Within the scope of sparse signal representation, we consider the problem of velocity ambiguity mitigation for wideband radar signal. We present a Bayesian robust algorithm based on a new sparsifying dictionary suited for range-migrating targets possibly straddling range-velocity bins. Numerical simulations on experimental data demonstrate the ability of the proposed algorithm in mitigating velocity ambiguity

Bidon, Stéphanie

Lasserre, Marie

Le Chevalier, François

Open Archive Toulouse Archive Ouverte

 Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr  Open Archive TOULOUSE Archive Ouverte (OATAO)  OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.  This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 16753 To cite this version: Lasserre, Marie and Bidon, Stéphanie and Le Chevalier, François Velocity ambiguity mitigation of off-grid range migrating targets via Bayesian sparse recovery. (2016) In: 2016 IEEE Statistical Signal Processing Workshop (SSP), 26 June 2016 - 29 June 2016 (Palma de Mallorca, Spain).  Official URL: http://dx.doi.org/10.1109/SSP.2016.7551791 VELOCITY AMBIGUITY MITIGATION OF OFF-GRID RANGE MIGRATING TARGETS VIABAYESIAN SPARSE RECOVERYMarie Lasserre, Ste´phanie BidonISAE/University of Toulouse, Toulouse, FranceFranc¸ois Le ChevalierMS3, Delft University of Technology,Delft, The NetherlandsABSTRACTWithin the scope of sparse signal representation, we considerthe problem of velocity ambiguity mitigation for widebandradar signal. We present a Bayesian robust algorithm based ona new sparsifying dictionary suited for range-migrating tar-gets possibly straddling range-velocity bins. Numerical sim-ulations on experimental data demonstrate the ability of theproposed algorithm in mitigating velocity ambiguity.Index Terms— Bayesian estimation, Sparse signal repre-sentations (SSR), wideband radar1. INTRODUCTIONA radar system transmitting a train of pulses at constant pulserepetition frequency (PRF) suffers from two types of ambi-guities: range ambiguities (if the PRF is too high) or veloc-ity ambiguities (if the PRF is too low) [1]. Moreover, onecannot increase the ambiguous range without decreasing theambiguous velocity, and reciprocally. Wideband radar sys-tems offer an alternative to the problem of ambiguity removal[2, 3]. More precisely, if we assume that a low PRF is used,there are no range ambiguities but many velocity ambiguities.Additionally, using a wide instantaneous bandwidth improvesrange resolution, so that fast moving targets are likely to mi-grate during the coherent processing interval (CPI) leading torange-velocity coupling. Unlike conventional Doppler phasemeasurement, this range walk phenomenon can give unam-biguous velocity measurement.Sparse signal representation (SSR) can be of particular in-terest when trying to remove velocity ambiguities since it al-lows the estimation of a sidelobe-less signal of interest (SOI)[4],[5, chap.5]. SSR relies on a sparsifying dictionary thatresults from a discretization of the range and velocity dimen-sions (in the case of this bi-dimensional problem) and hencedefine some analysis grid. In [6] an SSR algorithm was devel-oped within this framework and proven to mitigate velocityambiguities, provided that target’s features belong to the anal-ysis grid. Otherwise, as mentioned in [7, 8, 9, 10] a problemThe work of M. Lasserre and S. Bidon is supported by the DirectionGe´ne´rale de l’Armement via a DGA-MRIS scholarship.of grid-mismatch might arise and disrupt velocity ambiguitymitigation. In [7, 8] two SSR algorithms robustified towardsgrid mismatch are presented in the case of narrowband radarsignal where 1D and 2D Fourier analysis grids are used re-spectively (resp.). Here, the novelty lies into the new sparsi-fying dictionary that takes into account range migration, andthe corresponding estimation scheme; the algorithm is devel-oped within a Bayesian framework.The remaining of the paper is organized as follows. First,the wideband signal model is described in Section 2. Then,the proposed Bayesian hierarchical model and the associatedestimation scheme are presented in Section 3 and 4 resp. Theproposed algorithm is finally evaluated on experimental datain Section 5.2. SIGNAL MODELThe signal model used to describe migrating targets has beenintroduced in [6]. In this paper, a row-vectorized version ofthis model is usedy =N∑n=1αnan + n (1)where each vector has a length KM , K being the number ofsubbands and M the number of pulses corresponding to thereceived samples gathered in vector y. n is the noise vector;αn, an and N represent the complex amplitude and signatureof the nth scatterer, and the total number of scatterers in thescene. The scatterer signature a involved in (1) is the productof a conventional 2D-cisoid with cross-coupling terms thatmodel the range migration, i.e.,[a]m+kM = ej2pi−τ0BKk+2vfccTrm1+ BKfck(2)where fc is the carrier frequency, B the bandwidth, Tr thepulse repetition interval (PRI); τ0 is the initial round-trip de-lay of the scatterer and v its radial velocity. fd , 2vfcTr/cis the normalized Doppler frequency of the scatterer, whilefr , τ0B/K represents its normalized range frequency. Inthe following equations, the fractional bandwith per subbandµ = B/(Kfc) is used.Within the scope of SSR methods, the signal y is refor-mulated asy = Hx+ n (3)where x is a sparse vector that represents the targets ampli-tude andH is aKM×K¯M¯ sparsifying dictionary that stemsfrom a discretization of the range and velocity dimensions,K¯M¯ being the dimensions of reconstruction of the scene.Then, using (2) the i¯-th column of the dictionary H is ex-pressed as[hi¯]i =1√KMexp {−j2pifri¯k} exp {j2pifdi¯(1 + µk)m}where fri¯ and fdi¯ are the normalized range and Doppler fre-quencies from the range and velocity analysis grids. The ta-ble indices k, m, i and k¯, m¯, i¯ used to index the sparsifyingdictionary H are related as i = m + kM (k ∈ {0,K −1}, m ∈ {0,M − 1}) and similarly for i¯, k¯, m¯.The range analysis grid is divided into K¯ bins with a pos-sible zero-padding factor, so that K¯ = nrzpK. Then, the nor-malized range Doppler frequency fri¯ of a scatterer located atthe k¯-th range bin is expressed as fri¯ = k¯/K¯.In order to mitigate velocity ambiguity, the range-velocitymap is unfolded with a factor of nva on the velocity axis.Taking into account positive and negative velocities leads usto consider a velocity analysis grid that goes from −nvava/2to nvava/2, where va = c/(2fcTr) is the ambiguous velocity.This grid is divided into M¯ bins with M¯ = nvzpnvaMwhere nvzp represents the zero-padding factor per velocity am-biguity. The normalized Doppler frequency is then fdi¯ =nvam¯′/M¯ where (assuming M¯ is even) m¯′ = m¯ if m¯ ∈{0, ..., M¯/2− 1}; m¯′ = m¯− M¯ if m¯ ∈ {M¯/2, ..., M¯ − 1}.As explained in [7, 8], the range and velocity of the tar-gets present in the radar scene does not necessarily coincidewith that of the sparsifying dictionary H . Following the ap-proach proposed in [8] two mismatch vectors (εv, εr) are in-troduced to address the mismatch problem on the velocity andrange axis resp. They parametrize the sparsifying dictionaryH such thatfdi¯ =m¯′ + εvi¯M¯nva, fri¯ =k¯ + εri¯K¯. (4)Generally speaking, the problem (3) is ill-posed becauseK¯M¯ >> KM (especially when the range-velocity maps areunfolded, i.e., nva > 1); in this paper, the problem is regular-ized using a Bayesian approach.3. HIERARCHICAL BAYESIAN MODELA Bayesian framework is set up in order to estimate the pa-rameters of interest x and (εv, εr). The hierarchical Bayesianmodel adopted is that of [8] so it is briefly recalled in this sec-tion. However, note that the sparsifying dictionary in case ofmigrating targets will demand a new method for the samplingof the mismatch parameters (εv, εr).3.1. LikelihoodThe noise vector n is assumed white and distributed follow-ing a centered Gaussian distribution with power σ2. The like-lihood function is then expressed asf(y|σ2,x, εv, εr) =exp{−||y −H(εv, εr)x||2σ2}(piσ2)KM. (5)3.2. Prior model3.1.1) Target amplitude vector xThe elements xi¯ , [x]¯i of the amplitude vector are as-sumed independent and identically distributed (iid) accordingto the following mixed type probability density function (pdf)pi(xi¯|w, σ2x) = (1−w)δ(|xi¯|)+w1piσ2xexp{−|xi¯|2σ2x}. (6)The prior (6) is denoted as xi¯|w, σ2x ∼ BerCN(w, 0, σ2x). Itpromotes the sparsity of the radar scene, while decorrelatingsparsity level and target power via its mixed-type structure.3.1.2) Noise power σ2An inverse-gamma prior is chosen for the white noisepower σ2; it is denoted IG (γ0, γ1) and can be expressed aspi(σ2|γ0, γ1) ∝ e−γ1/σ2(σ2)γ0+1I[0,+∞)(σ2) (7)where γ0, γ1 are resp. the shape and scale parameters.3.1.3) Target signal power σ2xSimilarly to σ2, an inverse-gamma prior is chosen forthe target signal power σ2x and is denoted as σ2x|β0, β1 ∼IG (β0, β1).3.1.4) Level of occupancy wIf no information is available about the sparsity level ofthe target scene, a convenient prior is a uniform pdf over theinterval [0, 1], i.e., w ∼ U[0,1].3.1.5) Grid errors (εv, εr)The joint prior pdf of the grid error in the i¯th analysisbin (εvi¯, εri¯) is chosen conditioned to the magnitude of xi¯; weconsider that the (εvi¯, εri¯|xi¯) are iid with conditional joint pdfpi(εvi¯ , εri¯ |xi¯ = 0) =δ(εvi¯ )δ(εri¯ ) (8a)pi(εvi¯ , εri¯ |xi¯ 6= 0) =I[−0.5,0.5](εvi¯ )I[−0.5,0.5](εri¯ ) (8b)where IA(.) is the indicator function of the set A.4. BAYESIAN ESTIMATIONWe propose an estimation scheme of the target scene viathe parameters of interest x, (εv, εr) based on the Bayesianhierarchical model described in Section 3. More precisely,our objective is to obtain the minimum mean square error(MMSE) estimators of x and (εv, εr)xˆMMSE =∫xf(x|y)dx, (9a)(εˆv, εˆr)MMSE =∫(εv, εr) f(εv, εr|y)dεvdεr. (9b)However, (9) seems intractable to derive analytically, soa Monte-Carlo Markov Chain (MCMC) is implemented [11].More specifically, an hybrid Gibbs sampler [11, chap.10] isused, which simulates iteratively samples θi(t) according totheir conditional posterior distribution f(θi|y,θ−i) whereθ =[σ2, εvT , εrT ,xT , w, σ2x]Tand θ−i is the vector θwhose ith element has been removed. After a burn in timeNbi, the samples are distributed according to their posteriordistribution f(θi|y). When enough samples are acquired(Nr), the MMSE estimators can be built empirically asθˆiMMSE = N−1rNr∑t=1θi(t+Nbi). (10)The conditional posterior distributions are obtained from thejoint posterior pdf of σ2,x, εv, εr, w, σ2x|y. In particular, vec-tors x and (εv, εr) are sampled element-wise following theconditional posterior distributions of xi¯ and (εvi¯ , εri¯) that arederived from the conditional joint posterior distributionf(εvi¯ , εri¯ , xi¯|y, εv−i¯, εr−i¯,x−i¯, σ2, w, σ2x)∝ exp{− [|xi¯|2||hi¯(εvi¯ , εri¯ )||2 − 2Re{x∗¯ihi¯(εvi¯ , εri¯ )Hei¯}]σ2}× pi(xi¯|w, σ2x)pi(εvi¯ , εri¯ |xi¯) (11)where we used (5) along with the fact that y−H(εv, εr)x =ei¯ − hi¯xi¯ with ei¯ = y −∑l 6=i¯ hl(εvl , εrl )xl.4.1. Sampling of x, σ2, w, and σ2xFollowing [7], x is sampled element-wise; the i¯th element ofx follows the distribution BerCN (wi¯, µi¯, η2i¯ ) withη2i¯ =(σ−2 + σ−2x ||hi¯(εvi¯ , εri¯ )||2)−1(12a)µi¯ =σ−2η2i¯ hi¯(εvi¯ , εri¯ )Hei¯ (12b)wi¯ =w(1− w)σ2xη−2i¯ exp{−η−2i¯|µi¯|2}+ w. (12c)Note that with the sparsifying dictionaryH adopted||hi¯(εvi¯ , εri¯ )||2 = 1.The conditional posterior distributions of σ2, σ2x and ware given byσ2|y,x, εv, εr ∼ IG (γ0 +KM, γ1 + ||y −H(εv, εr)x||2)σ2x|y,x ∼ IG(||x||0 + β0, ||x||2 + β1)w|y,x ∼ Be (||x||0 + 1, K¯M¯ − ||x||0 + 1)where ||x||0 is the number of non-zero elements in x.4.2. Sampling of (εv, εr)The parameters (εv, εr) are jointly sampled element-wise.Using (11),(8) and ||hi¯(εvi¯ , εri¯ )||2 = 1 we obtainf(εvi¯ , εri¯ |y, εv−i¯, εr−i¯,x, σ2;xi¯ = 0) = δ(εvi¯ )δ(εri¯ ) (13a)f(εvi¯ , εri¯ |y, εv−i¯, εr−i¯,x, σ2;xi¯ 6= 0) (13b)∝ exp{KM−1∑i=1κi cos(2pi[εvi¯M¯nva(1 + µk)m− kεri¯K¯]− φi)}knowing that (εvi¯, εri¯) ∈ [−.5, .5]2 and where κi = |[zi¯]i|,φi = ∠[zi¯]i, with zi¯ = 2σ−2x∗¯i [h∗¯i (0, 0) ei¯].We can see that range and velocity indices are coupled indistribution (13), contrary to that of [8]. Besides, it does notbelong to any familiar class of distribution so a Metropolis-Hastings move is added to the Gibbs sampler in order to sam-ple jointly (εvi¯, εri¯). In the MH algorithm, samples are drawnfrom a proposal distribution and accepted or rejected with agiven acceptance ratio [11]. The shape of the target distribu-tion (13) depends on the signal-to-noise ratio (SNR): for highSNR values, it is peaked around the true values of (εvi¯, εri¯),whereas it is flat in the case of low SNR values. In this paper,we employ postprocessing SNR, defined as |xi¯|2/σ2. Thus,the choice of the proposal is chosen conditionally to the SNRvalue: i) a uniform proposal distribution for low SNR values,εvi¯, εri¯∼ U[−.5,.5]2 . ii) a Gaussian proposal distribution forhigh SNR values, εvi¯, εri¯∼ N (µε,Σε).This sampling procedure was already used in [8], but hereestimation of the mean and covariance matrix of the Gaussianproposal is adapted to the case of migrating targets due tocoupling of indices k and m in (13). Moreover, the samplingprocedure is improved wrt that of [8] in the sense that esti-mation of the parameters of the proposal distribution is moreaccurate (as described later). This ensures better sampling ofthe parameters (εv, εr), but at the cost of computational effi-ciency.Estimation of µε First, the mean is estimated as the valuesof (εvi¯, εri¯) giving the maximum of (13). To do so, (13) isreformulated asf(εvi¯ , εri¯ |y, εv−i¯, εr−i¯,x, σ2;xi¯ 6= 0) ∝ I[−.5,.5](εvi¯ )I[−.5,.5](εri¯ )exp{Re{KM−1∑i=1[zi¯]i exp {j2pi (frcik − fdci(1 + µk)m)}}}with frci = εri¯/K¯ and fdci = nvaεvi¯/M¯ . One can recog-nize in this last expression the coherent integration with mi-gration compensation of vector zi¯ [5, p.522]. Hence, µε canbe estimated as the location of the maximum of this coherentintegration restricted to the domain (εvi¯, εri¯) ∈ [−.5, .5]2.Estimation of Σε Then a second-order Taylor series ex-pansion of (13) around its peak (namely, µε) is conductedand (13) is approximated byεvi¯ , εri¯ |y, εv−i¯, εr−i¯x, σ2;xi¯ 6= 0 ∼ N (µε,Σε) .5. EXPERIMENTAL RESULTSIn this section, we depict the results obtained with the pro-posed algorithm on experimental data recorded with thePARSAX radar of TU Delft [12] . The data set consideredentails several echoes of vehicles on a freeway; data has beenpre-whitened via some ad-hoc procedure.It can be seen in Fig.1(a) that four targets are correctly 1estimated by the proposed algorithm and that velocity ambi-guities are mitigated. However, when using a non-robustifiedversion of the algorithm (i.e., mismatch is considered null,Fig.1(b)) the sidelobes of targets at range bin 6/7, 12 and 14are seen as “real” targets. Note that one target is still correctlyestimated (20 m/s, range bins 3/4), probably because its mis-match with respect to the range-velocity analysis grid is low.To finish, we depict the histogram of (εvi¯, εri¯)|y for the tar-get at range bin 6, -14.6 m/s; it is built using samples drawnby the hybrid Gibbs sampler described in Section 4. We cansee that, even with a uniform prior distribution, the empiri-cal posterior distribution is strongly peaked around the valueestimated by the SSR algorithm, namely (.32, .36).6. CONCLUSIONIn this paper, we present a Bayesian sparse recovery algo-rithm able to mitigate velocity ambiguities in the case of off-grid range migrating targets. More precisely, this algorithmrelies on a robustified sparsifying dictionary suited for range-migrating targets that possibly straddle range-velocity bins.Then, the parameters of interest are estimated thanks to aMonte-Carlo Markov chain algorithm. The proposed algo-rithm is successfully evaluated on experimental radar data andis proved to mitigate velocity ambiguities. In the future, thealgorithm could be extended to colored noise in order to dealwith diffuse clutter component.1Though ground truth is unknown, true values of non-ambiguous veloci-ties were confirmed by stationarity analysis based on two consecutive burstsof M pulses [13].(a) Proposed algorithm(b) No mismatch estimationFig. 1: Range-velocity map from PARSAX data estimated by proposed algo-rithm with or without mismatch estimation (circles). Coherent integration ofthe scene represented background. K = 16, M = 64, fc = 3.315 GHz,B = 100 MHz, va = 45 m/s, σ2 ≈ 1. K¯ = K, nva = 2, M¯ = 2M ,(mσ2x, stdσ2x) = (40, 10) dB, (mσ2 , stdσ2 ) = (0, 5) dB.Fig. 2: Prior and empirical posterior pdf of (εvi¯, εri¯) for target at range bin 6,-14.6 m/s. MMSE estimate depicted as circle.AcknowledgementThe authors would like to thank O. Krasnov at TU Delft forkindly providing the PARSAX experimental data.7. REFERENCES[1] M. Richards, Fundamentals of Radar Signal Processing.McGraw-Hill Electronic Engineering, 2005.[2] F. Le Chevalier, “Proce´de´ et dispositif de de´tection decibles pour radar Doppler a` impulsions non ambigu¨ a`large bande,” 1996, French Patent 96 08509.[3] A. Farina and F. A. Studer, “Detection with high reso-luton radar: great promise, big challenge,” MicrowaveJournal, vol. 34, no. 5, pp. 223–273, May 1991.[4] J. Tsao and B. D. Steinberg, “Reduction of sidelobe andspeckle artifacts in microwave imaging: the CLEANtechnique.” IEEE Transactions on Antennas and Prop-agation, vol. 36, no. 4, pp. 543–557, 1988.[5] W. Melvin and J. Scheer, Eds., Principles of ModernRadar: Advanced Techniques. Edison, NJ: Institutionof Engineering and Technology, 2012.[6] S. Bidon, J.-Y. Tourneret, L. Savy, and F. Le Cheva-lier, “Bayesian sparse estimation of migrating targets forwideband radar,” IEEE Trans. Aerosp. Electron. Syst.,vol. 50, no. 2, pp. 871–886, Apr. 2014.[7] M. Lasserre, S. Bidon, O. Besson, and F. Le Chevalier,“Bayesian sparse Fourier representation of off-grid tar-gets with application to experimental radar data,” SignalProcessing, vol. 111, pp. 261–273, 2015.[8] S. Bidon, M. Lasserre, O. Besson, and F. Le Cheva-lier, “Bayesian sparse estimation of targets with range-Doppler grid mismatch,” in Proc. of IEEE InternationalRadar Conference, Arlington, VA, May 2015.[9] Y. Chi, L. Scharf, A. Pezeshki, and A. Calderbank,“Sensitivity to basis mismatch in compressed sensing,”IEEE Trans. Signal Process., no. 5, pp. 2182–2195, May2011.[10] M. Duarte and R. Baraniuk, “Spectral compressive sens-ing,” Appl. Comput. Harmon. Anal., vol. 35, pp. 111–129, 2013.[11] C. Robert and G. Casella, Monte Carlo Statistical Meth-ods. Springer, 2004.[12] O. A. Krasnov, G. P. Babur, Z. Wang, L. P. Ligthart, andF. van der Zwan, “Basics and first experiments demon-strating isolation improvements in the agile polarimet-ric FM-CW radar – PARSAX,” International Journal ofMicrowave and Wireless Technologies, vol. 2, pp. 419–428, 8 2010.[13] F. Le Chevalier, O. Krasnov, F. Deudon, and S. Bidon,“Clutter suppression for moving targets detection withwideband radar,” in Proc. of European Signal Process-ing Conference (EUSIPCO), 2011, pp. 427–430.

Velocity ambiguity mitigation of off-grid range migrating targets via Bayesian sparse recovery

International audienceWithin the scope of sparse signal representation, we considerthe problem of velocity ambiguity mitigation for wideband radar signal. We present a Bayesian robust algorithm based on a new sparsifying dictionary suited for range-migrating targets possibly straddling range-velocity bins. Numerical simulations on experimental data demonstrate the ability of the proposed algorithm in mitigating velocity ambiguity

Archive Ouverte en Sciences de l'Information et de la Communication

https://hal.archives-ouvertes.fr/hal-01446111

Velocity ambiguity mitigation of off-grid range migrating targets via Bayesian sparse recovery

Abstract

Similar works

Full text

Available Versions

Open Archive Toulouse Archive Ouverte

Archive Ouverte en Sciences de l'Information et de la Communication