urst Mde Nn-Causal Decisin-Feedback Equalizer based n Sft Decisins Elisabeth de Carvalh and Dirk T.M. Slck Institut EURECOM, 2229 rute des Crêtes,.P. 93, 694 Sphia ntiplis Cedex, FRNCE Tel: +33 493263 Fax: +33 4932627 fcarvalh, slckg@eurecm.fr bstract: The Nn-Causal Decisin-Feedback Equalizer (NCDFE) is a decisin-aided equalizer that uses nt nly past decisins, like DFEs, but als future decisins, which usually cme frm anther, classical equalizer. When there are n errrs n the decisins, the NCDFE attains the Matched Filter bund (). In practice, it suffers frm the prpagatin f errrs. We prpse an implementatin f the NCDFE based n sft decisins where nly the mst reliable decisins are fed back: this decreases errr prpagatin and allws perfrmance clser t the matched filter bund. I Intrductin The principle f the Nn-Causal Decisin-Feedback Equalizer (NCDFE) was first prpsed by Prakis []: this equalizer uses past and future decisins in rder t cancel all the ISI present in the signal. Gersh and Lim [2] intrduced its MMSE design: the frward filter is prprtinal t the matched filter and the feedback filter applied t the past and future symbl decisins w.r.t. the symbl t be detected, is the cascade f the channel and the frward filter, withut the central cefficient. These past and future symbl decisins cme frm anther classical equalizer, linear r DFE (nte that the past decisins may cme frm the NCDFE itself). burst mde unbiased MMSE versin based n MLSE was als prpsed in [3]. When n errrs n the decisins are made, this equalizer attains the Matched Filter und () and is then ptentially mre pwerful than the ther equalizers, linear r DFEs. Hwever the errr prpagatin phenmenn can cause sme degradatins, like fr the classical DFE, and the NCDFE may bring nly a marginal imprvement. Our purpse is t build a nncausal decisin-feedback equalizer where nly the mst reliable symbls are fed back. Symbls are estimated thrugh several iteratins: each iteratin finds the linear MMSE estimate f the unknwn symbls and hard decisins are made nly n the mst reliable symbls which are then cnsidered as knwn. The estimatin f the remaining unknwn symbls is dne taking int accunt the new knwn symbls, which increase their estimatin perfrmance. We define a measure f reliability related t y The wrk f Elisabeth de Carvalh was supprted by Labratires d Electrnique PHILIPS under cntract Cifre 297/95. The wrk f Dirk Slck was supprted by Labratires d Electrnique PHILIPS under cntract LEP 95FR8 and by the EURECOM Institute the MMSE estimatin errr variance: it depends n the experimental cnditins, n the psitin f the symbl in the burst and n the presence f knwn symbls. II Prblem Frmulatin We cnsider a single-user multichannel mdel: this mdel results frm the versampling f the received signal and/r frm receptin by multiple antennas. Cnsider a sequence f symbls a(k) received thrugh m channels f length N and cefficients h(i): y(k) = N, X i= h(i)a(k,i) +v(k); () v(k) is an additive independent white Gaussian nise with rvv(k,i) = E v(k)v(i) H = 2 v I m ki. The symbl cnstellatin is assumed knwn. When the input symbls are real, it will be advantageus t cnsider separately the real and imaginary parts f the channel and received signal as: Re(y(k)) = Im(y(k)) N, X i= Re(h(i)) Re(v(k)) a(k,i)+ Im(h(i)) Im(v(k)) (2) Let s rename y(k) =[Re H (y(k)) Im H (y(k))] H,andidemfr h(i) and v(k), we get again (), but this time, all the quantities are real. The number f channels gets dubled, which has fr advantage t increase diversity. Nte that the mnchannel case des nt exist fr real input cnstellatins. ssume we receive M samples, cncatenated in the vectr Y M (k): Y M (k) =T M (h) M+N,(k) +V M (k) (3) Y M (k) =[y H (k,m +) y H (k)] H, similarly fr V M (k), and M (k) = a H (k,m,n +2) a H (k) H,where (:) H dentes Hermitian transpse. T M (h) is a blck Teplitz matrix filled ut with the channel cefficients gruped in the vectr h. We assume that sme symbls are knwn: k cntains the M k knwn symbls and u,them u unknwn symbls. We shall simplify the ntatin in (2) with k = M, t: Y = T (h) + V = T k (h) k + T u (h) u + V (4)
Influence f the knwn symbls The structure f the burst-mde multichannel classical equalizers has been established in [4]. We derived the linear and decisin-feedback equalizers in their Minimum-Mean- Squared-Errr (MMSE) Zer-Frcing (ZF), MMSE and Unbiased MMSE (UMMSE) versins when sme symbls in the burst are knwn, as well as expressins fr the utput SNRs. s an example, in figure, we shw the SNR at the utput f the MMSE Linear Equalizer (LE). In the fllwing we will use tw imprtant prperties f the burst mde equalizatin: The SNR depends n the psitin f the symbl n the burst. Fr a given symbl, the SNR is higher when there are knwn symbls in the burst and especially when the symbl is surrunded by knwn symbls. s seen in figure, when n symbls are knwn, perfrmance at the edges degrades: the middle symbls appear in N utputs whereas the symbls at the edges appear in strictly less than N utputs and thus there is less infrmatin abut them. When N, knwn symbls are present at each end f the burst, perfrmance is better fr the symbls lcated at the edges: after eliminating the cntributins f the knwn symbls, the utputs at the edges cntain strictly less than N symbls, s there is mre infrmatin n thse symbls, which are then better estimated. We als shw the case f and 5 knwn symbls dispersed all ver the burst: and can see the advantage f taking int accunt the presence f knwn symbls in the burst t estimate the unknwn symbls. 25 2 5 5 SNR at the utput f the MMSE linear equalizer + n knwn symbls N- knwn symbls at each end x dispersed knwn symbls * 5 dispersed knwn symbls 5 5 2 25 3 35 4 45 5 Unknwn symbls Figure : SNR at the utput f a MMSE LE as a functin f unknwn symbl psitin in the burst: influence f the presence f knwn symbls n the estimatin f the unknwn symbls urst Mde NCDFE The burst mde structure f the NCDFE was derived in [3] based n MLSE. Its structure is given in figure 2. The frward filter is the multichannel matched filter T H (h) fllwed by a ing t the nature f the equalizer, MMSE r UMMSE: fr the UMMSE NCDFE [3], D =, diag(t H (h)t (h)), (diag(:) is a diagnal matrix cntaining the diagnal f its argument). The nn-causal feedback filter cnsists in the frward filter withut the central cefficient. ^ may be the utput f anther equalizer r the utput f the burst mde NCDFE at a previus iteratin. If ^ cntains n errrs, the perfrmance f the NCDFE attains the. T (h) Channel IV V Y T H (h) Matched Filter D Scaling Factr DT H (h)t (h),diag(dt H (h)t (h)) Figure 2: urst Mde Nn Causal DFE NCDFE based n Sft Decisins MMSE LE gives as estimates f the unknwn symbls u based n the bservatins Y and the knwn symbls k : ^ u = C uy u C, Y uy u (Y,T k (h) k ) (5), = Tu H (h)t u(h) + 2 v I T H a 2 u (h)(y,t k(h) k ) (6) We recgnize a nn-causal decisin-feedback structure where nly the knwn symbls are fed back. ssume yu want t detect ne symbl in the burst and yu knw all the ther symbls, the slutin in (6) wuld give the utput f the MMSE NCDFE crrespnding t this symbl. We will cnsider the UMMSE. The MMSE equalizer is indeed biased: at each utput MMSE (k) f the MMSE equalizer, the term f interest is (k)a(k), with (k) <. Fr cnvenience, we prefer t wrk with unscaled quantities. The UMMSE equalizer is simply a rescaled versin f the MMSE equalizer giving fr each utput: ((k)), MMSE (k). Ithas a lwer utput SNR than the MMSE but ffers the advantage t give a lwer prbability f errr, except fr cnstant mdulus cnstellatins fr which the prbability f errr remains the same. Sft Decisins We will cnsider here nly a PSK; the principle f sft decisins culd be extended t ther cnstellatins thugh. Fr each utput f the UMMSE, (k) =a(k) +~a(k) where ~a(k) cntains intersymbl interference and nise, the sum f which can be apprximated by a centered Gaussian variable. The variance f the errr ~a(k) is [4]: (k) = v 2 Tu H(h)T u(h) + 2 v 2 I a, k;k d, (7) We will nt cnsider a hard decisin scheme as shwn in figure 3, but a sft decisin scheme that will give hard decisins
ten based n the tanh curve (see figure 3). Indeed, the MMSE estimate f = a +~a, with a taking with equal prbability the values + and, and ~a a centered Gaussian randm variable independent f a, hyptheses verified (with the Gaussian apprximatin) in ur prblem, is: = tanh (8) We envisage an iterative scheme with each iteratin cmpsed f tw steps. In the first step we perfrm linear estimatin f the symbls based n the received data and the symbl estimates frm the previus iteratin. The first step wuld crrespnd t the NCDFE if the symbl estimates were perfect. The secnd step perfrms element-wise nnlinear estimatin. The ptimal nnlinearity t be used in the secnd step is the tanh(:). Hwever, with such nnlinear symbl estimates, the design f the linear estimatr fr step ne in the next iteratin becmes nntrivial. Therefre, we prpse the fllwing simplified nnlinearity: jj < = f () = (9) sign() jj (k) gives the reliability f the symbl estimate and depends n : it is determined by searching the best MMSE estimate f f the frm f () shwn in figure 3: min E (a, f ()) 2 () clsed frm expressin fr culd nt be fund. Hwever a Hard decisins f (), tanh @ C 2 ~a Figure 3: Sft Decisin Curves linear apprximatin w.r.t. seemed t match well, especially fr lw : =:33 (see figure 4). The cmplete iterative scheme is depicted in figure 5. ^sft;i dentes the fr which jj <, whereas ^ hard;i dentes the fr which jj =. ^ hard (i) dentes the accumulatin fn ^ hard; ;:::; ^ hard;i. ^ lin;i is a linear cmbinatin f ^ i = ^sft;i ; ^ hard (i) and Y, i.e., ^lin;i is a linear estimate f the remaining undecided symbls in terms f the received data and sft decisins fr all symbls. One can bserve that ^ lin;i is in fact als a linear cmbinatin f nly ^ hard (i) and Y and since the ^ hard (i) are assumed t be errr-free, the MMSE design f ^ lin;i becmes tractable. ^ lin;i,.2.8.6.4.2 Linear apprximatin f..2.3.4.5.6.7.8.9 Figure 4: Linear apprximatin f f (:) ^sf t;i ^ hard;i ^ hard (i,) ^ hard (i) Figure 5: Iterative Sft Decisin Scheme NCDFE based n sft decisins Y ^ lin;i The implementatin f the NCDFE based n the sft decisins is nw as fllws:. Linear MMSE estimates f the unknwn symbls based n Y and k are cmputed by (6). 2. Fr each estimate (k), the reliability measure (k) is cmputed and the sft decisin strategy (9) is applied. The hard decisins are treated as knwn symbls. Steps -2 are reiterated until ^ hard;i is empty. Feeding back nly the mst reliable symbls allws t avid the phenmenn f errr prpagatin. Furthermre, as the presence f knwn symbls allws t increase the estimatin quality f the unknwn symbls, the feedback will help the detectin f symbls n which errrs culd have been made by using a simple linear equalizer. t the end f this prcess, the symbls that remain nn reliable even when using the feedback frm knwn symbls are decided upn. Few iteratins f the algrithm are necessary in general as will be seen in the simulatins. This strategy allws t autmatically adapt the reliability intervals t the experimental cnditins: The nise level: is all the larger as the nise level is large. The presence f knwn symbls will be reflected in the value f (k). Figure 6 shws the evlutinfthe reliability intervals frm ne iteratin t anther (fr a randmly chsen channel at 5 d): mst f the symbls that remain unknwn at the secnd iteratin are lcated at the edges where indeed perfrmance is lwer. t the secnd iteratin, the reliability n thse symbls increases due t the feedback f the knwn symbls. The reliability intervals have hwever fr disadvantage t be based n mean quantities, and it may happen that hard decisins n certain realizatins f (k) cnsidered as reliable are in
2.5 (k) at iteratin - =.5 (k) at iteratin 2 5 5 2 25 3 35 4 45 Psitin f the unknwn symbls Figure 6: Evlutin f the reliability intervals fact false: this prblem may arise especially at the first iteratin fr bad experimental cnditins like lw SNR r ill-cnditined channels. T alleviate this prblem, shuld be taken larger than the previusly given value that we will dente. In ur simulatins, we tk = =2 : this chice gave indeed better results fr bad experimental cnditins. In ur simulatins, we will cmpare the perfrmance f this NCDFE with the fllwing mre ptimal but als mre cmplex scheme:. We cmpute the utput f the NCDFE with feedback k as in (6). 2. We chse the utput with highest magnitude, i.e. the mst reliable symbl estimate, make a decisin n it and add it t the list f knwn symbls in k. Reiterate -2 until u is empty. Figure 7: Distributin f the symbl estimates at the utput f the UMMSE equalizer: reliable decisins such that j(k)j (example fr =) half a Gaussian - ~a(k) Reliable symbls a(k) - ~a(k) Nn-reliable symbls half a Gaussian Figure 8: Jint distributin f ~a(k) and a(k) fr the reliable (left) and nn reliable decisins (right) the cmputatin f the equalizer utput with a cmplexity f rder MN. Hwever Tu H (h)t u(h) + 2 v 2 I lses the quasia Teplitz prperty but is still banded: a cmplexity f rder M u N 2 can then be achieved. On-ging studies are trying t reduce this cmplexity t an rder f M u N. ut it has t be nted that M u is in general small (see the simulatin part) the cmplexity M u N 2 shuld nt be an bstacle. a(k) V Perspectives Changes in crrelatins The incrpratin f the sft decisins creates sme prblems. Indeed, it intrduces crrelatins between a(k) and ~a(k), and then, as ~a(k) cmbines nise and input burst cmpnents, between the nise V and the symbls, riginally independent, as well as between the elements f. Figure 8 shws the jint distributin f a(k) and ~a(k), fr the reliable and nn-reliable (k) fr a case where (k) =(see figure 7). a(k) and ~a(k) are crrelated in bth cases, and ~a(k)ja(k) isnt Gaussian anymre, nly marginal distributin f ~a(k) remains unchanged. The frmulatin in (6), valid nly when the nise and the symbls, as well the different symbls between themselves are independent, did nt seem t be very sensitive t the crrelatin changes. nd we kept this expressin in its riginal frm when using the sft decisins. The incrpratin f these changes are the subject f n-ging research. Cmplexity The matrix T H (h)t (h) + 2 v 2 a I being quasi-teplitz and banded, fast algrithms (based n the Schur algrithm) allw C Use f a Decisin-Feedback Equalizer t each step f the algrithm, the unknwn symbls are estimated by a MMSE linear equalizer. This equalizer culd be replaced by a DFE which gives in general better results: this DFE shuld hwever use amng its past decisins nly the mst reliable nes. VI Simulatins We tested the NCDFE based n sft decisins n tw channels with real cefficients. The first channel H (m = 2, N = 5) was randmly chsen, the secnd channel H 2 = has a nearly cmmn zer. We plt the ttal : number f errrs fr a input burst f length M +N,, with M = fr a given input burst averaged ver the burst length and Mnte-Carl runs f the nise. The SNR is at 7 d and d. N knwn symbls are initially present in the burst. We shw the different measures in the fllwing rder (frm left t right):. The ptimal scheme previusly described where nly the mst reliable decisin is fed back at each iteratin. The
hard decisins; tw iteratins f the hard NCDFE are dne. 2. The prpsed sft decisin strategy with fllwedby2 iteratins f the hard decisin NCDFE. 3. The prpsed sft decisin strategy with fllwedby2 iteratins f the hard decisin NCDFE. 4. The MMSE LE n which hard decisins are made fllwed by 2 iteratins f the hard decisin NCDFE. 5. The MMSE DFE n which hard decisins are made fllwed by 2 iteratins f the hard decisin NCDFE. The is als shwn as reference: it is cmputed by averaging the number f errrs at the utput f a NCDFE with feedback f the exact symbls. Fr bth channels, it first has t be nticed that the hard NCDFE imprves perfrmance significantly w.r.t. the MMSE LE r DFE, but this is nt always the case. Fr channel H (figure 9), we see that ur sft decisin strategy imprves perfrmance w.r.t. t the classical linear r DFE equalizer and w.r.t. t these same equalizers fllwed by the hard NCDFE. The sft decisin strategy using attains the at d. The use f r is apprximately equivalent. Few steps f the algrithm were required: 2.5 fr and 2.9 fr at 7 d and.5 fr and.8 fr at d. The experimental cnditins being gd, mst f the symbl estimates are cnsidered as reliable: nly apprximately 5% f the symbls remains unknwn after the first iteratin at 7 d, and nly 2% at d. Our prcedure remains hwever useful. 6.5 x 3 6 5.5 5 4.5 4 veraged number f errrs fr H at 7 d hard NCDFE mental cnditins are less favrable and the prbability f errr is higher. In this example, we see mre significantly the advantage f the sft decisin strategy, and the necessity f chsing an higher than : indeed with, false decisins were taken as reliable even at the first iteratin. Mre iteratins are dne than in the previus example: 3 fr,3.75fr at 7 d and 2. fr,2.2fr at d. Furthermre, at 5 d, 2% fr, 26% fr f the symbls remains unknwn after the first iteratin, and 7% fr, 2% fr at d. t d, we als see that a classical DFE perfrms better than ur sft decisin prcedure: this suggest the use f a DFE estimatin f the unknwn symbls instead f a linear ne as already mentined in previus sectin..25.2.5..5 veraged number f errrs fr H2 at 7 d.3 hard NCDFE.4.2..8.6.4.2 ptimal veraged number f errrs fr H2 at d hard NCDFE ptimal Figure : verage number f errrs fr channel H 2 at 7 d andd. 3.5 3 2.5 ptimal References.8 x 3.6.4.2.8.6.4.2 veraged number f errrs fr H at d hard NCDFE ptimal Figure 9: verage number f errrs fr channel H at 7 d and d. [] J. G. Prakis. daptive Nn Linear Filtering Techniques fr Data Transmissin. IEEE Symp. n daptive Prcesses, pages XV.2. 5, 97. [2]. Gersh and T. L. Lim. daptive Cancellatin f Intersymbl Interference fr Data Transmissin. ell Syst. Tech. J., 6():997 22, Nv. 98. [3] D. T. M. Slck and E. de Carvalh. urst Mde Nn- Causal Decisin-Feedback Equalizatin and lind MLSE. In Prc. GLOECOM 96 Cnf., Lndn, Great ritain, Nvember 996. [4] D. T. M. Slck and E. de Carvalh. Unbiased MMSE decisin-feedback equalizatin fr packet transmissin. In Prc. EUSIPCO 96 Cnf., Trieste, Italy, September 996.