Improved Performance of the MMSE Linear Equalizer in Turbo Equalization by the use of Extrinsic Feedback Information

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1 Improv Prformc of th MMS Lir qulizr i Turbo quliztio by th us of xtrisic Fbck Iformtio M.Y. Abul Gffr, H. Xu F. Tkwir Abstrct This ppr prsts grl structur for Miimum M Squr rror (MMS) qulizrs from which w c riv istcs such s th: two propos Lir qulizrs with xtrisic Fbck (L-F), Lir qulizr (L) cisio Fbck qulizr (F). Th xtrisic Iformtio Trsfr (XIT) chrt ws us to lyz th two propos MMS L-F rcivrs thir chrctristics ws compr to th xistig MMS L Mximum A Postrior (MAP) qulizrs us i turbo quliztio. W show by usig th Kolmogorov-Smirov (K-S) tst tht th Gussi ssumptio o th probbility sity fuctio (PF) of th priori qulizr iformtio ftr th first itrtio is ot vli. Simultio rsults mostrt tht th prformc of o of th MMS L-F rcivrs os pproch th MAP qulizr th MMS lir qulizr with prfct xtrisic fbck (L-PF) for log block lgths t high sigl to ois rtios (SNRs). usig th prior iformtio [3], whil w prst th MMS L-F (I) MMS L-F (II) rcivrs which comput th postcursor m stimts usig th postriori iformtio xtrisic iformtio rspctivly. Th XIT chrt [4] ws show i [5] to b o of th most ccurt msurs to urst th bhviour of th itrtiv coig procss. This smi-lytic tchiqu is us to compr th MMS L-F rcivrs with th xistig MAP MMS qulizrs i [3]. Th vliity of usig th Gussi ssumptio o th PF of th priori qulizr cor iformtio is lso ivstigt usig th K-S tst. W oly cosir th us of covolutiol cos for th cor th bit optiml MAP cor for th coig procss. Howvr, othr cos such s irrgulr LPC cos r ttrctiv ltrtiv [6]. II. SYSTM MOL Ix Trms turbo quliztio, lir quliztio with xtrisic fbck, MMS quliztio, XIT chrt. cor x Π y ISI Chl I. INTROUCTION h joit sig of quliztio coig origilly Tpropos by th uthors i [] hs rctly l to th us of MMS qulizrs i turbo quliztio. Th vtg of usig MMS qulizrs is th lowr computtio complxity compr with trllis bs qulizrs [, ] oly if th MMS qulizr filtr lgth is smllr th th lgth of th chl impuls rspos. It ws show i [3] tht th MMS L os pproch th prformc of th optiml bit rror rt (BR) MAP qulizr for log block lgths t high SNRs. Th mi objctiv of this ppr is to improv th BR prformc of th MMS L [3] by obtiig th postcursor m stimtios usig two othr mthos. Th MMS L computs th postcursor m stimt by oly Muscript rciv Ju 0, 003. M.Y. Abul Gffr is post-grut stut with th prtmt of lctricl, lctroic Computr girig, Uivrsity of Ntl, South Afric (pho , -mil: bulm@u.c.z). H. Xu is sior lcturr i th prtmt of lctricl, lctroic Computr girig, Uivrsity of Ntl, South Afric (-mil: xuh@u.c.z) F. Tkwir is profssor h of school i th prtmt of lctricl, lctroic Computr girig, Uivrsity of Ntl, South Afric (-mil: ftk@u.c.z) r SISO MMS/ MAP qulizr Π - L( x) L ( x ) Π MAP cor Fig.. A srilly coctt co t trsmissio systm showig th structur of th trsmittr rcivr Fig. shows block igrm of iscrt tim trsmissio mol. This ppr cosirs cohrt bu rt qulizrs usig BPSK moultio. A block of N b t bits, {0,}, is co with rt R covolutio cor to N = N / R co symbols c i b c x, =, N i, x {, ) icluig trllis trmitio. Th itrlvr, fi by o to o ix mppig fuctio, prmuts th co symbols x outputs N i symbols y, =, N i, y {,}. A block of N t triig symbols t, which r kow to th rcivr, N i t symbols r trsmitt ovr itrsymbol ˆ

2 itrfrc (ISI) chl. Th rl iscrt tim ISI chl of lgth L is rprst by lir filtr with th fiit impuls rspos (FIR) h k, k = 0, L-. Th rciv symbols, r, c b xprss s L = k k + k = 0 r h y, () whr th ois smpls r ipt iticlly istribut (i.i..) Gussi rom vribls with zro m vric σ. I rl commuictio systm th chl cofficits ois vric r rquir to b stimt. I this ppr w ssum prfct kowlg of th chl cofficits h k th ois vric σ. Th Soft-Iput Soft-Output (SISO) qulizr rcivs th ois corrupt triig t symbols computs th xtrisic log liklihoo rtio (LLR) L( y ) bs o ithr th MMS critri or th optiml BR MAP lgorithm. Th xtrisic LLRs of th qulizr r itrlv to provi th corrct orrig of th LLRs L( x ) to th iput of th cor. For coig w oly cosir th optiml MAP lgorithm, which uss th BCJR lgorithm [9]. Th LLRs L ( x ) r itrlv us s th priori LLRs L( y) to th qulizr for th xt itrtio. Th trsfr of xtrisic iformtio is crucil i th turbo pricipl to prvt prmtur covrgc urig th itrtiv procss. A trmitio critri or prtrmi umbr of itrtios stops th itrtiv coig systm. III. MMS QUALIZRS IN TURBO QUALIZATION A grl structur to riv th MMS qulizrs is show i Fig.. A soft itrfrc ccllr (SIC) is prform o th rciv sigl r follow by lir qulizr. A mppig trsforms th stimt output y ˆ ito its corrspoig xtrisic LLR vlu. Th MMS L, L-F (I) L-F (II) rcivrs c b obti by followig th rspctiv lgs i Fig.. rivig th MMS F [3] is lso possibl from th grl structur by pssig yˆ to hr cisio mppig, which is th us s o of th postcursor m stimts. Th MMS F structur is ot show i Fig. for clrity. Not tht th istcs tht r riv from th grl MMS qulizr structur oly iffr i th wy th postcursor m stimts r comput. Th rivtio of L( y) for th MMS L MMS F r show i [3]. Th MAP qulizr computs th xtrisic LLRs ftr rcivig th N t triig N i t symbols. Howvr, th MMS qulizrs clcult th xprssio for L( y ) by oly cosirig wiow of rciv symbols corrspoig to th FIR lgth of th MMS qulizr [3]. Th xt xtrisic LLR vlu is obti usig th sm r - SIC prcursor cursor postcursor Lir qulizr M stimts Soft cisio mppig yˆ Mppig to LLR Soft cisio mppig ly L-F (I) L-F (II) L Fig.. Grl structur for MMS qulizrs usig priori iformtio wiow siz but shiftig th wiow o positio to iclus th xt rciv symbol. Th MMS qulizrs c b thought of computig th xtrisic LLR vlus by usig sliig wiow o th rciv squc of symbols. Th MMS L-F rcivrs iffr i th wy th postcursor m stimts r comput,.g. po th(( L( y) + L( y)) / ) L-F(I), y th( L( y) / ) L-F(II), whil th prcursor cursor m stimts for both th MMS L MMS L-F rcivrs r comput by oly usig th priori iformtio,.g. y pr th( L( y )) c y th( L( y )). Th MMS L-F rcivr cosists of tim vryig FIR filtr w [ w, M... w, M] T of lgth M = M + M +. W fi th followig two vctors: whr ( M+ L-)x po po c pr pr T L M+ + + M ( M+ L-)x, () y [ y... y y y... y ], (3) s y (4) is M + L row vctor with ll s. Usig th MMS critri { y yˆ }, th cofficits of th qulizr s FIR filtr c b obti by [3] ( H ( c H = σ w M + + y ), w I HF H ) zz z (5) whr I M is th M x M itity mtrix, H is th M x (M + L - ) chl covolutio mtrix fi i [], T F ig( s ) z H[ 0 x ( L+ M ) 0 x M]. Th stimt output y ˆ c thrfor b xprss s H c yˆ = w ( r Hy + y z ), (6) ftr rcivig vctor of M symbols fi s

3 r [ r... r ]. Approximtig th stimt output yˆ T M + M to b Gussi istribut [3], th xtrisic LLR of th qulizr c b comput by th xprssio yˆ L( y ) =. H ( z w ) (7) Th computtiol complxity btw th MMS MAP qulizrs ws lso outli i [3] to show tht th computtiol complxity of th MMS qulizrs icrs lirly with rspct to th lgth of th ISI of th chl. IV. TH XIT CHART Th XIT chrt hs b show to b importt lysis tool to sig itrtiv coig schms [4] to prict th bhviour of th itrtiv procss. W fi I o I o s th mutul output iformtio from th qulizr cor rspctivly giv spcific iput mutul iformtio. Th xchg of xtrisic iformtio urig th itrtios is rprst by obtiig th trsfr fuctios Io =ϒ ( Ii, s / No) Io =ϒ ( Ii ) iptly giv spcific iput mutul iformtio of I i I i for th qulizr cor rspctivly. Th mutul iformtio rquir for th trsfr fuctios ϒ (.) ϒ (.) r comput by th followig xprssios [6] g ( l) l I g l g () l g () l y, o =, y ().log y {,} y, =+ + y, = (8) g ( l) l I g l g () l g () l x, o =, x ().log x {,} x, =+ + x, = (9) whr gy, () l gx, () l r th PFs of th xtrisic iformtio of th qulizr th cor rspctivly pig o th trsmitt symbol y x. I orr to obti xprssio for th PFs gy, () l gx, () l urig th itrtios is itrctbl. Th criticl ssumptio m for th XIT chrt is tht th PF of th priori qulizr cor iformtio r ssum to b Gussi istributio th LLRs PF obys th cosistcy hypothsis [4]. If ths coitios r vli th th PFs c b rprst by sigl prmtr whr th vric is twic th m. Howvr, i itrtiv coig procss ths ssumptios r oly pplicbl for lrg itrlvr block lgths fix umbr of itrtios. Th XIT chrt lysis tchiqu ws show i [3] to b succssfully xt to turbo quliztio systm. A ipt iticlly istribut log-liklihoo rtios (LLRs) usig orml istributio with m of xσ. / vric σ is grt for th prior iformtio of th qulizr which is th pss to th qulizr to comput th corrspoig xtrisic iformtio. Th xtrisic iformtio obti is th us I o =Ii 0. MAP q. L F (II) L MAP cor 0 0. I =Io i Fig. 3. XIT Chrt for MMS MAP qulizrs for th PF of gy, () l to clcult th output mutul iformtio xprssio i qutio (8). This mtho to comput th output mutul iformtio is cll th histogrm mtho. Similirly, th trsfr fuctio for th cor usig qutio (9) is lso corrspoigly obti by th bov procur. It ws show i [7] tht th mutul iformtio c b pproximt by tim vrg pproximtio if th PFs gy, () l gx, () l r both symmtric cosistt. I this ppr w oly l with th histogrm mtho. Th covrgc thrshol is trmi by th lowst SNR t which th two fuctios ϒ (.) ϒ (.) o o itrsct ch othr. Th xchg of xtrisic iformtio urig th ctul itrtiv coig procss, bs o mutul iformtio, c b visuliz by th systm trjctory i th XIT chrt. Th systm trjctory is comput usig th ctul PFs of g, () l g, () l for qutios (8) (9) rspctivly. y x V. RSULTS Th followig prmtrs wr st for th XIT chrt simultios. Th Prokis C chl of lgth L = 5 fi i [3] ws us. Th lgth of th t bits ws N b = Th t bits wr co usig rt R c = 0.5 systmtic rcursiv covolutiol co for th co G=[7 5] i octl form. A rom itrlvr of lgth N i = ws us. Th MMS filtr prmtrs wr st to M = 9 M = 5. A block of trsmitt symbols for th MAP qulizr cosist of Nt = L triig symbols follow by th N i = co t symbols. For th MMS qulizrs th triig bits wr st to Nt = M + L. Ths triig symbols wr to voi itr-block itrfrc to sur corrct iitiliztio of th lph mssgs for th MAP qulizr s wll s th postcursor m stimt for th MMS qulizrs.

4 I o =Ii L F (II) 0. MAP cor trj. trj. L F (II) I =Io i Fig. 4. Systm trjctory for MMS L-F rcivrs Th qulizr cor trsfr fuctios wr comput 7 usig 0 quiprobbl t symbols thir corrspoig LLRs ovr svrl blocks. Th sigl to ois rtio (SNR) fi s s / N 0 ws st to 4B. Th XIT Chrt for th vrious quliztio schms r show i Fig. 3. Fig. 3 shows tht th MMS L-F (I) hs suprior prformc compr with th MMS L-F (II). Th bttr prformc of th MMS L-F (I) compr with th MMS L-F (II) is u to th postcursor m stimts big mor rlibl wh both th xtrisic priori iformtio r us rthr th oly th xtrisic LLRs. W lso ot tht th MMS L-F (I) pprochs th MAP qulizr surprisig prforms bttr for Ii It is lso obsrv from Fig. 3 tht th prict trjctory for th MMS L-F (I) rquirs 3 itrtios i orr to rch covrgc. Fig. 4 shows th systm trjctoris of th itrtiv coig lgorithm usig th MMS L-F () MMS L-F (II) rcivrs. Th systm trjctoris, for both th MMS L-F lgorithms, follow th prict trjctoris for th iitil quliztio stg th strt to vit from th bous of th trsfr fuctios i th XIT chrt. Th XIT chrt lysis is thus oly ccurt urig for th iitil quliztio stg. Howvr, th systm trjctory i th XIT chrt rvls grphicl rprsttio of th rt of covrgc of th propos MMS L (I) MMS L (II) rcivr schms compr with th MMS L show i []. Th MMS L (I) is obsrv to hv th fstst covrgc mogst th MMS qulizrs. Th iffrc btw th systm prict trjctoris thrfor rquirs ivstigtio. Th PFs of L( x ) L( y ) wr ivstigt s this is thought to b whr th iccurcis occur i obtiig th trsfr fuctios for th XIT chrt. W oly cosir th PFs of L ( x ) for th symbols corrspoig to x =+ y =+, i.. gy, =+ () l gx, =+ () l. Usig th systm trjctory w c prict th xpct vric σ v m σ v /, from th corrspoig mutul iformtio, if w ssum tht th coitiol PFs of L( y) L( x ) r Gussi istribut [4]. W c th rtificilly grt two coitiol PFs, with il Gussi istributio, i.. v, =+ ( γ σ ) y vx, =+ ( γ σ v) to compr with th rspctiv coitiol PFs of L( y ) L( x ). A gooss of fit msur o th ctul coitiol PFs of th priori qulizr cor iformtio c thrfor b us to tst th vliity of th Gussi ssumptio for umbr of itrtios util covrgc. Th K-S tst [8] is simpl o-prmtric mtho to tst for gooss of fit, which rquirs trmiig th mximum bsolut istc (for th two si tst) btw th two cumultiv istributio fuctios i qustio. Th mximum bsolut istc obti for th K-S tst is comput by th xprssio rg mx( v y ( γ σ ) g y ( l) ) (0), =+ v, =+ for th prior iformtio of th qulizr. Similrly, th vlu for th prior iformtio of th cor is lso clcult. Fig. 5 shows th rsults from th K-S tst o th vrious qulizrs. Th itgr th hlf umbr of itrtios i Fig. 5 corrspo to th K-S tst prform o gy, =+ () l gx, =+ () l rspctivly. Th rsults of th K-S tst wr obti by usig 0 7 LLRs for L( y ) L( x ) ovr svrl blocks. Th K-S tst ws lso vrg ovr 00 PFs for v, =+ ( γ σ ) v, =+ ( γ σ ). vlu y v MAP q. L F (II) L x umbr of itrtios Fig. 5. K-S tst prform o th coitiol PFs of th priori qulizr cor iformtio v v

5 ρ MAP q. 0.3 L F (II) L umbr of itrtios Fig. 6. Corrltio btw th priori xtrisic LLRs urig quliztio coig It is itrstig to obsrv i Fig. 5 tht th Gussi ssumptio is mor vli ftr quliztio th ftr coig. Th vlu obti i th K-S tst for th MAP qulizr th MMS L rmis lmost costt util thr itrtios but icrss i th fourth itrtio. Th rltiv high vlu for th MAP qulizr, urig th first itrtio, c b ttribut to th iitil skwss of th coitiol PFs [9]. Howvr, th MMS L-F rcivrs xhibit icrs i th vlu ovr th itrtios oly ftr th coig stg urig th first itrtio. This xplis why th trjctoris of th MMS L-F rcivrs o ot rch th bous of th trsfr fuctios i Fig. 4. Th XIT Chrt ws scrib i [4] to b oly vli wh th corrltio btw th xtrisic iformtio of th cors is rltivly low. W thrfor sk to ivstigt this proprty i turbo quliztio systm. Th corrltio vlu ρ is fi s th corrltio btw th priori xtrisic iformtio ftr quliztio or ftr coig usig thir rspctiv hr cisios. Fig. 6 shows th plot 7 of th corrltio vlu usig 0 LLRs corrspoig to t symbols oly ovr svrl blocks. Th itgr hlf umbr of itrtios r fi s ftr coig quliztio rspctivly. Th high corrltio vlu for th MMS L th MAP qulizr ftr 4 itrtios rsults i icrs i th vlu s otic from Fig. 5. It c b ot, wh th MMS L-F rcivrs is us, tht th corrltio vlu ftr coig urig th first itrtio is rltivly low. This suggsts tht th corrltio btw th priori xtrisic iformtio is ot th rso why th MMS L F rcivrs hv o Gussi istributio of gy, =+ () l ftr th first itrtio. Th BR prformc of th MMS MAP qulizrs r show for 4 4 itrtios i Fig. 7() (b). Th BR prformc of th MMS L (II) rcivr is omitt u to its ifrior prformc. Th prformc of oprco turbo quliztio systm is limit by th mtch filtr (MF) bou [3], which w ot s o ISI i Fig. 7() (b). Th MMS L-PF is il qulizr whr th xtrisic fbck is prfct similr to th i of F whr th fbck cisios r prfct. Th MMS L-PF os ot pproch th MF bou t low SNRs u to th poor qulity of th prcursor m stimts, which ps o th prior iformtio obti from th cor. A systm usig th MMS L-F (I) os yil similr prformc compr to th MAP qulizr ftr just 4 itrtios t 4B to th MMS L-PF for SNR B ftr 4 itrtios. Th MMS L-F (I) hs bttr prformc compr with th MMS L, provig tht xtrisic iformtio fbck os yil improvmts i th BR prformc for turbo quliztio systm. VI. CONCLUSION Th propos MMS L-F (I) ws show to hv suprior prformc compr with th xistig MMS L. Th coitiol PF of th priori qulizr iformtio for th MMS L-F rcivrs wr fou to ot hv Gussi istributio ftr th first itrtio ipt of th corrltio btw th priori xtrisic iformtio of th cor. W thik this u to th fbck tur of th MMS L-F rcivrs, which cuss hi rror propgtio ffct hc ffcts th subsqut MAP coig stg. Th MMS L-F (I) ws show to succssfully mitigt th rror propgtio phom wh compr to th MMS L-PF rcivr t 4 itrtios for high SNRs. A sourc of futur work is to urst combt th hi rror propgtio ffct i th MMS L-F (I) rcivr urig th rly itrtios. RFRNCS [] C. ouillr t l., Itrtiv corrctio of itrsymbol itrfrc: turbo quliztio, urop Trs. o Tlcommuictios, vol. 6, pp , Sptmbr-Octobr 995. [] G. Buch, H. Khorrm, J. Hgur, Itrtiv quliztio coig i mobil commuictio systms, i Proc. of th urop Prsol Mobil Comm. Cof., pp , Spt/Oct 997. [3] M. Tüchlr, R. Kottr A. Sigr, Turbo quliztio: pricipls w rsults, I Trs. o Comm., vol 50, pp , My 00. [4] S. t Brik, Covrgc bhviour of itrtivly co prlll coctt cos, I Trs. o Comm., vol 4, pp , Oct 00. [5] M. Tüchlr, S. t Brik, J. Hgur, Msurs for trcig covrgc of itrtiv coig lgorithms, i Proc. 4 th I/ITG Cof. o Sourc Chl Coig, Brli, Grmy, pp , J 00. [6] N. Vric A. Kvcic, Optimiz LPC cos for prtil rspos chls, i Proc. Itrtiol Symposium o Iformtio Thory, pp. 97, Lus, Switzrl, July 00. [7] M. Tüchlr J. Hgur, XIT chrts of irrgulr cos'', i Proc. Cof. o Iformtio Scics Systms, Pricto, U.S.A., Mrch 00.

6 0 0 4 itrtios itrtios BR 0 3 BR o ISI MAP q. L PF L s /N o i B () o ISI MAP q. L PF L s /N o i B (b) Fig. 7() (b) BR prformc of th MAP MMS qulizrs [8] W. il, Appli o prmtric sttistics, Houghto Miffli Compy, 978. [9]. ivslr, S. olir F. Pollr, Itrtiv turbo cor lysis bs o sity volutio, I Slct Ars i Comms., vol. 9, pp , My 00. M. Y. Abul Gffr rciv his BScg gr i lctroic girig from th Uivrsity of Ntl i 00. H is currtly M.Scg stut i th School of lctricl, lctroic Computr girig t th Uivrsity of Ntl. His rsrch itrsts iclu quliztio coig thory. H. Xu rciv th BSc gr i 984 from th Uivrsity of Guili Tchology th MS gr from th Istitut of Tlcotrol Tlmsur i 989, Th Ph gr from Th Bijig Uivrsity of Aroutics Astroutics i 995. His rsrch itrsts r i th r of igitl commuictios igitl systms. F. Tkwir rciv th BSclcg gr i 98 from th Uivrsity of Mchstr th Ph gr from th Uivrsity of Cmbrig i 984. At prst h is th Profssor of igitl Commuictios H of th School of lctricl, lctroic Computr girig t th Uivrsity of Ntl, South Afric.