arXiv:1509.06290v1 [stat.ML] 21 Sep 2015 pisteaglrrgo into Bayesian region a angular using framework the BCS tracked The splits (BCSKF). a compress then far filter Bayesian Kalman of and the sensing a compressed using framework from (DOA) made array (BCS) arrival is sensor sensing estimate of initial a The on direction field. impinging dynamic signal the narrowband estimating of aeasga rsn hnw a osdrtepolmfrom problem will the DOAs consider we potential can the we if of then Instead, present few signal a snapshots. a only time have that different fact the at receiv consider sensor signals the each form problem by estimated the is solve matrix covariance to This required complexity incre thus computational required, the is some Secondl matrix present. covariance need are the that of we evaluation signals Firstly, of number drawbacks: However, the of [5]. two knowledge [4], have ESPRIT methods and [3] this these solving [2], of array MUSIC methods an are: used problem on Commonly [1]. impinging from signal arrived has a direction which determining amnfitr yai O,DAtracking DOA DOA, dynamic filter, Kalman nipoeeti siainacrc spsil ihu a without possible complexity. is that computational accuracy show in increase They estimation significant method. in estimation improvement based an wi BCS made comparisons traditional distrib and the Gaussian provided are a scenarios as modelling test signal signals Example by received received done sparse sparse is estimated expected the This the change. between DOA difference pred the known the match a will signals given estimated the signals that associat belief proble belief a sparse this to BCS traditional tackle To accurat the DOAs. changing to In the propose array. struggle in we be changes the non-zero can track can of methods a and endfire There current estimate the have framework. region the approaches angular same will track DOA this to the the DOAs when used using the issue be DOA an then of the can few in BCSKF change A a present. only signal valued that belief a aeinCmrse esn amnFle for Filter Kalman Sensing Compressed Bayesian A Abstract ieto farvl(O)etmto stepoesof process the is estimation (DOA) arrival of Direction ne Terms Index I hsppr elo oadesteproblem the address to look we paper, this —In a DAetmto,Bysa opesdsensing, compressed Bayesian estimation, —DOA eateto uoai oto n ytm niern,U Engineering, Systems and Control Automatic of Department { .ae,l.s.mihaylova m.hawes, ate Hawes Matthew .I I. b nttt ie eeo/eeo il,CItLURCR 91 CNRS UMR CRIStAL Lille, Telecom/Telecom Mines Institute NTRODUCTION ieto fArvlEstimation Arrival of Direction c N eateto niern,CmrdeUiest,C P,U 1PZ, CB University, Cambridge Engineering, of Department oeta Osadenforces and DOAs potential a ydiaMihaylova Lyudmila , } sefil.cu,[email protected] s [email protected], @sheffield.ac.uk, dwith ed ution. and s asing icted ely ive m, ed th y, . a rnosSeptier Francois , siae sn C,hnetetr aeincompressed Bayesian from term estimate the the hence as BCS, using taken (precisio of hyper-parameters estimated is the mean and iteration iteration predicted previous each the the at where signals [20], the signals KF. sparse the of dynamic iteration each at t array reevaluate signals the to array having of of vector measured stage steering the additional with an work introduces and directly to being advantag able the region being removes from estimate this angular However, DOA iteration. the the previous on the closely narrow more in and focus to (KF) considered filter of Kalman tracking a and estimation or DOA [18]. filters [17], of [16], particle areas [15], the use sources in to used filters is These been filters. option (PHD) density One hypothesis probability DOA. dynamic a approach of case this on the [14]. in based results shown methods encouraging based offer been that (RVM) has estimation machine form DOA It static [13]. vector probabilistic [12], relevance a [11], a into approach also using is problem It solve this estimates. and DOA convert non-zero the the to as have used possible that then directions approximati are Those signals acceptable valued output. an with array gives DOAs the still of of that number present minimum signal the a finding as formulated hr inl culyipneo h ra rmonly from array the on impinge into First ( actually interest [10]. [9], of signals [8], region where estimation angular DOA applie the be of split can problem This the [7]. solve [6], measurements to methods fewer traditional from by signals used some than recover to possible is directly work signals. and received (CS) the sensing with compressive of point view the Kalman Filter (BCSKF). There are still some issues d M =(M−1) d with this method when applied to the problem of DOA estimation of a dynamic far field source using uniform linear ∆ d1= d arrays (ULAs). Namely when the DOA approaches the endfire region (i.e. the signal arrives parallel/close to parallel to the θ θ θ array), the estimation accuracy can degrade. This means that it is possible to initially have an accurate estimate and then not track the changes in DOA properly. In this paper, to solve this problem, we propose modifying the BCSKF to include information about what the received array signals would be expected to be at a given time snapshot. This is done by changing the distribution used to model the re- y ceived signals. Now instead of assuming a zero-mean Gaussian k hierarchial prior we assume that the mean is instead centred Fig. 1. Linear Array structure being considered, consisting of M sensors at the value that is expected given the previous snapshot’s with a uniform adjacent sensor separation of ∆d. estimate and the expected change in DOA. As a result we derive a new posterior distribution and marginal likelihood function that can be used to solve the DOA estimation problem namely the sensor locations and the delay required for a signal by following a similar framework as used for the RVM. to reach a given sensor. We term this framework the modified RVM, which is used The output of the array, yk, at time snapshot k is then given to find an initial estimate of the DOA at the first snapshot by and then at each subsequent snapshot to optimise the noise yk = Axk + nk, (2) variance estimate as well as the hyperparameters required by T CN×1 the BCSKF. where xk = [xk,1, xk,2, ..., xk,N ] ∈ gives the received T CM×1 The remainder of this paper is structured in the following signals, nk = [nk,1,nk,2, ..., nk,M ] ∈ the sensor CM×N manner: Section II gives details of the proposed estimation noise and A = [a(Ω,θ1), a(Ω,θ2), ..., a(Ω,θN )] ∈ is method. This includes the array model being used (II-A), the the matrix containing the steering vectors for each angle of modified RVM framework for BCS (II-B) and the BCSKF interest. (II-C). In Section III an evaluation of the effectiveness of the B. Bayesian Compressed Sensing for DOA Estimation proposed method is presented and conclusions are drawn in First split the angular range that is being monitored into Section IV. N potential DOAs. Each direction can then be considered II. PROPOSED DESIGN METHODS as having a signal present. However, only L << N of the A. Array Model received signals are non-zero valued, with the directions of these L signals giving the actual DOAs. A narrowband array structure consisting of M sensors is Now we split (2) into real and imaginary components in the shown in Fig. 1. The sensors are assumed to be omnidirec- following way tional with identical responses. A plane-wave signal model ˜ is assumed, i.e. the signal impinges upon the array from the y˜k = Ax˜k + n˜k (3) far field at an angle θ as shown. In this work we assume R(yk) R(A) −I(A) R(xk) R(nk) that 0◦ ≤ θ ≤ 180◦. The distance from the first sensor to = + , I(yk) I(A) R(A) I(xk) I(nk) subsequent sensors is denoted as d for m = 1, 2,...,M,        m where R(·) and I(·) give the real and imaginary components with d1 = 0, i.e. the distance from the first sensor to itself. respectively. Note, the difference between y and y˜ is that y Note, these values are multiples of a uniform adjacent sensor k k k has been split into its real and imaginary components in y˜ . separation of ∆d. k As a result the dimensions of y˜ are larger than for y , but The steering vector of the array is given by k k we are now only considering real valued data. Similar is true −jµ2Ω cos θ −jµM Ω cos θ T a(Ω,θ) = [1,e ,...,e ] , (1) when comparing A and A˜, and xk and x˜k and nk and n˜k. The aim is to now find a solution for x˜k that gives the where Ω= ωTs is the normalised frequency with Ts being the minimum l0 norm, i.e. the minimum number of non-zero sampling period, µ = dm for m = 1, 2,...,M, and {·}T m cTs valued signals. This is done by evaluating the following denotes the transpose operation. Note, the steering vector of 2 an array gives contains information about the array geometry, x˜k,opt = max P(x˜k, σ , p|y˜, xe), (4) where σ is the variance of the Gaussian noise n, p = Note, the maximum of (10) is the posterior mean µ. T 2 [p1,p2, ..., p2N ] ] contains the hyperparameters that are to be Similarly to [12], the probability P(σ , p|y˜k) can be repre- estimated and xe holds the expected values of x˜k. sented in the following form: To do this first obtain the following from (2): 2 2 2 P(σ , p|y˜k) ≈P(y˜k|p, σ , xe)P(p)P(σ ) (13) 1 P(y˜ |x˜ , σ2)=(2πσ2)−M exp − ||y˜ − A˜x˜ ||2 . (5) k k 2σ2 k k 2 and the second two terms on the right of (13) become constant n o with uniform scale priors, then maximising P(σ2, p|y˜ ) is Now exert a belief about the values of x˜k that are expected k 2 by enforcing roughly equivalent to maximising P(y˜k|p, σ , xe). This can be achieved by a type 2 maximisation of its logarithm, which −N 1/2 P(x˜k|p, xe) = (2π) |P| (6) is given by 2 1 T × exp − (x˜k − xe)P(x˜k − xe) , 2 2 1 1 1 2 L(p, σ ) = log (2πσ )−M |Σ| 2 |P| 2 exp − (14) n o ( 2 where |P| indicates the determinant of P, where P = diag(p).  It is also necessary to define the hyperparameters over p y˜T By˜ xT Cx σ2y˜T A˜ΣPx and σ2. There are various possibilities for the structuring of × ( k k + e e − 2 k e) )  the priors on p, which represent mixing parameters in a scale 1 = − 2M log(2π)+2M log σ2 − log |Σ|− mixture of normals representation of the marginal distribution 2 of xk, which will here be in the Student-t family, see e.g. [21].  −2 ˜ 2 T log |P| + σ ||y˜k − Aµ||2 + µ Pµ One possibility would be to treat the complex components T T +xe Pxe − xe Pµ , of xk as complex Student-t distributed, as detailed in [22], [23]. Here though we treat the real and imaginary components T  where B = (σ2I + AP˜ −1A˜ )−1 and C = P − PT ΣP. of x as independent Student-t distributed random variables, k To do this (14) is differentiated with respect to p and σ−2 and hence have independent Gamma priors for the mixing n to obtain the update expressions 3 variables pn over all real and imaginary components of xk: new γn 2N pn = 2 2 , (15) µn + xe,n − xe,nµn P(p)= G(pn|β1,β2). (7) th n=1 where γ =1 − p Σ , Σ is the n diagonal element of Y n n nn nn 2 Σ and A Gamma prior can also be used for σ ˜ 2 2 ||y˜k − Aµ||2 2 −2 σnew = . (16) P(σ )= G(σ |β3,β4), (8) 2M − γn n where β1,β2,β3 and β4 are scale and shape priors. Note, when The maximisation is then achievedP by iteratively maximising T xe = [0, 0, ..., 0] then (6) reverts to the traditional hierarchial (11) and (12) and (15) and (16) until a convergence criterion T prior used in BCS [11]. is met [11], [12]. Note that when xe = [0, 0, ..., 0] the We know that update expressions match that of the traditional RVM. The 2 2 2 final estimate of the received signals is then given by P(x˜k, σ , p|y˜k, xe)= P(x˜k|y˜k, σ , p, xe)P(p, σ |y˜k) (9) T T ˜ ˜ −1 ˜ 1 A A A y˜k and x˜k,opt = 2 + Popt 2 + Poptxe (17) 2 σopt σopt 2 P(y˜k|x˜k, σ )P(x˜k|p, xe)     P(x˜k|y˜ , p, σ , xe) = (10) 2 T k 2 where σ and P = diag([p 1,p 2, ..., p 2 ] ) are P(y˜k|p, σ , xe) opt opt opt, opt, opt, N = (2π)−N |Σ|−1/2 the result of optimising the noise estimate and hyperparameters respectively. 1 T −1 × exp − (x˜k − µ) Σ (x˜k − µ) , The final estimated signals are then given by ( 2 ) x =x ˜ + jx˜ + . (18) where the covariance matrix is given by k,opt,n k,opt,n k,opt,N n T Thresholding can then be applied to keep the L˜ most sig- Σ = (σ−2A˜ A˜ + P)−1, (11) nificant signals as in [14]. To do this find the total energy and the mean given by content of the estimated received signals and then sort them. A threshold value, η, is then defined as a percentage of the Σ −2 ˜ T µ = (σ A y˜k + Pxe). (12) 2See Appendix B 1See Appendix A 3See Appendix C TABLE I energy content that is to be retained. Starting with the most PERFORMANCESUMMARYFORTHEENDFIREREGIONEXAMPLE. significant estimated signal, the estimated signals are summed until the threshold is reached and the remaining signals set Mean RMSE Mean Computation Method (degrees) Time (seconds) to be equal to 0. The remaining non-zero valued signals then RVM 11.03 0.35 give the DOA estimates and L˜ as an estimate of the number Modified RVM 0.98 0.55 of far field signals impinging on the array.

C. Bayesian Compressed Sensing Kalman Filter sparse then subsequent estimates are likely to not be sparse. In order to track the changes in the DOA estimates at As a result, care should be taken when choosing the initial each time snapshot the BCS based DOA estimation procedure parameter values and determining the likely DOA change. detailed above is combined with a BKF, giving a BCSKF for DOA estimation. Here, the signal model described above is III. PERFORMANCE EVALUATION again used along with the prediction In this section a comparison in performance of the tradi- x˜ x˜ ∆x Σ Σ P−1 k|k−1 = k−1|k−1 + k|k−1 = k−1 + k tional RVM and the proposed modified RVM will be made. ˜ y˜k|k−1 = Ax˜k|k−1 y˜e,k = y˜k − y˜k|k−1 (19) Three example scenarios will be considered. One where the initial DOA starts outside of the endfire region and then moves and update steps into it, one where the DOA remains out of the endfire region Σ ˜ Σ x˜k = x˜k|k−1 + Kky˜e,k k|k = (I − KkA) k|k−1 and finally one where the initial DOAs and signal values are T 2 T −1 randomly generated. All of the examples are are implemented Kk = Σ 1A˜ (σ I + A˜Σ 1A˜ ) (20) k|k− k|k− in Matlab on a computer with an Intel Xeon CPU E3-1271 of the BKF. Here, k|k −1 indicates prediction at time instance (3.60GHz) and 16GB of RAM. k given the previous measurements and ∆x is determined by The performance of each method will be measured using the expected DOA change. For example, if we sample the the root mean square error (RMSE) in DOA estimate. This angular range every 1◦ and the the DOA increases by 2◦ then is given by ∆ then x will be selected to increase the index of the non- Q ˆ 2 zero valued entries in x˜k−1|k−1 by two to give the index of |θ − θ| v q=1 the non-zero valued entries in x˜k|k−1. In this work we have RMSE = u , (22) u P Q assumed that there will be a constant change in the DOA. u t At each time snapshot it is necessary to estimate the where θ is the actual DOA, θˆ is the estimated DOA and Q noise variance and hyperparameters in order to evaluate the is the number of Monte Carlo simulations carried out, with prediction and update steps of the BCSKF. This is done by Q = 100 being used in each case. considering the log likelihood function given by The array structure being considered is a 20 sensor ULA 2 1 2 with an adjacent sensor separation of λ . We assume the actual L(pk, σ ) = − 2M log(2π)+2M log σ − (21) 2 2 noise variance is given by σ2 =0.4 and an initial estimate of Σ −2 ˜ 2 n log | |− log |P| + σ ||y˜e,k − Aµ||2 2 the noise variance of σinit = 0.1 used when initialising the T T T +µ Pµ + x˜k|k−1Px˜k|k−1 − x˜k|k−1Pµ , RVM and proposed modified RVM. which can be optimised by following the procedure described A. Endfire Region in Section II-B. Here we have used the Kalman filter prediction ◦ x˜k|k−1 as the expected estimate values xe. For this example the initial DOA of the signal is θ = 20 , It is worth noting that the continued accuracy of the pro- which then decreases by 1◦ at each time snapshot. The signal posed BCSKF relies on the accuracy of the initial estimate value at each snapshot is set to be 1. Table I summarises the and the parameter values selected. If the initial estimate performance of the two methods for this example. Here we can (made using the framework described in Section II-B and see that there is in an improved accuracy in the DOA estimates, T xe = [0, 0, ..., 0] ) of the received signals is accurate and as the mean RMSE has decreased for the proposed modified sparse, then the priors that are enforced will ensure this RVM method. This is also supported by the overall RMSE continues to be the case. However, an inaccurate initial DOA values as illustrated in Fig. 2. It is also worth noting that the estimate or poorly matched expected DOA change can lead mean computation times show that this improvement has not to the introduction of inaccuracies in subsequent estimates. been at the expense of a significant increase in computational Similarly, if the initial estimate of the received signals is not complexity. 35 TABLE III RVM Modified RVM PERFORMANCESUMMARYFORTHERANDOMINITIAL DOA EXAMPLE. 30 Mean RMSE Mean Computation 25 Method (degrees) Time (seconds)

20 RVM 10.98 0.34 Modified RVM 3.52 0.43

15 RMSE (degrees)

20 10 RVM 18 Modified RVM 5 16

0 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time Snapshot 12

10 Fig. 2. RMSE values for the endfire region example. 8 RMSE (degrees)

6

TABLE II 4 PERFORMANCESUMMARYFORTHENON-ENDFIRE REGION EXAMPLE. 2

0 Mean RMSE Mean Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Method (degrees) Time (seconds) Time Snapshot RVM 5.59 0.33 Fig. 4. RMSE values for the random initial DOA example. Modified RVM 0.36 0.41

B. Non-Endfire Region obtained an improved accuracy without a significant increase In this instance the initial DOA is θ = 100◦ with the DOA in computational complexity. increasing by 1◦ at each time snapshot, with the signal value IV. CONCLUSIONS remaining constant at -1. The performance of the two methods is summarised in Table II, with the RMSE values illustrated This paper proposes a BCSKF to estimate the DOA of in Fig. 3. Again this illustrates the improved performance a single signal impinging on a ULA from the far field. A offered by the modified RVM has not been at the expense new posterior distribution and marginal likelihood has been of a significant increase in computation time. found and unlike traditional BCS the expected values of the estimates are accounted for. This is done to combat the C. Random Initial DOA problem of inaccurate DOA estimates when the actual DOA Finally, we consider the case where the initial DOA is approaches the endfire region of the angular range. Then a randomly chosen from the entire angular range and increased similar optimisation framework to what is used in the RVM by 1◦ at each time snapshot. The signal value is randomly is applied to find the optimal hyperparameters and noise selected as ±1 for each simulation and remains constant as variance estimate, which are then used to estimate the received the DOA changes. As for the previous two examples Table. array signals. Example test scenarios have shown the proposed III and Fig. 4 indicate that the proposed modified RVM has modified RVM is more accurate in not only the endfire region, but also in the entire angular region as a whole. This is also without a significant increase in computational complexity. 12 RVM Modified RVM

10 APPENDIX

8 A. Derivation of Posterior Distribution From Bayes’ rule we know that 6

RMSE (degrees) 2 2 4 P(x˜k|y˜k, p, σ , xe)P(y˜k|p, σ , xe) = (23) 2 P(y˜ |x˜k, σ )P(x˜k|p, xe), 2 k 2 0 where P(y˜k|x˜k, σ ) and P(x˜k|p, xe) are known from (5) and 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time Snapshot (6), respectively. Now following the method suggested in [12] carry out the Fig. 3. RMSE values for the non-endfire region example. multiplication on the right hand side, collect terms in x˜k in the exponential and complete the square. This gives the marginal likelihood as

2 2 1 1 1 −M Σ 2 2 −2 ˜ T ˜ P(y˜k|p, σ , xe) = (2πσ ) | | |P| (30) − σ (y˜k − Ax˜k) (y˜k − Ax˜k)+ (24) 2 1 T T h T × exp − [y˜k By˜k + xe Cx˜e − (x˜k − xe) P(x˜k − xe) 2 n 2 2σ y˜T A˜ΣPx ] , 1 −2 T −2 T ˜ −2i T ˜ T k e = − σ y˜k y˜k − σ y˜k Ax˜k − σ x˜k A y˜k + 2 where B and C are defined as in Section II-B.o The log −2h T ˜ T ˜ T T T σ x˜k A Ax˜k + x˜k Px˜k − x˜k Pxe − xe Px˜k likelihood is then given by T +xe Pxe 2 2 1 1 L(p, σ ) = log (2πσ )−M |Σ| 2 |P| 2 (31) 1 T Σ−1 iT Σ−1 = − (x˜k − µ) (x˜k − µ) − µ µ + n 1 T T 2 × exp − [y˜k By˜k + xe Cx˜e − h 2 2 σ− y˜T y˜ + xT Px k k e e n 2 T ˜Σ 2σ y˜k A Pxe] i where Σ and µ are given by (11) and (12), respectively. This oo 1 = −M log(2π) − M log σ2 + log |Σ| + then gives the posterior distribution as (10), with the remaining 2 1 1 exponential terms log |P|− [y˜T By˜ + xT Cx˜ − 2 2 k k e e 1 2σ2y˜T A˜ΣPx ]. − σ−2y˜T y˜ + xT Px − µT Σ−1µ . (25) k e 2 k k e e " # Using the Woodbury matrix inversion identity we have T T B. Derivation of Marginal Likelihood B = σ−2I − σ−2A˜(P + σ−2A˜ A˜)−1A˜ σ−2, (32) From (23) we know that which means we have 2 2 P(y˜k|x˜k, σ ), P(x˜k|p, xe) T T −2 T −2 −2 ˜ P(y˜k|p, σ , xe)= 2 , (26) y˜k By˜k = y˜k σ y˜k − y˜k (σ I − σ A (33) P(x˜k|y˜ , p, σ , xe) k −2 ˜ T ˜ −1 ˜ T −2 × (P + σ A A) A σ )y˜k meaning the term in the exponential will be (25) where T −2 T −2 ˜Σ˜ T −2 = y˜k σ y˜k − y˜k σ A A σ y˜k T Σ−1 −2 ˜ T T ΣT Σ−1 −2 T ˜ −2 T ˜Σ µ µ = (σ A y˜k + Px˜e) (27) = σ y˜k (y˜k − Aµ)+ σ y˜k A Pxe Σ −2 ˜ T −2 T ˜ 2 T −2 T ˜Σ × (σ A y˜k + Pxe) = σ ||y˜k − Aµ||2 + µ Pµ + σ y˜k A Pxe. T T −2 ˜ T −2Σ˜ Σ T = (σ A y˜k + Pxe) (σ A y˜k + Pxe) Also, we know that P = P as P is a real valued diagonal −4 T ˜Σ˜ T −2 T ˜Σ matrix. This means = σ y˜k A A y˜k + σ y˜k A Pxe + T σ−2xT PT A˜ y˜ xT PT ΣPx . T T Σ e k + e e xe Cxe = xe [P − P P]xe (34) xT Px xT PΣPx Therefore the exponential term is given by = e e − e e T T −2 T ˜Σ = xe Pxe − xe Pµ + σ y˜k A Pxe, 1 −2 T T −4 T ˜Σ˜ T − σ y˜k y˜k + xe Pxe − σ y˜k A A y˜k − (28) which then gives the log likelihood function in (14). 2" −2 T ˜Σ −2 T T ˜ T C. Derivation of Update Expressions for Modified RVM σ y˜k A Pxe − σ xe P A y˜k − Firstly, differentiating with respect to pi gives T T Σ xe P Pxe # 1 1 2 2 − Σnn − + µn + xe,n − xe,nµn (35) 2 pn 1 T −2 −4 ˜Σ˜ T T T Σ h i = − y˜k [σ − σ A A ]y˜k + xe [P − P P]xe and equating to zero gives 2" 1 Σ − + µ2 + x2 − x µ = 0 (36) −2 T ˜Σ −2 T T ˜ T nn p n e,n e,n n −σ y˜k A Pxe − σ xe P A y˜k n # 2 2 1 − pnΣnn − pnµn − pnxe,n + pnxe,nµn = 0 2 2 The term outside of the exponential is given by γn − pn[µn + xe,n − xe,nµn] = 0

2 −M −N 1/2 (2πσ ) (2π) |P| 1 1 2 −M 2 2 1 = (2πσ ) |Σ| |P| . (29) (2π)−N |Σ|− 2 which leads to (15). Now collect the terms with σ in to give [7] D. Donoho, “Compressed sensing,” IEEE Transactions on Information 1 Theory, vol. 52, no. 4, pp. 1289–1306, 2006. 2 Σ −2 ˜ 2 [8] D. Malioutov, M. Cetin, and A. Willsky, “A sparse signal reconstruction − 2M log σ − log | | + σ ||y˜k − Aµ||2 (37) 2 perspective for source localization with sensor arrays,” IEEE Transac- and then defineh τ = σ−2 giving i tions on Signal Processing, vol. 53, no. 8, pp. 3010–3022, 2005. [9] M. Hyder and K. Mahata, “A robust algorithm for joint-sparse recovery,” 1 −1 2 IEEE Signal Processing Letters, vol. 16, no. 12, pp. 1091–1094, 2009. − 2M log τ − log |Σ| + τ||y˜ − A˜µ||2 (38) 2 k [10] Q. Shen, W. Liu, W. Cui, S. Wu, Y. Zhang, and M. Amin, “Group 1h i sparsity based wideband DOA estimation for co-prime arrays,” in In = − − 2M log τ − log |Σ| + τ||y˜ − A˜µ||2 . 2 k 2 Proc. 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