UNDERWATER ACOUSTIC CHANNEL ESTIMATION USING STRUCTURED SPARSITY

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1 UNDERWATER ACOUSTIC CHANNEL ESTIMATION USING STRUCTURED SPARSITY Ehsan Zamanizadeh a, João Gomes b, José Bioucas-Dias c, Ilkka Karasalo d a,b Institute for Systems and Robotics, Instituto Suerior Técnico, Technical University of Lisbon, , Lisbon, Portugal, {ezamanizadeh, jg}@isr.ist.utl.t c Instituto de Telecomunicações, Instituto Suerior Técnico, Technical University of Lisbon, , Lisbon, Portugal, jose.bioucas@lx.it.t d Aeronautical and Vehicle Engineering, Marcus Wallenberg Laboratory, KTH Royal Institute of Technology, SE , Stockholm, Sweden, ilkkak@kth.se Abstract: A novel framework for collaborative estimation of multile underwater acoustic (UWA) communication channels, considering the high correlation of received signals in a linear hydrohone array, is develoed and evaluated. Throughout this work, the channel is assumed to be a time-variant linear system and is reresented by its 3D delay-dolerdeth function (DDDF), an extension of the 1D imulse resonse of LTI systems. To jointly estimate the DDDF coefficients for all hydrohones in a linear receiver array, sarse estimation techniques such as Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP) are used. A new structured dictionary matrix is introduced to encomass the linear structure of energetic arts of the DDDF corresonding to wavefronts iminging uon the array. The over-comlete structured dictionary is built by concatenating small blocks, each defining a candidate wavefront in the 3D DDDF image. Blocks are swet according to a grid of sloes, delay and Doler shifts. To reduce the roblem size and increase the erformance, lines are relaced by Gaussian tubes in the structured dictionary. These can accommodate wavefronts that deviate slightly from the linear assumtion. For near-field scenarios where wavefronts are more markedly nonlinear, an additional ste is alied to fully retrieve the evolution attern in the DDDF by solving a fast BP roblem with narrow delay/doler suort. The collaborative framework makes it ossible to leverage the satial dimension to detect arrival atterns across the array whose estimation would be too unreliable based on a single hydrohone, while retaining reasonable comutational comlexity. The ractical feasibility of the develoed schemes is assessed for various channel configurations using both simulations and exerimental data collected during the CALCOM 10 sea trial in Faro, Portugal, 010. Keywords: Underwater Acoustic Communication, Channel Estimation, Structured Sarsity, Grou Matching Pursuit, Basis Pursuit, Delay Doler Deth Function (DDDF) 1375

2 1. INTRODUCTION Time-varying channel resonses may be modeled by (D) Delay-Doler Sread Functions (DDSF), a generalization of the concet of time-invariant channel imulse resonse to the time-frequency lane that introduces a Doler dimension [1]. The received signal is considered as a sum of relicas of the transmitted signal, each associated with a given delay and Doler shift that are assumed ersistent over an averaging san []. Nearly all DDSFs of ractical UWA are sarse with most of the energy localized in several small regions []. Exloiting sarsity is a key insight for attaining a generic delay- Doler reresentation of underwater channels with manageable comlexity using sarse estimation methods. This work resents a framework for identifying UWA time-varying channels using a 3D delay-doler-deth function (DDDF). Fig. a shows thresholded coefficients of a DDDF in a volumetric lot which may be viewed as the satio-temoral skeleton of the acoustic field, comrising several wavefronts that iminge uon a receiver array, after interacting with the surface and/or bottom. Such functions are reresented by a otentially large set of coefficients, but sarsity ensures that most of them are zero, excet for a small subset that exlains how the observed channel oututs are roduced from an inut signal. The main contribution of this work is develoing collaborative aroaches for channel estimation of multile UWA communication links, focusing on high correlation of received signals in a linear array. Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP) are used to jointly estimate the DDDF coefficients for all hydrohones. A new concet for building the over-comlete dictionary matrix is resented to leverage the linear structure of energetic arts of the DDDF corresonding to wavefronts iminging uon the array. In addition to fast channel estimation, this aroach detects the key skeleton of the acoustic field contained in DDDFs and contributes to imrove the robustness of subsequent detection algorithms that utilize that structure.. TIME-VARYING CHANNEL MODEL A samled baseband reresentation of a time-varying channel, for the transmitted signal, x(n), and the received signal, y(n), is adoted with the following discrete-time inut-outut model, y( n) u ( ),,, x n k k l k l l x ( n) x( n) e l j n l, (1) where the samling frequency, fs, is a multile of the inut signal bandwidth and l l /( Tfs ) for an inut block of duration T. The DDSF, whose samles u k,l aear in (1), is the Fourier transform of the channel imulse resonse along the time variable. In a multiath channel the (continuous) DDSF comrises a set of imulses in delay-doler, N U (, ) ( ) ( ). () 1 The channel model (1) is linear in the DDSF coefficients, and may be written in matrix form as y = Xu, where y denotes a vector of M observed samles, u holds the DDSF 1376

3 coefficients to be determined, and X is the known dictionary matrix. To address the collaborative channel estimation roblem, the channel model is reresented as Y = XU, where Y and U are, resectively, the matrix of M received samles and the matrix of unknown DDSF coefficients for all hydrohones. (a) (b) Fig. 1: (a) A samle Delay-Doler-Deth Function (DDDF) for an array of 16 hydrohones with equal sacing. (b) Arrival time delay vs. hydrohone deth..1. Individual Channel Estimation Through Basis Pursuit Methods Basis Pursuit (BP) techniques are used to find sarse aroximate solutions to large underdetermined linear systems of equations. According to the original BP rincile, a signal is decomosed into a suerosition of highly redundant dictionary signals, and an otimal set of weights is found such that the resulting coefficient vector has minimum l1 norm. Among several variations of BP that have been roosed [3], we are mainly interested in solving unconstrained l-l1 otimization roblems of the form 1 min u y-xu +τ u 1, (3) where the first term measures how well the candidate solution fits the observed data while the second one acts as a regularizer for setting zero for small coefficients. The regularization arameter τ controls the relative weight of the two terms. Sarse Reconstruction by Searable Aroximation (SaRSA) and Two-ste Iterative Shrinkage/Thresholding (TwIST) [4] are two elegant methods for solving unconstrained l-l1 otimization roblems with comlex variables and data [5]. 3. STRUCTURED CHANNEL ESTIMATION Matching Pursuit iteratively decomoses a signal into a linear exansion of waveforms that are selected from a redundant dictionary. Both MP and OMP sequentially select dominant tas of the DDSF that maximize the rojection of the residual observation vector onto the corresonding symbol vector and then calculate ta coefficients. The difference is that MP calculates each ta coefficient directly from the rojection, while OMP derives a joint LS solution for the coefficients of all the selected tas []. To jointly estimate the DDSF for all hydrohones in the receiver array, a new Grou Matching Pursuit aroach is introduced. This is more efficient and accurate than indeendently rocessing each hydrohone in the array to obtain the set of (correlated) 1377

4 channel estimates. In this scheme the structured dictionary matrix encomasses the linear structure of energetic arts of the DDDF corresonding to wavefronts iminging uon the array (see Fig. 1). As Fig. illustrates, the over-comlete structured dictionary, D, is built by concatenating small blocks, each defining a candidate wavefront in the 3D DDDF image. Blocks are swet according to a grid of sloes, delay and Doler shifts. To reduce the roblem size and increase the erformance, lines are relaced by Gaussian tubes in the structured dictionary. These can accommodate wavefronts that deviate slightly from the linear assumtion. Fig. : (a) Matrix of M observed samles for all hydrohones. (b) 3D delay-doler-deth matrix. (c) Structured dictionary matrix The roosed joint channel estimation technique consists of two main stes. At each iteration, the first ste detects the active wavefront from the structured dictionary matrix D that correlates best with the aroximation residual from the revious iteration, Rt-1, t H Ds Rt 1 arg max, (4) s I t 1 Ds where It-1 is the index set of all reviously selected blocks. The initial residual is the observation vector, R0 = Y. At each iteration, the residual matrix, R, is built by attaching the residual vectors, r, comuted for each receiver. When the same structured dictionary matrix is reeatedly alied to different observation vectors, the online comutation of inner roducts can be eliminated by recomuting a table with all inner roducts with blocks of the dictionary matrix. Estimation of the unknown coefficients is done in the second ste of this Grou Matching Pursuit (GMP) aroach. For near-field scenarios where wavefronts are more markedly nonlinear, an alternative ste is alied to fully retrieve the evolution attern in the DDDF by solving a fast BP/OMP roblem with narrow delay/doler suort, defined by a local over-comlete dictionary matrix for each receiver. This is termed here Grou Matching/Basis Pursuit (GMBP). The stoing criterion can be based on evaluating t at each iteration, comaring to 0 for the first selected wavefront. The whole roosed collaborative channel estimation scheme can be summarized as follow: 1. Initialize R0 = Y. Select a block from the structured dictionary matrix as in (4) 3. Estimate the coefficients for each hydrohone using narrow delay/doler suort alying the BP method as described in section.1 with local dictionary matrix Xl. 4. Udate I t I t 1 and rt rt 1 X lu for each receiver. 1378

5 5. If the stoing criterion is not met go to the nd ste In [6] a robust aroach is resented to deal with wavefront classification and assign the aroriate number of surface and bottom bounces to a roagation ath detected in the 3D channel resonse, considering ossible omission or dulication of some aths. Fig. 3: Simulation results on channel estimation. (a) Individual channel estimation using BP methods (TwIST). (b) Grou Matching Pursuit (GMP). (c) Grou Matching/Basis Pursuit (GMBP). 4. PERFORMANCE ASSESSMENT Performance evaluation of the collaborative estimation of sarse DDDF in singlecarrier (QPSK) transmissions over simulated and real underwater channels is resented in this section. Simulation results are obtained using an underwater acoustic simulator develoed by the University of Algarve. The transmission uses 5.5 khz carrier frequency, 4.5 khz bandwidth, root-raised-cosine (RRC) ulse shaes, and total duration 1 s. The baseband received signal is samled at 4 times the symbol rate, fs=1 khz. Fig. 3 shows the DDDF estimation results for this channel using simulated data. Comaring to individual channel estimation resented in Fig. 3.a it is seen that both structured sarsity techniques detect the major structure of the DDDF. However, GMBP rovides better channel estimates than GMP. The exerimental results are based on data collected during the CALCOM'10 sea trial which was conducted south of Faro, Portugal, on June 010. The receiver was a vertical drifting array with 16 uniformly-saced hydrohones from 6 m to 66 m deth and the source was attached to the boat at 10 m deth. We focus on QPSK ackets at 5.6 kbit/s, with 4.5 khz bandwidth, 5.5 khz carrier frequency (refer to [6] for more details). As the exerimental channel estimation results of Fig. 4 illustrate, GMBP (Fig. 4.b) catures the effective suort region for the DDDF reasonably well. In this case, due to the high noise level in the exerimental data, GMBP fails to detect the tomost wavefront shown in Fig.4.a. Also due to the same noise issue in ractical situations, GMP was found to rovide oor erformance for collaborative channel estimation in CALCOM 10 data. 5. CONCLUSION This work addresses the roblem of collaborative channel estimation of multile underwater acoustic (UWA) communication channels in a linear receiver array, alying sarse estimation techniques such as Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP). 1379

6 Fig. 4: Channel estimation results using exerimental data. (a) Individual channel estimation using BP methods (TwIST). (b) Grou Matching/Basis Pursuit (GMBP). Comaring to individual channel estimation, the resented collaborative scheme makes it ossible to leverage the satial dimension to detect arrival atterns across the array whose estimation would be too unreliable based on a single hydrohone, while retaining reasonable comutational comlexity. Using both simulated and real data, the erformance of the develoed collaborative channel estimation aroach is comared to the individual Basis Pursuit aroach which has been roosed reviously for similar uroses. Not only does this aroach lead to fast channel estimation, but the induced structured sarseness also detects the skeleton of the acoustic field contained in DDDFs and thereby contributes to imrove the robustness of subsequent detection and estimation algorithms that exloit that structure. Acknowledgements: This work was suorted by Fundação ara a Ciência e a Tecnologia (roject PTDC/EEA-CRO/10443/008 and ISR/IST lurianual funding), and EU roject MORPH (grant agreement no ) under the 7 th Framework Programme. REFERENCES [1] P. A. Bello, Characterization of randomly time-variant linear channels, IEEE Transactions on Communications Systems, vol. CS-11, , December [] W. Li and J. C. Preisig, Estimation of raidly time-varying sarse channels, IEEE Journal of Oceanic Engineering, vol. 3, no. 4, , October 007. [3] A. Bruckstein, D. Donoho, and M. Elad, From sarse solutions of systems of equations to sarse modeling of signals and images, SIAM Review, vol. 51, no. 1, , February 009. [4] J. Bioucas-Dias and M. Figueiredo, A new TwIST: Two-ste iterative shrinkage/thresholding algorithms for image restoration, IEEE Trans. on Image Processing, vol. 16, no., , December 007. [5] E. Zamanizadeh, J. Gomes, and J. Bioucas-Dias, Identification of sarse time-varying underwater channels through basis ursuit methods, 10 th Euroean Conference on Underwater Acoustics (ECUA 10), Istanbul, Turkey, July 010. [6] E. Zamanizadeh, J. Gomes, and J. Bioucas-Dias, Wavefront Segmentation and Classification for Model-Based Underwater High-Frequency Tomograhy, OCEANS 1 MTS/IEEE, Virginia Beach, USA, October