EXIT CHART ANALYSIS APPLIED TO ADAPTIVE TURBO EQUALIZATION

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1 EXIT CHART ANALYSIS APPLIED TO ADAPTIVE TURBO EQUALIZATION Roald Otes UiK Uiversity Graduate Ceter PO box 7, N-7 Kjeller, Norway Michael Tüchler Muich Uiversity of Techology Arcisstr., D-89 Muich, Germay ABSTRACT I this paper we cosider iterative chael estimatio, equalizatio, ad decodig, or adaptive Turbo equalizatio, as a receiver techology for digital commuicatio systems where the chael imposes time-varyig itersymbol iterferece. We show how a semiaalytical techique called EXIT charts, origially developed by S. te Brik for aalysis of Turbo codes, ca be used to predict the performace of the iterative receiver for such systems. To demostrate the usefuless of the techique, we use EXIT charts to address iterestig questios about adaptive Turbo equalizatio for time-varyig chaels: Which patter of traiig sequeces should be used, what is the performace differece betwee optimal MAP equalizatio ad suboptimal liear equalizatio, ad betwee the case of a kow chael ad the case of a estimated chael, ad what ca be gaied by usig a recursive precoder i cojuctio with the symbol mapper.. INTRODUCTION I this paper, we address the problem of digital commuicatios over a chael imposig time-varyig itersymbol iterferece (ISI) as well as additive white Gaussia oise. We assume a trasmitter cosistig of a errorcorrectig code (ECC), a iterleaver shufflig the code bits, a symbol mapper mappig the iterleaved code bits oto a M-ary sigal costellatio (PSK or QAM), ad modulatio oto a sigle carrier frequecy. A block diagram of such a system is show i Fig. icludig the receiver, which is described below. High frequecy (HF, 3-3 MHz) or mobile commuicatios systems (GSM or EDGE) are examples of such system structures. For the give system ad chael model, covetioal receivers perform adaptive equalizatio icludig (soft) symbol demappig, deiterleave the (soft) iformatio o the code bits (e.g. log-likelihood ratios, LLRs, []) from the demapper, ad perform (soft-i) decodig of the ECC. Commoly used equalizers for such receivers are the trellis-based maximum likelihood sequece estimator (MLSE) [] or the decisio feedback equalizer (DFE) [3]. A optimal receiver would joitly perform chael estimatio, equalizatio, ad decodig, which is a extremely complicated problem especially whe a iterleaver is preset. Recet work attacks this problem usig iterative receiver algorithms usually referred to as Turbo equalizatio or iterative equalizatio ad decodig. This priciple to approach the performace of joit equalizatio ad decodig was first itroduced i [4] for a kow chael impulse respose (CIR) as a extesio of iterative decodig of cocateated codes. I a receiver employig Turbo equalizatio, the equalizer ad decoder are soft-i soft-out () modules, which were iitially based o the maximum a posteriori probability (MAP) [5] or MLSE algorithm [] (see for example [4, 6]). Due to the large computatioal complexity ad memory requiremets of these algorithms, suboptimal equalizers have bee itroduced later, e.g., based o soft ISI cacellatio ad liear filters [7 ]. Whe the CIR is ukow ad timevaryig, iterative equalizatio ad decodig ca be exteded to iterative equalizatio, estimatio, ad decodig. However, i existig work [ 3], the ecessary joit equalizer ad estimator is very complex. We attempt to reduce the computatioal burde of these schemes by separatig out equalizer ad estimator ad carefully perform iterative equalizatio, estimatio, ad decodig over three receiver modules. We cosider a separate estimator providig estimates of the CIR ad a separate equalizer usig the CIR estimates to obtai estimates of the trasmitted, see Fig.. The advatage of this costructio is a high flexibility i choosig estimatio ad equalizatio algorithms. Joit equalizatio, estimatio, ad decodig is performed by iteratig equalizatio ad decodig tasks, where the estimator improves its CIR estimates by icorporatig fed back iformatio from the decoder by trasformig the code bit LLRs L d (c k ) ito soft traiig symbols ȳ = E{y } (the expectatio give the LLRs L d (c k ) about the ukow trasmitted symbols y ). The improvemet of the CIR estimates over the iteratios has bee aalyzed i [4 6]. Moreover, this work shows that icorporatig E{y } for estimatio is better that usig hard-decisios ŷ derived from the L d (c k ). Clearly, better CIR estimates improve the performace of the equalizer ad fially that of the etire system performace. For liear equalizers, a alterative to separate estimatio ad equalizatio is to update the equalizer coefficiets directly usig commo algorithms for adaptive equalizatio, icorporatig soft iformatio fed back from the decoder [7]. I this case, the coefficiet update rule exteds the algorithm to compute a approximate versio of the optimal liear equalizer i [] to the case that the CIR is time-varyig. We ote that re-

2 a m Ecoder Traiig sequece t e k d k Iterleaver Deiterleaver w y c k Iterleaver ISI chael c x Time-varyig Adaptive k Symbol z mapper h equalizer e k d k decoder ^am estimate Figure : Block diagram of a commuicatios system employig adaptive Turbo equalizatio i the receiver. z d k t equalizer (ay type) LLR symbol coversio h ^- y ^ w x h ^- Soft iput chael estimatio e k Figure : Adaptive equalizer usig separate (soft-i) chael estimatio ad equalizatio. lated algorithms for Turbo equalizatio give ukow, time-varyig chael characteristics have bee proposed, e.g., for HF commuicatios [8 ], for GSM [], for GPRS [, 3], ad for EDGE [7]. Aalytical expressios for the performace of a receiver employig Turbo equalizatio are ot kow, ad the task of derivig aalytical expressios is eve more complicated whe chael estimatio is icluded i the iterative process. Therefore, the performace of Turbo equalizatio for a particular system is usually studied via time-cosumig simulatios. Alteratively, a semiaalytical approach called extrisic iformatio trasfer charts (EXIT charts) [4] ca be used, which obtais mutual iformatio trasfer fuctios separately for each module i the receiver via off-lie simulatios. These trasfer fuctio map a parameter of the distributio of the LLRs eterig a module, the mutual iformatio to the correspodig iput o the trasmitter side, to the parameter of the distributio of the LLRs exitig a module. Because the output of oe module is iput to the other module ad vice versa, the two trasfer fuctio ca be draw i a two-dimesioal chart ad the performace of the receiver is give by a trajectory betwee the two curves [4]. The beefits of this method are to predict the system performace without actually simulatig trasmissio ad to gai isight about the covergece of the iterative procedure. The beefit of usig EXIT charts to aalyze the performace of Turbo equalizatio was preseted i [] for the case of a kow, time-ivariat chael. I this paper, we show how EXIT charts ca be applied to receivers performig iterative equalizatio, estimatio, ad decodig, which is referred to as adaptive Turbo equalizatio i the sequel. e. SYSTEM MODEL Cosider the system model i Fig.. A block of bits a m is ecoded with a rate-r c covolutioal code to N it code bits c k {+, } icludig trellis termiatio. The bits c k are iterleaved to c k, which are mapped to N it /Q symbols y from a Q -ary sigal costellatio. The symbols y are multiplexed with N t traiig symbols t kow to the receiver to form a block of N block = N it /Q + N t symbols x. The overall code rate icludig codig ad traiig overhead is R = R c N it /(N it +QN t ). The x are trasmitted over the time-varyig ISI chael with equivalet discrete-time CIR h =[h, h,m ]. The legth of the CIR is M symbol itervals. The received symbols z are give by z = w + M h,lx l, l= where the w are complex white Gaussia oise samples with variace σw, i.e., their probability desity fuctio (pdf) is p(w)=exp( w /σw)/(πσ w). The (adaptive) equalizer processes the received symbols z, the traiig symbols t, ad the a priori LLRs L d (c k ) fed back from the decoder. Durig first-time equalizatio, all L d (c k ) are zero. The equalizer outputs the LLRs L e (c k ), which, deiterleaved to L e (c k ), are iput to the decoder. The decoder computes estimates â m of the trasmitted bits ad outputs the LLRs L d (c k ), which, iterleaved to L d (c k ), are used as a priori LLRs by the equalizer i the ext iteratio. 3. EXIT CHARTS FOR ADAPTIVE TURBO EQUALIZATON The EXIT chart describes the covergece of the iterative receiver algorithm by ivestigatig the exchage of mutual iformatio betwee the equalizer ad the decoder. Observed is the mutual iformatio = I(L e (c k ); c k ) at the equalizer output ad = I(L d (c k ); c k ) at the decoder output, where the mutual iformatio is defied as I(L; C)= p(l c) p(l c)log p(l +)+p(l ) dl, c {+, } where p(l c) deotes the pdf of the LLRs modelled as outcomes of the radom variable (r.v.) L coditioed o its correspodig trasmitted bit c represeted by the r.v. C. The mutual iformatios ad ca be foud by calculatig the itegral above umerically from histograms of c k L e (c k ) ad c k L d (c k ), respectively. These histograms are used to estimate p(l +) ad p( l ) [4], which

3 a m Ecoder c k c k y Symbol x mapper t N( L L /, ) (a) Decoder Time-varyig ISI chael h e k d k decoder w ~ N(, w ) z Adaptive equalizer ^am estimate e k d k 4 db db db 8 db 6 db E b /N N( L L /, ). (b) Equalizer Figure 3: Simulatio setup for geeratig mutual iformatio trasfer fuctios.. Memory Memory Bit error rate Figure 4: Trasfer fuctio (solid lies) from (iput) to (output), ad bit error rate (dashed lies) as fuctio of, of a MAP-based decoder for rate-/ covolutioal codes with memory or 6. ca usually be assumed equal. The rage of ad is [, ], where =or =correspod to o or perfect kowledge about c k give L d (c k ). The evolutio of ad over the iteratios i a real receiver is called the system trajectory of the iterative algorithm. The purpose of the EXIT chart is to predict the system trajectory from mutual iformatio trasfer fuctios, without performig simulatios o the complete iterative receiver. The simulatio setup to geerate the trasfer fuctio of the decoder is show i Fig. 3 (top). Code bits c k are geerated by ecodig a block of idepedet bits a m with the ECC. The iput LLRs L e (c k ) to the decoder are draw idepedetly from a ormal distributio with average value c k σl / ad variace σl. This is a commo distributio of LLRs i a iterative receiver [4], ad a oe-to-oe relatioship exists betwee σl ad. I this paper we use the opti-. Figure 5: Mutual iformatio trasfer fuctio of the liear equalizer receivig CIR estimates from a LMSbased chael estimator. mal MAP algorithm [5] for decodig. A umber of blocks is simulated for differet values of σl i order to estimate the output mutual iformatio ad the bit error rate of the bit estimates â m. The result is show i Fig. 4 for two differet rate-/ covolutioal codes: a memory- code with geerator polyomials +D ad +D+D, ad a memory-6 code with geerator polyomials +D+D 3 +D 4 +D 6 ad +D 3 +D 4 +D 5 +D 6. Note that the stroger code (memory-6) has a steeper trasfer fuctio tha the weaker code, ad therefore requires a lower iput i order to produce high-quality output LLRs ad low bit error rates. A icrease/decrease i code rate R c would move all curves i Fig. 4 towards the right/left. The trasfer fuctio of the equalizer is similarly geerated usig the setup show i Fig. 3 (bottom). A block of N it idepedet bits c k is passed through all blocks i Fig. from the symbol mapper to the received symbols z. The ECC geeratig the c k is omitted ad depedecies withi the iterleaved bits c k, which exist i a real system, are eglected. I a parallel path, the c k are multiplied by idepedet samples from a ormal distributio with average value σl / ad variace σ L, to form the iput LLRs L d (c k ). Sice the adaptive equalizer processes also received symbols z, its trasfer fuctio will deped o E b /N i cotrast to the decoder. Fig. 5 shows the trasfer fuctio for the followig example: The block of trasmitted symbols x cosists of 48 iitial kow traiig symbols, followed by 56, 3 traiig, 56, 3 traiig symbols ad so o, util the ed of the block. The sigal costellatio is Gray-coded 8-PSK, ad the CIR is of the form h =[h, h,5 ]. The two ozero taps h, ad h,5 are draw accordig to idepedet Rayleigh fadig with a Gaussia Doppler spectrum [5] havig a σ Doppler spread (fadig rate) of /(4T s ), where T s is the symbol period. The adap-

4 Bit error rate Figure 6: Left: EXIT chart for the example i the text at db E b /N. The predicted system trajectory is show as dashed vertical ad horizotal lies, ad the actual system trajectory obtaied from simulatios o the complete iterative receiver is show as solid vertical ad horizotal lies. Right: Bit error rate out of the decoder as a fuctio of from Fig. 4 (with flipped axes), showig the predicted bit error rate after covergece of the iterative receiver. Bit error rate E /N b Figure 7: Simulated bit error rate as fuctio of E b /N for the example i the text, after (upper lie) util 6 (bottom lie) iteratios. tive equalizer uses separate soft-i chael estimatio ad equalizatio as show i Fig., where the least mea squares (LMS) algorithm [3] is used to geerate a time-varyig chael estimate ad the MMSE-optimal liear equalizer of [9, 5] is used for equalizatio. The step size of the LMS algorithm is.. The EXIT chart i Fig. 6 (left) combies the equalizer trasfer fuctio at db E b /N ad the decoder trasfer fuctio for the memory- code. Sice the output LLRs from the equalizer are iput to the decoder ad vice versa, both trasfer fuctios are draw i the same plot with the axes beig flipped for the decoder trasfer fuctio. The system trajectory of the iterative receiver ca be predicted by represetig each equalizatio task as a vertical lie ad each decodig task as a horizotal lie, boucig betwee the two trasfer fuctios (the dashed lies i Fig. 6). The trajectory starts i the (,) poit ad approaches a fixed poit, which is the leftmost crossig poit of the two trasfer fuctios. Each iput mutual iformatio to the decoder is associated with a bit error rate as show i Fig. 4 ad (with flipped axes) i Fig. 6 (right). We fid from the EXIT chart i Fig. 6 that at db E b /N, iterative decodig coverges after 4 times equalizatio ad decodig to a bit error rate of 3 4 correspodig to = i the fixed poit. To verify the predicted performace, we track the trajectory of a real system by calculatig the mutual iformatio at the output of equalizer ad decoder after each iteratio. We use a iterleaver of legth N it = The real system trajectory at db E b /N is show as solid lies i Fig. 6. We see that the actual performace is close to the predicted performace for the first few iteratios, but they depart at later iteratios. We fid that i reality, equalizatio ad decodig tasks must be performed 5 or 6 times i order for the receiver to coverge, i.e. oe or two times more tha the predictio of 4. The simulated bit error rate is show i Fig. 7. At db E b /N the bit error rate after covergece is 7 4, which is a bit more that the predicted bit error rate. The EXIT chart aalysis assumes that all LLRs are idepedet, ad is therefore i theory valid oly for a ifiite iterleaver legth N it. This is the reaso for the slight differeces betwee predicted ad actual performace i the example discussed above, ad we fid that the differeces become smaller whe N it is icreased. For time-varyig chaels, N it should be large compared to the fadig rate, i.e., there should be several fades withi each iterleaver block, i order to esure that all blocks have the same statistical properties such that the EXIT chart aalysis becomes exact. For smaller N it, the performace predicted by the EXIT chart ca be viewed

5 Memory 6 code Memory code 5. Figure 8: EXIT chart for differet frame patters, havig 64, 8, 56, ad 5 symbols betwee each traiig sequece of legth 3 symbols. E b /N =db for all curves. as a practical boud o the achievable performace. 4. USING THE EXIT CHART Now that we have demostrated by a example how EXIT charts ca be applied to adaptive Turbo equalizatio, we are ready to discuss their beefits, some of which are. The EXIT chart is a ice way to visualize what happes i a iterative receiver, which may otherwise seem magic to people ot familiar to the subject.. We ca ivestigate the effect of modifyig oe of the two algorithms without performig simulatios o the complete iterative receiver. 3. We ca see how well the ECC/decoder ad chael/equalizer ( outer ad ier code) are fitted to each other. 4. The trasfer fuctio of the equalizer ca be used to calculate the maximum achievable code rate for specific chael coditios. The first poit was demostrated thoroughly by the example i the previous sectio. To demostrate the remaiig poits, we give some further examples. I Fig. 8 we show the effect of chagig the patter of traiig symbols. Whe the iterval betwee traiig sequeces is icreased, the overall code rate R will icrease, causig the curves to move upward at high. However, the chael estimate will be poorer whe is low because it fails i trackig the chael variatios betwee traiig sequeces, ad the curves with the most traiig have the highest i the left part of the EXIT chart. Fidig the optimal traiig patter is therefore a trade-off betwee iitial covergece at low E b /N ad error rates after covergece at high E b /N, ad for this example 8 or 56 symbols betwee each traiig sequece seem to be MAP, estimated chael LE, estimated chael. MAP, kow chael LE, kow chael MAP, kow precoded ch.. Figure 9: Equalizer trasfer fuctio for differet scearios: Liear equalizatio (LE) vs optimal MAP equalizatio, kow vs estimated chael, ad a system usig a recursive precoder, at E b /N =8dB. good choices. The trasfer fuctio for the two rate-/ codes discussed earlier are also show i this EXIT chart. Note that the memory- code has a trasfer fuctio almost parallel to most equalizer trasfer fuctios, while the memory-6 code has a kee at low which may itroduce a fixed poit at low whe E b /N is decreased compared to this plot. Therefore, a strog outer code is ot always a good choice whe usig Turbo equalizatio i the receiver: Whe E b /N is decreased, the stroger code will fail before the weaker code i terms of iitial covergece because the weaker code is better matched to the equalizer. It is possible to costruct a code matched to a give equalizer trasfer fuctio accordig to some optimizatio criterio, e.g., to obtai values of ad as high as possible after a certai umber of iteratios [6]. I Fig. 9 we have compared the equalizer trasfer fuctio for differet scearios. The example is the same as i Sec. 3, except that the chael legth has bee reduced to M =3, h =[h, h, ], to make MAP equalizatio feasible. We see that at =, MAP equalizatio has the same performace as the much simpler approach of soft ISI cacellatio combied with liear equalizatio (LE), for the case of a kow chael as well as for a estimated chael. Thus, the performace usig LE is similar to the performace usig MAP equalizatio, after covergece of the iterative receiver. The major differeces betwee LE ad MAP equalizatio are therefore the umber of iteratios eeded before covergece, ad the iitial covergece at low E b /N. We also fid that the trasfer fuctios whe the chael is kow at all times are much flatter tha whe the chael is estimated, idicatig that there is much more to gai from doig iteratios whe the chael estimate is improved over the iteratios. At =, the curves for a kow ad for a estimated chael are

6 quite close. We have i Fig. 9 also ivestigated the use of a recursive precoder i cojuctio with the symbol mapper. I [7,8] is show that Turbo equalizatio ca have performace similar to Turbo-coded systems whe a recursive precoder is used: A Turbo cliff at a certai E b /N, above which the bit error rate is very small (egligible compared to the error rates i Fig. 7)). Here we have used the simple precoder defied by s,i = s,i c 3+i, i {,,}, s,i {+, }, where three precoded bits s,i are mapped oto each symbol y usig a 8-PSK costellatio. Note that the trasfer fuctio goes to the (,) poit whe usig a precoder (here, the chael is kow). This explais the Turbo cliff behavior: Whe the iterative decodig does coverge, it coverges to the (,) poit where error rates are egligible. The price to pay compared to the oprecoded case is that is decreased at low, such that the E b /N required for covergece is icreased if the same ECC is used. Also, o-precoded Turbo equalizatio ca be used as a receiver techology i preset-day commuicatio systems desiged without Turbo equalizatio i mid, whereas the itroductio of recursive precodig requires chages to the trasmitter. Future work i precoded Turbo equalizatio icludes complexity reductio by derivig algorithms based i liear filters, ad the iclusio of chael estimatio. It ca be show [9, 3] that the maximum rate of a outer code which stays udereath the trasfer fuctio of the equalizer, such that the iterative receiver ca coverge, is equal to the area uder that trasfer fuctio. A iterestig poit to ote is that the area uder the curve for a precoded system is exactly the same as the area uder the curve for the correspodig o-precoded system [6], such that the same overall rate ca be achieved for both systems if the ECC is optimized. 5. REFERENCES [] J. Hageauer ad P. Hoeher. A Viterbi algorithm with softdecisio outputs ad its applicatios. I Proc. IEEE Global Telecomm. Cof., pages , 989. 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