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1 Iterative Equalization and Decoding in Mobile Communications Systems Gerhard Bauch, Houman Khorram and Joachim Hagenauer Department of Communications Engineering (LNT) Technical University of Munich WWW: Abstract In iterative equalization and decoding the \turbo{principle" is used for iterative detection of coded data transmitted over a frequency selective channel. We view channel encoder and ISI{channel as a serially concatenated coding scheme and use the \Symbol by Symbol MA { Algorithm" for both equalization and decoding. erforming some iterations like in turbo{decoding can improve the bit error rate considerably. Furthermore, we present some methods to reduce the average number of iterations needed for the same BER performance by determining when further iterations achieve no improvement. I. Introduction The so called \turbo" principle rst used in [] for iterative decoding of parallel concatenated codes can be used in a wide variety of receiver detection and decoder tasks. The basic idea is to use a maximum a posteriori (MA) detector, which is able to accept not only channel values but also a priori information about the symbols to be detected. This a priori information can be obtained in various ways, i.e. from the feedback of an outer or parallel decoder. This is the \turbo" component of the iterative detection scheme. For binary symbols the a priori information is used as a log{likelihood ratio: Output{Viterbi{Algorithm" (SOVA) which delivers soft information for the coded bits. Therefore, we call the outer decoder a COD-MA or a COD-SOVA. The extrinsic information of the outer decoding is the information about the current bit stemming only from the other bits connected by the parity constraint. It is interleaved and used as the a priori information for the MA{equalizer. This iterative equalization/decoding scheme is a serial \Turbo" scheme similar to the parallel scheme used in [] and [5]. Finally, we will present methods to reduce complexity by stopping the iterations when we recognize that the decoding result of a block cannot be improved by further iterations. II. rinciple of Iterative ("Turbo") Equalization and Decoding A mobile radio channel with intersymbol interference (ISI) can be regarded as a time{varying convolutional code with complex valued coded symbols, given by the propagation conditions (Figure ). If we know the co- T T T L(x) = log (x = +) (x =?) () We consider coded transmission in a mobile system. A mobile multipath channel including the transmit and receive lters can be modelled as tapped delay line digital system. In the new scheme, rst discussed in [2], we view the channel as an inner encoder of a concatenated scheme and replace the ML{equalizer for the channel by an optimum\symbol-by-symbol Maximum A osteriori (MA)" equalizer/decoder [3], [4]. Furthermore, the equalizer/decoder must be able to accept a priori information for the coded bits which are transmitted over the channel. It also delivers soft values and therefore is called a soft{in/soft{out equalizer. The outer Viterbi decoder is replaced by a modied MA Symbol-by-Symbol decoder or \Soft{ Fig. : Time{discrete channel model ecients g i (t) of the time{discrete channel model, we can decode the channel by the means of maximum a posteriori (MA) symbol estimation or sequence estimation. From this point of view channel encoder and ISI{ channel form a serially concatenated scheme which can be decoded by an iterative algorithm. The performance of serially concatenated coding systems can be improved by using soft{in/soft{out decoders (Figure 2). However, in a \turbo scheme" it is even necessary to use

2 soft{in/soft{out decoders that besides the channel values accept further a priori information about the bits to be decoded and deliver soft values of the decoded bits. In iterative equalization it is also necessary to obtain soft values about the coded bits at the output of the channel decoder. It is useful to compute soft values in the form of log{likelihood ratios (L{values) as dened in (). input log - likelihoods a priori values for information bits channel values for code bits L(^x) Le(^x) a posteriori values extrinsic values for code bits 6 6for code bits L(u) Le(^u) - Soft-In/ - Soft-Out L(~x) or - L(^u) Decoder - Lcy or y output log - likelihoods extrinsic values for information bits a posteriori values for information bits Fig. 2: Soft-In/Soft-Out - Decoder which are again passed to the decoder. Repeating this procedure a few times can improve the bit error rate. It is important to feed back only the extrinsic part L D e (^x 0 ) of L D (^x 0 ) because the correlation between the a priori information used by the equalizer and previous decisions of the equalizer should be minimized. Ideally the a priori information would be an independent estimation. For the same reason we have to subtract the a priori information L D e (^x) from the a posteriori L{values L E (^x) at the output of the equalizer before passing them to the decoder. Thus, as the full soft output of the equalizer L E (^x 0 ) consists of channel information, a priori information and extrinsic information, the decoder is supplied only with channel information and extrinsic information. As indicated in Figure 3, an independent a priori information L(u), which i.e. could be known from a source decoder, might be used in the turbo scheme. We can now describe the principle of iterative equalization (see Figure 3): : Interleaver _ : Deinterleaver Fig. 3: Iterative equalization _ MAequalizer COD-MAdecoder At the receive lter output we observe the complex valued sequence y. The equalizer delivers L{values L E (^x) about the coded bits. After deinterleaving, the channel decoder delivers L{values L D (^u) about the information bits and L{values L D (^x 0 ) about the coded bits. The L{values at the output of the decoder consist of an extrinsic and an intrinsic part. The extrinsic part is the incremental information about the current bit obtained through the decoding process from all information available for the other bits. It can be calculated by subtracting bitwise the L{values at the input of the decoder from the corresponding L{values at the output: L(u) L D e (^x 0 k) = L D (^x 0 k)? L E (^x 0 k) (2) The extrinsic information L D e (^x 0 ) is interleaved and fed back to the equalizer where it is used as a priori information in a new decoding attempt constituting the rst iteration. Using this a priori information the equalizer is supposed to deliver less erroneous decisions III. "Symbol-by-Symbol" Maximum A osteriori (MA) Algorithm for Equalization and Channel Decoding We use the "Symbol-by-Symbol" maximum a posteriori algorithm ("Symbol-by-Symbol" MA) [3] for both equalization and channel decoding. In terms of bit error rate the "Symbol-by-Symbol" MA is an optimal block orientated symbol estimator which delivers the a posteriori probabilities p(u k jy) of an information bit u k given the received sequence y. Information from all symbols in a block to be decoded is taken into account. In [4] the "Symbol-by-Symbol" MA was described as a soft{in/soft{out algorithm that accepts L{values at the input and delivers L{values at the output (Figure 2). The L{value L(^u k ) of an information bit can be calculated according to p(s 0 ; s; y) L(^u k ) = log (u k = +jy) (u k =?jy) = log u k =+ u k =? p(s 0 ; s; y) ; (3) where s 0 and s denote the states of the trellis at level k? and k, respectively (Figure 4). The sums in the numerator and in the denominator are to be taken over all existing transitions in the trellis from state s 0 to state s labelled with the information bit u k = + or u k = {, respectively. Using a COD-MA-decoder, we analogously compute for the coded bits: L(^x k; ) = log (x k; = +jy) (x k; =?jy) = log x k; =+ x k; =? p(s 0 ; s; y) p(s 0 ; s; y) (4)

3 states with forward probabilities α k- (s ) s k s states with backward y = (y... y ) probabilities k k, k,n β k (s) u =+ (x...x ) k k, k,n u =- (x...x ) k k, k,n Fig. 4: Trellis buttery The joint probability p(s 0 ; s; y) is the product of three factors, two of which can be calculated by recursive formulae: p(s 0 ; s; y) = p(s 0 ; y j<k) ) (sjs 0 ) p(y k js 0 ; s) k? (s 0 ) k? p(y j>k js) k (s) (5) k (s) = X s 0 k (s 0 ; s) k? (s 0 ) (6) k? (s 0 ) = X s k (s 0 ; s) k (s) (7) The trellis has to be of nite duration. Assuming the zero state as start and end state, we can initialize and as follows: start (0) = and start (s) = 0 8s 6= 0 (8) end (0) = and end (s) = 0 8s 6= 0 (9) For the existing transitions the branch transition probability k (s 0 ; s) can be expressed as the product of a priori probability (u k ) and transition probability p(y k js 0 ; s): k (s 0 ; s) = p(y k js 0 ; s) (u k ) (0) The only dierence in the metrics of the equalizer and of the channel decoder appears in the calculation of the branch transition probability k. For the MA{equalizer for a transition labelled with x can be calculated according to (s 0 ; s) = (s 0 ; s) exp 2 x L(x ) () where (s 0 ; s) = exp MX 2? 2 2 y? g i (t) x?i A : i=0 (2) M denotes the memory of the discrete{time channel model with complex valued time{varying coecients g i (t) (Figure ). The complex valued symbol y was received at time and the equalizer uses an a priori information L(x ). 2 is the variance of the additive white Gaussian noise in inphase and quadrature component. For the decoder we obtain: k (s 0 ; s) = exp NX = 2 L(~x k;) x k; + 2 u k L(u k ) (3) where N denotes the inverse of the code rate. L(u k ) is the a priori information for the information bit u k. L(~x k; ) is the input log-likelihood value L(~x k; ) = L(x k; jy k; ) = log (x k; = +jy k; ) (x k; =?jy k; ) (4) for the code bit x k; given the received symbol y k;. In the serially concatenated scheme we use the output loglikelihoods of the equalizer as an estimation of L(~x k; ). In transmission over an AWGN{channel without ISI, L(~x k; ) would be 4 N Es 0 y k; = L c y k;. For a systematic code the output L(^u k ) of the decoder can be expressed as a sum of a priori information L(u k ), channel information L(~x k; ) and extrinsic information L e (^u k ): L(^u k ) = L(u k )+L(~x k; )+log where e k(s 0 ; s) = exp NX =2 u k =+ u k =?! e k (s0 ; s) k? (s 0 ) k (s) e k (s0 ; s) k? (s 0 ) k (s) L e(^u k ) (5)! 2 L(~x k;) x k; : (6) Analogously the output of the COD{MA{decoder for the coded bits is: L(^x k; ) = L(~x k; )+log where ec 0 B k (s0 ; s) = exp@( x k; =+ x k; =? ec k (s0 ; s) k? (s 0 ) k (s) ec k (s0 ; s) k? (s 0 ) k (s) NX = 6= L e(^x k; ) (7) 2 L(~x k;) x k; ) + 2 u k L(u k ) (8) C A

4 As the ISI{channel is a non-systematic non-binary code we cannot separate channel information and extrinsic information in the output of the equalizer: L(^x ) = L(x ) + log x =+ x =? with (s0 ; s) according to (2). (s0 ; s)? (s 0 ) (s) (s0 ; s)? (s 0 ) (s) (9).0e+00.0e-0.0e-02.0e-03.0e db 0.9 db 0 Iterations Iteration 2 Iterations 3 Iterations Decoded AWGN-channel IV. Simulation Results For channel coding we used a rate /2 recursive systematic convolutional code (RSC{Code) with memory m c = 4. We used BSK modulation and evaluated the performance of iterative equalization for transmission over a time{invariant worst case channel with ve taps as given in Figure 5 [6] and for a time{varying channel model with ve taps of equal average power with fast uncorrelated Rayleigh fading. The channel coecients were supposed to be known. Before transmission the coded bits were interleaved by a random interleaver of size L = 4096 bits. Figure 5 shows the bit error rate for the time{invariant channel model. At a bit error rate of 0?4 the gain after the rst iteration is 2.3 db. The bit error rate can be signicantly improved up to the 5th iteration, where the total gain is 5.4 db at a BER of 0?4. By further iterations we only achieve a marginal gain. For comparison, the BER for MA { decoding of encoded data transmitted over an AWGN{ channel without intersymbol interference is also shown in Figure 5. After the 8th iteration the degradation due to intersymbol interference is only 0.8 db..0e+00.0e-0.0e-02.0e-03.0e-04.0e i Iterations Iteration 2 Iterations 3 Iterations 4 Iterations 5 Iterations 8 Iterations Decoded AWGN-channel 5.6 db Fig. 6: Bit error rate for a channel with 5 taps and fast Rayleigh fading the channel coecients are supposed to be known, the system makes the most of the diversity. Hence, the result of the rst decoding is already very good and an improvement in BER is only achieved for the rst iteration. Nevertheless, we also almost reach the BER achieved for the AWGN channel without intersymbol interference. An improvement in BER is only achieved if, compared to the previous iteration, the equalizer corrects other symbols than those corrected by the decoder in the previous iteration. Finally, equalizer and decoder nd one common path through the trellis resulting in the same bit error rate at the output of the equalizer and at the output of the decoder. V. Reducing Complexity by Applying Stop Criteria for the Iterative Algorithm In iterative equalization not all transmitted data blocks need the same number of iterations to be detected with the minimal number of errors possible. Some blocks will be error free after the rst decoding, others after a few iterations. There will be also blocks where no improvement in bit error rate can be achieved, i.e. if the whole block was transmitted during a deep fade. Therefore, we should be able to notice whether we can improve decoding by further iterations. If not, we can stop iterating and thus reduce complexity. In the following section we will present three stop criteria. Fig. 5: Channel taps and bit error rate for a time{ invariant channel In Figure 6 simulation results for transmission over a channel with fast Rayleigh fading are shown. As A. Cross{Entropy Moher [7] has schown that the cross-entropy is a useful criterion for iterative decoding. In [4] cross{ entropy was introduced as a stop criterion in "turbo" decoding. We now adopt it to "turbo" equalization.

5 The cross{entropy of two distributions (^x) and Q(^x) is given by: Let us dene: log (^x) Q(^x) = X k log (^x k) Q(^x k ) R = L 0 L X k= L(^u k )>0 p(u k =?jy) + LX k= L(^u k )<0 p(u k = +jy) C A < threshold with Q (^x) = LE(i?) (^x) + Le D(i) (^x) L (i) L (i) (^x) = LD(i) (^x) + L E(i) (^x) e L (i) (^x)? L(i) Q (^x) = LE(i) (^x)? L E(i?) (^x) = L E(i) (^x) (i) represents the ith iteration (i=0,, 2,...), that is Le D(i) (^x k ) is the extrinsic information obtained by the decoder after the (i+)-th decoding. For each iteration we calculate L E(i) (^x) and jl (i) Q (^x)j and stop iterating if T (i) = X k L E(i) (^x) 2 < threshold exp(jl (i) Q (^x)j) A suitable value for threshold is T (i = ) 0?3. B. Observing Hard Decisions A very simple stop criterion that works nearly as well as the criterion of cross{entropy is just to observe the hard decisions at the output of the decoder in two successive iterations. If the hard decisions do not change, we stop iterating. However, applying this criterion, we have to execute two iterations with exactly the same decoding result concerning hard decisions before we can stop iterating. for a block size L and stop iterating when R remains under a certain threshold. We found 5 0?5 to be a suitable value for the threshold. D. Simulation Results Figures 7 { 9 show the average number of iterations performed using the stop criteria described above and the resulting bit error rates. We set the maximum number of iterations to 20. Simulation conditions were the same as for the simulation shown in Figure 5. For evaluation we compare each of the criteria with a genius stop criterion that stops the iterations when a block is decoded error free, this means the criterion is genius in terms of error detection..0e+00.0e-0.0e-02.0e-03.0e-04.0e Cross-entropy Genius C. Risk { Function This criterion is not based on measuring the dierence of L{value distributions in successive iterations but takes into account that the L{values at the output of the channel decoder correspond to error probabilities and therefore measure the reliability of the decisions. If the L{values become large, this means that the reliability of the decisions is high, the decisions will not change any more. Therefore, we can stop iterating when the reliability of the decision measured over a block reaches a certain amount or, in other words, the risk of the decision becomes small. Calculating the a posteriori probabilities p(u k jy) from the L{values at the output of the decoder, we can evaluate the risk function Fig. 7: Bit error rate and average number of iterations using the stop criterion \cross{entropy" compared to a genius stop criterion Figure 7 shows the results when applying the criterion of cross-entropy. For bad channels no improvement in BER can be achieved by iterations. As no block can be detected error free for SNR 3 db the maximal number of 20 iterations is performed for every block when applying the \genius" criterion. The criterion of cross-entropy recognizes that no improvement can be achieved and stops iterating after 2-3 iterations on average, resulting in the same BER. However, the performance of a stop criterion at such high bit error rates

6 is not of practical interest. A stop criterion should be applied for signal to noise ratios where error free detection of some blocks is possible. In our example that is for SNR > 3 db. For those signal to noise ratios applying the stop criterion of cross-entropy results in a slightly higher average number of iterations and a marginal degradation in bit error rate compared to the genius stop criterion. The other criteria show similar results. In terms of computational requirements cross{ entropy is more complex than evaluating a risk function or observing the hard decisions..0e+00.0e-0.0e-02.0e-03.0e-04.0e Hard decisions Genius Fig. 8: Bit error rate and average number of iterations when stopping iterations according to changes in the hard decisions compared to a genius stop criterion.0e+00.0e-0.0e-02.0e-03.0e-04.0e Risk function Genius Fig. 9: Bit error rate and average number of iterations when evaluating a risk function as stop criterion compared to a genius stop criterion Iterative equalization is a suitable method to reduce the eects of intersymbol interference in a coded system. Only a few iterations are sucient to improve the bit error rate result considerably. Iterative equalization does not require any modications in standards as it is implemented only at the receiver. Moreover, we presented some criteria to reduce complexity by avoiding iterations when no improvement in bit error rate can be achieved. Using the criterion of cross-entropy as given in section V.A may serve as an eective method for this task. VII. References [] C. Berrou, A. Glavieux, and. Thitimajshima, \Near shannon limit error{correcting and decoding: Turbo{codes ()," in International Conference on Communication (ICC), pp. 064{070, IEEE, May 993. [2] C. Douillard, M. Jezequel, C. Berrou, A. icart,. Didier, and A. Glavieux, \Iterative correction of intersymbol interference: Turbo{equalization," European Transactions on Telecommunications, vol. 6, pp. 507{5, September{October 995. [3] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, \Optimal decoding of linear codes for minimizing symbol error rate," IEEE Transactions on Information Theory, vol. IT-20, pp. 284{287, March 974. [4] J. Hagenauer, E. Oer, and L. apke, \Iterative decoding of binary block and convolutional codes," IEEE Transactions on Information Theory, vol. IT{ 42, pp. 425{429, March 996. [5] J. Lodge, R. Young,. Hoher, and J. Hagenauer, \Separable map \lters" for the decoding of product and concatenated codes," in International Conference on Communication (ICC), pp. 740{745, IEEE, May 993. [6] J. G. roakis, Digital Communications. Singapore: McGraw Hill, 989. [7] M. Moher, \Decoding via cross-entropy minimization," in Globecom, pp. 809{83, IEEE, December 993. [8] H. Khorram, \Iterative Entzerrung und Kanaldecodierung bei frequenzselektiven Kanalen mit dem MA { Algorithmus," diploma thesis, Department of Communications Engineering, TU Munich, January 997. (in German). VI. Conclusions

A rate one half code for approaching the Shannon limit by 0.1dB

100 A rate one half code for approaching the Shannon limit by 0.1dB (IEE Electronics Letters, vol. 36, no. 15, pp. 1293 1294, July 2000) Stephan ten Brink S. ten Brink is with the Institute of Telecommunications,