Markov Chain Monte Carlo Detection for Underwater Acoustic Channels

Transcription

1 Markov Chan Monte Carlo Detecton for Underwater Acoustc Channels Hong Wan, Rong-Rong Chen, Jun Won Cho, Andrew Snger, James Presg, and Behrouz Farhang-Boroujeny Dept. of ECE, Unversty of Utah Dept. of ECE, Unversty of Illnos at Urbana-Champagn Appled Ocean Physcs and Engneerng, Woods Hole Oceanographc Insttute Abstract In ths work, we develop novel statstcal detectors to combat ntersymbol nterference for frequency selectve channels based on Markov Chan Monte Carlo () technques. Whle the optmal maxmum a posteror (MAP) detector has a complexty that grows exponentally wth the constellaton sze and the memory of the channel, the detector can acheve near optmal performance wth a complexty that grows lnearly. Ths makes the detector partcularly attractve for underwater acoustc channels wth long delay spread. We examne the effectveness of the detector usng actual data collected from underwater experments. When combned wth adaptve least mean square (LMS) channel estmaton, the detector acheves superor performance over the drect adaptaton LMS turbo equalzers (LMS-TEQ) for a majorty of data sets transmtted over dstances from 6 meters to 1 meters. I. INTRODUCTION Underwater acoustc (UWA) channels pose unque challenges due to the low speed of sound, lmted communcaton bandwdth, and the ntrnsc moton due to waves and currents [1]. Such channels feature large delay spread, frequency-dependent Doppler shft, and hgh tme varablty. To overcome these challenges, turbo equalzaton technques have been appled to UWA channels [2], whch demonstrate mproved performance over the phase-coherent recevers n [3]. The LMS-TEQ turbo equalzer developed n [2] are based on the least mean square (LMS) algorthm and t s shown to converge rapdly to the optmal equalzer wthout the need for channel estmaton. In ths work, we study the applcaton of statstcal detectors based on Markov Chan Monte Carlo () technques to UWA channels. Such detectors offer a low-complexty approxmaton to the maxmum-lkelhood sequence estmaton (MLSE) and are used to drectly perform data detecton wthout channel equalzaton. The detectors have been studed prevously n [4] [6] for both multple-nput multpleoutput (MIMO) frequency-flat channels and for frequency selectve channels wth nter-symbol nterference [7]. Under the assumpton of perfect channel state nformaton (CSI) at the recever, the detectors developed n these work demonstrate excellent performance at low-complexty. In ths paper, we apply detectors to UWA channels and show that when combned wth adaptve channel estmaton, the detectors demonstrate excellent performance when compared wth the adaptve LMS-TEQ equalzer of [2]. For UWA channels wth sparse channel mpulse response, we demonstrate that the varable step sze LMS (VSLMS) algorthm provdes superor channel trackng and hence yelds mproved performance than that of the standard LMS algorthm. Expermental results are provded for multple data sets measured over 6 meter to 1 meter dstances from the recent SPACE 8 experment conducted off the coast of Martha s Vnyard, MA. II. CHANNEL MODEL AND SYSTEM SETUP We frst descrbe the structure of the transmtter, the UWA channel model, and the structure of the recever. At the transmtter sde, we pass every group of N b nformaton bts to the channel encoder to generate a sequence of N c coded bts. The coded bts are nterleaved and mapped nto a sequence of complex symbols. We then nsert plot symbols and the resultng symbol sequence {x n } s dvded nto multple packets, each contanng N p plot symbols and N d data symbols. Assumng that channel codng s performed across every I packets, we must have N b = IN d (log 2 M c )R, wherer s the code rate andm c s the sze of the constellaton. The sequence{x n } s passed through a pulse shapng flter and transmtted by a transducer through the UWA channel. Assume that a total of K recevng hydrophones are used. The receved sgnal at tme n at the k-th receve hydrophone can be expressed as n = x n l +v (k) n, (1) where l =,,L s the ndex of l-th path, denotes the channel gan of the l-th path at tme n, between the transducer and the k-th receve hydrophone. The recever structure s llustrated n Fg.1, whch employs teratve jont channel estmaton, data detecton, and channel decodng over every groups of I packets. We dvde each packet nto multple blocks and perform channel estmaton and detecton block-wse. Wthn each packet, we frst estmate the channel from the plot block. These channel estmates are then passed to the detector for data detecton over the frst data block. We then refne the channel estmates based on the output of the detector, and

2 use these to perform data detecton for the second block. The soft outputs of the detector from all the blocks wthn the group of I packets are passed to the channel decoder for decodng. For the next teraton, soft nformaton from the channel decoder s fed back to the detector for data detecton. Detals of the channel estmaton and detecton wll be presented n the followng sectons. Fg. 1. Flowchart of the recever III. CHANNEL ESTIMATION We consder two adaptve channel estmaton algorthms: the LMS and the VSLMS. The latter algorthm s shown to outperform the LMS when the channel mpulse response s sparse. A. Least mean square (LMS) channel estmaton For the LMS channel estmaton, the channel gans are updated accordng to n+1,l = ĥ(k) + µ L = x n 2e(k) n ˆx n l. (2) For the plot block, each ˆx n equals the plot symbol. For each data block wthn a packet, ˆx n denotes the soft value at the output of the detector, e.g., ˆx n = qp(x n = q), q where q =,1,,M c 1 s an arbtrary constellaton pont. The resdual error e (k) n n (2) s computed as e (k) n = n L ˆx n l. We run the LMS algorthm for multple passes wthn each block of symbols to ensure convergence and the step-sze µ s set to be.2. B. Varable step sze LMS (VSLMS)channel estmaton For the VSLMS, nstead of usng the same step sze µ to update all L channel taps, we allow the step szes to vary for dfferent taps. Let µ (k) denote the step sze for the l-th channel tap. We update the channel estmates as follows [8] n+1,l = ĥ(k) +µ(k) e(k) n ˆx n l. (3) At the tme n, µ (k) µ (k) φ (k) can be updated from µ(k) n 1,l as = µ (k) n 1,l +ρ Re{e(k) n ˆx n l φ(k) }, (4) = αφ (k) n 1,l +e n 1ˆx n l 1, (5) where we set ρ =.5,α =.95. Then the vector of step szes s scaled such that 1 L L µ(k) = µ =.2 to guarantee stablty. Ths wll guarantee the same msadjustment for both LMS and VSLMS algorthms [9]. To gve some nsghts to the mechansm of the VSLMS algorthm, we note that the sgn of φ (k) ndcates the average drecton of the stochastc gradent n the past. Accordngly, f the present gradent, e n 1ˆx n l 1 φ (k), has the same sgn as φ (k), then we assume that the VSLMS algorthm has not converged yet. Thus, we should ncrease the step-sze parameter µ (k). Otherwse, t should be decreased. IV. DETECTION The basc prncples of the detector for ISI channels wth perfect CSI are presented n [7]. The man dea s to use the Gbbs sampler to generate a small set of most lkely transmtted sequences, based on whch the loglkelhood-rato (LLR) of each transmtted bt s computed. Assume that each data block wthn a packet ncludes n d data symbols, correspondng to B = (log 2 M c ) n d transmtted bts b = (b,,b B 1 ). The Gbbs sampler s a statstcal procedure used to draw one bt at a tme. Consder bt b m, where m B 1. Let λ m denote the LLR of b m, provded by the channel decoder. Let y = (y (1),,y (K) ), where denotes the receved sgnal sequence from the k- th hydrophone. We run the Gbbs sampler over I teratons to generate a set of I most lkely transmtted sequences, denoted by {b (1),,b (I) }. Detals of the Gbbs sampler are descrbed n Algorthm 1. Algorthm 1: Gbbs sampler generate an ntal b () for n = 1 to I from dstrbuton P(b = a b (n 1) 1,b (n 1) 2,,b (n 1) B 1,y,λ ) 1 from dstrbuton P(b 1 = a b (n 1),b (n 1) 2,,b (n 1) B 1,y,λ 1). B 1 from dstrbuton P(b B 1 = a b (n 1),b (n 1) 1,,b (n 1) B 2,y,λ B 1) end for Note that when updatng b (n) m durng the n-th teraton, we condton upon updated samples (b (n), b(n) m 1 ) obtaned durng the same teraton, and samples (b (n 1) m+1, b(n 1) B 1 ) obtaned from the prevous teraton. Let b m = (b (n),,b(n) m 1,b(n 1) m+1,,b(n 1) B 1 ).

3 We draw sample b (n) m based on the condtonal probablty dstrbuton P(b m = a b m,y,λ m ), where a =,1. For each a, we defne b a = {b (n),,b(n) m 1,a,b(n 1) m+1,,b(n 1) B 1 }, and letx a denote the symbol vector correspondng tob a. Also, let x j L:j = (x j L,x j L+1,,x j ). Frst, we assume that the channel gans { j,l } are perfectly known, and the nose {v n (k) } s whte wth a complex Gaussan dstrbuton of zero mean and varance of σk 2. Assume that bt b m s mapped to symbol x. Then we obtan P(b m = a b m,y) K p( x a )P(x a ) k=1 K +L p( j x a j L:j )P(b m = a) { K +L = C exp ( 1 σk 2 j j,l xa j l 2)} P(b m = a), (6) where P(b m = a) can be computed from λ m and C s a scalng constant to ensure that P(b m = b m,y) +P(b m = 1 b m,y) = 1. For channels wth mperfect CSI, we replace j,l n (6) by the estmated channel ĥ(k) j,l. We wll also replace σk 2 n (6) by ˆσ2 k to take nto account both channel estmaton error and the varance of channel nose. To be specfc, assume that The receved sgnal can be wrtten as Let ñ (k) = = = v (k) + L,l = ĥ(k),l +e (k),l. (7),l x l +v (k),l x l +v (k) + e (k),l x l. (8) e (k),l x l and σ 2 k = Var(ñ(k) ). We then estmate σ k 2 from the plot block such that σ k 2 1 N p N p y(k) =1,l x l where x, = 1,,N p are plot symbols. Fnally, we rewrte (6) as P(b m = a b m,y) (1) { K +L ( exp 1ˆσ k 2 j j,l xa j l 2)} 2 (9) P(b m = a). (11) To obtan better performance, we run Q Gbbs samplers n parallel wth I teratons each. Hence, a maxmum of Q I most lkely transmtted sequences are generated by the, whch are used to compute the output LLRs followng the procedure gven n [7]. A. Experment setup V. NUMERICAL RESULTS The experment was conducted off the coast of Martha s Vnyard, MA durng Oct. 14th - Nov. 2nd, 28. Durng the experment, there s no movement of the transmtter and recever. There s a sngle transducer, and a vertcal hydrophone array deployed at 6, 2, and 1 meters away from the source. The hydrophone array contans 12 elements spaced apart by 12cm. Epochs of data, each contanng multple data fles for varous modulaton schemes, are transmtted every two hours. Every data fle wthn an epoch contans 42 data packets wth the same modulaton scheme (e.g. 4QAM, 16QAM, or 64QAM). Each packet conssts of N p = 4 tranng symbols and N d = 12 data symbols. The data symbols wthn each packet are dvded nto T = 3 blocks, and each block contans n d = 12/3 = 4 symbols. The channel codng s across every I = 6 packets. The carrer frequency s 13 khz, and the symbol rate s 9.77k sym/sec. The data bts are encoded by a rate 1/2 recursve systematc convolutonal (RSC) encoder wth the generator polynomal (23, 35). A square-root rased cosne flter wth a roll-off factor.2 s used at both the transmtter and the recever. For each data set, a preamble of 1 symbols s nserted before data transmsson to facltate data synchronzaton. Estmaton of the channel length L s performed after the synchronzaton process s complete. For the data sets consdered here, we fnd L to be n the range of 6-8. B. Expermental results We frst compare performance of the detector wth the LMS-TEQ [2] over a set of 22 data fles for the 1 meter dstance. Each fle s from a dfferent epoch and thus s transmtted two hours apart. For the 1 meter dstance, the estmated channel s not very sparse, as shown n Fg. 2, and hence the advantage of the VSLMS over the LMS s not evdent. For ths settng we use the LMS for channel estmaton. Frst, we note that wth 4QAM modulaton, due to lower date rates, both detectors obtan excellent performance, achevng error-free decodng results for almost all 22 data sets. Hence, we present our results only for the hgher order modulatons 16QAM and 64QAM. In Fg. 4 (for 16 QAM) and Fg 5 (for 64QAM), the x-axs represents a total of 22 fles. The y-axs represents the average number of errors n nformaton bts per packet. Fg. 4 shows that for 16QAM (each packet has 12 4/2 = 24 nformaton bts), the s much better than the LMS-TEQ after one teraton of detecton/equalzaton and decodng. After seven teratons, the LMS-TEQ stll has more than 35 bt errors per packet for fles In comparson, the has fewer errors

4 and only fles 1 and 18 have more than 35 bt errors per packet. The overall number of bt errors usng s only one thrd of LMS-TEQ. For the same symbol rate of 9.77k sym/sec, the bt rate for 64QAM s hgher (each packet has 12 6/2 = 36 nformaton bts), whch yelds more bt errors than that of the 16QAM. We observe from Fg. 5 that the performs better than the LMS-TEQ for all the fles after the frst teraton. After seven teratons, even though the number of errors s stll hgh for most fles, t s clear that the performance of s ether much better than LMS-TEQ, e.g., fles 1-8, 12-18, 21,22, or comparable to LMS-TEQ, e.g., fles 9,1,11,19,2. We also compare performance of the detector wth the LMS-TEQ for the 6 meter dstance. The channel estmaton for s done usng ether LMS or VSLMS. Here we use a total of 4 receve hydrophones. As n the case of 1 meters, the performs better than the LMS-TEQ for most cases after one teraton. After seven teratons, t s clear that the detector wth ether LMS or VSLMS outperforms the LMS-TEQ for the majorty of data sets, e.g, fles 5-9, Also, snce the UWA channel s more sparse for the 6 meter dstance, as shown n Fg. 3, the VSLMS outperforms the LMS for the majorty of data sets, e.g., 1, 5-9, and VI. CONCLUSION In ths paper, we studed detecton for UWA channels. Through actual expermental data we have demonstrated the effectveness of the detectors for both 6 meter and 1 meter transmssons. Usng LMS or VSLMS channel estmaton, the detector acheves superor performance to the LMS-TEQ for the majorty of the data sets that we have examned. The VSLMS algorthm s shown to provde better channel estmaton than the LMS algorthm for sparse UWA channels. Delay (channel taps) Fg meter, 16QAM Symbol tme x 1 4 Estmated channel mpulse response for the 1 meter dstance Delay (channel taps) Fg meter, 16QAM Symbol tme x 1 4 Estmated channel mpulse response for the 6 meter dstance 1 meters, 16QAM teraton meters, 16QAM teraton Fg. 4. Performance comparsons between LMS-TEQ and over 22 data sets for the 1 meter dstance. Assume 16QAM constellaton and LMS channel estmaton. K = 1 receve hydrophones are used. REFERENCES [1] A. C.Snger, J. K.Nelson, and S. S.Kozat, Sgnal processng for underwater acoustc communcatons, IEEE Communcatons Magazne, pp. 9 96, Jan. 29. [2] J. W. Cho, R. J.Drost, A. C.Snger, and J. Presg, Iteratve multchannel equalzaton and decodng for hgh frequency underwater acoustc channels, Sensor Array and Multchannel Sgnal Processng Workshop, July 28. [3] M. Stojanovc, J. A.Catpovc, and J. G.Proaks, Phase-coherent dgtal communcatons for underwater acoustc channels, IEEE Journal of Oceanc Engneerng, vol. 19, pp , Jan [4] B. Farhang-Boroujeny, H. Zhu, and Z. Sh, Markov chan Monte Carlo algorthms for CDMA and MIMO communcaton systems, IEEE Trans. Sgnal. Process., vol. 54, no. 5, pp , May 26. [5] H. Zhu, B. Farhang-Beroujeny, and R. R. Chen, On performance of sphere decodng and Markov Chan Monte Carlo methods, IEEE Sgnal Processng Letters, vol. 12, no. 1, pp , Oct. 25. [6] R.-R. Chen, R. Peng, A. Ashkhmn, and B. Farhang-Beroujeny, Ap-

5 meters, 64QAM teraton 1 1 meters, 64QAM teraton Fg. 5. Performance comparsons between LMS-TEQ and over 22 data sets for the 1 meter dstance. Assume 64QAM constellaton and LMS channel estmaton. K = 1 receve hydrophones are used. # of bt errors per packet # of bt errors per packet meters, 16QAM teraton 1 6 meters, 16QAM teraton VSLMS +VSLMS Fg. 6. Performance comparsons between LMS-TEQ and over 22 data sets for the 6 meter dstance. Assume 16QAM constellaton. Assume LMS or VSLMS channel estmaton. K = 4 receve hydrophones are used. proachng MIMO capacty usng btwse Markov Chan Monte Carlo detecton, To appear IEEE Trans. Commun., Feb. 21. [7] R. Peng, R.-R. Chen, and B. Farhang-Beroujeny, Low complexty markov chan monte carlo detector for channels wth ntersymbol nterference, To appear: IEEE Trans. on Sgnal Processng, 21. [8] W. Ang and B. Farhang-Boroujeny, A new class of gradent adaptve step-sze lms algorthms, IEEE Trans. on Sgnal Processng, vol. 49, pp , Aprl. 21. [9] B. Farhang-Boroujeny, Adaptve flters: theory and applcaton. Chchester, U.K.: John Wley & Sons, 1998.