ITERATIVE MULTI-CHANNEL EQUALIZATION AND DECODINGFOR HIGH FREQUENCY UNDERWATER ACOUSTIC COMMUNICATIONS

Jun Won Choi*, Robert J. Drost*, Andrew C. Singer*, and James Preisig**

*University of Illinois at Urbana-ChampaignCoordinated Science Laboratory

1308 West Main St. Urbana, IL 61801, USA**Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA

ABSTRACT

In this paper, an iterative multi-channel equalization anddecoding technique is introduced to improve system perfor­mance in underwater acoustic communications. The turboprinciple is applied to the existing canonical receiver struc­ture including fractionally spaced decision feedback equal­ization with phase synchronization. The performance ofmulti-channel equalization and adaptive weight update al­gorithms are aided by soft information delivered from thechannel decoder. The complexity of the proposed schemeis shown to be reasonable for highly reverberant underwaterchannels since it has linear complexity by the use of the leastmean square (LMS) algorithm. Experimental results for datatransmission at 41.67k symbol/s with a carrier frequency oflOOk Hz demonstrate the feasibility of the proposed algo­rithm.

Index Terms- iterative, underwater, equalization, turbo,lms

1. INTRODUCTION

Bandwidth-efficient communications over underwater acous­tic channels suffer from channels of poor quality with longreverberation times and dynamic channel variation in time. Avariety of techniques have been proposed to overcome suchdifficulties. The success of phase-coherent communicationswas reported in [1], where a digital phase locked loop anda decision-feedback equalizer were coupled and an adaptivefast recursive least square (FRLS) algorithm was employed totrack time-varying channels. Such phase-coherent receivershave been extended to multi-channel receivers to exploit spa­tial diversity [2].

Turbo equalization [5] has been shown to provide signif­icant performance gains even for severe intersymbol interfer­ence (lSI) through iterative equalization and decoding. Re­cently, such iterative receiver schemes were applied to under­water communications in [3]. Due to the long reverberationtime (delay spread) of underwater channels, high computa­tional complexity is required for effective equalization. The

computational complexity of optimal minimum mean squareerror (MMSE) equalization with soft-inputs for turbo equal­ization as proposed in [5] is prohibitive when applied to un­derwater channels with such long delay spread. In addition,the need for a channel estimator makes its performance de­pend on the quality of channel estimation. The adaptive formof the turbo equalizer in [4] is therefore better suited to thisapplication.

In this paper, we use an adaptive turbo equalization tech­nique based on the least mean square (LMS) algorithm forunderwater acoustic communications. The proposed tech­nique, called iterative multi-channel equalization and decod­ing (IMED) does not need the aid of a channel estimator andconverges rapidly to the optimal equalizer via the adaptivealgorithm, which accounts for the soft information deliveredfrom the channel decoder in the weight update. Experimentalresults are provided for data measured on two research vesselsover 100m to 530m distances in Buzzards Bay off the coastofMassachusetts, USA.

2. SYSTEM MODEL

In this section, the basic system model is described. The in­formation bits, {bk} are encoded by a convolutional channelencoder, producing code symbols {Ck}. The coded bits areinterleaved yielding {Ck} and then mapped to the sequenceof M -ary symbols {Xk}. The sequence of symbols are thenmodulated to carrier frequency fe, transmitted through theacoustic channel and received by an L-channel hydrophonearray. After demodulation, the signal received at the lth arrayelement can be modeled by

yz(t) = L ht,z (t - kTs ) ej()z Xk + nz(t), (1)k

where ht,z(t), nz(t) and ()z are the channel response, ambientnoise, and phase distortion associated with the lth array el­ement, respectively. Temporal variations of the channel arealso included in (1).

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mate of Xk. We let Wl be the length-(Nb + Nf + 1) feed­forward weight vector for the lth array element and v be thelength-(Kb+ K f ) feedback weight vector.

The filter weights are found adaptively by the LMS algo­rithm. After converging approximately to an MMSE solution,the equalizer output can be written

(

1 A A 12

)E exp _xn-gox.L(el) = In xEArn,l a~ (6)

n,m '"'" ( IXn -gox I2 ) ,L...,.xEArn,o exp - a~

where B is the block length, xn = Q(xn ), and Q(-) is afunction that maps an input to the nearest constellation point.The set, {aI, ... ,aM} is the finite alphabet used for M -arysignaling. The variance a; of the residual interference is esti­mated as

Let M' ~ log2 M (interleaved) coded bits, denoted as{Cn,b .. " Cn,M/}, be mapped to a symbol Xn. Let Am ,l andAm,o be the set of all possible symbol values such that themth coded bit is 1 and 0, respectively. Using the estimate go,the soft information on cn,m is represented in the form of alog-likelihood ratio (LLR) [5] as

(5)

(4)

B

2 1 ~ I" "v 12(J'9 = B L..J X n - goxn .

i=l

where L~~~ = InP(xnlcm = 1) - InP(xnlcm = 0). Thecomputation of (6) can be simplified by the log-max approx­imation. These LLRs are deinterleaved and delivered to thechannel decoder, where they play the role of a priori LLRs.The channel decoder generates another set of extrinsic LLRs,

L~~~. These are interleaved and used to obtain the soft sym­bol estimate xn as

where {gk} is the impulse response including the channel, re­ceive filter and equalizer. It has been shown in [5] that theresidual interference term in the bracket in (3) is well ap­proximated by a zero mean Gaussian random variable withvariance go (1 - go). To avoid the use of a channel estima­tor, go should be estimated from the sequence xn . Given thatgo = E [xn IX n] / X n, a simple estimator might be the one thatdetermines X n by hard-decision and estimates go as

3.2. Iterative multi-channel equalization and decoding

Let the received signal vector, Yl,n be [Yl ((2n - Nb)T) , ... ,

Yl ((2n + Nf )T)]T and nl,n be defined from the noise sam­ples similarly. The estimate of X n at the output of the equal­izer is written

Fig. 1. IMED system block diagram.

3. ITERATIVE MULTI-CHANNEL EQUALIZATIONAND DECODING TECHNIQUE

3.1. Receiver structure

The proposed IMED system is depicted in Fig. 1. To make thereceiver robust to symbol timing misalignment and Dopplershift, the received signal at each array element is fractionallysampled, i.e., the sampled signal is given by {Yl (kT) }, whereT = Ts / F. In the sequel, we let F = 2. The IMED scheme

corrects phase distortion by multiplying by e-jOj , where OJ isthe estimate of ()j. Then, the feedforward and feedback filtersare applied to obtain an estimate of the transmitted symbolXk. In the IMED scheme, the multi-channel equalizer (in­cluding the phase recovery block) and the channel decoderexchange soft information in an iterative fashion. Specifi­cally, at the output of the equalizer, the soft input-soft output(SISO) demapper computes soft information on (interleaved)coded bits based on the symbol estimate Xk. This soft in­formation is deinterleaved and delivered to the maximum aposteriori (MAP) channel decoder. In addition to providingthe decoded output, the channel decoder also computes softinformation on the coded bits, which is interleaved and con­verted to a soft estimate of the symbols by the SISO mapper.These soft symbol estimates are used to aid the operation ofthe multi-channel equalizer and its adaptive weight update al­gorithm. Note that the information exchange is performed ona block-by-block basis.

L" ~ H -jOl H [Xn - 1]X n = L..J wl Yl,n e +v X '

l=l n+1(2)

M'

xn = L x II ~ (1 + (2cm - 1) tanh (L}i,~)), (7)xEA m=l

where we define Xn-1 and Xn+1 as [Xn-Kb' ... ,Xn_1]T and

[Xn+1"" ,Xn+Kj]T, respectively, and Xk is the soft esti-

where A denotes the set of all possible values that a symbol

can take and L~;2 the a priori LLR delivered by the channel

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decoder. The estimation error, x n - xn is denoted by Un, andits variance, denoted by a~n' is given by

M'

a~n = L IxI 2 II ~ (1 + (2cm - 1) . tanh (L~;{))xEA m=l

4. EXPERIMENTS

4.1. Environment

where the superscript x(n) denotes a realization of the nthsymbol time, and J.L is the step size. Note that the update al­gorithm for el can be implemented by a second-order digitalphase locked loop (PLL) [1].

In training mode, X n , Xn-l and Xn+l can be replaced bytraining symbols. In decision-directed mode, X n is replacedby xn (== Q(xn)) in the first turbo iteration and by Q(xn ) be­ginning in succeeding iteration. By using the hard decisionof the soft estimate in order to train the filter weights, theprobability that· the adaptive algorithm converges to the op­timal equalizer increases with more iterations. To facilitatetracking of the LMS algorithm without requiring substantialtraining overhead, the adaptation of the equalizer is repeatedover each block until the filter weights are stabilized. For ev­ery repetition (called a 'pass'), the step size J.L is decreased bythe factor of p < 1, since an initially high J.L helps the LMSalgorithm accommodate the initial uncertainty in the trans­mitted data and as the data estimates become better, J.L can bereduced to help the algorithm converge. Note that the equal­izer tap estimates at the end of each pass are used to initializethe equalizer at the start of the next pass. In addition, J.L isweighted depending on the reliability of the training symbolQ(xn ). As such, the step size for the pth pass is given by

J.L == J.Lopp-l (1 - ~~n I{#O}) , (16)amax

where I {.} is an indicator function, and a~ax is the varianceof Un when no a priori information is available. Hence, wereduce the impact of an unreliable training symbol on theweight adaptation.

(9)

(8)

where Xn+l and Xn-l contain the transmitted symbols.Since Xk == Xk + Uk, we can substitute (Xn-l + Un-I) and(Xn+l + Un+l) for Xn-l and Xn+l, where Un-l and Un+l

aredefinedas[un-Kb ,··· ,Un_l]T and [Un+l'··· ,Un+KfJT

,respectively. Then, the gradient of £ with respect to Wl, (h,and v, are given by

3.3. LMS weight update algorithm

The LMS weight update algorithm allows the receiver to findthe optimal filter weights, Wl and v, and phase estimate, {)jadaptively. The LMS algorithm approximately minimizes the

mean square error (MSE), denoted by E [Ixn - XnI2]. It

would be advantageous to define a new cost function by re­placing Xn+l and Xn-l with Xn+l and Xn-l in (2), resultingin a cost function

where 1m(.) is the imaginary part and COy ( .) is the covari­ance matrix of the input vector. The signals Un-l and Un+lare assumed to be uncorrelated with x n -1 and x n+1 as wellas with X n - xn by the orthogonality principle. Note thatthe contributions of soft information are taken into account in(12). It can be shown that the covariance of [U~-l' U~+lJ T

is the diagonal matrix :E with diagonal entries {a~n_Kb'

. .. a2 a2 . . . a2 } . Such a covariance term, Un-I' Un-Kb' , Un-I

allows the LMS algorithm to weight the update equation cor­responding to each feedback filter tap differently based on thereliability of the soft symbol estimates. From (10) through(12), the LMS weight update algorithm is given by

win +1) == win) + J.L (Xn - xn )* Yl,ne-jOl (13)

oin+1) = oin) + J.LIm { (Xn- Xn)' wfIYl,ne-jii,} (14)

v(n+l) == (I - J.L~) v(n) + J.L (Xn - xn)* [~n-l], (15)Xn+l

The environment is described as follows: • Location and time: Buzzards Bay, MA, USA, August, 2007, • Carrier frequency: 100 kHz, • Symbol rate: 41.67k symbolls, • Water depth: 47 ft, • Array deployed depth: 20-25 ft (transmitter), 20ft (receiver) • Array spacing: 0.2m, • Distance between re­search vessels: 100m, 180m, 330m, 420m, 530m.

4.2. Data description

In the experiments, one transmitter and four receiver array el­ements are used. A rate 1/2 convolutional channel code witha (117,155)8 generator polynomial is used for forward errorcorrection. In symbol mapping, BPSK, QPSK, and 4-PAMare employed and pulse shaping is performed by a root raisedcosine filter with roll-off factor 0.2. A total of 6600 symbolsare transmitted for each signaling setup. A total of 600 sym­bols are training symbols, which are used for symbol synchro­nization and equalizer initialization. An ambient noise signalwas separately recorded for the purpose ofperformance eval­uation over various signal to noise ratios (SNR).

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Fig. 2. BER versus SNR for QPSK signaling in 530m data

Table 1. Performance results for several data setsDist. Mod. SNR LMS FRLS IMED

BPSK 27.08 dB 0.0 0.0 0.0100m QPSK 24.43 dB 0.0 0.0 0.0

4-PAM 28.37 dB 0.0 0.0 0.0

BPSK 13.21 dB 0.0 0.0 0.0180m QPSK 10.80 dB 0.0 0.0 0.0

4-PAM 20.95 dB 0.39 0.0001 0.0

BPSK 12.02 dB 0.0 0.0 0.0330m QPSK 9.62 dB 0.0 0.0 0.0

4-PAM 15.51 dB 0.49 0.023 0.0

BPSK 9.87 dB 0.0 0.0 0.0420m QPSK 8.54 dB 0.0051 0.0001 0.0

4-PAM 14.78 dB 0.45 0.008 0.0

BPSK 6.87 dB 0.0 0.0 0.0530m QPSK 3.50 dB 0.0022 0.0017 0.0

4-PAM 12.30 dB 0.47 0.45 0.51

0::wm

-1 1SNR(dB)

4

4.3. Performance results

The performance of three equalization schemes are evaluatedand compared: LMS, FRLS [1], and IMED algorithms. Forfair comparison, the number of filter weights for the three al­gorithms are set equal. We set the size ofthe data block to 200symbols. In the LMS and IMED algorithms, the weight adap­tation is repeated at least 10 times over each block to contendwith slow convergence. The specific (Nb' Nf' Kb' K f) val­ues used in the IMED scheme are as follows: 1) (20,30,10,3)for the 100m-180m BPSK, QPSK data, 2) (30,40,14,4) forthe 100m-180m 4-PAM data and 330m-420m BPSK, QPSKdata, 3) (40, 50, 14,6) for the 330m-420m 4-PAM data, and4) (40, 60, 14, 6) for the 520m data. The parameters p and /-Laare set to 0.8 and 10-3 , respectively.

The bit error rate (BER) values measured by counting er­rors in 6000 information bits are tabulated in Table 1. The cor­responding receive SNRs are also provided. Unfortunately, ingenerating the transmitted data set used in this at-sea exper­iment, interleaving coded bits was not considered. Hence,we applied the IMED algorithm ignoring the use of an inter­leaver and limiting it to only two iterations. Nevertheless, theIMED scheme made no errors in detecting 6000 informationbits, except for the one case 4-PAM signaling in 530m data.Meanwhile, the LMS and FRLS algorithms were unsuccess­ful in the detection of the QPSK and 4-PAM signals in the420m and 530m data. These results indicate the potential per­formance gain of the IMED scheme, which should achieveeven better performance with the aid of an interleaver.

In Fig. 2, BER graphs are provided in terms of SNR forthe three algorithms. The recorded ambient noise is added tothe 530m QPSK received data to obtain the desired SNRs. Itis shown that the IMED algorithm yields a 2 dB SNR gain at

10-3 BER over the FRLS method [1].

5. CONCLUSIONS

An iterative multi-channel equalization technique based onthe turbo principle has been applied to high frequency un­derwater acoustic communications. The IMED scheme suc­cessfully decoded most of the data sets collected during the41.67k symbolls data transmission experiment, suggestinggreat promise for such high frequency underwater applica­tions.

6. REFERENCES

[1] M. Stojanovic, 1. Catipovic, and 1. G. Proakis, "Phase­coherent digital communications for underwater acousticchannels," IEEE Journal ofOcean Engineering, vol. 19,pp. 100-111,Jan. 1994.

[2] Q. Wen and J. A. Ritcey, "Spatial diversity equalizationapplied to underwater communications," IEEE Journal ofOceanic Eng., vol. 19, pp. 227-241, April 1994.

[3] T. Oberg, B. Nilsson, N. Olofsoon, M. L. Nordenvaad, E.Sa~gfelt, "Underwater communication link with iterativeequalization," IEEE Oceans 2006, pp. 1-6, Sep. 2006.

[4] C. Laot, A. Glavieux, and 1. Labat, "Turbo equalization:adaptive equalization and channel decoding jointly op­timized," IEEE Journal on Selected Areas of Commun.,vol. 19,pp. 1744-1751,Sep.2001.

[5] M. T. Tuchler, R. Koetter, and A. C. Singer, "Turbo equal­ization: principles and new results," IEEE Trans. Com­mun., vol. 50, pp. 754-767, May 2002.

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