An Improved SIC based Turbo Equalizer

Transcription

1 Mauscript received July 5, 7; revised Oct. 6, 7 A Improved SIC based Turbo ualizer ARUNA TRIPATHY, SANT SARAN PATHAK, SASWAT CHAKRABARTI 3 Departmet of Electroics ad Electrical Commuicatio Egieerig GS Sayal School of Telecommuicatios Idia Istitute of Techology Kharagpur INDIA-7 3 arua@ece.iitgp.eret.i Abstract: --This paper ivestigates a ew soft iterferece caceller (SIC) based turbo equalizer. The SIC is cofigured as a decisio feedbac (DF) equalizer that uses two liear trasversal filters. Both the filters update their taps as per the least mea square (LMS) algorithm. This receiver is differet from the receivers discussed i literature i the sese that it uses the soft outputs of both the equalizer ad the decoder i order to provide a improved estimate of the post cursors. It also uses the variace of the soft data bit estimates i its computatio of the log-lielihood ratios (LLR). Probability distributio fuctio for the soft data bit estimate is derived. This SIC based turbo equalizer is show to result i a faster covergece ad improved error floor at the decoder output. This study is carried out for a two typical highly frequecy selective chaels i order to establish the better performace of the proposed receiver. Key-Words: - SIC, LLR, SCS, LMS, Soft Output Itroductio The turbo equalizer is a base bad digital receiver that carries out equalizatio of data trasmitted over multipath time-dispersive chaels ad decodig of the equalized data i order to tae care of additive white Gaussia oise (AWGN) i a iterative maer. The approaches to turbo equalizatio are broadly classified as belogig to three techiques, amely, the LMS based SIC [], the miimum mea square error (MMSE) Wieer filter [-3] ad trellis based structures for the equalizer [4]. The secod approach requires a matri iversio of size equal to the filter taps while the third techique eeds computatio of certai probabilities ad thus requires a epoetially large computatioal effort. Thus, the SIC type is cosidered to be more suitable tha the other two approaches for implemetig a turbo equalizer i this paper. I [5], a LMS filter ad full decoder output were used where the decoder was of a Ma-Log-MAP type. A LMS based SIC to serve as the equalizer that uses a improved estimate of the post cursors has bee itroduced i this paper. This may be cosidered to some etet, i lie with the receiver proposed i [5]. A low compleity SIC has bee discussed i [6] for possible applicatio i a high data rate wireless applicatio that uses a variable threshold techique. The issue of real time implemetatio of the SIC based turbo equalizer has bee cosidered i [7]. A aalysis of the SIC TEQ through BER trasfer ad etrisic iformatio trasfer (EXIT) charts has bee attempted i [8]. A optimizatio of equalizatio ad iterleavig tass is carried out i [9]. The suitability of a iterleaver for SIC based turbo equalizatio of frequecy selective ad time selective chaels for higher order modulatios has bee ivestigated i []. However, i this paper, the soft outputs from the equalizer ad the decoder sum for a oliear operatio that produces the post cursors. The precursors are computed by applyig the same oliear fuctio but oly to the decoder outputs. Problem Formulatio The system model is a serial cocateated system (SCS) that cosists of a outer forward error correctig (FEC) ecoder of give parameters ad a ier ecoder represeted by the itersymbol iterferece (ISI) chael. These two ecoders are coected to each other by meas of a iterleaver of suitable type ad size. The ISI chael is the discrete time equivalet of the modulatio chael which cosists of the trasmit pulse shapig filter, the actual physical chael, the receive matched filter ad the sampler. The ISI chael, hece, is 867

2 represeted as a fiite impulse respose (FIR) filter that has a equivalet fiite state machie (FSM) represetatio. The baud-rate sampled output of the matched filter provides a set of sufficiet statistics for detectio. The -th time domai matched filter output samples are epressed as L z = h + w, =,,... N () l l= The 's are the iterleaved ecoded bits, the L chael taps are represeted as hl, =,... L ad w is a i.i.d. AWGN sample with distributio w N (, σ w).the z s are iput to the SIC that is also served by the soft outputs comig from the chael decoder. The SIC produces soft output o all N bits, where N is the total umber of ecoded bits ad it is also the size of the iterleaver. The soft output correspodig to the equalizer is computed by assumig the SIC output to be Gaussia distributed. The log lielihood ratio (LLR) for all the bits is computed from this approimatio. The receivers, i geeral use the soft output of oly of the decoder to compute the estimates. The SIC is a combiatio of a liear forward filter ad a DF filter. The coefficiets of the forward filter are foud out by optimizig the mea square errorε betwee ad its estimate ˆ as { ˆ } ε = E () where [.] E is the esemble average. I [4-5], the DF is used as iterferece cacellatio filter. The trasfer fuctio for the forward filter correspodig to the optimum coditios is * H ( f ) P( f ) = β (3) opt hh ad the bacward filter that is desiged to cacel the effect of the iterferig symbols has a trasfer fuctio H( f ) Q( f ) = β (4) opt hh This is due to the fact that, the cetral coefficiet of the bacward filter is q =, so as ot to cacel the l desired symbol from the forward filter output. The weightig coefficiet β is defied as σ hh d β = ( σ dhh + σ w) (5) where σ d is the symbol eergy, which for a costellatio of uit eergy ad ormalized chael taps becomes β = (6) + σ w The SIC output at a give time istat is epressed as t t s = Pz Q (7) where t deotes traspose of a vector. The forward filter coefficiets are give as ( )... ( )... ( ) P = p L p pl (8) The boldface deotes a vector. The iput to the feedforward filter comes from the matched filter outputs that is epressed as t z = z+ L... z... z L (9) It is obvious that, the forward filter has a legth of L + with the cetral tap correspodig to the desired bit. Similarly, the DF filter coefficiets are give as Q ( )... ( ) = q q L L The iput to the DF filter is a vector of soft bit estimates that is epressed as t = + L L () Thus, the legth of the bacward filter is L +.Each term i the RHS of () is computed by applyig a oliear fuctioal mappig to the etrisic iformatio of the decoder which becomes λdec, = tah () This is assumed to be a approimate value for ˆ due to the tah (.) fuctio. The term soft comes from the fact that, each elemet i RHS of t t 868

3 () is obtaied as a floatig poit umber as show by () istead of taig a hard decisio. Here, λ is the etrisic iformatio produced by the dec, soft-i soft-out (SISO) decoder for the th bit. We ote that, i literature, () is used for estimatig the precursors as well as the postcursors, while the equalizer has updated the soft output for the previous bits correspodig to the curret bit i the et iteratio. The receiver operatio begis with the traiig of the forward filter coefficiets. This is doe by sedig a pseudo-oise sequece (PNS). The taps of the filters are updated by the LMS algorithm as * P = P µ z ( s ) () * ad Q = Q + µ ( s ) where < µ < is the step size. (3) The soft output of the equalizer is geerated by approimatig the equalizer output as Gaussia distributed ad it is epressed as λ e where = s,,,... σ + σ = N (4) ( ) σ res w res is the variace of the residual ISI. We may ote that, its value decreases as more cofidet reliability measures are available from the decoder ad asymptotically it approaches a value of zero. It is defied as σ res L = vi L (5) i= The variace of the data estimates is i ( ) v = (6) i It is iterestig to derive the asymptotic SIC output from (7), which for perfect iterferece cacellatio becomes L * s = β + hlw+ l (7) hh l= It is oted from (7) that, this SIC receiver is capable of attaiig the matched filter boud. This is possible due to the improved data estimates comig from the decoder at each iteratio. The process of turbo equalizatio yields a equivalet AWGN chael at the iput of the decoder. We further ote that, a idetical epressio has bee obtaied for the trellis based equalizers correspodig to this asymptotic case i [] for a turbo equalizer. This brigs out the immediate advatage of a SIC over the trellis based equalizers i the sese that, the former is computatioally less demadig tha the latter. The SISO decoder accepts these λ ( ) e after suitable deiterleavig ad computes a set of updated LLR values as ( = λ, : λ, ) ( = λ, : λ, ) p ( ) e e N Λ = log, Dec p e e N (8) where λ : λ deotes a N e, e, N dimesioal vector of etrisic iformatio at the equalizer output. The decoder computes (8) by startig with the computatio of the brach metrics. The brach metric for the decoder desiged for a rate / code is computed as γ ep( λ ( )) ( ) = e, i i=, i = ep ( λ, ), ep( λ (, )) e i i j i e j, j e, i, i e, i, i e, j, j i= j i (9) It has bee show i [] that, a soft output Viterbi algorithm (SOVA) [3] based turbo equalizer is more suitable from the implemetatio poit of view with a acceptable performace. Hece, SOVA has bee used as the SISO decodig algorithm i this wor. As the SOVA decoder wors i the log domai, (8) becomes equal to ( ) ( ) ( ) lγ = λ = λ + λ () The soft output that is further passed o to the SIC is the etrisic message. This becomes, for the decoder, ( ) ( ) ( ) λ = Λ λ () dec Dec e The soft data estimate as defied i () depeds o the etrisic iformatio of the decoder output. The 869

4 Fig. Bloc Schematic of Proposed SIC asymptotic values of the soft bit estimates approach ±. It is oted from the soft bit epressio that, the same epressio is used for estimatig the precursors as well as the postcursors.both deped o the etrisic iformatio of the decoder. However, as the equalizer produces a updated etrisic iformatio i the et iteratio, this ca be used alog with the decoder's etrisic iformatio to provide a better post cursor estimate. This is because, the mai tap is cosidered to be the cetral tap of the bacward filter ad hece the improved post cursor estimates feed the taps of the bacward filter correspodig to these postcursors. For a particular data bit, we have a improved postcursor estimate by summig the decoder ad equalizer outputs. It is oted from () that, if both terms i the argumet of tah. are of the same sig, the the () soft bit estimates would approach their true biary values faster compared to whe we cosider oly oe argumet for this fuctio... SIC Receiver Structure The bloc schematic of the proposed receiver is illustrated i Fig.. The equalizer produces a updated etrisic iformatio i the et iteratio. The ew post cursor estimate, amely the part of the DF filter correspodig to,... L for the th bit becomes λ λ dec, i e, i = tah +, i L This maes the ( ) () tah. fuctio approach ± faster as compared to if oly oe variable is used as the argumet as (). It is oted from () that, if both terms i the argumet of tah. are of the same sig, the the soft bit estimates would approach their true biary values faster compared to whe we cosider oly oe argumet for this fuctio. If turbo equalizatio is covergig, the the LLR produced by the equalizer is higher tha that of the equalizer correspodig to a give bit ad the summatio as i () helps i achievig faster covergece. If the argumets are of opposite sig, the the particular bit would have a little cotributio o the output of the DF filter ad it would ot affect the rate of covergece. () 87

5 It may be oted that, for chaels with a log delay spread, the proposed scheme produces sigificat improvemets as it has to tae care of a loger strig of post cursors. We assume the decoder's etrisic iformatio to be Gaussia distributed defied as follows ( y, y) y N m σ (3) where y λ λ dec, i e, i = + (4) It may be oted that, this assumptio has bee used i literature [4]. We are iterested i fidig the distributio of the soft bit estimate. This is doe by writig (3) as e = tah ( y) = e y y + Doig a polyomial divisio, we obtai for (4), e e e (5) y 4y 6y = (6) It is observed that, (6) represets a covergig series ad hece, by the cetral limit theorem, the statistics of the soft bit is observed to follow a Gaussia distributio. We ote from (6) that, the higher order terms rapidly decay for higher values of y ad we may approimate (6) by the first three terms. The resultig distributio is show to have a approimated mea of my my m = e + e (7) ad a variace of my my my ( ) ( ) 7my 6my ( e ) e 8 3 4m y σ = + e e + e e + (8) This has bee derived by assumig the symmetry property for each epoetial of (6) for the sae of coveiece. For large values of y, the soft bit saturates to. ± The symmetry property of the Gaussia distributio of the etrisic iformatio gives the SNR associated with the etrisic iformatio as SNR m m m = e, e, e e, σ = e, m =, e, (9) The mea of the Gaussia distributio is give by (7) whe the soft bit is cosidered as (). However, whe we cosider (), we ote that, if the equalizer is covergig, the the two argumets tah. are of same sig. As both are of the ( ) Gaussia radom variables, the resultig mea is the sum of the two meas ad similar argumet is valid for the variaces. Thus, similar to (9), we epress, the SNR correspodig to (4) as m e, SIC, SNR e, SIC, = (3) As m > m e, SIC, e, because the meas of two idepedet Gaussia distributios add. This is true due to the presece of the iterleaver. Thus, we defie the figure of merit for the SIC receiver as SIC gai which is epressed as SIC m e, SIC, gai = > (3) me, This shows that, the proposed receiver has a better SNR tha the receiver proposed i literature ad hece, the probability of average bit error is less for this receiver. This follows from the fact that, a better SNR at the equalizer output yields a better performace at the decoder output whe the receiver is covergig... Soft Bit Estimatio Error Behaviour for SIC Receiver The soft bit formed as per () has bee show to be a Gaussia radom variable. The average value of () is computed by averagig it over a Gaussia statistics with the symmetry property. For the sae of coveiece, the soft outputs have bee modeled as Gaussia distributed with a variace that is twice its mea. Uder the assumptio of all-zero trasmissio ad the represetatio of zero by +, we ote that, the error i data estimate becomes y e= = tah (3) We ote that (3) is a radom variable whose average is obtaied by averagig it over the 87

6 statistics of y. This is regarded as the error i the estimatio of.we, thus obtai, for the average value of this error ( ) y m y e = ep tah dy, m> 4 4m R π m (33) Usig a result of [5], after some algebraic maipulatios, we obtai the followig bouds o the average value of the error π m 3 π m ep < e < ep +, m> m 4 m m 4 7m (34) This gives the upper ad lower bouds o the estimatio error i a SIC based turbo equalizer. Net, we compare the BER of the two receiver schemes as give by () ad (). We have established that, the soft bit i either case is represeted by a Gaussia radom variable. As for a stadard Gaussia radom variable, the average probability of error is epressed as ( ) ( ) P =.5 p y' dy' + p y' + dy' b e (35) where z ' = λdec for the receiver correspodig to (). Similarly, b Pe =.5 p( y ) dy+ p( y+ ) dy variable, p( y' ) p( y) (36) Ad from the property of Gaussia radom < as y' > y whe the receiver is covergig. Thus, the differece i (35) ad (36) is a positive umber that establishes that, the average probability of error of the receiver correspodig to () is smaller tha that obtaied by the use of (). It is oted from Fig. that, the average value of the error i the data bit estimatio teds to zero for icreasig values of the mea of the etrisic iformatio. As show by [6], this mea icreases with respect to the iteratios ad covergece of the turbo equalizer is achieved whe the esemble average of the radom compoet of estimatio teds to zero with icrease i the umber of iteratios. It is also oted from Fig. that, the estimatio error attais a value approimately equal to zero whe the mea of the etrisic iformatio approaches a value of 7. Soft Bit Estimatio Error Lower Boud Upper Boud Mea of Erisic Iformatio Fig. Aalytical Bouds o the Soft Bit Estimatio Error This is true whether we cosider the lower or the upper boud. It may be iterpreted that, whe the receiver is iteratig, the mea of the etrisic iformatio should be approimately 7 for covergece to the true data values. 3 Problem Solutio We evaluate the performace of our SIC by cosiderig the followig simulatio parameters. The outer FEC is a rate ½ recursive systematic covolutioal (RSC) code with geerator polyomials [7,5] for costrait legth 3 ad the ISI chael cosidered is the 5-tap Proais-C chael epressed as [ ]. A radom iterleaver of size bits has bee used for each simulatio ru. The feedforward filter has taps ad iitialized to zeros ecept the cetral tap beig equal to. The feedbac filter has similarly taps with all the taps iitialized to zero. The filters are traied by a traiig sequece of a 55 legth pseudo oise sequece (PNS) at the begiig of receiver operatio. The cetral tap of the feedbac filter is forced to zero after every iteratio of adaptatio algorithm. The step size for the forwrad filter is.8 ad its value is.5 for the bacward filter. This may be attributed to the fact that, the feedbac filter is made isured to coverge, however at a slower rate tha that of the feedforward filter. It may be oted that a similar loop gai combiatio has bee obtaied i 87

7 literature. The decoder is the tracebac SOVA decoder that uses a tracebac depth of 3 bits i order to compute the LLR o each coded bit. The result as obtaied for iteratios is show i Fig.3. The left most curves represet the theoretical lower boud plotted by usig (6). BER - - Proposed Scheme Laot Scheme Theoretical Boud BER - - Proposed Scheme Laot Scheme Theoretical Boud Eb/No i db -3 : Proposed Scheme, : Laot Scheme 3: Theoretical Performace Eb/No i db : Proposed Scheme, : Laot Scheme 3: Theoretical Performace Fig.3 Performace of tap SIC i Proais- C Chael I almost all cases, the performace gai is ot sigificat upto 6 db SNR. The SIC receiver performs better givig a gai of aroud.5 db SNR from 7 to 9 db SNR. The receiver performace is better tha the reported performace by about.-.5 db. It is observed that, the performace improvemet becomes sharper for SNRs higher tha 7 db. The performace of this receiver has bee evaluated for the Proais-B chael [ ] also ad is illustrated i Fig.4. The proposed receiver seems to offer a SNR improvemet of.-.4 db over the reported result. It has bee observed that, the receiver performace is better at 3. db over the scheme reported i literature. A compariso of Fig.3 ad Fig.4 reveal that, for a chael with higher SNR loss (Proais-C), the gap of the proposed receiver from the theoretical performace is more proouced while the gap is about.5 db for a chael with a comparatively small SNR loss. Fig.4 Performace of tap SIC i Proais-B Chael 4 Coclusio A receiver structure proposed here provides improved estimates of the post cursors by summig up the etrisic iformatio of the equalizer at each iteratio alog with the decoder correspodig to the previous iteratio. A aalytical result o the estimatio error shows the covergece of the receiver to the true data values whe the mea of the etrisic iformatio is above a threshold. Refereces: [] C. Laot, A.Glavieu ad J.Labat, "Turbo ualizatio: Adaptive ualizatio ad Chael Decodig Joitly Optimized", IEEE Joural o Selected Areas i Commuicatios, Vol.9, No.9 Sep., pp [] M.Tuchler, R.Koetter ad A.C. Siger, "Turbo ualizatio: Priciples ad New Results", IEEE Tras. o Comm., Vol.5,No.5, May pp [3] M.Tuchler, R. Koetter ad A.C. Siger, " Miimum Mea Squared Error ualizatio Usig A Priori Iformatio", IEEE Tras. o Sigal Processig,Vol.5,No.3, March, pp [4] C.Douillard, A.Picart, P.Didier, M.Jezquel, C.Berrou ad A.Glavieu, "Iterative correctio of itersymbol iterferece: Turbo equalizatio," Eur. 873

8 Tras. Telecommu, Vol.5, Sept. 995, pp [5] F.Vogelbruch ad S.Haar,"Improved Soft ISI Cacellatio for Turbo ualizatio usig Full Soft Output Chael Decoder's Iformatio", GLOBECOM,3,Dec.-5, SaFracisco, Califoria, pp [6] D.Ampeliotis ad K.Berberidis, Low Compleity Turbo ualizatio for High Data Rate Wireless Commuicatio, EURASIP Jou. O Wireless Commuicatio ad Networig, pp.-, vol. 6 [7] R.Le Bida, C.Laot ad D. Lerou, Real-Time MMSE Turbo-ualizatio o the TMS3C559 Fied-Poit DSP, Techical Report of Sigal ad Commuicatios Dept., GET/ENST Bretage, BREST Cede, Frace, applicatios," i Proc., IEEE Globecom Cof. Dallas, TX, Nov.989, pp [4] S.Te Bri," Covergece behavior of iteratively decoded parallel cocateated codes," IEEE Tras. o Comm., vol.49, No.,Oct.999, pp [5] S Y. Chug, T.J Richardso ad R.L. Urbae, "Aalysis of Sum-Product Decodig of Low-Desity Parity-Chec Codes Usig a Gaussia Approimatio", IEEE Tras. o Ifo. Theory, Vol. 47, No., Feb., pp [6] D. Divsalar, S.Dolier ad F.Pollara, "Iterative Turbo Decoder Aalysis Based o Desity Evolutio", IEEE Joural o Selected Areas i Commuicatios, Vol.9, No.5, May, pp [8] S. -Ju Lee, Adrew C. Siger ad Naresh R.Shabag, Liear Turbo ualizatio Aalysis via BER Trasfer ad EXIT Charts, IEEE Tras. o Sigal Processig, Vol.53, No.8, pp , Aug 5 [9] C. Laglais ad M. Helard, Optimizatio of the ualizatio ad the Iterleavig for a Frequecy-Selective Fadig Chael, IEEE Sigal Processig worshop, Norway, pp [] C.Laglais, P-J. Bouvet, M. Helard ad C.Laot, Which Iterleaver For Turbo-ualizatio System O Frequecy ad Time Selective Chaels For High Order Modulatios?, 4 th IEEE Worshop o Sigal Processig Advaces i Wireless Commuicatios (SPWAC'3) pp [] A. Tripathy, S.S.Patha ad S. Charabarti," Turbo ualizer as a Maimal Ratio Combier", Proc. of d It. Cof. o Wireless Comm. ad Sesor Networs (WCSN), 7-9 th Dec.,6,IIIT,Allahabad, Publisher McMilla Idia, Advaced Research Series, p33-39 [] A. Tripathy, S.S.Patha ad S. Charabarti," SISO based Turbo ualizatio for Wireless Chaels corrupted with ISI", d IEEE It. Cof. o Computig,Theory ad Applicatios(ICCTA 7), ISI, Kolata, 5-7 th Mar, 7, pp.8-86, Published by IEEE CS [3] J.Hageauer ad P. Hoeher," A Viterbi algorithm with soft decisio outputs ad its 874