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Physical Communication 1 (2008) 146–161www.elsevier.com/locate/phycom

Full length article

Efficient processing of acoustic signals for high-rate informationtransmission over sparse underwater channelsI

M. Stojanovic∗

Massachusetts Institute of Technology, Cambridge, MA 02139, United States

Abstract

For underwater acoustic channels where multipath spread is measured in tens of symbol intervals at high transmission rates,multichannel equalization required for bandwidth-efficient communications may become prohibitively complex for real-timeimplementation. To reduce computational complexity of signal processing and improve performance of data detection, receiverstructures that are matched to the physical channel characteristics are investigated. A decision-feedback equalizer is designedwhich relies on an adaptive channel estimator to compute its parameters. The channel estimate is reduced in size by selecting onlythe significant components, whose delay span is often much shorter than the multipath spread of the channel. Optimal coefficientselection (sparsing) is performed by truncation in magnitude. This estimate is used to cancel the post-cursor ISI prior to linearequalization. Spatial diversity gain is achieved by a reduced-complexity pre-combining method which eliminates the need fora separate channel estimator/equalizer for each array element. The advantages of this approach are reduction in the number ofreceiver parameters, optimal implementation of sparse feedback, and efficient parallel implementation of adaptive algorithms forthe pre-combiner, the fractionally-spaced channel estimators and the short feedforward equalizer filters. Receiver algorithm isapplied to real data transmitted at 10 kbps over 3 km in shallow water, showing excellent results.c© 2008 Elsevier B.V. All rights reserved.

Keywords: Underwater acoustic communications; Adaptive channel estimation; Sparse equalization; Phase synchronization; Reduced-complexitycombining

1. Introduction

Underwater wireless (acoustic) communicationshave traditionally relied on noncoherent modula-tion/detection methods to avoid the problems of longmultipath and time variations encountered in the ma-jority of underwater environments, especially on thehorizontal transmission channels. While noncoherentsignaling provides robustness to channel distortions,

I This work was supported in part by the ONR MURI grantN00014-07-1-0738.

∗ Tel.: +1 617 253 7136; fax: +1 617 258 5730.E-mail address: [email protected].

and still represents the preferred method in commer-cially available acoustic modems, it lacks the band-width efficiency necessary for achieving high-ratedigital communications over severely band-limited un-derwater acoustic channels. Bandwidth-efficient under-water acoustic communications can be achieved byemploying spatial diversity combining and equalizationof PSK or QAM signals. The receiver structure thathas been found useful in the majority of present ap-plications is a multichannel decision-feedback equalizer(DFE) [1]. This receiver consists of a bank of adaptivefeedforward filters, one per array element, followed bya decision-feedback filter. It has been implemented inthe prototype high-rate acoustic modem developed at

1874-4907/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.phycom.2008.02.001


M. Stojanovic / Physical Communication 1 (2008) 146–161 147

the Woods Hole Oceanographic Institution, and shownto perform well in a variety of sea trials [2]. Due to thecomplex nature of the propagation channel, whose im-pulse response may extend over several tens or even ahundred milliseconds, causing severe intersymbol in-terference (ISI) at high transmission rates, the signalprocessing required by this receiver may become pro-hibitively complex in certain situations, ultimately lim-iting the achievable bit rate, as well the receiver’s appli-cability to difficult channels. It is of interest for such sit-uations to develop alternative processing methods thatprovide the same or similar performance at a lower com-putational complexity.

Reduction in computational complexity can beachieved in two ways: (1) by using efficient adaptivealgorithms, and (2) by altering the conventional receiverstructure to obtain one with fewer elements that needto be adjusted adaptively. The work in this area todate has focused on using a class of low-complexityLMS algorithms with improved tracking properties [5]and on several techniques for reducing the size ofadaptive filters. In particular, these techniques include(1) reducing the size of the spatial combiner [3]; (2)employing time reversal to perform adaptive matchedfiltering prior to equalization [4], and (3) reducing thesize of the adaptive equalizer through sparsing [6,7].The major focus of the present paper is on the lastform of complexity reduction. Reduced-complexityspatial combining method of [3] is used with adecision-feedback equalization method that is basedon sparse channel estimation. In doing so, our goalis to consider a receiver structure that is matched tothe physical characteristics of the propagation channel.In particular, by tracking the channel explicitly, andnot implicitly through the coefficients of a largeequalizer, it is possible to design an adaptive processingmethod that takes into account only the significantchannel components. The composite time span of thesecomponents is often much shorter than the overallmultipath spread, leading to the desired reduction incomplexity.

Sparse, or tap-selective equalization is a methodthat has been considered for communications overhorizontal underwater acoustic channels [5,6], forbroad-band wireless radio channels [8,9] and forterrestrial HDTV systems [10]. An ad hoc sparseDFE used for equalization of underwater acousticchannels [5] determines the positions of significantequalizer taps by computing the full-size solutioninitially, but keeping only those taps whose magnitudeexceeds a pre-determined threshold. The coefficientsof the feedforward and the feedback filter are updated

as a single-coefficient vector based on minimization ofthe mean-squared error in data symbol estimation. Thistap selection method is not optimal because the inputsignal to the equalizer is not white. Selection of optimaltap locations in such conditions is a difficult probleminvolving an exhaustive search. This fact serves as amotivation for developing a channel-estimation-basedequalizer: the input to the channel estimator is a datasequence of uncorrelated symbols, and, thus, optimaltap selection can be accomplished simply by truncationin magnitude [6].

General methods for estimation of sparse commu-nication channels [11] simultaneously search for thelocation of significant channel taps and their magni-tudes. Casting the channel identification problem intothe framework of on–off keying (a channel tap is ei-ther on or off), these methods employ techniques suchas the Viterbi algorithm or sphere decoding to performeither joint or iterative detection of the channel taps.Their performance was investigated through simulationand analysis, both in terms of the mean-squared channelestimation error, and in terms of the bit error rate thatresults when the channel estimates are used to imple-ment a correlation receiver. The results quantify the ben-efits of tap-selective solutions over the conventional,full-size ones. A method based on the technique ofmatching pursuits, specifically modified for identifica-tion of sparse underwater acoustic channels, was also re-cently proposed [7]. In this method, the iterative searchfor tap locations and magnitudes is extended to includean unknown frequency offset associated with each sig-nificant tap. It was applied to real data, showing supe-rior performance in the cases of extreme difficulty, atthe price of additional computations needed to identifythe significant tap locations.

In this paper, we focus on a method for decision-feedback equalization based on adaptive sparse channelestimation, targeting reduction in computational com-plexity and simplicity of implementation. The equalizerperforms a two-step procedure. First, it uses a channelestimate and previous symbol decisions to determinethe ISI resulting from the post-cursors. The post-cursorISI term is subtracted from the input signal, and theso-obtained signal is equalized by a linear feedforwardfilter. Symbol decisions are then made, which are alsoused for channel estimation in a decision-directed mode.Thus, the feedback filter is not implemented explicitly,but through the channel estimation feedback. The op-timal DFE solution obtained by this method in a sta-tionary case is identical to the one obtained by solvingfor the equalizer coefficients directly. However, in prac-tice this method has several advantages. Decoupling


148 M. Stojanovic / Physical Communication 1 (2008) 146–161

of channel estimation from equalization allows parallelimplementation of efficient adaptive algorithms for thechannel estimator and the feedforward equalizer. Spars-ing of the channel estimate results in the optimal selec-tion of feedback taps which eliminates the unnecessarynoise in data detection, thus improving the receiver per-formance.

In the majority of underwater communicationscenarios, the SNR observed on a single channelis not sufficient for reliable equalizer performance.Hence, it is imperative that some form of multichannelprocessing be used. Decision-feedback equalization viachannel estimation can readily be extended to themultichannel case. In a commonly used multichannelimplementation, every element of the receiving arrayis accompanied by an adaptive feedforward equalizerfilter. To avoid the computational demands of suchimplementation, the front section of the receiver ismodified. It uses a spatial pre-combiner, which reducesthe number of adaptive filters required per inputchannel, but preserves the multichannel processinggain [3].

The paper is divided into three parts. The first part,given in Section 2, outlines the concept of equalizationvia channel estimation. For simplicity, the discussionis limited to the single-channel case. Adaptiveimplementations are discussed and a computationallyefficient channel estimator is proposed. The problemof tap selection is addressed, and illustrated through anumerical example. The second part, given in Section 3,is devoted to the multichannel receiver. This sectionincludes receiver parameter optimization based on aminimum mean-squared error (MMSE) criterion, anda complete algorithm for adaptive implementation ofthe pre-combiner and the equalizer. The algorithm alsoincorporates a multichannel decision-directed carrierrecovery scheme. The third part, given in Section 4,contains the results of experimental data processing.Receiver performance is demonstrated on real datatransmitted using QPSK at 10 kilobits per second over3 km in shallow water, showing excellent results. Theconclusions are summarized in Section 5.

2. Channel-estimation-based equalization

The method for determining the equalizer coeffi-cients from a channel estimate is based on an alternativeinterpretation of the classical MMSE DFE.

2.1. DFE: An alternative interpretation

Let the input signal to the equalizer be the phase-synchronous baseband signal, coarsely aligned intime:

v(t) =

∑n

d(n)h(t − nT )+ w(t) (1)

where d(n) is an i.i.d. sequence of unit-variance datasymbols transmitted at times nT ; h(t) is the overallchannel response, including transmitter and receiverfiltering, and w(t) is the additive noise. This signalis sampled at the Nyqist or higher rate. Without lossof generality, we assume a sampling rate of 2/T fora signal band-limited to 1/T . The signal samples arearranged in a column vector:

v(nT + M1T/2)...

v(nT + T/2)v(nT )

v(nT − T/2)...

v(nT − M2T/2)

=

∑k

h(kT + M1T/2)...

h(kT + T/2)h(kT )

h(kT − T/2)...

h(kT − M2T/2)

×d(n − k)+

w(nT + M1T/2)...

w(nT + T/2)w(nT )

w(nT − T/2)...

w(nT − M2T/2)

(2)

or shortly,

v(n) =

∑k

h(k)d(n − k)+ w(n). (3)

The reference channel vector h(0) has a time span(−M2T/2,M1T/2) which is chosen to capture all ofthe channel response h(t). Coarse alignment is normallyperformed such that the magnitude of cross-correlationbetween the sequences v(nT + lT/2) and d(n) ismaximal for l = 0, in which case the reference elementh(0) is the channel coefficient with maximal amplitude.

The DFE with feedforward filter coefficients akarranged in a vector a, and feedback coefficients bk ,estimate the data symbol as

d(n) = a′v(n)−

∑k>0

b∗

k d(n − k) (4)

where d(n) is the correct data symbol in the trainingmode or the symbol decision in the decision-directedmode. Prime denotes conjugate transpose, and all thevectors are defined as column vectors. In the absenceof decision errors, the optimal choice of the feedbacktaps is the one for which post-cursor ISI is completely



cancelled:

b∗

k = a′h(k), k > 0. (5)

For this choice, the decision variable is expressed as

d(n) = a′

[∑k≤0

h(k)d(n − k)+ w(n)

]= a′v f (n). (6)

This expression is used to determine the MMSEsolution for the feedforward filter in terms of thechannel vector shifts h(k) and the noise covariance N =

E{w(n)w′(n)}. The solution is given by

a = R−1f h(0) (7)

where R f is the covariance of the term v f (n), and, forindependent data symbols, it is given by

R f = E{v f (n)v′

f (n)} =

∑k≤0

h(k)h′(k)+ N. (8)

Expressions (8), (7) and (5) define the classical MMSEDFE. They also provide insight into the needed sizes ofequalizer filters.

A usual approach to implementing an adaptive DFEis to group the feedforward and the feedback filtertaps in a composite vector, and apply a least-squaresalgorithm to compute this vector recursively from theinput signal vector v(n) and the previous decisionsd(n − k), k > 0, using the data estimation error e(n) =

d(n)− d(n). The so-obtained equalizer taps are used tofilter the received signal, and subtract the post-cursorsISI term according to the expression (4).

An alternative implementation is based on theexpression (6). For this implementation, the equivalentfeedforward signal v f (n) has to be obtained first.However, this signal cannot be measured directly.Instead, it can be reconstructed. Reconstruction is basedon the fact that this signal can be represented as

v f (n) = v(n)− vb(n) (9)

where

vb(n) =

∑k>0

h(k)d(n − k) (10)

is the equivalent feedback signal that can be obtainedfrom the previous decisions and a channel estimate.Hence, the modified DFE implementation is defined asfollows:

(1) Using a channel estimate h(0), its shifts h(k) fork > 0, and previous decisions, determine theequivalent feedforward input signal as

v f (n) = v(n)−

∑k>0

h(k)d(n − k)

= v(n)− vb(n). (11)

(2) Apply an adaptive linear equalizer to this signal toobtain the data symbol estimate

d(n) = a′(n)v f (n). (12)

Any adaptation algorithm may be considered.

In this approach, the feedback filter taps bk arenot computed explicitly at all. The feedforward filteroperates on the equivalent signal, from which the post-cursor ISI has been removed. In this sense, adaptivefeedforward filtering is different from the originalapproach in which the filter operates directly on thereceived signal.

A feature worth noting is that the estimate of thefeedback signal, vb(n), obeys a shifting law:

vb(n) =↓ vb(n − 1)+ h(1)d(n − 1) (13)

where ↓ indicates shifting of the vector downward bytwo elements (in general, by as many elements as thereare samples per one symbol interval) and filling the topby zeros. This property eliminates the need to carry outthe entire summation for determining the post-cursorISI every time a new data decision becomes available.It was shown originally in Ref. [5] for a T -spacedequalizer.

2.2. Adaptive channel estimator and equalizer imple-mentation

There are many possibilities for implementing thechannel estimator and the equalizer adaptively. Eachshould be chosen according to the channel at hand.Some scenarios are the following:

1. Fixed channel estimate/adaptive equalizer. Channelis estimated from the packet preamble, and thisestimate is frozen for the duration of the packet.Short feedforward equalizer is adapted throughoutthe packet. This approach is suitable for high-speedradio receivers, where computational complexity isof paramount importance, but the channel can safelybe assumed constant over the packet duration [8,9].

2. Adaptive channel estimate/adaptive equalizer. Chan-nel estimator and a short equalizer are updatedthroughout the packet. Channel estimation is inde-pendent of equalization, except that it relies on sym-bol decisions in the decision-directed mode. Thisapproach is the first choice for underwater acous-tic communications that use packets long enoughto support significant channel changes. The receiver



structure also allows arbitrary choice of updatingintervals.

The equalizer and the channel estimator are updatedseparately. Thus, they may use different adaptivealgorithms. For channel estimation, a number ofadaptive algorithms can be devised based on themodeling Eq. (1). For instance, the estimates of channelcoefficients h(nT ) can be obtained from the coefficientsof an adaptive filter that uses the data sequence d(n) asits input to estimate the signal v(nT ). For a fractionalspacing of T/2, two estimators are needed to generatea T/2-spaced channel response. Since computationalcomplexity is the focal point of this work, and thechannel responses are expected to be of considerablelength as measured in the number of samples M =

M1 + M2 + 1, it is of interest to use a computationallysimple adaptive algorithm for channel estimation. Thealgorithm proposed below is based on the fact that theinput signal can be modeled as in the expression (3).From this expression it follows that

h(0) = E{v(n)d∗(n)}. (14)

To obtain an estimate of the above vector, a simplestochastic approximation can be used:

h[n] = (1 − λch)

n∑i=0

λn−ich v(i)d∗(i) (15)

where h[n] denotes the estimate of h(0) obtained inthe nth iteration, i.e., h[n] = h(0, n), and λch is anexponential forgetting factor. The scaling factor (1 −

λch) ensures an asymptotically unbiased estimate. Thisexpression gives way to a very simple recursion:

h[n] = λch h[n − 1] + (1 − λch)v(n)d∗(n). (16)

The estimate h[n] is chosen to span all of thesignificant channel response, but it suffices to keep onlythe channel coefficients with significant amplitude. Inthis regard, a key feature of the algorithm is that thechannel coefficients are updated independently. Whilesuch a method is not optimal if there is correlationbetween the channel coefficients, it allows fast trackingof the channel response.

Note that there is a choice when it comes to channelestimation: to update the entire channel estimate vector,or to update only the selected taps. Updating the entirevector provides constant on-line monitoring of the timevariations in the channel (even though all of the tapsare not used for post-cursor ISI suppression). Thisapproach may be useful if tap migration is present due toresidual Doppler effect (note that this may be differentfor different taps [7]). Updating only the selected taps

provides reduction in computational complexity. Thechoice should be made based on the properties of aparticular channel, and the desired performance. Thesimplicity of the algorithm presented above serves tosomewhat reconcile the two requirements.

The adaptive algorithm for the equalizer is com-monly chosen from the LMS or the RLS family. A least-squares algorithm for the equalizer vector a will oper-ate on the equivalent input signal v f (n), driven by theerror e(n). There is also a possibility that the feedfor-ward equalizer be calculated from the channel estimate,making use of the relation (7). The matrix inverse inthis case would be computed at the beginning of a datapacket, and possibly frozen for the packet duration onthe grounds that the channel covariance changes moreslowly than the channel realization. However, the ad-vantage of updating the equalizer independently is thatit may compensate to some degree for the channel esti-mation errors.

2.3. Tap selection

Decomposing the decision-feedback equalizationinto the channel estimation part and the feedforwardequalization part eases the difficult problem of optimalDFE tap selection. As the feedback is implementeda priori through the channel estimate, feedback tapselection is obviously made by choosing only thesignificant coefficients of the channel estimate. Aslong as the input to the channel estimator is a datasequence of uncorrelated symbols, optimal feedbacktap selection is performed by choosing only thechannel estimate coefficients whose amplitude is abovesome threshold. In addition, this choice can be madeadaptively throughout the packet. (Note the differencebetween sparsing the channel estimate and sparsing thecoefficients of the equalizer.)

Selection of the best feedforward filter taps, however,remains a difficult optimization problem. In general, notall the M coefficients will be updated in the equalizervector, as it is desired to use only a short feedforwardfilter with N < M nonzero taps. The problem offeedforward equalizer tap selection is investigatedbelow. Three approaches come to mind:

1. Optimally sparsed filter. The search is conductedamong all the sparsing patterns S to find the one forwhich the minimum mean-squared data estimationerror obtained with the filter of given size N isminimal:

Emin(N ) = minSM×N

{1 − h′(0)S[S′R f S]−1S′h(0)}.(17)

The selection matrix S is of size M×N . Each columncontains exactly one 1 among all zeros. Its meaning



is that of selecting the elements of the vector v f (n) to

form an input signal v(s)f (n) = S′v f (n) for the sparsefeedforward filter with N ≤ M nonzero taps.

Note from the expression (17) that tap selectionaccording to magnitude would be optimal only ifR f ∼ I, i.e. for uncorrelated input signal. However,this is not the case in the equalization problem.Finding the optimal sparsing pattern in generalinvolves an exhaustive search whose complexity maybe prohibitive for a practical implementation.

2. Approximation of the optimization criterion to allowfor a more efficient search.

One such approximation is investigated inRef. [6]. In this reference, a T -spaced feedforwardfilter is considered and the approximation is madeby searching among all the single-tap filters andselecting those N which result in the lowest MMSEs.In other words, only the M selection vectors s of sizeM × 1 are considered. Each vector contains a single1 at a different location, and results in the MMSE

E(1) = 1 − h′(0)s[s′R f s]−1s′h(0). (18)

The M values of this single-tap MMSE arecomputed, and the N lowest ones are identified. Thecorresponding taps are chosen to form the equalizervector.

3. An ad hoc method. For example, a feedforwardfilter can be chosen with contiguous taps, butof total length much shorter than the lengthof the channel estimate. (Ideally, the number ofcontiguously-spaced feedforward taps will be chosenat least equal to the number of channel taps.) Inreference to the channel vector, the feedforwardfilter coefficients of the DFE are arranged as a′

=

[0 · · · 0 a(−N1) · · · a(N2)0 · · · 0]∗. The channels of

interest are characterized by large M , and inparticular by large M1, the part of the responseresponsible for post-cursor ISI. In this case, it willbe of interest to use a short feedforward filter withN1 < M1 taps. This approximation is justified by thefact that post-cursor ISI has been cancelled from thefeedforward filter input. It seems to be effective forchannels with decaying multipath intensity profile aslong as N is large enough to capture sufficient signalenergy. Extending the feedforward span N1 improveson the matched filtering, but increases the irreduciblepre-cursor ISI.

Another ad hoc method would be to assignfeedforward filter taps to match the significant taps ofthe channel estimate h(0). However, its effectiveness isnot obvious except in special cases of distinctly sparsechannels with comparable energy of multipath arrivals.

Fig. 1. Performance of the channel-estimation-based DFE. ad hocsparsing of the feedforward filter. N = 3,M = 19, Nt = 38,G0 =

1/6, λeq = 0.999, λch = 0.99, SNR = 20 dB.

2.4. Example

To demonstrate the design concepts, and to analyzethe various tap selection methods, a numerical exampleis constructed. A two-path time-invariant channel isconsidered, with path amplitude ratio 2/1 and relativepath delay of 4.25 symbols. The transmitter filter isa spectral raised cosine with roll-off factor 0.25 andtruncation length of ±4 symbol intervals. (This is theactual filter response used to generate experimentalsignals of Section 4.) The channel introduces additivewhite Gaussian noise (AWGN). The SNR is 20 dB. Thechannel estimator looks 2 symbols to the left (M2 = 4taps) and 7 symbols to the right (M1 = 14 taps) ofthe reference tap. The feedforward filter uses 3 tapsaround the main arrival (N1 = 1, N2 = 1) andan RLS algorithm. The DFE performance is shown inFig. 1. The figure shows the estimated MSE, the outputscatter plot, the tap magnitudes of the feedforwardfilter, the true channel response (dotted) and the sparsechannel estimate (solid). The receiver is trained duringthe first 38 data symbols, and then switched into thedecision-directed mode. There are no decision errors.The channel taps whose magnitude is less than 1/6 ofthe main tap magnitude are set to zero. In this manner,out of the total of 19 taps, 5 are kept to be used for post-cursor ISI cancellation. An ad hoc choice of few taps



Fig. 2. Theoretical performance of the optimally sparsed DFE. “Bestselection MMSE” is the value of Emin(N ), the MMSE obtained withthe optimally selected positions of N feedforward taps. SNR of 20dB and uncorrelated noise are assumed for evaluating the MMSE.The channel is truly sparse, and shown in the upper right corner. Theoptimal feedforward tap magnitudes for several sizes N are shownin the lower right corner. Lower left corner shows the MSE obtainedby a single-tap equalizer, which is used in tap selection based on anapproximation of the optimal criterion. The best choice in this case istap # 3 (this choice results in E(1) = Emin(1)). The second best istap #2, then 4, 12, 11, etc.

around the main arrival works very well in this type ofchannel.

The problem of optimal tap allocation is analyzedin Fig. 2. This figure shows the theoretical results forthe sparse MMSE DFE. The channel is truly sparse,as shown in the upper right corner. The total channellength is set to M = 15. Shown in the upper left corneris the MMSE versus the number of equalizer taps whichare selected optimally (tap selection according to rule1 of Section 2.3). Naturally, the MMSE decreases asthe total number of taps increases, and the improvementobtained after a certain size becomes negligible. It isinteresting to compare the magnitudes of the optimallychosen feedforward filter taps to the channel responsemagnitude. The feedforward equalizer tap magnitudesare shown below the channel response for several totalequalizer lengths N . The equalizer size N = 5 is thefirst whose optimal selection includes taps outside ofthe main arrival region. Although as N increases, theoptimally selected taps tend to include the region ofthe second arrival, they are not limited to it. Tap

Fig. 3. SNR at the equalizer output as a function of the thresholdused for sparsing the channel estimate. The input SNR is determinedby the AWGN power. Threshold is given relative to the absolutevalue of the maximum channel coefficient: a threshold of 0 indicatesthat all coefficients are kept in the channel estimate used for post-cursor ISI calculation, while a threshold of 1 indicates that only one(the strongest, or the reference coefficient) is kept. The feedforwardequalizer uses 3 contiguously-spaced taps around the reference tap;the channel estimate has 19 coefficients. The curves are obtained byaveraging 500 independent simulation runs.

selection according to the approximate rule proposed inRef. [6] (rule 2 of Section 2.3) is illustrated in the lowerleft corner. Interestingly, if this rule were applied todetermine the best 5 taps, those taps would correspondto the nonzero elements of the channel vector (taps 2, 3,4, 11 and 12, as numbered in the figure). Application ofthe optimal rule, however, gives a different solution inwhich more taps are allocated to the region of the mainarrival (taps 2, 3, 4, 5, and 12). These observations stressthe difficulty of choosing the tap selection criterionfor a practical implementation. However, the fact thatboth rules contain taps in the region of the main arrivalprovides an encouragement for choosing the taps inan ad hoc manner (criterion 3 of Section 2.3). Thisrule, which requires no computations, is used in theexperimental data processing described in Section 4.

While the number of feedforward taps is determinedin advance and their location chosen according to one ofthe rules described, the sparse feedback is implementedby truncating the channel estimate. The choice oftruncation threshold obviously influences the receiverperformance. This effect is illustrated in Fig. 3 whichshows the output SNR as a function of the truncationthreshold. Simulation results demonstrate the existenceof an optimal threshold for which the output SNR is



Fig. 4. Reduced complexity multichannel DFE incorporates a K to P pre-combiner and a P-channel DFE. The DFE is based on channel estimationand sparsing.

maximized. Corresponding to the optimal threshold isthe number of channel coefficients that are kept forevaluating the post-cursor ISI. As expected, this numberis less than the total number of coefficients used torepresent the channel response. At threshold 0, all 19coefficients are used. The number of coefficients keptat threshold 0.1 is 5–6 (exact value depends on thenoise realization), and decreases to 2–3 at threshold0.5. After the threshold increases above 0.6, only thestrongest coefficient is kept in the channel estimate.The value of the optimal threshold obviously dependson the particular channel response and the input SNR.While this dependence may be difficult to evaluateanalytically, a threshold close to optimal can easilybe found in practice by tuning the receiver whilemonitoring the average-squared error during training.

3. Multichannel receiver

In the previous section, the principles of channel-estimation-based DFE were described and demon-strated on a simple example generated by computersimulation. In order to apply these principles to realdata, however, a multichannel processing gain is re-quired in the majority of underwater acoustic commu-nication scenarios. Hence, the sparse DFE needs to becast in the framework of a multichannel receiver.

3.1. Receiver structure

A multichannel receiver in its conventional formconsists of a number of input channels, K , to eachof which an adaptive feedforward filter is assigned.This receiver structure is effective for various types

of underwater acoustic channels; however, for a largenumber of input channels, its computational complexitybecomes high, because each of the K channelsrequires an adaptive filter of length that is oftenmeasured in tens of coefficients. To alleviate thisproblem, a computationally efficient LMS algorithmwas considered in Ref. [2] as an alternative to a fastRLS, but the inevitable trade-off in convergence ratewas found to limit its usefulness.

Further reduction in complexity is possible by reduc-ing the size of the front spatio-temporal processor [3].In this approach, the K spatially distributed input chan-nels are first combined into a smaller number of chan-nels, P . No temporal processing (filtering) is used indoing so, but only weighted combining using adaptivelydetermined weights. The resulting P channels are thenequalized. The approach of pre-combining, or reduced-complexity combining has proven to be very effective inprocessing different types of real data. What is interest-ing to bear in mind is that a small value of P , usuallyonly two or three channels (but rarely one) is suffi-cient for extracting the multichannel processing gain.This property stems from the broad-band nature of un-derwater acoustic communication signals. The reduced-complexity multichannel structure is found to be ap-propriate for use with sparse, channel estimation aidedDFE, for reasons that will become apparent shortly. Theresulting receiver structure is shown in Fig. 4.

In addition to multichannel combining, a practicalreceiver must incorporate phase tracking. Specifically,let the input signals be given by

vk(t) =

∑n

d(n)hk(t − nT )e jφk (t) + wk(t),



k = 1, . . . , K (19)

where, as before, d(n) denotes the transmitteddata sequence and wk(t) is the additive noiseobserved at the kth receiving element, but the phasedeviation is included explicitly as φk(t), rather thanbeing incorporated into the (complex-valued) channelresponse hk(t). This is done so as to separate the rapidtime-variation of the phase from that of the channelwhich usually varies much more slowly. While the rapidvariation must be tracked by a phase-locked loop (PLL),the slow one can be handled by the channel estimator.There are several choices for implementing a decision-directed PLL. Phase tracking can be performed beforecombining, i.e., individually for all K channels; afterpre-combining, i.e., using only P phase estimates, orafter linear equalization using a single-phase estimate.In what follows, the second choice will be used, on thegrounds that the K phases φk(t) are usually correlated,but the P signals obtained after pre-combining maycontain independently varying components. If this is notthe case, and there is strong correlation between the Pphase estimates, modification of the phase-locked loopfrom P channels to a single channel is straightforward.Since P is usually chosen to be small, there is not muchto be gained in complexity reduction by eliminatingthese additional phase estimates.

The question that arises in the multichannel receiverdesign is how many channel estimates are needed, or,equivalently, at which point in the receiver should thepost-cursor ISI be removed. If the channel responseshk(t) from the transmitter to each of the receivingelements were known, this could be done directly onthe K input signals vk(t). However, we already knowthat a single-input signal does not yield sufficient SNRfor data detection (hence multichannel processing), andconsequently, it is not suitable for individual channelestimation either. Thus, a processing gain must beextracted before channel estimation is attempted. Theapproach of pre-combining is precisely suited for thispurpose. As shown in Fig. 4, the channel estimates areformed after extracting the pre-combining gain.

3.2. Parameter optimization

Receiver parameters that need to be determinedinclude the K × P pre-combiner weights, arranged in amatrix C; the feedforward filter coefficients arranged inP vectors denoted by ap; the P phase estimates θp, andthe P channel estimates denoted by fp, p = 1, . . . , P .These parameters are optimized jointly in a manner thatminimizes the MSE in data detection.

The signals obtained after pre-combining are givenby

x p(t) =

K∑k=1

c∗

p,kvk(t) = c′pv(t), p = 1, . . . P (20)

where cp is the pth column of the pre-combining ma-trix C. The signals are sampled once every Ts seconds.(As before, we may assume Ts = T/2.) Let V(n) de-note the matrix of all signal samples used for processingat time instant nT , associated with detection of the nthdata symbol d(n):

V(n)

=

v1(nT + M1Ts ) · · · v1(nT ) · · · v1(nT − M2Ts )

.

.

....

.

.

.

vK (nT + M1Ts ) · · · vK (nT ) · · · vK (nT − M2Ts )

.(21)

The signals at the output of the pre-combiner can nowbe arranged in a matrix

X(n) = C′V(n)

=

x1(nT + M1Ts ) · · · x1(nT ) · · · x1(nT − M2Ts )

.

.

....

.

.

.

xP (nT + M1Ts ) · · · xP (nT ) · · · xP (nT − M2Ts )

=

xT1 (n)· · ·

xTP (n)

. (22)

The signal xp(n) is modeled as

xp(n) =

∑i

fp(i)d(n − i)e jθp(n) + zp(n) (23)

where

fp(i) =

f p(iT + M1Ts)

...

f p(iT )...

f p(iT − M2Ts)

(24)

represents the i-shift of the vector fp(0) which is cen-tered so as to capture all of the channel response (asseen at this point in the receiver); θp(n) represents thephase deviation, and zp(n) represents the additive noise.It is the set of channel responses fp(0), p = 1, . . . , Pthat will be estimated and used for post-cursor ISI sup-pression, which is thus performed after pre-combining.

Phase correction is performed using the estimates θpof the phases θp. This operation yields the signals

xpθ (n) = xp(n)e− j θp(n), p = 1, . . . , P (25)

which are the inputs to the P-channel DFE.



The analysis of the multichannel DFE that useschannel estimates fp(i) closely follows that of a single-channel DFE given in Section 2.1 In a conventionalmultichannel DFE representation, with feedback taps{bi }, i > 0, the data symbol estimate is obtained as

d(n) =

P∑p=1

a′pxpθ (n)−

∑i>0

b∗

i d(n − i). (26)

Substituting for the signals xpθ (n), and assuming for themoment that the phase estimates are correct, one obtainsthe condition for perfect post-cursor ISI cancellation:

b∗

i =

P∑p=1

a′pfp(i), i > 0. (27)

Using the channel estimates for the lack of true valuesin the above expressions, the data symbol estimate iscomputed as

d(n) =

P∑p=1

a′p[xpθ (n)−

∑i>0

fp(i)d(n − i)]. (28)

In the last expression, we recognize the backward signal

xpb(n) =

∑i>0

fp(i)d(n − i) (29)

and the forward signal, i.e., the input to the pthfeedforward filter

xp f (n) = xpθ (n)− xpb(n). (30)

The estimation error is given by

e(n) = d(n)− d(n). (31)

This error is used to obtain the MMSE solution forthe receiver parameters. In a practical implementation,the MMSE solution is computed recursively tosuit the time-varying nature of the channel. Theresulting adaptive algorithms for the pre-combiner, theequalizers, the PLL and the channel estimates are givenbelow.

The pre-combiner. To arrive at the solution for thepre-combiner weights, we use the fact that

xp(n) = VT(n)c∗pe− j θp(n). (32)

The data estimate d(n) is then written as

d(n) = [c′

1 . . . c′

P ]

V(n)a∗

1e− j θ1(n)

...

V(n)a∗

P e− j θP (n)

−[a′

1 . . . a′

P ]

x1b(n)...

xPb(n)

= c′y(n)− a′xb(n) (33)

where we have defined the composite vectors c and aof the pre-combiner and the equalizers, respectively, aswell as the equivalent pre-combiner input signal y(n).Having expressed the estimated quantity as an innerproduct of the coefficient vector to be determined andthe equivalent input signal (plus a fixed term that doesnot influence the optimization), the MMSE solutionfor the pre-combiner coefficients is obtained as theusual Wiener filter. The coefficients can be obtainedrecursively as

c(n + 1) = c(n)+ A1[y(n), e(n)] (34)

where A1[u(n), e(n)] denotes an adaptive algorithmA1, such as LMS or RLS, that operates on the inputsignal vector y(n) and the estimation error e(n). Forexample, for K = 8 and P = 2, which are the valuesthat will be used in the following section when real-data processing is described, there is a total of 16 pre-combiner coefficients. This is a value small enough topermit even the use of a standard RLS algorithm.

The equalizer. To obtain a recursion for updating theequalizer coefficients, the data estimate is expressed as

d(n) = [a′

1 . . . a′

P ]

x1 f (n)...

xP f (n)

= a′x f (n). (35)

The composite feedforward filter vector is updated as

a(n + 1) = a(n)+ A2[x f (n), e(n)] (36)

where A2 is an adaptive algorithm not necessarily thesame as A1. Regarding the equalizer size, a word ofcaution is in order. Although the vectors ap in ourtreatment so far have been assumed to be of sizeM , only N ≤ M of these elements are actually used.The others are set to zero, and, thus, do not need tobe updated. The choice of the N elements has beenaddressed in Section 2.3 for the single-channel case.In the multichannel case, of course, all the channels donot need to have the same selection of N taps, but thesame general principles apply. In the experimental dataprocessing described in the following section, all theequalizers will use an ad hoc method, with the same Ncontiguous taps selected around the reference channeltap.

The PLL. To obtain the algorithm for updating thephase estimates θp, the data symbol estimate is written



as

d(n) =

P∑p=1

αp(n)e− j θp(n) −

P∑p=1

αpb(n) (37)

where

αp(n) = a′pxp(n) (38)

αpb(n) = a′pxpb(n). (39)

The phase estimate is chosen to minimize the MSEin data estimation, and is computed recursively usinga second-order stochastic gradient approximation. Theinstantaneous gradient of the MSE with respect to thepth phase estimate is given by

∂|e2(n)|

∂θp= 2Re

{∂e(n)

∂θpe∗(n)

}

= −2 Im{αp(n)e− j θp e∗(n)}. (40)

From this expression follows the second-order update:

ψp(n) = Im{αp(n)e− j θp(n)e∗(n)}

θp(n + 1) = θp(n)+ K f 1ψp(n)+ K f 2

n∑i=0

ψp(i) (41)

where K f 1 is the proportional tracking constant, andK f 2 is the integral tracking constant, often chosen asK f 2 = K f 1/10.

The channel estimator. At last, it remains to specifythe channel estimation algorithm. The estimate of fp(0),generated at the nth iteration is denoted as fp(0, n) =

fp[n]. The estimates of vectors fp(1, n), needed todetermine the post-cursor ISI, are computed by shiftingthe vectors fp[n]. Since the model (23) implies that

fp(0) = E{xp(n)e− jθp(n)d∗(n)}, p = 1, . . . , P (42)

the channel estimation algorithm described in Sec-tion 2.2 is applicable to the multichannel case as well.The only difference is that the phase correction needs tobe taken into account. Thus, the phase-corrected signalsare used for channel estimation:

fp[n] = λch fp[n − 1] + (1 − λch)xpθ (n)d∗(n),

p = 1 . . . , P. (43)

Each of the estimated vectors fp[n] is of length M ,which may be large if it is to span all of the channelresponse. However, not all of the M coefficients areused for post-cursor ISI estimation. Sparsing of thechannel estimates is performed in an optimal mannerby truncation.

A rapidly varying channel necessitates trackingof more than just the selected taps. An underwaterchannel with a moving transmitter/receiver is such anexample. Changes in the propagation path length causedrifting of the channel taps which can be capturedby constant monitoring of all the taps. Alternatively,the position of the strongest channel tap could becontrolled by performing explicit bit synchronization,i.e., by incorporating a delay-locked loop (DLL) into thereceiver. Once the locations of significant taps do notchange with time, adaptive channel estimation can beconfined to those taps only. For the present application,fractional spacing of the equalizer suffices to extract thecorrect bit timing.

Finally, it has to be emphasized that the post-cursorISI term xpb(n) is given by (29) in terms of a sumthat involves all the positive shifts of the estimatedchannel vector. However, the summation does not needto be carried out explicitly, because the shifting propertystill holds in the multichannel case. Computation of thepost-cursor ISI term is greatly simplified by using thefollowing recursion:

xpb(n + 1) =↓ xpb(n)+ fp(1, n)d(n),

p = 1, . . . , P (44)

where ↓ means shifting downwards by the oversam-pling factor (2, for a T/2 fractional spacing), andfp(1, n) =↓ fp[n] is the shifted version of the currentchannel vector estimate. We note again that not all ofthe M elements of xpb(n) are actually computed, butonly those N that are needed for equalization.

In summary, the K input channels are pre-combinedinto P ≤ K , which are used for channel estimation andequalization. A total of M signal samples per channelare used to obtain the M channel coefficients, but anumber of those coefficients, say Moff, are set to zerowhen evaluating the post-cursor ISI. At the same time,only N ≤ M signal samples are used for equalization.The positions of the Moff taps are determined on-line, by thresholding. The positions of the N equalizertaps could theoretically be calculated on-line from thechannel estimate; however, this calculation is extremelycomplex, and, thus, the N taps will be selected a prioriin a practical implementation.

3.3. Algorithm summary

The algorithm for adaptive multichannel DFE,based on spatial pre-combining, channel estimation andsparsing, is defined by the following steps carried out ateach iteration n:



(1) Form the matrix of input signals V(n). This matrixis of full-size K × M .

(2) Compute the pre-combiner output X(n) =

C′(n)V(n). This matrix is of size P × M . (P ≤ K ,i.e., size of spatial processing is reduced.)

(3) Compute the phase-corrected signals:

Xθ (n) =

e− j θ1(n) · · · 00 · · · 0

0 · · · e− j θP (n)

× X(n) =

xT

1 (n)e− j θ1(n)

...

xTP (n)e

− j θP (n)

=

xT1θ (n)...

xTPθ (n)

. (45)

This matrix is of size P × M .(4) According to a tap selection rule for equalization,

take N elements from each row of Xθ (n), andarrange them in a matrix X(s)θ (n) of size P × N .(N ≤ M , i.e., size of temporal processing isreduced.)

(5) Compute X(s)b (n) by selecting the elements ofXb(n), the matrix whose rows are the post-cursorISI terms xT

pb(n).

(6) Compute X(s)f (n) = X(s)θ (n) − X(s)b (n), the inputsignals to the feedforward equalizer filters.

(7) Form the K × N matrix V(s)(n) of selected signalcomponents, and determine the equivalent pre-combiner input

y(n) =

V(s)(n)a(s)∗1 (n)e− j θ1(n)

...

V(s)(n)a(s)∗P (n)e− j θP (n)

. (46)

(8) Compute αp(n) = a(s)′

p (n)x(s)pθ (n), and αpb(n) =

a(s)′

p (n)x(s)pb(n). The quantities with superscript (s)are of length N .

(9) Compute the data symbol estimate as in (37).(10) After training (n > Nt ), compute the data symbol

decision d(n) from the estimate d(n), using thedecision regions appropriate to the modulationmethod used. Any linear modulation applies.

(11) Compute the error as in (31).(12) Update the phase estimates as in (41).(13) Update the pre-combiner weight vector

c(n + 1) = c(n)+ A1[y(n), e(n)]. (47)

(14) Update the equalizer vectors

a(s)(n + 1) = a(s)(n)+ A2[x(s)f (n), e(n)] (48)

where x(s)f (n) is formed by arranging the rows of

X(s)f (n) into a vector.

(15) Update the matrix of channel estimates

F[n] = λchF[n − 1] + (1 − λch)Xθ (n)d∗(n). (49)

(16) Truncate the channel estimates

F(t)[n] = F[n]|elements less than threshold set to 0. (50)

(17) Compute the post-cursor ISI term

XTb (n + 1) =↓ XT

b (n)+ ↓ F(t)T [n]d∗(n). (51)

Initially, the values of the equalizers, the channelestimates, the phase estimates and the post-cursor ISIterms are all set to zero, and the pre-combiner weightsare initialized such that arbitrarily chosen P inputchannels are passed to the equalizers unchanged, whilethe remaining channels are cut off. Updating of the pre-combiner is delayed with respect to the equalizer bya certain number of iterations Ntc. In the examples tofollow, both the pre-combiner and the equalizers usethe form II RLS [12]. This algorithm is of quadraticcomplexity; however, the two algorithms A1 and A2are run in parallel, which increases the overall speedof computations. At the same time, it allows the pre-combiner and the equalizer to use different forgettingfactors.

4. Experimental results

The algorithm described in the previous section wasimplemented in Matlab and used for off-line processingof experimental data. The experimental data, providedby the Woods Hole Oceanographic Institution, werecollected in the Continental Shelf region near the coastof New England. The water depth was between 100m and 200 m. The signals were transmitted usinga carrier frequency of 25 kHz, over the range of3 km. The modulation format was QPSK, and thesignals were transmitted at 10 kilobits per second.The vertical receiver array consisted of eight omni-directional hydrophones, spaced by 0.03 m. In thissection, results of signal processing are demonstratedusing an average quality data set.

The channel responses, recorded at four of the inputsensors, are shown in Fig. 5. These responses representsnapshots obtained from the channel probe transmittedbefore the actual data. They can be used to determineroughly the extent of multipath; however, they do notprovide information about the time variability of thechannel. The channel response consists of the principalcluster of arrivals whose delay spread is of the order of15 symbol intervals. In addition, there is a distant clusterof arrivals, approximately 225 symbol intervals away(the distant cluster is quite apparent in the response



Fig. 5. Recorded channel responses.

of the second sensor). Within each cluster, there areseveral distinct peaks of the response magnitude. Therelative energy of multipath arrivals varies with thesensor location.

As a reference, Fig. 6 shows the results ofsignal processing using a conventional multichannelequalizer in a full-complexity configuration. The inputto the receiver is the raw received signal, brought tobaseband using the nominal carrier frequency, sampledat 2/T , and frame-synchronized using a 13 elementBarker code. No phase synchronization or bit-timingadjustment is performed on this signal. The receiveruses eight feedforward filters, each of size 16. Thefeedback filter is implemented explicitly, using 25 taps.The total of 8×16+25 = 153 equalizer parameters areupdated as a single vector using a numerically stablefast RLS [13] with a forgetting factor 0.99. There are2000 symbols in a data packet, 300 of which are usedfor training. A single-phase estimate is used for all eightchannels. The details of tracking parameters are givenin the figure. The performance of this receiver is ofmoderate quality, as seen by the output scatter plot andthe estimated error probability (fraction of erroneousbits in the data packet). Note that if it were desiredto extend the feedback section to take into accountmore distant arrivals, the size of the feedback filterwould become a limiting factor in the efficiency of thealgorithm that updates all the equalizer coefficients as asingle-parameter vector.

In the absence of noise and time variability ofthe channel, the performance of the full-complexity

Fig. 6. Results of full-complexity multichannel processing. DFE isimplemented explicitly. Figure shows the MSE, the phase estimateand the output scatter plot of the estimated data symbols. Equalizercoefficients are updated using the fast RLS algorithm [8].

receiver represents a bound on the performance of thereduced-complexity receiver. However, in a practicalimplementation this may not be the case. Fig. 7shows the results of data processing using the reduced-complexity channel-estimation-based DFE. The eightinput channels are pre-combined into two channels,which are then equalized, i.e., processed by the twochannel estimators that accompany the two feedforwardfilters. The channels are estimated over a total lengthof M = 25 taps. The truncation threshold for channelestimate sparsing is set to G0 = 1/6, i.e., all thechannel taps whose magnitude falls below 1/6 ofthe strongest tap magnitude (computed individuallyfor each of the two estimators) are set to zero. Thisprocedure results in Moff = 18 taps being turnedoff in each of the channel estimators. The remaining7 taps are used in post-cursor ISI cancellation. Thefeedforward filters have N = 17 taps per channel.The forgetting factors of the equalizer, the pre-combinerand the channel estimators are indicated in the figuretogether with the PLL tracking constants. The pre-combiner (16 taps) and the two-channel equalizer (34taps) each use a standard RLS algorithm, while thechannel estimators are updated using the algorithm (43).Adaptation is carried out continuously throughout thedata packet. The two-channel PLL tracks a Doppler shift



Fig. 7. Results of reduced-complexity multichannel processing. DFEis implemented via channel estimation. Figure shows the MSE, thephase estimates and the output scatter plot of the estimated datasymbols. The pre-combiner and the equalizer coefficients are updatedusing standard RLS algorithms; the channel estimates are updatedusing the algorithm (43).

of approximately −15 Hz with good accuracy. The MSEindicates steady convergence, and the scatter plot showsno errors in the detection of the data block.

Increasing the number of equalizers (pre-combineroutputs) to more than two does not result in performanceimprovement on this channel. This fact illustrates thepower of reduced-complexity multichannel combining.The pre-combiner succeeds in suppressing the distantmultipath arrivals (more than 200 symbols away), asevidenced by the fact that only a short feedback isneeded for suppression of ISI from within the firstcluster. The small number of feedforward coefficientsallows efficient processing without unnecessary noiseenhancement.

The channel estimates obtained during data process-ing are shown in Fig. 8. Recall that these estimatesrepresent the channel responses as seen after pre-combining. Shown in the figure are both the completechannel estimates and their values after truncation. TheISI extends over 10 to 15 symbols. The channels exhibita fair degree of time variability. The variation occurs inboth the amplitudes and arrival times of signals prop-agating over different paths. The latter is evident fromthe variation in position of the selected channel coeffi-cients: the position of the strongest coefficient changes

from the reference value 0 at the beginning of the datablock to a lag of 4 (2 symbol intervals) at the end of thedata block. For this reason, all the coefficients in thechannel estimates need to be continuously monitoredand updated. The adaptive fractionally-spaced equaliz-ers can then successfully compensate for the motion-induced drifting of the channel taps, as long as theresulting Doppler spread is relatively low.

As it was pointed out earlier, one advantage ofthe channel-estimation-aided DFE is that the channelestimator operates separately from the rest of thereceiver, and thus provides the possibility to monitorthe channel response without affecting the equalizersize. This point is illustrated in Fig. 9, which shows theresults of processing the same data packet with the spanof channel estimators extended to include the distantmultipath arrivals. The receiver performance in thiscase is again very good, with no detection errors. (Theincrease in the MSE occurs at the time when the latearrival becomes visible to the equalizer.) The channelestimators capture the distant arrivals, 225 symbolsaway from the main arrival. This example demonstratesa powerful advantage of sparse channel estimation:the overall performance is not affected by the largeextent of the feedback, because only the significanttaps are used for ISI suppression. It is thus possibleto constantly monitor the channel changes withoutaffecting the equalizer size (the feedforward filters inthis example have 9 taps only). If the feedback sectionof a conventional DFE were extended to cover 225 post-cursors, this would significantly slow the convergence,and also restrict the tracking speed (by constrainingthe choice of the LMS step size or the fast RLSforgetting factor). As different channel taps are updatedindependently in the algorithm proposed, convergenceremains fast, channel estimation is not overly sensitiveto the choice of forgetting factor λch , and this factor isnot restricted by the estimator size.

5. Conclusions

Reduction of computational complexity is a problemof paramount importance for real-time implementationof high-speed underwater acoustic receivers whichrequire sophisticated multichannel equalizers. Thisproblem was addressed from the viewpoint of reducingthe size of adaptive filters needed to perform spatialand temporal signal processing. In particular, themethod of channel-estimation-aided decision-feedbackequalization was used in conjunction with multichannelspatial diversity pre-combining. By reducing thenumber of channels that are equalized, and by



Fig. 8. Channel estimates after pre-combining. Shown on the left are the full-channel estimates. Shown on the right are their truncated values whichcontain only the significant coefficients used for equalization.

selecting only the significant components of thechannel estimates, not only is the efficiency of signalprocessing increased, but its performance is improvedby eliminating the unnecessary noise from the detectionprocess.

The ideas that govern the receiver implementationare:

(1) Pre-combining of a larger number of input channelsinto a few that will be equalized.

(2) Adaptive channel estimation and sparsing.(3) Subtraction of post-cursor ISI using the channel

estimate feedback prior to adaptive feedforwardequalization.

The advantages of this method are the following:

(1) Preservation of full-multichannel processing gainwithout temporal processing in each channel.

(2) Optimal selection of the channel taps by simpletruncation in magnitude (rather than direct selectionof the equalizer taps).

(3) Efficient post-cursor ISI calculation using theshifting proposition (44).

(4) Use of a short feedforward equalizer (possibly muchshorter than the multipath spread).

(5) Computationally efficient estimation of thefractionally-spaced channel response using the re-cursion (43).

(6) Continuous channel monitoring.(7) Flexibility to increase the feedback length on

demand without affecting feedforward equalization.(8) Parallel implementation of adaptive algorithms

for channel estimation, equalization and pre-combining.

The algorithm that incorporates the above principles,together with a method for data-directed phase tracking,was successfully demonstrated on real data.

Further research in this area will likely concentrateon highly-mobile underwater acoustic communications,with relative transmitter/receiver velocities of the order



Fig. 9. Results of reduced-complexity multichannel processing. DFEis implemented via channel estimation. Figure shows the estimatedmean-squared error, the channel estimates and truncation thresholds(approximately same for the two estimates) at the end of the datapacket, and the output scatter plot. In this case, the distant arrivalsare taken into account.

of tens of meters per second. As the speed of acousticsignal propagation underwater (1500 m/s) is verylow, Doppler effects become a major limitation inmobile underwater channels. Signal processing basedon channel estimation provides a natural framework forthe development of algorithms capable of dealing withextreme motion-induced signal distortions.

References

[1] M. Stojanovic, J. Catipovic, J. Proakis, Adaptive multichannelcombining and equalization for underwater acoustic communi-cations, J. Acoust. Society Amer. 94 (3) (1993) 1621–1631. Pt.1.

[2] L. Freitag, M. Grund, S. Singh, J. Partan, P. Koski, K. Ball,The WHOI micro-modem: An acoustic communications andnavigation system for multiple platforms, in: Proc. IEEE OceansConf. 2005.

[3] M. Stojanovic, J. Catipovic, J. Proakis, Reduced-complexitymultichannel processing of underwater acoustic communicationsignals, Journal of the Acoustical Society of America 98 (2)(1995) 961–972. Pt. 1.

[4] H.C. Song, W. Hodgkiss, W. Kuperman, M. Stevenson,T. Akal, Improvement of time-reversal communications usingadaptive channel equalizers, IEEE J. Oceanic Eng. 31 (2) (2006)487–496.

[5] M. Johnson, L. Freitag, M. Stojanovic, Efficient equalizer updatealgorithms for acoustic communication channels of varyingcomplexity, in: Proc. IEEE Oceans Conf, 1997.

[6] M. Stojanovic, L. Freitag, M. Johnson, Channel-estimation-based adaptive equalization of underwater acoustic signals, Proc.IEEE Oceans Conf. (1999).

[7] W. Li, J. Preisig, Estimation of rapidly time-varying sparsechannels, IEEE J. Oceanic Eng. 32 (4) (2007) 927–939.

[8] S. Ariyavisitakul, L. Greenstein, Reduced-complexity equaliza-tion techniques for broad-band wireless channels, IEEE Trans.Commun. (1997) 5–15.

[9] S. Ariyavisitakul, N. Sollenberger, L. Greenstein, Tap-selectabledecision feedback equalization, IEEE Trans. Commun. (1997)1497–1500.

[10] I. Fevrier, S. Gelfand, M. Fitz, Reduced complexity decisionfeedback equalization for multipath channels with large delayspreads, IEEE Trans. Commun. (1999) 927–937.

[11] C. Carbonelli, S. Vedantam, U. Mitra, Sparse channel estimationwith zero tap detection, IEEE Trans. Wireless Commun. 6 (5)(2007) 1743–1753.

[12] S. Haykin, Adaptive Filter Theory, 2nd edn, Prentice Hall, 1991.[13] D. Slock, T. Kailath, Numerically stable fast transversal filters

for recursive least squares adaptive filtering, IEEE Trans. SignalProc. (1991) 92–114.

Milica Stojanovic graduated from the Uni-versity of Belgrade, Serbia, in 1988, and re-ceived the M.S. and Ph.D. degrees in elec-trical engineering from Northeastern Univer-sity, Boston, MA, in 1991 and 1993. She iscurrently a Principal Scientist at the Mas-sachusetts Institute of Technology, and alsoa Guest Investigator at the Woods HoleOceanographic Institution. Her research in-terests include digital communications theory,

statistical signal processing, communications networks, and their ap-plications to underwater acoustic communication systems.

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Author's personal copymillitsa/resources/pdfs/mchdfe-elsevier.pdf · adaptive lters. In particular, these techniques include (1) reducing the size of the spatial combiner [3]; (2)