arXiv:1912.11526v1 [eess.SP] 24 Dec 2019 ..Ofieo aa eerhudraadnmesN00014-13- numbers award (e-m under USA Research N00014-18-1-2415. su 02747 Naval and research N00014-17-1-2397 of MA upon Office Dartmouth, based is U.S. Rd, material of Westport This University Old [email protected]). Engineering, Computer 285 and Dartmouth, Electrical of ment siain Snapshots-limited Direction-o Spat estimation, enumeration, averaging, Source resampling, Periodogram correlation matrix, covariance Augmented loih nsaso iie scenarios. focu limited subspace snapshot DOA incoherent in and the algorithm th enumeration over source both source performances improve that estimation algorithms for show SCR algorithm simulations and MDL-gap, MUSIC Numerical AP description narrowband estimation. minimum termed DOA standard new for criteria, the a and of based enumeration input (ACM). 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Pop- each sensor are divided into L segments. Applying the discrete ular broadband focusing algorithms include rotational signal Fourier transform (DFT) to each segment forms multiple non- subspace focusing matrix (RSS, [23]), steered covariance ma- overlapping narrow frequency bands, from which we extract trices (STCM, [24]), DFT projection [25], weighted average of the frequency domain phasors at the frequencies of interest signal subspaces (WAVES [26]), beamforming invariance [27] f1, ..., fM ∈ [fmin,fmax] [13]. The segment duration is and auto-focusing [28]. Many of these focusing algorithms assumed much longer than the signal correlation time, such require preliminary estimates of the number of sources and that the different DFT bins are statistically uncorrelated. The their DOAs, which increase the computation cost and bias to vector of DFT coefficients (or complex phasors) for all N the final estimates. Moreover, these algorithms were primarily sensors and the lth snapshot at frequency fm is developed in the context of ULAs and do not apply directly x A s n to sparse array data. l(fm)= (fm) l(fm)+ l(fm), m = 1, ..., M This paper extends two broadband focusing algorithms orig- l = 1, ..., L, (2) inally proposed for ULAs to sparse arrays: spatial periodogram where xl(fm) is the N × 1 DFT coefficients vector, A(fm) is averaging [29] and spatial resampling [30]. Neither of these the N × D array manifold matrix at temporal frequency fm extensions require preliminary DOA estimates for broadband and sl(fm) is the D × 1 source amplitudes vector. The array focusing and can be applied on any sparse array geometry manifold corresponding to the nth element and the ith source with a contiguous coarray region, including MRAs, CSAs at frequency fm is and nested arrays. Constructing the ACM using the corre- j(2πfmdn/c)ui lations estimated from the spatial periodogram and spatially [A(fm)]n,i = e , (3) resampled correlations offers processing gains for both source where d is the location of the nth element with respect to the enumeration and localization over the ISS approach in [21], n array phase center and c is the field propagation speed. The especially in low SNR and few snapshots scenarios. source signal amplitudes are assumed uncorrelated zero-mean The rest of this paper is organized as follows. Section II and circular complex Gaussians s (f ) ∼ CN(0, σ2 ),i = discusses the broadband signal model and briefly reviews the i m i,m 1, ..., D and uncorrelated from the noise. The additive noise ISS method for broadband sparse array processing. Section is assumed zero-mean, white, and circular complex Gaussian III proposes the periodogram averaging (AP) and spatial n ∼ CN(0, σ2 I ). correlation resampling (SCR) based algorithms for broadband n N focusing. The focused ACMs from these algorithms are then the inputs for the new MDL-gap source enumeration algorithm B. Incoherent signal subspace method for sparse arrays and the standard narrowband MUSIC DOA estimator. The For the broadband signal model in (2), the ISS method performances of the proposed algorithms are compared with applies narrowband subspace processing to each frequency extensive numerical simulations in Section IV. Section V band and combines the estimation results across all bands concludes this paper. for the final estimate [14] [15]. The source enumeration and DOA estimation algorithms are often based on the eigenvalues II.INCOHERENT SPARSE ARRAY PROCESSING and eigenvectors of the sample covariance matrices (SCM) computed from each of the complex phasors data in (2) for a This section first describes the array signal model for ULA. broadband sources impinging on a sparse linear array and [21] extended the ISS method to sparse linear nested arrays. then reviews the incoherent method for broadband sparse array We review here its data processing procedures in the context processing. of finite snapshots, as shown in Fig. 1(a). For any particular frequency band fm, the narrowband SCM averaged over L A. Wideband signal model snapshots follows Assume a sparse linear array with N sensors and D L 1 H broadband planewave signals impinging on the array from Rxx,m = xl(fm)x (fm), (4) L l the far field with different DOAs within the visible region Xl=1 u1,u2, ..., uD ∈ [−1, 1]. Here we use the directional cosine where (·)H denotes Hermitian transpose. Reconstructing the o o u = cos(θ) to indicate the source DOA, where θ ∈ [0 , 180 ] SCM to obtain the (2P −1)×1 correlation vector correspond- is the angle-of-arrival with respect to the array endfire. The ing to the contiguous region of the difference coarray signal received by the nth sensor at time t can be modeled as 1 r (k)= [R ] , (5) D m η(k) xx,m n1,n2 ( 1 2X) ∈ ( ) xn(t)= si(t − τn(θi)) + nn(t), n =1, ..., N (1) n ,n ζ k Xi=1 where [R]n1,n2 selects the (n1,n2)th element of matrix R. where τn(θi) is the propagation time delay for the ith signal mThe set ζ(k) collects every sensor pair (n1,n2) separated by arriving at the nth sensor and nn(t) is the measurement the difference coarray index k = n1 − n2 ∈ [1 − P, P − 1] and noise at that sensor. We assume both the signals and noise η(k) = |ζ(k)| is the co-array weight equal to the cardinality measured by the sensors are samples of wide-sense stationary of the set ζ(k). Note for different sparse array geometries, the and ergodic complex Gaussian processes. The time series at co-array span P will be larger than the number of sensors N 3 by different amounts. To exploit fully the degrees-of-freedom A. Periodogram averaging (DOFs) offered by the co-array, apply spatial smoothing (SS) The spatial periodogram averaging for sparse arrays extends to construct a full-rank and positive semi-definite ACM by [4], Hinich’s broadband beamformer for undersampled ULAs to P nonuniform sparse arrays. This approach exploits the fre- 1 i i H Rss,m = v (v ) , (6) quency diversity obtained through the scanned responses P m m Xi=1 across the signal bandwidth while processing a single ULA i [29]. As Fig. 1(b) shows, the array frequency snapshot data where vm is a P ×1 vector containing the (P −k+1)th through (2P −k)th element of rm(k). The spatially smoothed ACM for for each band fm,m = 1, ..., M are conventionally beam- each frequency band goes through eigenvalue decomposition. formed independently via FFT and averaged over all snapshots The eigenvalues are used to compute the information criteria to estimate the narrowband spatial periodogram tm(u). The for each frequency band, which are then averaged across estimated spatial periodogram tm(u) is the Fourier transform all bands for source enumeration. The ISS method takes of the estimated spatial auto-correlation function rm(k) in (5), the source enumeration estimate and computes narrowband that is routinely used for ACM construciton [4] [6], weighted spatial pseudo-spectra for each frequency band, which are by the coarray weights η(k). Specifically, the narrowband then averaged to obtain a broadband pseudo-spectra used to periodogram follows estimate the DOAs. L 1 H 2 Eq. (6) indicates that SS exploits the fourth-order statistics tm(u)= w (u)xl(fm) = Fm(rm(k)η(k)). (7) L m of the propagating field by averaging the covariance matrices Xl=1 computed from the overlapping subarrays of the co-array where wm(u) is the conventional beamforming weights vector correlations. For infinite snapshots, the SS eigenvalues are for frequency fm at steering direction u (equal to the column proportional to the squares of the ensemble eigenvalues for a vector of the steering matrix A in (2) for direction u) and Fm P -element ULA [4] [31]. Thus, information criteria for source is the spatial Fourier transform operator accounting for the enumeration developed for ULA SCM eigenvalues, which are different temporal frequencies fm. In broadband processing, second moments, are more appropriately applied to the square only the true source peaks remain fixed in directional cosine root of the SS-ACM eigenvalues, and not the eigenvalues u across different frequency bands, while the grating lobes themselves as in [21]. and sidelobes change their locations in u as the temporal fre- For both fully populated and sparse linear arrays, the ISS quency varies. Averaging the periodograms across frequencies method works relatively well for broadband sources in high constructively reinforces the energy at the true source locations SNR and snapshot rich scenarios [21] [22]. However, the while other sidelobes are relatively attenuated source enumeration and localization performance suffers in 1 M low SNR scenarios, for sources with gaps in spectral energy t(u)= t (u). (8) M m such as harmonic sources, and in snapshot limited scenarios. mX=1 To address these issues, the following section proposes two Note that the broadband periodogram in ((8)) has the same coherent broadband focusing algorithms for sparse array pro- functional form as the steered covariance matrix estimate cessing. (STCM), which has attractive statisical features expressed in terms of a Wishart characteristic function [24]. The inverse spatial Fourier transform of the spatial periodogram t(u) III. PROPOSED COHERENT WIDEBAND SPARSE ARRAY estimates the spatial correlation function after normalizing for FOCUSING ALGORITHMS the coarray weights F−1(t(u)) ˜r(k)= , k = −(P − 1), ..., (P − 1) (9) This section proposes two broadband focusing algorithm for η(k) coherent correlation estimations: spatial periodogram averag- The estimated broadband correlation function ˜r(k) then popu- ing (AP) and spatial correlation resampling (SCR). The spatial lates the diagonals of a Hermitian Toeplitz ACM, as given in correlation estimates from either of these two algorithms Section III-C. then populate the diagonals of Hermitian Toeplitz ACMs for The covariance focusing through periodogram averaging subspace processing. The proposed approaches are coherent simplifies the coherent broadband processing algorithm while in the sense that they combine the observed data across all maintaining its advantages in low SNR and limited snapshot frequency bands to estimate a single broadband ACM from scenarios. Processing broadband data in the beamspace avoids which the number of sources and their DOAs are estimated. In the complexity of constructing focusing matrices that are this sense, the frequency averaging occurs with the narrowband commonly required in the coherent algorithms. Substituting spatial correlation functions, which still includes phase terms, (7)-(8) into (9), the correlation estimates can be written as in contrast with the incoherent approach which averages only the real-valued information criteria and pseudo-spectra. Both −1 1 M Fc M m=1 Fm (rm(k)η(k)) these two algorithms can be applied to any sparse array ˜r(k)=  , (10) P η(k) geometry based on a pruned ULA as long as a contiguous −1 coarray region exists. where Fc is the inverse spatial Fourier transform operator 4

] Fig. 1: Block diagrams for the incoherent signal subspace method (ISS, panel a) and the proposed periodogram averaging (AP, panel b) and spatial correlation resampling (SCR, panel c) algorithms for broadband sparse array source enumeration and DOA estimation. The N × L matrix X = [x1(fm), ..., xN (fM )] includes the DFT coefficients for all N sensors and L snapshots for frequencies f1, ..., fM corresponding to the central frequency within the bandwidth. bands, it is possible to obtain (nearly) the same array man- This notation implies that estimating the broadband spatial ifold vector at different frequencies. Spatial resampling for correlation function through inverse Fourier transform of broadband processing approaches the performance of the nar- the averaged spatial periodograms does not account for the rowband scenario with a comparable time-bandwidth product temporal frequencies mismatch between frequency bands. To [30]. account for this mismatch, it is in general a good practice to perform the inverse Fourier transform at the central frequency At first glance, the spatial resampling algorithm previously of the sources’ bandwidth. This is similar to choosing the applied to ULAs cannot be directly applied to sparse array data focusing frequency as the central frequency to reduce DOA due to the gaps in the spatial sampling. However, the important estimation bias, as suggested in [32]. insight is that the sparse arrays still provide contiguous and uniformly sampled difference co-array functions. This insight B. Spatial correlation resampling allows us to extend the application of the spatial resampling technique to sparse arrays. Rather than directly resampling the Another approach for coherent broadband focusing is array data, we resample the estimated second-order statistics through spatial resampling [30]. Spatial resampling exploits as a function of spatial lag. To make this insight precise, the the structural characteristic that the array manifold in (3) spatial correlation between the signals received by sensors depends on the source temporal frequencies and the element located at d and d for a single source with amplitude positions only through their product. By adjusting the spatial n1 n2 s from direction u can be expressed as sampling intervals of the frequency-domain snapshots as a i ∗ ∗ −j(2πfm/c)(dn1 −dn2 )ui function of the temporal frequency for each of the frequency E{x1(fm)x2(fm)} = ss e (11) 5

for frequency band fm. This implies that the array manifold The spatial resampling procedures are essentially the same corresponding to the coarray depends on the product of the as time domain resampling, as described in Fig. 4.28 [?], source temporal frequency fm and the inter-element spacing adapted to spatial correlation functions. It is worth to note dn1 − dn2 . Since the contiguous region of the coarray is uni- that, in theory, the focus frequency can be any value equal to form in spatial lag k, applying spatial resampling to the spatial or below the array design frequency to avoid spatial aliasing. correlations corresponding to this region for all frequency However, for practical implementation, we choose to focus bands will realign the coarray manifolds. The resampling at the minimum frequency in band such that f0 = f1. changes the spatial correlation sampling interval for the mth Resampling in this case corresponds to an interpolation or band from d to dm = df0/fm, where d is the physical inter- spatial sampling rate increase by a factor of Km/Lm at the sensor spacing and f0 is the focus frequency. mth frequency band. This makes sure that no extrapolation Unlike the periodogram averaging algorithm discussed in is needed in Step 5) to guarantee enough correlation samples Section ??, broadband focusing through spatial correlation to decimate in Step 6) in order to maintain the same coarray resampling explicitly accounts for the coarray manifold mis- support for spatial correlation estimates as before resampled. matches between frequency bands due to different temporal frequencies. Fig. 1(c) demonstrates the data processing pro- C. Augmented covariance matrix construction cedures for the SCR algorithm for broadband focusing. Each An alternative approach to SS for ACM construction is snapshot data at each frequency band fm goes through the following procedures: through lag redundancy averaging (LRA) [10]. This technique 1) Compute the spatial auto-correlation function by aver- exploits the coarray redundancies by averaging all repeated aging all L snapshots, take the portion corresponding to the estimates of the spatial correlation function at any given lag contiguous region of the coarray and normalize it by the from different sensor pairs and then replacing the individual coarray weights η(k) for unbiased narrowband correlation estimates at that lag by their average [33] [34]. As a result, the constructed ACM is populated with correlation estimates estimates rm(k) in (5). 2) Since the correlation estimate is even conjugate sym- with reduced variances. The LRA-ACM is populated with the metric about the coarray center, we apply spatial resampling spatial correlation estimates from either AP (9) or SCR 14 only to the right half side of the correlation estimate such that following zm(k)= rm(k =0, ..., P − 1) to save computation. 3) Choose integers Km and Lm appropriately such that ˜r(0) ˜r(−1) ··· ˜r(1 − P ) Km/Lm = fm/f0, where Km and Lm are both integers.  ˜r(1) ˜r(0) ··· ˜r(2 − P )  4) Upsample z by inserting zeros in between RLRA = . . . . . (15) m(k) (Km −1)  . . .. .  each sample of the correlation estimate such that    ˜r(P − 1) ˜r(P − 2) ··· ˜r(0)    ′ zm (k/Km) , for k =0,Km, ..., (P − 1)Km The LRA approach constructs a Hermitian Toeplitz ACM zm(k)=  0 , otherwise. from the correlation estimates, although the ACM is positive (12) indefinite. Compared against the SS-ACM, populating the 5) Filter the upsampled correlation function z′ (k) by a m LRA-ACM is more computationally efficient. For the same linear phase finite impulse response low pass filter with cut- sparse array data, note that the LRA-ACM exploits the second- off frequency of min(π/K ,π/L ) to obtain the interpolated m m order statistics, whereas the SS-ACM exploits the fourth-order correlations z′ (k). Shift or re-index z′ (k) to obtain the m,intp m,intp statistics of the propagating field. For finite snapshots, the SS- correct set of correlations by accounting for the group delay ACM in (6) can be shown explicitly related to the LRA-ACM due to linear phase filtering. ′ by [31] 6) Decimate zm,intp(k) by a factor of Lm such that ˜zm(k)= 2 ′ RSS = RLRA/P. (16) zm,intp(Lmk) to obtain the focused spatial correlation function. 7) Make up for the left half side of the resampled correlation This implies that RSS and RLRA share the same eigen space estimates using the even conjugate symmetry property such and the eigenvalues of RSS are proportional to the square of that the eigenvalues of RLRA. For infinite snapshots, the LRA- ∗ ACM approaches the ensemble covariance matrix of a fully ˜zm(−k), for k = −(P − 1), ..., −1 ˜rm(k)= (13) populated ULA with probability 1 [35]. This implies it is  ˜zm(k), for k =0, ..., P − 1 more reasonable to use the eigenvalue magnitudes and the The procedures above are repeated for all snapshots at all eigenvectors of the LRA-ACM rather than the SS-ACM for frequency bands before averaging across all M frequencies to source enumeration and DOA estimation. obtain the coherently combined spatially correlation estimates

M 1 D. Source Enumeration and DOA estimation ˜r(k)= ˜r (k). (14) M m mX=1 The ACM constructed in (15) goes through eigenvalue decomposition, with the eigenvalues sorted in descending The estimated correlation function ˜r then populates the (k) order by their magnitudes [31] diagonals of a Hermitian Toeplitz ACM as given in Section III-C. |λ1| ≥ |λ2|≥ ... ≥ |λk|≥ ... ≥ |λP |, (17) 6

before computing the information criteria for source enumer- IV. COMPARATIVE SIMULATION RESULTS AND ation. Rissanen proposed estimating the number of sources PERFORMANCE ANALYSIS as the model order that yields the minimum code length This section compares the performance of the proposed over a range of possible number of sources [17] [18]. The AP and SCR based broadband focusing algorithms for source proposed MDL criterion is the sum of the log-likelihood of enumeration and DOA estimation in numerical simulations. the maximum likelihood estimator of the model parameters These approaches are compared against the ISS processing and a bias correction term penalizing over-fitting of the model in scenarios with relatively few snapshots. All simulations order in this section model the source amplitudes as uncorrelated, (P −q)L gq 1 complex Gaussians with equal power occupying a bandwidth MDL(q)= − log + q(2P − q) log L, (18) of 40 Hz around the central frequency of 100 Hz. The aq  2 broadband sources are decomposed evenly into 41 narrowband for the possible number of sources q =0, ..., P −1. The func- components via FFT within the bandwidth. As a benchmark, tions P 1/(P −q) and 1 P gq = j=q+1 |λj | aq = P −q j=q+1 |λj | we compare all simulations against the narrowband (NB) case are, respectively,Q the geometric and arithmeticP mean of the with comparable time-bandwidth product to the broadband P −q smallest eigenvalues of the Wishart distributed SCM. The sources. This means the narrowband sources has 41 times estimated number of sources is qˆ = arg minq MDL(q). Since more snapshots than the broadband sources. This comparison the ACM in (15) does not follow Wishart distribution, there with the narrowband case makes clear the performance cost is no theoretical guarantee that the MDL criterion achieves paid by the focusing operations where the proposed broadband an accurate estimate of the number of sources, especially in algorithms combine information across the frequency band. under-determined scenarios. For demonstration purposes, we compare a MRA with [36] modified the standard MDL criterion in (18) and ex- 6 sensors at locations [1, 2, 5, 6, 12, 14]d. This array offers tended its application to the LRA-ACM for enumerating more a contiguous coarray region spanning k ∈ [−13, 13]. The sources than sensors using narrowband sparse arrays.The new fundamental inter-element spacing of the MRA is d = λ/2, information criterion, termed MDL-gap, is defined as the first- where λ is the spatial wavelength at the central frequency order backward difference of the MDL criterion normalized by f = 100 Hz. The sensor SNR level is defined as the ratio the number of snapshots such that between the power of each source signal to the noise power at a single sensor. The noise is assumed both temporally and spatially white and complex Gaussian occupying the same MDL-gap(q) = (MDL(q) − MDL(q − 1))/L (19) bandwidth as the sources, uncorrelated among the sources P −q+1 and also uncorrelated between each pair of sensors. The (aq−1) P − q + 1/2 = − log P −q + log L, following simulations consider two scenarios focusing on  |λq |(aq)  L different perspectives. The first is an over-determined scenario for the possible number of sources q =1, ..., P − 1. The de- to demonstrate the proposed algorithms’ capability to resolve tected source number is qˆ = arg minq MDL-gap(q). Since the closely spaced sources. The second is an under-determined MDL-gap criterion showed improved performance over MDL scenario to demonstrate the proposed algorithms’ capability in enumerating more sources than sensors in the narrowband to enumerate and localize more sources than sensors. scenarios [36], we here extend its application to the LRA- ACM in (15) for broadband sources. A. Resolving two closely-spaced sources Assuming the number of sources is accurately estimated, the In the two-source scenario, we first evaluate the performance DOA estimation is performed by directly applying the standard of the 4 approaches for source enumeration using the MDL narrowband spectral MUSIC algorithm [20] to the coherently and MDL-gap criteria. Fig. 2 compares the sample realizations constructed ACM. Specifically, the eigenvectors corresponding of the information criteria as a function of possible number to the P − D least significant eigenvalues of the ACM are of sources. All information criteria are normalized by their extracted to estimate the noise subspace maximum magnitudes respectively for demonstration purpose. ⊥ All simulations use 3 snapshots/sensor for the broadband V = [v , v , ..., v ]. (20) coh D+1 D+2 P approaches and equivalently, 123 snapshots/sensor for the Since the source manifold vectors at the focused frequency narrowband sources. There are two sources D = 2 arriving from directions u = [0, 0.06] for the left column and u = a(u ) = [1, ..., ej(2πf0kd/c)ui , ..., ej(2πf0 P d/c)ui ] (21) i [0, 0.3] for the right column. The true number of sources for each source i = 1, ..., D are orthogonal to the noise D = 2 is indicated by orange vertical dashed lines in all ⊥ subspace spanned by Vcoh, the MUSIC spectra computed as panels. For simplicity, all sources are assumed equal power with sensor level SNR = 0 dB. For the two-source case, all 1 Pcoh(u)= , (22) information criteria show minima at D = 2, which implies that H ⊥ ⊥ H a(u) Vcoh(Vcoh) a(u) all algorithms are able to estimate the true number of sources. will show D peaks at the source locations. The source DOAs The sample realization results indicate all algorithms struggle are then estimated by searching for the highest D peaks in the to enumerate closely spaced sources but start to enumerate coherently estimated MUSIC spectra. correctly when the sources are further separated. 7

Assuming the number sources are accurately estimated, Fig. 3 compares the MUSIC pseudo-spectra of AP, SCR, ISS and u = 0.06 u = 0.3 the equivalent NB scenario in panels (a-d) for two closely 1 1 spaced uncorrelated sources with DOAs at u = [0, 0.06], AP which are within the Rayleigh resolution limit ∆u = 0.13 SCR 0.5 0.5 calculated based on the MRA co-array aperture. The two ISS MDL NB sources are assumed equal power with sensor level SNR = 0 dB and 3 snapshots/sensor for each of the 41 narrow bands. 0 0 0 5 10 0 5 10 The equivalent NB case uses 123 snapshots/sensor. Note that the AP, SCR and NB MUSIC spectra all show two discernible 1 1 peaks near the true DOAs indicated by vertical orange dashed lines. However, the ISS approach fails to resolve these two sources, showing only one unique peak in between the true 0 0 DOAs instead. The MUSIC spectra imply the proposed AP MDL-gap and SCR approaches are more capable of resolving closely -1 -1 spaced sources than the ISS approach. 0 5 10 0 5 10 q q To characterize rigorously the resolvability and DOA esti- mate errors of the two closely spaced sources, we compare Fig. 2: Comparing the sample realizations of MDL and MDL- the 4 approaches on their probabilities of resolution [37] and gap criteria for the AP, SCR, ISS, and equivalent narrowband average root mean square errors (RMSE) for all estimated scenarios for 2 uncorrelated sources. The 2-sources are sep- DOAs against the source separation ∆u, source SNR and arated by ∆u = 0.06 on the left column and ∆u = 0.3 on number of snapshots/sensor. The DOA estimate performance the right column. All simulations assume equal power sources is characterized by the with sensor level SNR = 0 dB using 3 snapshots/sensor for broadband sources and 123 snapshots/sensor for narrowband D J 2 sources. The results imply all algorithms struggle to enumerate RMSE = v (ˆud(j) − ud) /DJ, (23) u closely spaced sources but start to enumerate correctly when uXd=1 Xj=1 t the sources are further separated. where uˆd(j) is the estimated DOA using the MUSIC algorithm for the d-th source in the j-th Monte Carlo trial with d = 1, ..., D and j =1, ..., J. All simulations results are averaged over J = 500 independent Monte Carlo trials. Fig. 4(a) compares the probability of resolution of these 4 approaches as a function of the spacing ∆u between two sources for (a) AP-MUSIC (b) SCR-MUSIC 100 100 SNR = 0 dB and 5 snapshots/sensor. AP and SCR have close performance in resolving two closely spaced sources, which are both worse than the NB case. However, the AP and SCR approaches outperform the ISS approach in their ability to resolve more closely spaced sources. Fig. 4(b) compares the 10-5 10-5 RMSE for the DOA estimates. For all approaches, the RMSEs

(c) ISS-MUSIC (d) NB-MUSIC decrease as the separation between the two sources increases. 100 100 The AP, SCR and NB have similar RMSEs, which are lower than the RMSE using the ISS approach. Normalized amplitude (dB) B. Enumerating/Localizing more sources than sensors 10-5 10-5 -0.5 0 0.5 -0.5 0 0.5 One major advantage that sparse arrays offer over the fully u = cos( ) u = cos( ) populated arrays is the capability of localizing more sources Fig. 3: Comparing the (a) AP, (b) SCR, (c) ISS, and (d) equiv- than sensors [10]. This section explores the advantages of alent narrowband MUSIC pseudo-spectra for two uncorrelated the proposed AP and SCR approaches in enumerating and sources with DOAs u = [0, 0.06] indicated by vertical dashed estimating more broadband sources than sensors over the ISS lines. All simulations assume equal power sources with sensor approach. We again use the same 6-element MRA as in the level SNR = 0 dB and 3 snapshots per sensor for each of the previous section, but with 9 uncorrelated equal power sources: 41 frequency bands. The equivalent narrowband case uses 123 1 at broadside, 4 uniformly spaced in θ = (90o, 135o] and the snapshots/sensor. The MUSIC spectra imply the proposed AP other 4 uniformly spaced in u = (0, 0.7]. and SCR approaches are more capable of resolving closely We first evaluate the performance of the 4 approaches for spaced sources than the ISS approach. source enumeration using the MDL and MDL-gap criteria. Fig. 5 compares the sample realizations of the criteria as a function of possible number of sources. All information criteria are 8

3 snapshots/sensor 10 snapshots/sensor (a) (b) (a) 1 1 1 AP AP SCR 0.5 ISS 0.5 SCR MDL NB ISS 0.5 0 0 NB 0 5 10 0 5 10 (c) (d) 1 1

Probability of Resolution 0 0 0.02 0.04 0.06 0.08 0.1 0 0

(b) MDL-gap 0 -1 -1 10 0 5 10 0 5 10 p p −1 10 Fig. 5: Comparing the sample realizations of MDL and MDL- AP gap criteria for the AP, SCR, ISS, and equivalent narrowband scenarios for 9 uncorrelated sources. All simulations assume RMSE −2 SCR 10 equal power sources with sensor level SNR = 0 dB and 3 ISS snapshots per sensor on the left column and 10 snapshots −3 NB 10 per sensor on the right column. The results imply that MDL 0 0.02 0.04 0.06 0.08 0.1 struggles to enumerate more sources than sensors regardless of ∆ u the number of snapshots available. However, using the MDL- gap criteria, the proposed AP and SCR approaches require Fig. 4: Comparing (a) the probability of resolution and (b) fewer snapshots than ISS for correct source enumeration. the RMSE of DOA estimates as a function of the spacing between 2 uncorrelated equal power sources with SNR = 0 dB. One source is fixed at broadside and the other source SCR approaches are capable of enumerating more sources than is located away from broadside by ∆u between [0.01, 0.1]. sensors in relatively few snapshots using MDL-gap. However, The simulations for broadband sources use 5 snapshots/sensor at least in this example, the ISS approach requires relatively and the equivalent narrowband sources use 205 snapshots large number of snapshots to achieve an accurate enumeration per sensor. The results indicate the proposed AP and SCR of more sources than sensors using MDL-gap. approaches are capable of resolving more closely spaced To quantify rigorously the performance of the proposed sources and achieving higher DOA estimate precision than the AP and SCR approaches in enumerating more sources than ISS approach. sensors, Fig. 6(a) compares the probability of correct enu- meration using MDL-gap against snapshots/sensor and Fig. 6(b) against sensor level SNR. The detection probability is normalized by their maximum magnitudes respectively for calculated as the number of Monte Carlo trials correctly demonstration purpose. The simulations in the left column of estimating Dˆ = 9 sources, normalized over a total of 500 panels (a,c) use 3 snapshots/sensor for the broadband source trials. The sensor level SNRs are the same of 0 dB for all 9 and equivalently, 123 snapshots/sensor for the narrowband sources for the simulations in panel (a). The simulation results source. The simulations in the right column of panels (b, d) show that the detection probabilities using all approaches use 10 snapshots/sensor for the broadband source and equiv- increase as the numbers of snapshots increase. In particular, alently, 410 snapshots/sensor for the narrowband source. For the AP and SCR approaches require much lower numbers of all panels, the true number of sources D = 9 is indicated by snapshots than the ISS approach to achieve a high detection vertical orange dashed lines. For simplicity, all sources are probability. AP has higher detection probability than SCR for assumed equal power with sensor level SNR = 0 dB. Panel less than 2 snapshots/sensor, but doesn’t converge to 1 as (a) shows that when the number of sources D = 9 exceeds the fast as SCR. In contrast, ISS requires 6 snapshot/sensor to number of sensors N = 6, none of the approaches exhibits a start detecting all sources and 10 snapshots/sensor to achieve minimal MDL value at Dˆ = 9. Panels (c) shows that the AP, a detection probability above 90%. Panel (b) evaluates the SCR and NB approaches show minimal MDL-gap values at detection probability as a function of sensor level SNR. The Dˆ = 9. However, the ISS approach is not able to estimate Dˆ = number of snapshots/sensor is fixed as 5 for the broadband 9 using either criteria at the modest snapshots level. When the source and 205 for the equivalent NB source. The simulation number of snapshots increases, panel (b) shows the MDL still results show that the NB approach requires the lowest SNR fails to estimate Dˆ = 9 for all four methods of constructing the level to start correctly detecting all sources. The AP and SCR ACM. However, panel (d) shows that all approaches using the approaches require SNR of -9 dB to start detecting all sources. MDL-gap criterion are able to correctly estimate the source ISS is not able to enumerate all sources for all SNRs with only number Dˆ = 9. These simulations imply that the AP and 5 snapshots per sensor available. 9

1 snapshot/sensor 10 snapshot/sensor (a) MDL(a)¡ gap (1a) AP-MUSIC (2a) AP-MUSIC 1 100 100

0.8 10-5 10-5 0.6 (1b) SCR-MUSIC (2b) SCR-MUSIC 100 100 AP 0.4 SCR

0.2 ISS 10-5 10-5 Detection Probability

Detection probability Detection NB (1c) ISS-MUSIC (2c) ISS-MUSIC 0 100 100 0 2 4 6 8 10 number of snapshots per sensor

-5 -5 (b) SORTE Normalized amplitude (dB) 10 10 (b) 1 (1d) NB-MUSIC (2d) NB-MUSIC 100 100 0.8

10-5 10-5 0.6 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 AP u = cos( ) u = cos( ) 0.4 SCR Fig. 7: Comparing the (a) AP, (b) SCR, (c) ISS, and (d) 0.2 ISS Detection Probability NB equivalent narrowband MUSIC pseudo-spectra for 9 broad-

Detection probability Detection 0 0 2 4 6 8 10 band sources with 1 snapshots/sensor (left column) and 10 number of snapshots per sensor snapshots/sensor (right column) for the broadband sources. The AP and SCR MUSIC spectra show sharper peaks than SNR (dB) the ISS MUSIC spectra for the same number of snapshots, Fig. 6: Comparing the probability of correctly enumerating the indicating more precise DOA estimation. number of sources using the MDL-gap criterion for different approaches (a) as a function of the number of snapshots per

£ 1 sensor for fixed sensor level SNR = 0 dB and (b) as a function 10 of sensor level SNR for a fixed 5 snapshots per sensor. There AP are 9 equal power sources impinging on the 6-element MRA. SCR The results indicate the AP and SCR approaches require fewer ISS snapshots and lower SNR than the ISS approach for source

¢ 2 NB enumeration. 10 RMSE

Assuming the number of sources is correctly estimated, we

3 10 explore the DOA estimation performances of the AP and SCR 0 2 4 6 8 10 approaches for scenarios with more sources than sensors. Fig. number of snapshots/sensor 7 (1a − 1d) compare the MUSIC pseudo-spectra of AP, SCR, 0 10 ISS and the equivalent NB approaches for 9 sources with AP DOAs indicated by vertical orange dashed lines. All sources SCR are assumed equal power with sensor level SNR = 0 dB and ISS

1 1 snapshots/sensor for each of the 41 frequency bands. The 10 NB equivalent narrowband case uses 41 snapshots/sensor. Note

that the AP, SCR and NB MUSIC spectra all show discernible RMSE

© 2 peaks near the true DOAs. However, the ISS approach shows 10 very shallow (smeared) peaks in its MUSIC spectra and misses detecting some sources. Panels (2a − 2d) compare the

¨ 3

MUSIC pseudo-spectra of AP, SCR, ISS algorithms for 10 10

¥ ¦ § ¤ 20 15 10 5 0 5 10 snapshots/sensor for the broadband sources and equivalently, SNR (dB) 410 snapshots/sensor for the narrowband scenario. When the Fig. 8: Comparing the RMSE of DOA estimates against (a) number of snapshots increases, the MUSIC spectra for all snapshots/sensor with fixed SNR = -5 dB (a) and against algorithms have sharper peaks at the true DOAs. However, (b) SNR with fixed 1 snapshots/sensor for 9 uncorrelated the MUSIC spectra for ISS algorithm is still shallower than equal power sources. The results indicate the AP and SCR the other 3 algorithms. algorithms achieve lower RMSE than the ISS algorithm in Fig. 8(a) compares the RMSEs of all approaches aver- low snapshots and SNR scenarios. aged over 500 Monte Carlo trials against number of snap- 10

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