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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2010 proceedings.

Stochastic Expectation Maximization Algorithm for Long-Memory Fast-Fading Channels Hong Wan∗ , Rong-Rong Chen∗ , Jun Won Choi∗∗ , Andrew Singer∗∗ , James Preisig† , and Behrouz Farhang-Boroujeny∗ ∗

Dept. of ECE, University of Utah Dept. of ECE, University of Illinois at Urbana-Champaign † Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institute ∗∗

Abstract—In this paper, we develop a novel statistical detection algorithm following similar principles to that of expectation maximization (EM) algorithm. Our goal is to develop an iterative algorithm for joint channel estimation and data detection in channels that have a long memory and are fast varying in time. At each iteration, starting with an estimate of the channel, we combine a Markov Chain Monte Carlo (MCMC) algorithm for data detection, and an adaptive algorithm for channel tracking, to develop a statistical search procedure that finds joint important samples of possible transmitted data and channel impulse responses. The result of this step, which may be thought as E-step of the proposed algorithm, is used in an M-step that refines the channel estimate, for the next iteration. Excellent behavior of the proposed algorithm is presented by examining it on real data from underwater acoustic communication channels. Keywords: Turbo Equalization, Markov Chain Monte Carlo Techniques, Underwater Acoustic Channels

I. I NTRODUCTION The increasing demand for high-speed tetherless (electromagnetic or acoustical) products has motivated a significant amount of research to combat the intersymbol interference (ISI) resulting from multipath transmission. Early developments date back to 1960s and 1970s when linear and decision feedback (DF) equalizers were developed, [1], [2]. These traditional methods face the problem of noise enhancement, in the case of linear equalizers, or error propagation, in the case of DF equalizers. Furthermore, these methods are based on hard decisions and thus cannot benefit from the modern coding techniques that can approach the channel capacity by making use of soft information. To achieve near capacity performance, modern communication systems operate based on the principle of the maximum a posteriori (MAP) detection. The detection is performed in two steps. In the first step, pilot symbols are used to estimate the channel impulse response. A MAP detector is then used to find the best bit/symbol stream that matches the received signal. The result is an optimal receiver in the sense of minimizing the probability of bit errors. However, the computational complexity of this method, even with the use of efficient implementations such as the BCJR algorithm [3] is exponential with respect to the length of the channel impulse response and the size of symbols constellation and thus may be prohibitive in many cases. To resolve this problem, low complexity equalizers/detectors have been developed. Most of these works, that

are generically referred to as turbo equalizers, operate based on the turbo principles where soft information is exchanged between a soft-in soft-out (SISO) equalizer/detector and a channel decoder. There are two approaches that one may take in realization of SISO detector. The first approach follows the principle based on which the classical linear and DF equalizers were built. These classical equalizers are modified to accept the soft information from the channel decoder and their outputs are manipulated to obtain soft information (log likelihood values) of the coded bits. Examples of this approach are the SISO minimum mean square error (MMSE) equalizer of Tuchler et al., [4], [5] and the soft interference cancelation (SIC) method of Glavieux et al., [6]. The second approach attempts to implement some simplified versions of the maximum likelihood sequence estimation (MLSE). The first work that has made an attempt to reduce the complexity of the SISO MLSE is due to Douillard et al., [7], where a softoutput Viterbi algorithm (SOVA) is used. On the other hand, methods that prune insignificant branches of trellis in BCJR detectors have been successfully adopted to develop detection algorithms whose complexity does not grow exponentially with the channel memory. Examples of such algorithms are the M-best BCJR (M*-BCJR) and threshold-based BCJR (T-BCJR) of [8], and the list-sequential (LISS) algorithm of [9]. These algorithms have more flexibility in controlling the complexity. However, a good performance cannot always be guaranteed. The above works make the assumption that the channel impulse response is known a priori or is estimated at the beginning of each data packet, using a pilot preamble. In certain applications, such as underwater acoustic (UWA) channels, the time variation of the channel is so fast that the preamble pilot can be only used to obtain an initial estimate of the channel. This initial estimate will subsequently be used along with a detector to track channel variations. The traditional algorithms operate in a decision directed mode where past decisions are used to estimate the channel for future detection of data. Also, channel estimate is refined along with the detected data in a turbo equalization loop. Among different works reported in this area, those algorithms that operate based on the expectation maximization (EM) technique are theoretically more solid and in most cases perform better [10], [11]. However, the complexity of EM-based turbo equalization

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algorithms grows exponentially with channel memory. Inspired by the idea of the EM technique and low complexity of Markov Chain Monte Carlo (MCMC) detection algorithms that we have developed recently, in this paper we develop a novel algorithm for joint detection of data and channel. Although the proposed algorithm is general and applicable to all kinds of channels, including inter-symbol interference (ISI) and multiple-input multiple output (MIMO) channels, our specific interest here is its application to underwater acoustic (UWA) channels. UWA channels feature large delay spread, frequency-dependent Doppler shift, and high time variability [12]. In fact, one may claim that UWA links are the most challenging communication channels and, thus, if an algorithm can handle UWA channels, it will be capable of handling all other types of channels in other applications as well. This paper is organized as follows. In Section II, we present a model of the communication system that is considered in the rest of the paper. A brief review of the standard EM algorithm for joint data detection and channel estimation is presented in Section III. Our proposed stochastic EM algorithm combined with MCMC is presented in Section IV. The experimental results that report the performance of the proposed algorithm in UWA channels are presented in Section V. The concluding remarks are presented in Section VI. II. S YSTEM M ODEL Fig. 1 presents a block diagram of the communication system that is considered in this paper. The vector b contains a sequence of (uncoded) information bits. This is passed to a channel encoder that adds redundant bits to b. The coded bits are passed through an interleaver block whose output is the interleaved coded bit vector c. The modulation block converts c to a vector of complex-valued symbols x for transmission. The communication link is a single-input multiple-output (SIMO) channel characterized by the impulse responses h1 , h2 , · · · , hK , where K is the number of receive elements. Each hi , i = 1, · · · , K is a column vector of length L, representing the channel impulse response between the transmit element and the i-th receive element. Here L is the channel memory. We combine {h1 , · · · , hK } to obtain the channel matrix H = h1 h2 · · · hK . (1) A set of pilot symbols is appended at the beginning of each packet for initial estimation of H. The received signal is a matrix Y whose columns are obtained by convolving x with columns of H. The detector, at the receiver, generates log-likelihood ration (LLR) values of coded bits and these are passed, after de-interleaving, to the channel decoder. Extrinsic information λde and λce are exchanged between the detector and the channel decoder, over successive iterations, to improve the quality of the decoded information. During each iteration the soft information from coded bits is also used to improve the quality of the channel estimates. Although, for brevity, it is not indicated explicitly in Fig. 1, the channel matrix H, in general, varies with time. We assume

b

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λce

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Channel Estimator Fig. 1.

System model.

that the channel variation over time is slow enough so that it can be approximated by a constant matrix over periods of time that each has a duration longer than the channel memory, a necessary condition for the validity of the assumed linear model of the channel. Accordingly, we divide Y in a number of segments, say, Y0 , Y1 , Y2 , · · · , with Y0 being the segment associated with the pilot symbols of the packet, and develop algorithms based on the flow diagram shown in Fig. 2. The preamble portion of each packet is used to obtain an initial ˆ 0 is fed to a processing block that ˆ 0. H estimate of H, say, H finds the LLR values of the coded bits of the first segment ˆ 1 , is constructed and accordingly a new estimate of H, say, H for processing of Y2 . This procedure extends sequentially over successive segments of Y. After one round of processing of all segments within a packet, the generated extrinsic LLR values are processed by the channel decoder and as also presented in Fig. 1 a turbo loop is formed. III. S TANDARD EM A LGORITHM The EM technique, when applied to the joint channel estimation and data detection problem of interest in this paper, leads to the following algorithm that is executed for each packet. We let xl denote the transmitted signal vector for the l-th segment of the packet. Assume that the constellation size is Mc , and each segment of the packet contains nd data symbols. Then the total number of possible xl equals Mcnd . The standard EM algorithm described below involves an exhaustive search over all possible xl . 1) Use the first segment of the packet that contains only ˆ 0 . Let m = 0, pilots to obtain an initial estimate of H, H ˆ 0 , and l = 1. H(0) = H 2) Use H(m) for the l-th segment of the packet, and (i) a) E-step: For all combinations of xl , denoted by xl , (i) for i = 1, 2, · · · , Mcnd , find P (xl |Yl , H(m) , λce ). b) M-step: Find (i) pi log p(Yl |xl , H) (2) H(m+1) = arg max

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H

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Data Detector + Channel Estimator Fig. 2.

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Receiver flow diagram.

(i)

where pi = P (xl |Yl , H(m) , λce ). 3) Let m = m + 1. Repeat step 2 until the algorithm ˆ l. converges. Denote the converged channel estimate by H 4) Compute the LLRs of the coded bits carried in xl . ˆ l to initialize the channel estimate for the 5) Use H(0) = H next segment. Let m = 0, l = l + 1. Repeat Steps 2-5 until all segments have been processed. After all the segments within a packet have been processed, we pass the LLRs of the coded bits generated in step 4 to the channel decoder to generate λce , the new extrinsic information of the coded bits. Subsequently, λce will be passed to the EM algorithm (Step 2) for further refinement of channel estimates and detected bits. Exact implementation of the above algorithm to the case of interest to us, i.e., UWA channels, leads to a prohibitive complexity. To resolve the problem of complexity, one may adopt a low complexity search method that picks a small, but important set of samples for xl , and use this sample set to compute the LLRs of the coded bits and to perform the M-step. A good candidate for such a search is MCMC technique. In [13], a number of different versions of Gibbs sampler (core part of MCMC) are used as search engine to find sample vectors of xl . We have adopted the socalled bitwise Gibbs sampler for the results presented in this paper. Application of the above algorithm to UWA data that are available to us, unfortunately, did not lead to satisfactory performance. The algorithm’s convergence rate, in particular, was too slow. These experiments and additional studies led us to the development of the algorithm discussed in the following section.

Our experiments with the above EM algorithm and, particularly, its slow convergence, convinced us that a more effective algorithm will be developed if one searches for joint samples (i) of xl and H(i) that match well with the received signal Yl . With this concept in mind, we propose the following algorithm. We call this algorithm statistical EM (SEM). 1) Use the first segment of the packet that contains only ˆ 0 , m = 0 and l = 1. ˆ 0 . Let H(0) = H pilots to obtain H (m) for the l-th segment of the packet, Yl , and 2) Use H a) E-step: Using the channel estimate H(m) , the prior from the channel decoder λce , the Gibbs sampler finds a (1) (1) sample of xl , say, xl . Using xl , find a new estimate of the channel, say, H(1,m) that minimizes the difference between Yl and the matrix formed by convolving (1) each column of H with xl . While different cost functions may be used for this minimization, the most trivial cost function is the mean square error difference. We have adopted this for our experiments. Using the (1) results of this step, we form the pair (xl , H(1,m) ). b) Using the channel estimate H(1,m) , the Gibbs sampler (2) generates the next sample of xl , say, xl and ac(2) (2,m) cordingly forms the pair (xl , H ). Also, continue (i) this process until sufficient pairs of (xl , H(i,m) ), for i = 1, 2, · · · , are collected. c) M-step: Find pi ||H − H(i,m) ||2F (3) H(m+1) = arg min H

i

(i)

where pi = P (xl |Yl , H(i,m) , λce ) and || · ||F denotes Frobenius norm. It can be shown that (3) has the following simple solution pi H(i,m) (m+1) = i . (4) H i pi 3) Let m = m + 1. Repeat step 2 until the algorithm converges. 4) Compute the LLR of the coded bits carried in xl . ˆ l denote the estimated sequence based on the LLR 5) Let x computed from step 4. Run the least-mean-square (LMS) ˆ l−1 and algorithm using the initial channel estimate H ˆ ˆ ˆ l to ˆ l }. Use H ˆ l to obtain Hl = LMS{Hl−1 , Yl , x x initialize the channel estimate for the next segment. Let m = 0, l = l + 1. Repeat Steps 2-5 until all segments have been processed. Details of the Gibbs sampler that are used in the E-step are presented in Algorithm 1. Here we let cl represent the length B bit vector corresponding to xl for the l-th segment of a packet. The Gibbs sampler is a statistical sequential procedure used (i) to generate a set of most likely transmitted vectors cl , i = 1, · · · , I given Yl and prior from the channel decoder. Here we incorporate the channel estimation into the Gibbs sampler (i) to find most likely pairs of (cl , H(i,m) ) jointly, based on an (m) initial channel estimate H .

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Algorithm 1: Gibbs sampler for MCMC-SEM (0)

generate an initial c for i = 1 to I (i) generate c0 from distribution (i) (i−1) (i−1) (i−1) , c2 , · · · , cB−1 , Yl , λce , H(i−1,m) ) P (c0 = a|c1 a = 0, 1 (i) generate c1 from distribution (i) (i) (i−1) (i−1) , · · · , cB−1 , Yl , λce , H(i−1,m) )) P (c1 = a|c0 , c2 .. . (i)

generate cB−1 from distribution (i) (i) (i) (i) P (cB−1 = a|c0 , c1 , · · · , cB−2 , Yl , λce , H(i−1,m) )) (i) (i) map bit sequence cl to symbol sequence xl (i) H(i,m) = LMS{H(m) , Yl , xl } end for

In Algorithm 1, H(i,m) is obtained by running the LMS (i) algorithm over sequence xl using the initial channel estimate (i) H(m) . We run multiple passes over xl until the algorithm (i) (i) converges. For brevity, we let cj denote the j-th bit of cl with the subscript l suppressed. To obtain better performance, we run Q Gibbs samplers in parallel with I iterations each. Hence, a maximum of Q·I most likely transmitted sequences are generated by the MCMCSEM, which are used to compute the output LLRs following the procedure given in [13]. Computation of the LLRs from (1) (Q·I) } is similar to that of [13] and the samples {xl , · · · , xl [14] with the modification that the true channel H is now replaced by H(i,m) . V. N UMERICAL R ESULTS In this section, we examine the effectiveness of the MCMCSEM detector using experimental data collected from actual underwater experiments. A. Experiment setup The experiment was conducted off the coast of Martha’s Vineyard, MA during Oct. 14th - Nov. 2nd, 2008. During the experiment, there is no movement of the transmitter and receiver. There is a single transducer, and a vertical hydrophone array deployed at 60, 200, and 1000 meters away from the source. The hydrophone array contains 12 elements spaced apart by 12cm. Epochs of data, each containing multiple data files for various modulation schemes, are transmitted every two hours. Every data file within an epoch contains 42 data packets with the same modulation scheme (e.g. 4QAM, 16QAM, or 64QAM). Each packet consists of Np = 400 training symbols and Nd = 1200 data symbols. The data symbols within each packet are divided into T = 3 segments, and each segment contains nd = 1200/3 = 400 symbols. The channel coding is across every I = 6 packets. The carrier frequency is 13 kHz, and the symbol rate is 9.77k sym/sec. The data bits are encoded by a rate 1/2 recursive systematic convolutional (RSC) encoder with the generator polynomial (23, 35). A

square-root raised cosine filter with a roll-off factor 0.2 is used at both the transmitter and the receiver. For each data set, a preamble of 1000 symbols is inserted before data transmission to facilitate data synchronization. Estimation of the channel length L is performed after the synchronization process is complete. For the data sets considered here, we find L to be in the range of 60-80. The readers may notice that in the presentation of the algorithms in Sections III and IV, we assumed that the algorithms operate over all the bits within each packet. In our experimental setup, however, coding in done across every 6 consecutive packets. Adoption of this change to our system is clear. B. Experimental results In this section, we compare performance of the proposed MCMC-SEM detector with the least mean-square turbo equalizer (LMS-TEQ) [15], and the MCMC-LMS, and MCMC variable step-size LMS (MCMC-VSLMS) detectors of [14]. The latter two MCMC detectors adopt the LMS and VSLMS algorithm, respectively, for channel estimation. It is reported in [14] that for the 60 meter distance where the channel is more sparse, the VSLMS facilitates better channel tracking than the LMS, hence MCMC-VSLMS outperforms the MCMC-LMS. We examine performance of these four detectors over a set of 22 data files for the 60 meter distance, using 16QAM modulation and K = 4 receive hydrophones. Each file is from a different epoch and thus is transmitted two hours apart. In Fig. 3, the x-axis represents a total of 22 files. The y-axis represents the average number of errors in information bits per packet (each packet has 1200 ∗ 4/2 = 2400 information bits). As shown in Fig. 3, the best performance is achieved by the MCMC-SEM. After seven iterations, the MCMC-SEM has noticeable errors only for data sets 11 (contains 24 bit errors) and 12 (contains 48 bit errors), corresponding to bit-error-rate (BER) of 1% and 2%, respectively. Compared to the MCMCVSLMS, the MCMC-SEM achieves much superior performance for data sets 9-13, 15. This is because the MCMC-SEM yields better channel estimation by combining MCMC with the SEM algorithm. For the MCMC-SEM, we use parameters Q = 20 and I = 10 for the Gibbs sampler and only two SEM iterations are used (m = 0, 1). Further iterations do not improve performance of the MCMC-SEM. The results for the MCMC-LMS, and MCMC-VSLMS detectors are generated using parameters Q = 10 and I = 10 because increasing the values of Q or I does not improve performance for these two detectors. In Fig. 4, we plot the BER curves of different detectors versus the number of turbo iteration. The BER is obtained by averaging over all 22 data sets. We compare performance of the MCMC-SEM with K = 4 receiving hydrophones with other detectors with K = 4 or K = 8. The gap between the MCMC-SEM and the other detectors are clearly shown. In particular, the MCMC-SEM with K = 4 outperforms all the other detectors with K = 8. We note that the BER of the

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LMS−TEQ K=4 MCMC−LMS K=4 MCMC−VSLMS K=4 LMS−TEQ K=8 MCMC−LMS K=8 MCMC−VSLMS K=8 MCMC−SEM K=4

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MCMC-SEM is still slightly above 10−3 , mostly due to the errors occurred in data sets 11 and 14. We have also made comparisons with the channel estimation based MMSE detector of [15], and find that its performance is comparable to that of the LMS-TEQ. Hence, the results are not included here. We are currently investigating performance of the MCMC-SEM for data transmissions over longer distances, e.g., 1000 meters. Since the SNR is lower for longer transmission distances, the channel noise, as well as the channel estimation error both contribute to detection error. The effectiveness of the proposed MCMC-SEM and its variants needs to be further explored for such settings.

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VI. C ONCLUSION In this paper, we proposed a novel MCMC detector based on a stochastic EM algorithm. The effectiveness of the proposed MCMC-SEM detector is examined using actual experimental data for UWA channels with ISI that extends as many as 80 symbols. For 60 meter SIMO UWA channels, the MCMCSEM significantly outperforms the LMS-TEQ detector, the MMSE detector, and previous versions of the MCMC detectors that perform LMS channel estimation. Our results demonstrate that the statistical detection approach developed here provides a low-complexity mean to approximate the standard EM algorithm, and is highly efficient for joint data detection and channel estimation. In-depth study of this novel algorithm for longer distance UWA channels and for terrestrial MIMO/ISI channels is under further exploration.

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Fig. 3. Performance comparisons between LMS-TEQ and MCMC detectors over 22 data sets for the 60 meter distance. Assume 16QAM constellation and K = 4 receive hydrophones.

R EFERENCES [1] R. Lucky, “Automatic equalization for digital communication,” Bell System Tech. J., pp. 547 – 588, April 1965. [2] M. E. Austin, “Decision feedback equalization for digital communication over dispersive channels,” MIT Lincoln Lab., Lexington, MA, Tech. Rep.

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Fig. 4. Bit error rate comparisons between LMS-TEQ and MCMC detectors averaged over 22 data sets for the 60 meter distance. Assume 16QAM constellation and K = 4 or 8 receive hydrophones.

[3] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. 20, pp. 284–287, 1974. [4] M. Tuchler, R. Koetter, and A. Singer, “Turbo equalization: principles and new results,” IEEE Trans. Commun., vol. 50, no. 5, pp. 754–767, May 2002. [5] M. Tuchler, A. C. Singer, and R. Koetter, “Minimum mean squared error equalization using a priori information,” IEEE Trans. Signal Process., vol. 50, no. 3, pp. 673–683, March 2002. [6] A. Glavieux, C. Laot, and J. Labat, “Turbo equalization over a frequency selective channel,” in Proc. Int. Symp. Turbo Codes, Related Topics, Brest, France, pp. 96–102, Sept. 1997. [7] C. Douillard, M. Jezequel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, “Iterative correction of intersymbol interference: Turbo equalization,” Eur. Trans. Telecommun., vol. 6, pp. 507–511, Sept.-Oct. 1995. [8] V. Franz and J. B. Anderson, “Concatenated decoding with a reducedsearch bcjr algorithm,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 186–195, Feb. 1998. [9] J. Hagenauer and C. Kuhn, “The list-sequential (liss) algorithm and its application,” IEEE Trans. on commun., vol. 55, no. 5, pp. 918–928, May 2007. [10] M. Kobayashi, J. Boutros, and G. Caire, “Successive interference cancellation with SISO decoding and EM channel estimation,” IEEE Journal on Selected areas in Communications, vol. 19, no. 8, pp. 1450– 1460, August 2001. [11] Y. Xie and C. N. Georghiades, “Two EM-type channel estimation algorithms for OFDM with transmitter diversity,” IEEE Transactions on Communications, vol. 51, no. 1, pp. 106–115, Jan. 2003. [12] A. Singer, J. Nelson, and S. Kozat, “Signal processing for underwater acoustic communications,” IEEE communication magazine, vol. 47, no. 1, pp. 90–96, Jan. 2009. [13] R. Peng, R.-R. Chen, and B. Farhang-Beroujeny, “Low complexity markov chain monte carlo detector for channels with intersymbol interference,” IEEE Trans. on Signal Processing, vol. 58, no. 4, pp. 2206–2217, April 2010. [14] H. Wan, R.-R. Chen, J. W. Choi, A. Singer, J. Preisig, and B. FarhangBoroujeny, “Markov Chain Monte Carlo detection for underwater acoustic channels,” in Proc. Information theory and applications workshop(ITA)’10, 2010. [15] J. W. Choi, R. J.Drost, A. C.Singer, and J. Preisig, “Iterative multichannel equalization and decoding for high frequency underwater acoustic channels,” Sensor Array and Multichannel Signal Processing Workshop, July 2008.

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