Iterative Equalizatioflecoding of TCM for Frequency-Selective Fading Channels *

Transcription

1 Iterative Equalizatioflecoding of TCM for Frequency-Selective Fading Channels * Achilleas Anastasopoulos and Keith M. Chugg Communication Sciences Institute Electrical Engineering-Systems Dept. University of Southern California Los Angeles, CA Voice: (213) FAX: (213) achillea@zeus.usc.edu,chugg@milly.usc.edu Abstract The severity of frequency-selective fading channels necessitates the combining of multiple diversity sources to achieve acceptable performance. Traditional techniques often perform the combining of different sources of diversity separately which may result in a significant perjformance degradation (uncoded systems, for example, outperform TCM codes when used in an interleaved frequencyselective fading channel with separate decoding and equalization). Recently, it was demonstrated that soft decision equalization techniques are necessary and sufjicient for the application of TCM techniques over such channels. This enabling feature was obtained with relatively simple, lowlatency, non-iterative algorithms. In this paper we investigate the applicability of more complex iterative equalizatioddecoding algorithms. Several conjigurations are examined using simulation and further improvements are demonstrated. 1. Introduction Transmission channels in mobile radio systems are often characterized as time-varying fading multipath channels corrupted by additive white Gaussian noise (AWGN). In order to cope with these severe channel conditions, three main diversity mechanisms are employed: (i) time diversity achieved by interleaving and error-correcting codes, (ii) frequency diversity achieved by the frequency-selective fading, slow frequency hopping, or multicarrier transmission and (iii) spatial diversity achieved via antenna arrays. The need for diversity can be appreciated from the fact that over a Rayleigh fading channel without diversity the error rate decays as (SNR)-l, where SNR is the average signal-tonoise ratio. However, with N orders of diversity the error *This research has been funded in part by the Integrated Media Systems Center, a National Science Foundation Engineering Research Center, with additional support from the Annenberg Center for Communication at the University of Southem California Trade and Commerce Agency $ IEEE 177 rate decays roughly as (SNR)-N [ 11. Furthermore, it has been shown that as N + CO, the performance of the system approaches that of the AWGN channel with the same average SNR [2]. Traditional diversity combining techniques often perform the combining of different sources of diversity separately. For example a Viterbi Equalizer (VE) [3] is used to combine the frequency diversity from the channel, providing hard estimates of the coded symbols; these hard estimates are deinterleaved and passed to a Viterbi Decoder (VD) that combines the time diversity of the code. The limitations of such separate combining are evident when coded modulation is used. Specifically, TCM codes fail to provide coding gain when they operate on an interleaved frequencyselective fading channel [4]. A non-interleaved system may be used to overcome this, by allowing the use of ajoint Maximum Likelihood Sequence Detection (MLSD) receiver, but this approach suffers from a significant reduction of time diversity [4]. This is attributed to the fact that the outer code in conjunction with the interleaver constitute a powerful long overall code with built in time diversity. On the other hand, due to the presense of interleaving, the overall model constitutes a Hidden Markov Field, for which no efficient recursive algorithm exists for MLSD (or MAP) detection. This fact was pointed out in [5], where near-optimal data detection for an analogous 2D IS1 channel was considered. Early attempts to implement some form of joint processing were centered around joint Decision Feedback Equalization (DFE), deinterleaving and decoding [6]. More recently, techniques that calculate A- Posteriori Probabilities (APP s) of the coded symbols conditioned on a single column observation, which are then deinterleaved and used by the outer VD, have been derived [7,8,9, 101. These aproaches are based on several soft decision algorithm developed in the literature ([ l l] and references therein). Among these, the Minimum Sequence Metric (MSM) algorithm [I21 which can be interpreted as a more efficient implementation of the SSA algorithm of [ 1 I], is an attractive candidate for practical implementation, both

2 for its relatively low complexity and its moderate performance degradation with respect to the APP scheme. These algorithm when applied in the particular problem of TCM in interleaved /frequency-selective fading channels result in significant performance gains with respect to the traditional techniques [7,9, IO]. In an effort to further approach the optimumperformance as dictated by MLSD theory, a technique first applied in the decoding of Turbo Codes [ 131 is considered. In particular, by viewing the overall system as a serial concatenation of two Finite State Machines (FSM) through an interleaver, the iterative decoding procedure proposed in [ 141 can be implemented. In this paper we investigate the performance enhancement of iterative techniques. Several receiver configurations are examined and the effect of different parameters in performance is quantified. Consideration of the issue of channel estimation, which is not addressed in this paper (refer to [15,16] forjoint channel estimation and soft decision algorithms), is expected to also affect the design trade-offs for practical systems. The remainder of the paper is structured as follows: Section 2 contains a description of the transmitted signal and channel model. The iterative receiver is presented in Section 3, followed by simulation results in Section 4. Concluding remarks are made in Section Basic Assumptions and Signal Model Figure 1. Overall system block diagram Consider the communication system depicted in Figure 1, consisting of a memoryless source which outputs symbols b, from the alphabet { BI,..., BN~}, with a bit rate Rb (bitdsec). The source symbol stream is encoded by a rate R, convolutional code with memory length L, bits. The code rate R, is such that each output symbol is mapped on a finite symbol alphabet { U1,..., U,,}, resulting in a sequence with rate R, = Rb/(R,log, NU) (symbols/sec). The trellis-coded symbols are interleaved using a size J x K block interleaver, pulse-shaped, transmitted through a frequency selective fading channel, employing in general Ndivth order diversity, and are observed in white noise. At the receiver side the sequence is match-filtered with the known pulse shapelchannel and sampled at the symbol rate. These samples provide sufficient statistics for further processing, when the channel is assumed known [17]. Issues concerning optimal pre-processing in the case of unknown channel will not be addressed (refer to [ 181 for a complete treatment of this issue); rather, we illustrate the concepts here based on such symbol-spaced sampling. The equivalent discrete-time model for the above scenario consists of the convolution of the coded symbols with an (L + 1)-tap FIR, time varying channel: L 2; = c + n: i = 1, ' '., NdiV (I) n=o where hi,n is the nth tap of the channel at time IC, for the ith diversity branch, Uk is the coded symbol and ni is a white complex Gaussian noise with E{ Ink 12} = No. The channel and noise on each diversity branch are assumed to be independent and identically distributed. The source symbols and the channel taps are normalized such that average signal energy per diversity branch is E, /Ndiv. A frequency selective fading model which is widely assumed is the wide sense stationary, uncorrelated scatter (WSSUS) model of Bello [19]. Under the WSSUS assumption the channel taps are modeled as uncorrelated complex Gaussian processes with correlation given by [20]. d(k) = JO(2Tvd IC) (2) where Vd = fdt, is the normalized Doppler spread of the channel. In reality the physical channel taps are independently fading and should be modeled as fractionally spaced. This results in overall channel coefficients that are correlated [4,21]. However, we adopt this simplified model to improve simulation efficiency. Regarding the interleaver design, the depth is chosen such that successive coded symbols, which are actually transmitted J symbols apart, are independently faded, while the width of the interleaver li' is chosen to separate any LD + l successive symbols as far as possible, where LD is the decoding depth of the code [4]. These design constraints are met with J > 1/(2Vd) and I( > 7Lc [ Iterative Soft-Decisions Equalization and Decoding Two separate tasks are performed at the receiver. The first - referred to as inner equalization - consists of combining the diversity provided by the uncorrelated branches with the implicit diversity provided by the frequency selectivity of the channel, while the second - referred to as outer decoding - involves the combining of the information made available by the previous task, with the time diversity of the code. Three different receiver structures that carry out the 178

3 Final Final Final A Decisions 4 bn - n MLSE f- Deinterleaver Viterbi viterbi Equalizer APP or MSM and no extra gain can be harnessed. Although the two softdecision modules have similar structures, the outer soft algorithm differs in two main aspects: (i) it is designed to handle TCM codes that are realized with feedback systematic encoders, as well as code trellises having parallel transitions [22] and (ii) it outputs soft-decisions on both the coded symbols U,, and the source symbols b, (the latter are hard quantized to provide the final estimates when the iterative procedure has converged). The choice of the particular soft algorithm depends on performance, complexity and latency requirements for the specific application. 4. Simulation Results Figure 2. Block Diagram for receiver types A, B and C above tasks are considered in Figure 2. Receiver A consists of a concatenation of a VE with a VD through a deinterleaver. The inner equalizer passes hard decision estimates on the symbols Uk to the outer decoder which produces the final decisions for the source symbols b,. In receiver B, a more sophisticated equalization algorithm replaces the VE, providing appropriate soft estimates on the coded symbols Uk. The recently introduced full-record (FR) or fixed-delay (FD) (i.e., type I and I1 [ 111) APP and MSM algorithms [ which are equivalent to the OSA and SSA algorithms [ 1 I] respectively, with approximately Nu times less complexity - are used for the inner equalization. An obvious drawback of the two previous receiver structures is that the inner equalizer treats the coded symbols Uk as i.i.d., disregarding their inherent structure due to the outer encoder. A possible remedy is to build an MLSD receiver based on the concatenated FSM (outer convolutional encoder/interleaver/inner IS1 channel). Although this is a valid approach, the presense of interleaving results in tremendous complexity due to the fact that the overall channel spans in length almost as long as the total interleaver size. In order to approach the optimum performance, while maintaining reasonable complexity, the type B receiver can be modified in the following manner: the outer VD is replaced by a Soft- Decision algorithm (APP or MSM) providing soft-decisions on both the source symbols b, as well as the coded symbols U,. The APPs of the coded symbols are then fed back - through the interleaver - to the inner equalizer, supplying it with the additional information of the outer code structure. These APPs on the symbols uk are used by the inner softequalizer as a-priori information on { u k} to perform another repetition, resulting in an iterative scheme which terminates as soon as the decisions on the input symbols b, mature Extensive simulation results have been obtained for a variety of system configurations in order to quantify the performance of the iterative receiver. For all the systems, a 500 x 50 interleaver and a 3-tap equal-power static or fading channel was used and the rate 2/3-8PSKTCM codes are the ones described in [22, pp The simulation results showed that the performance of the APP and MSM algorithms is almost identical. Based on the above observations, only the MSM results are presented. * 32-rlate ode a 0 P Eb/No (db) Figure 3. Iterative receiver (type C, with FR- MSM for equalizatioddecoding) in 3-tap static channel In Figures 3 and 4 the performance of receiver C is plotted for two TCM codes with 1, 2 and 3 iterations, in a 3- tap static and fading channel respectively. A FR-MSM algorithm is running both for the inner as well as the outer detector. We conclude from these Figures that the performance gain from iterative decoding is moderate and most of the gain is achieved in the second iteration. In particular, the difference between the first and last iteration for the static channel is approximately 1.5dB and IdB for the 4-state and 179

4 j - - Iteration Iteration sratecode IO II Eb/No (db) 1 1,.-. J Figure 4. Iterative receiver (type C, with FR- MSM for equalizatioddecoding) in 3-tap fading channel 12 ference for all 4 receiver configurations. The 32-state code, though, experiences a performance loss of approximately 0.5 to 1 db when FD-MSM is used as a decoder. We conclude that as long as the outer decoder is Full Record (or Fixed Delay with sufficiently large delay) the iterative receiver does not experience any loss in performance. In an attempt to gain more insight in the behavior of the iterative algorithm, and in particular its rapid convergence properties, we examined the quality of the soft decisions on the coded symbols Uk at the output of the equalizer and decoder at each iteration. In figure 6, the average entropy of the APP is plotted as a function of the iteration number for different SNR values for the 32-state code. The fact that the lo i 32-state code respectively. For the fading channel the differences are further suppressed (IdB and 0.5dB for the 4-state and 32-state code respectively). IO" I I 0.5 I Iteration # Figure 6. APP entropy for receiver C. (iteration numbers that are odd multiples of 0.5 denote the output of the inner equalizer, while whole numbers correspond to the decoder output.) 7 a 9 10 II I2 ELVNo (db) Figure 5. Iterative receiver (type C, with FR- MSM or FD-MSM for equalizatioddecoding) in 3-tap fading channel Figure 5 shows the performance of receiver C with the four possible combinations resulting from the choice of FR- MSM and FD-MSM for the inner and outer detectors. The performance of these receivers after the second iteration is shown. A delay value of twice the FSM memory is used for the cases of FD-MSM algorithms. The results indicate that for the 4-state code there is no notable performance dif- APP entropy is almost zero at the output of the equalizer in the second repetition results in no soft-decision gain for the decoder. In other words, after the second repetition, only hard decisions are circulating between the inner and outer stages of the receiver. This can be contrasted to the serial and parallel concatenated coding case, where as many as 10 or 15 iterations are required for the iterative algorithm to converge [13]. In addition, it can be contrasted with the results obtained for the case of equivalent iterative process in the presense of 2D-ISIr.51, where a gain of almost 3dB is obtained at the third iteration (with respect to the first iteration) for a severe 3 x 3 channel. This behavior can be attributed to two possible reasons: (i) the performance of receiver B is very close to the optimal MLSD performance, so any further improvement (realized with iterative processing) is insignificant, or (ii) the iterative algorithm does not converge to the MLSD solution. 180

5 5. Conclusions The main conclusion that can be drawn form the results in the previous section is that, although soft-decision equalization offers a great advantage over traditional receiver structures, iterative equalization and decoding provides a relatively small further improvement. In addition, this moderate advantage can be realized with only two iterations. The results presented in this paper lead directly to many interesting open problems. For example, the reason for iterative decoding being so beneficial in the cases of turbo coding and 2D-ISI, while it offers a small improvement in the present case, is unclear. References [I] J. G. Proakis, Digital Communications, 3rd ed., McGraw Hill, New York, [2] J. Ventura-Traveset, G. Caire, E. Biglieri, and G. Tanico, Impact of Diversity Reception on Fading Channels with Coded Modulation. Part I: Coherent Detection, IEEE Trans. Commun., May 1997, pp [3] G. D. Fomey, The Viterbi Algorithm, Proc. of IEEE, vol. 61, March 1973, pp [4] G. L. Stuber, Principles of Mobile Communication, Kluwer Academic Press, [SI X. Chen and K. M. Chugg, Near-Optimal Data Detection for Two-Dimensional ISVAWGN Channels using Concatenated Modeling and Iterative Algorithms Algorithms, ICC 98, submitted August [6] M. V. Eyuboglu, Detection of Coded Modulation Signals on Linear, Severely Distorted Channels Using Decision Feedback Noise Prediction with Interleaving, IEEE Trans. Communications, Jan. 1988, pp [7] P. Hoeher, TCM on Frequency-Selective Fading Channels: a Comparison of Soft-Output Probabilistic Equalizers, Proc. Globecom 1990, Nov. 1990, pp [8] R. Mehlan, J. Wittkopp, and H. Meyr, Soft Output Equalization and Trellis Coded Modulation for Severe Frequency- Selective Fading Channels, Proc. ICC 1992, 1992, pp [9] A. Anastasopoulos and K. M. Chugg, TCM for Frequency- Selective, Interleaved Fading Channels Using Joint Diversity Combining, ZCC 1998, submitted August [lo] A. Anastasopoulos and K. M. Chugg, Iterative and Non- Iterative EqualizatiodDecoding of TCM for Mobile Radio Systems, IEEE J. Selected Areas in Commun., submitted September [ 111 Y. Li, B. Vucetic and Y. Sato, Optimum Soft-Output Detection for Channels with Intersymbol Interference, IEEE Trans. Information Theory, vol. IT-41, No. 3, May 1995, pp [12] K. M. Chugg and X. Chen, Efficient A-Posteriori Probability (APP) and Minimum Sequence Metric (MSM) Algorithms, IEEE Trans. Communications, submitted August [13] C. Berrou, A. Glavieux, and P. Thitimajshima, Near Shannon Limit Error-Correcting Coding and Decoding: Turbo- Codes, Proceedings if ICC 93, Geneva, Switzerland, pp ,May [ 141 S. Benedetto, G. Montorsi, D. Divsalar and E Pollara, A soft-input soft-output maximum a posteriori (MAP) module to decode parallel and serial concatenated codes, tech. rep., TDA Progress Report , Nov [ 151 A. Anastasopoulos and A. Polydoros, Soft-Decisions Per- Survivor Processing for Mobile Fading Channels, Proc. of VTC 97, May 1997, pp [16] A. Anastasopoulos and A. Polydoros, Adaptive Soft- Decision Algorithms for Mobile Fading Channels, European Trans. on Communications, submitted May [ 171 G. D. Fomey, Maximum-Likelihood Sequence Estimation of Digital Sequences in the Presence of Intersymbol Interference, IEEE Trans. Information Theory, vol.it- 18, May 1972, pp [18] K. M. Chugg and A. Polydoros, MLSE for an Unknown Channel - Part I: Optimality Considerations, IEEE Trans. Communications, vol. 44, July 1996, pp [19] P. A. Bello, Characterization of Randomly Time-Variant Linear Channels, IEEE Trans. Communication Systems, vol. 11, 1963,~~ [20] R. Clarke, A Statistical Theory of Mobile Radio Reception, Bell System Tech. J., vol. 47, 1968, pp [21] K. M. Chugg and A. Polydoros, MLSE for an Unknown Channel - Part 11: Tracking Performance, IEEE Trans. Communications, vol. 44, August 1996, pp [22] S. H. Jamali, and T. Le-Ngoc, Coded-Modulation Techniques for Fading Channels, Kluwer Academic Press, Boston, [23] R. Steele, Mobile Radio Communications, Pentech Press, London,