () can b rwrittn as follows: y = Hx + n, whr x = (x L,...,x T ) T is th vctor of transmittd symbols, n = (n 0,...,n T ) T is th Gaussian nois vctor, y = (y 0,..., y T ) T is th vctor of th output symbols and H is a T (T + L ) Toplitz matrix dfind as: h L... h h H = L... h () h L... h 0 W assum that th channl is normalizd: L i=0 h i =. In this sction, w considr th cas whr no a priori information is providd to th qualizr. Th data stimat according to th MAP squnc critrion (or to th Maximum Liklihood (ML) critrion, sinc thr is no a priori) is givn by: x = argmin u ( y Hu : u A (T +L ) ), whr A is th symbol alphabt. An rror occurs if th stimatd squnc x is diffrnt from th tru squnc x. Lt us dnot = x x th rsulting rror vctor. A subvnt ξ of th rror vnt is that x is bttr than x in th sns of th ML mtric: ξ : y H x y Hx, W now procd to driv a lowr bound on th of th qualizr. This can b don by computing th xact rror probability of a gni aidd qualizr that has som sid information about th snt squnc. Mor prcisly, w considr th cas whr th rcivr has sid information (from a gni) that on of th two squncs x or x was transmittd. Whn th gni aidd qualizr has this sid information, th pairwis rror probability that it chooss x instad of x is givn by [3]: ( ε P x, x = Q ), (3) 007 EURASIP 357

2 whr ε = H, and Q(α) = ( ) 5th Europan Signal Procssing Confrnc π xp y (EUSIPCO 007), Poznan, dy. Proposition Poland, Sptmbr Suppos3-7, w hav 007, acopyright transmission by EURASIP of M-PSK symbols ovr a frquncy slctiv channl dfind by a Toplitz ma- α trix H and with a complx AWGN of varianc σ. Considr that th MAP qualizr has sid information (from a gni) that on of th two squncs x or x was transmittd and has a priori obsrvations modld as outputs of an AWGN channl with varianc. Thn, th pairwis rror probability that it chooss x instad of x is: Lt us assum that th gni tlls th rcivr that it has to choos btwn th tru squnc x and anothr squnc x, such that (3) is maximal i.. such that th distanc ε is minimal. W now procd to comput th xact rror probability for this gni aidd rcivr. Lt E min b th st of all achiving th minimum valu of ε. Lt π b th probability that th input squnc x will b such that x = x+ is an allowabl input squnc for this. π can b intrprtd as th probability that th data squnc x is compatibl with. Thn, th probability that th gni aidd qualizr maks an rror is min ε π Q E min In ordr to obtain a lowr bound on th, w assum that is mad of m() non-zro conscutiv symbols, th othr symbols bing zro. Othr saying, w assum that thr xists an intrval of siz m() such that all th symbols of x ar diffrnt from th corrsponding symbols of x whil th prcding symbol and th following on ar th sam. m() is rfrrd to as th wight of th rror squnc in th following. Morovr, w rmark that thr is at last on rronous bit pr rronous symbol. Thrfor, whn th qualizr has no sid information, th P is lowr boundd by th probability achivd by th gni aidd qualizr: min ε P m()π Q E min As will b shown by simulations, this lowr bound is a good approximation of th rror probability at high and for a Gray mapping. This can b xplaind by th fact that at high and for a Gray mapping, th union bound (uppr bound) and th abov drivd lowr bound ar qual. Our goal is to find an approximation of th at th output of th MAP qualizr with a priori information. 3. PERFORMANCE ANALYSIS In this sction, w propos to valuat th impact of th a priori information on th MAP qualizr prformanc whn a M-PSK modulation with Gray mapping is usd. Th study will b don for th qualizr using th MAP squnc critrion. It holds for th MAP symbol qualizr using th BCJR algorithm [4] sinc th two qualizrs hav almost th sam prformanc as obsrvd in [5,pag 84]. W assum that channls ar prfctly known at th rcivr. Morovr, w suppos that a priori obsrvations at th input of th qualizr ar modld as outputs of an additiv whit Gaussian nois (AWGN) channl. This is a usual assumption in th analyss of itrativ rcivrs [6]. Ths a priori obsrvations on bits b l, for q( L) l qt, ar: z l = b l + w l, whr w l ar indpndnt random variabls of a ral whit Gaussian nois with zro man and varianc with normal pdf N(0, ), whr N ( α,σ ) dnots a ral Gaussian distribution with man α and varianc σ. Thn, th a priori Log Liklihood Ratios (LLRs) ar: LLR(b l ) = log P(z l b l = ) P(z l b l = ) = z l. Thus, ths LLRs can b modld as indpndnt and idntically ( distributd ) random variabls with th conditional pdf N bl, 4 (with b l th transmittd bit). 007 EURASIP 358 P x, x = Q ( ε + m b ()µ ), (4) whr ε = H, = x x is th symbol rror vctor, m() is th wight in symbols of, m b () is th wight in bits of (m() m b () qm()) and µ = σ σ a. Th proof of proposition is givn in th Appndix. Hr again w assum that th gni always chooss a squnc ˆx that maximizs th pairwis rror probability (4). LtE min b th st of all achiving th minimum valu of ε + m b ()µ, π b th probability that th input squnc x will b such that x = x + is an allowabl input squnc. Thn, th probability that th gni aidd qualizr chooss ( an allowabl squnc x instad of x is Emin π Q (min ε + m b ()µ )). Corollary W considr a M-PSK modulation with Gray mapping. Th at th output of th MAP qualizr without gni, using th a priori information can b approximatd by: ( P m()π Q (min ε + m()µ )), (5) E min whr m() is th numbr of (conscutiv) non zro symbols in. Proof of corollary : For th MAP qualizr without gni, th is lowr boundd by th probability achivd by th gni aidd qualizr: ( m b ()π Q (min ε + m b ()µ )). E min In addition, sinc w us a Gray mapping, w furthr assum that thr is on rronous bit pr rronous symbol, s.t. m b () m(). Simulations show that this assumption bcoms tru at high. Morovr, at high th union bound (uppr bound) and th lowr bound ar qual. Thrfor, th lowr bound of th bcoms an approximation as writtn in (5). In ordr to valuat th xprssion (5), an xhaustiv sarch has to b prformd ovr all possibl rror squncs. W propos to rduc this xhaustiv sarch and to rstrict ourslf to th cas of rror squncs of symbol wight or. Simulations will show that this provids rliabl approximations of th ovrall. In th following, w considr diffrnt cass according to th valu of µ. Corollary Lt d = min ε and d = min ε. If th : m()= : m()= a priori information is unrliabl (i.. d > d and µ µ lim = d d ), th at th output of th MAP qualizr can b approximatd by: ( P π Q d ), + 4µ (6)

3 whr π 5th is Europan th probability Signal Procssing that w can Confrnc find pair(eusipco of squncs 007), Poznan, Poland, Sptmbr 3-7, 007, copyright by EURASIP compatibl with rror squncs of (symbol) wight. Mor Tabl 3: µ lim valus for 8-PSK modulation prcisly π = E π and E is th st of (symbol) wight channl3 channl4 channl3c squncs s.t. ε = d. µ lim On th othr hand, if th information is rliabl (i.. d > d 0 and µ > µ lim or if d d ), th at th output of th MAP qualizr can b approximatd by: ( P Q d ), + µ (7) Proof of corollary : Th valu of µ lim rsults from th quality btwn quations (6) and (7). Whn th a priori information at th input of th MAP qualizr is unrliabl (i.. µ µ lim ), rrors occur in bursts. Thus, w do not considr isolatd rrors and w look for th maximum of (5) for m(). Simulations show that th minimum valu of ε + m()µ is obtaind for rror squnc of wight (i.. m() = and ε = d ). Thrfor th is approximatd by (6). Th valus of π for th diffrnt modulations can b found by an xhaustiv sarch and ar givn in Tabl. In th cas of rliabl a priori information (i.. µ > µ lim ), a priori obsrvations hav mor influnc on th dtction than channl obsrvations and most of th rrors ar isolatd. Thrfor, m() = and ε = d. In this cas for all modulations, π = {: m()= and ε =d } π is qual to, which lads to (7). Dpnding on µ, w gt ithr (6) or (7). W dtrmin a thrshold µ lim as th valu of µ whn (6) quals (7). It follows that µ µ lim corrsponds to th cas with unrliabl a priori and µ > µ lim to th cas with rliabl a priori. Tabl : π valus whn th a priori information at th input of th MAP qualizr ar unrliabl BPSK QPSK 8-PSK 6-PSK π / 3/4 3/8 3/6 4. SIMULATION RESULTS In this sction, w propos to validat th analysis by simulations. In th simulations, th modulation usd is ithr a 8-PSK or a 6-PSK using Gray mapping and th channl is assumd to b constant. W provid th MAP qualizr with a priori information on th transmittd bits b l modld as indpndnt and idntically distributd random variabls with th conditional pdf N( b l, 4 ). W considr diffrnt channls. Figurs, and 3 rprsnt rspctivly th with rspct to th at th output of th MAP qualizr with 8-PSK modulation, for channl3, channl4 and channl3c with impuls rsponss: channl3 = (0.5, 0.7, 0.5), channl4 = (0.37,0.6, 0.6,0.37) and channl3c = ( j, j, j). Tabl givs th valus of d and d for ach channl. For all ths channls, w hav d > d. Th limit valus µ lim ar givn in Tabl 3. Tabl : d and d valus for 8-PSK modulation channl3 channl4 channl3c d d Figurs 4 and 5 rprsnt th with rspct to th at th output of th MAP qualizr with 6-PSK modulation rspctivly for channl3 and channlcomp with impuls rsponss: channl3 = (0.5, 0.7, 0.5) and channlcomp = Figur : vrsus : comparison of th qualizr prformanc curvs) for channl3 and 8-PSK modulation ( j, j, j). Tabl 4 givs th valus of d and d for ach channl. For all ths channls, w hav d > d. Th limit valus µ lim ar givn in Tabl 5. Tabl 4: d and d valus for 6-PSK modulation channl3 channlcomp d d In all figurs, ach curv is obtaind whil th ratio µ is kpt constant. Th solid lins indicat th qualizr prformanc givn by simulations. Th dottd lins ar obtaind by considring th analytical approximat xprssions calculatd in th prvious sction. W notic that th thortical curvs approximat wll th. This approximation bcoms bttr at high (for low valus of ). W can also dduc that th approximation is mor accurat for 8-PSK than for 6-PSK, sinc th assumption that thr is on rronous bit pr rronous symbol is bttr validatd for a 8-PSK modulation. 5. CONCLUSION In this papr, w considrd a transmission of M-PSK symbols ovr a frquncy slctiv channl. W proposd to study analytically th impact of a priori information on th MAP qualizr prformanc. W gav an approximation of th rror probability at th output of th MAP qualizr. Simulation rsults showd that th analytical xprssions giv a good approximation of th qualizr prformanc. Th aim of this work is to prform in th futur th analytical convrgnc analysis of itrativ rcivrs with MAP qualization. 6. APPENDIX: PROOF FOR PROPOSITION W rcall that th output of th channl during a burst is th T complx vctor y = (y 0,...,y T ) T dfind as: y = Hx + n, 007 EURASIP 359

4 0 5th Europan Signal Procssing Confrnc (EUSIPCO 007), Poznan, 0Poland, Sptmbr 3-7, 007, copyright by EURASIP Figur : vrsus : comparison of th qualizr prformanc curvs) for channl4 and 8-PSK modulation Figur 4: vrsus : comparison of th qualizr prformanc curvs) for channl3 and 6-PSK modulation Figur 3: vrsus : comparison of th qualizr prformanc curvs) for channl3c and 8-PSK modulation Figur 5: vrsus : comparison of th qualizr prformanc curvs) for channlcomp and 6-PSK modulation whr x = (x L,...,x T ) T is th (T + L ) vctor of transmittd symbols, n = (n 0,...,n T ) T is a complx whit Gaussian nois vctor with zro man and varianc σ and H is th T (T + L ) channl matrix givn by (). Th ral vctor of a priori obsrvations z = (z q( L),...,z (qt ) ) T on th transmittd bits is dfind as: z = b + w, whr w = (w q( L),...,w (qt ) ) T is a ral whit Gaussian nois vctor with zro man and varianc and b = (b q( L),...,b (qt ) ) T is th vctor of transmittd bits. Taking into account th a priori information, th a postriori Tabl 5: µ lim valus for 6-PSK modulation channl3 channlcomp µ lim probability of th squnc x is givn by: P ( x y,z ) ( y Hx ) ( ) z b xp σ xp a. As in th cas of no a priori information, w considr an rror vnt ξ. This rror vnt is that x is bttr than x in th sns of th MAP squnc mtric: ξ : y H x + σ a z b y Hx + σ a z b, (8) Lt = x x and a = b b b rspctivly th symbol rror 007 EURASIP 360

5 vctor and 5th theuropan bit rrorsignal vctor. Procssing W hav: Confrnc (EUSIPCO 007), Poznan, sinc th Poland, numbr Sptmbr of rronous 3-7, 007, bit copyright pr rronous by EURASIP symbol is b- y H x y Hx twn and q. Th vctor a has m b () componnts qual to (9) = y Hx H ± and th othrs qual to zro. Thus, w can rplac a by y Hx 4m b () and w obtain: = H R( y Hx,H ( ) = ε P x, x = Q ε + m b ()µ ). R( n,ε ), whr ε = H. In addition: z b z b = z b a z b = a z b,a W obtain using (8), (9) and (0): = a w,a. (0) ξ : ε + µ a R( n,ε ) + µ w,a, whr µ = σ σ a. W not χ = R( n,ε ) + µ w,a. W can writ that χ N (0, ( ε + µ a )). Thus, whn th gni aidd qualizr is providd with a priori information, th pairwis rror probability that it chooss x instad of x is givn by: ( ) P x, x = Q ε + µ a. Lt m b () b th wight of th bit rror vctor, and m() th wight of th symbol rror vctor with m() m b () qm() REFERENCES [] C.Douillard, M.Jzqul, C.Brrou, A.Picart, P.Didir, A.Glaviux, Itrativ corrction of intrsymbol intrfrnc: Turbo-qualization, Europ. Trans. On Tlcomm., vol. 6, pp , 995. [] N.Sllami, A.Roumy, I.Fijalkow, Th impact of both a priori information and channl stimation rrors on th MAP qualizr prformanc, IEEE Transactions on Signal Procssing, vol. 54, No 7, pp , July 006. [3] G.D.Forny, Jr., Lowr bounds on rror probability in th prsnc of larg intrsymbol intrfrnc, IEEE Transactions on Communications, pp , Fbruary 97. [4] L.R.Bahl, J.Cock, F.Jlink, J.Raviv, Optimal dcoding of linar cods for minimizing symbol rror rat, IEEE Trans. Inf. Thory, vol.it-3, pp , March 974. [5] S.Bndtto, E.Bigliri, Principls of digital transmission with wirlss applications, kluwr/plnum, Nw York 999. [6] S.Tn Brink, Convrgnc of itrativ dcoding, IEEE Elctronic Lttrs, vol. 35, pp , May EURASIP 36