Doa Estimation of Sparsely Sampled Nonstationary Signals

DOA ESTIMATION OF SPARSELY SAMPLED NONSTATIONARY SIGNALS

Liang Guo †‡, Yimin D. Zhang ‡, Qisong Wu ‡, and Moeness G. Amin‡

† School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China ‡ Center for Advanced Communications, Villanova University, Villanova, PA 19085, USA

ABSTRACT [14–17] for nonstationary signals. Traditional TF analysis The paper deals with sparsely sampled nonstationary signals and STFD applications assume that the nonstationary signals in a multi-sensor array platform. We examine direction-of- are uniformly sampled with a rate satisfying the Nyquist arrival (DOA) estimation using sparsity-based time-frequency criterion. In many real-world applications, however, nonsta- signal representation (TFSR). While conventional time- tionary signals are often observed with missing samples due frequency analysis techniques suffer from noise-like artifacts to fading, obstruction, and/or impulsive noise. The emerging due to missing data samples, high-fidelity time-frequency compressive sensing (CS) techniques also allow intentional signatures can be obtained by applying kernelled processing random undersampling for reduced system complexity [18]. and sparse reconstruction. Since the signals received at Missing samples generally yield noise-like artifacts that different sensors occupy the same time-frequency regions and spread over the entire TF domain [19]. It is shown that share a common nonzero support, the recovery of TFSRs can effective artifact mitigation can be achieved by applying TF be cast as a group sparse reconstruction problem. The recon- smoothing kernels [19, 20]. In multi-sensor systems, artifact structed auto- and cross-sensor TFSRs enable the formation mitigation can also be accomplished by averaging the TFSRs of the spatial time-frequency distribution (STFD) matrix, or ambiguity functions (AFs) corresponding to the different which is used, in turn, to propose the sparse time-frequency available sensors [21–23]. MUSIC (STF-MUSIC). The proposed STF-MUSIC method In this paper, inspired by the TF-MUSIC developed achieves effective source discrimination capability, leading to in the STFD context [8, 9, 11–13] and the sparsity-based improved DOA estimation performance. TFSR reconstruction as recently introduced in [22–25], we Index Terms— Time-frequency analysis, DOA estima- examine the sparsity-based STFD reconstruction of sparsely tion, compressive sensing, sparse sampling sampled nonstationary signals in a multi-sensor platform, and propose the sparse time-frequency MUSIC (STF-MUSIC) for effective DOA estimation. We exploit the fact that the 1. INTRODUCTION auto- and cross-sensor TFSRs share the same nonzero TF A large number of nonstationary signals encountered in support to cast their sparse reconstruction as a block-sparse radar, sonar, communications, and biomedical applications or group-sparse compressive sensing problem [26–30]. To are frequency modulated (FM) and can be characterized by mitigate the effect of the artifacts due to missing samples their instantaneous frequencies (IFs) [1–3]. Time-frequency in TF kernel optimizations, the well-known data-dependent signal representations (TFSRs) of such signals enable ex- adaptive optimal kernel (AOK) [6] is modified such that the ploitation of signal sparsity and local signal behaviors. The kernel operates on multi-sensor observations and applied to Cohen’s class defines a general bilinear TFSR framework [4], the AF averaged over all available sensors [23]. within which reduced interference distributions employ time- Once the kernel is designed, the corresponding TFSRs can frequency (TF) kernels to mitigate the effect of undesirable be obtained using two-dimensional (2-D) Fourier transform cross-terms [5–7]. of the kernelled AF. CS techniques may, however, replace For multi-sensor array applications, incorporation of the the Fourier transform with sparse parametric estimation and signal TF signatures into parameter estimation and signal achieve enhanced resolution in the TF domain [24]. By recovery was accomplished within the framework of spatial utilizing the one-dimensional (1-D) Fourier transform rela- time-frequency distributions (STFDs). The STFD has ef- tionship between the instantaneous auto-correlation function fectively been used to serve both problems of direction-of- (IAF) and the TF distribution (TFD), in lieu of the 2-D Fourier arrival (DOA) estimation [8–13] and blind source separation transform relationship between the AF and TF domains, CS reconstruction not only reduces the computation complexity, The work of L. Guo was supported in part by the National Natural but also improves the performance as a result of the presence Science Foundation of China (Grant No. 61107006), by the Fundamental Research Funds for the Central Universities (Grant No. K5051305005 and of local sparsity over each time instant. Considering that the No. NSIY171412), and by the China Scholarship Council. TFSRs across different array sensors share the same nonzero support, a group sparse compressive sensing algorithm is 2.2. The MUSIC Algorithm adopted. The reconstructed auto- and cross-sensor TFSRs Conventional MUSIC algorithm uses the correlation matrix enable formation of the STFD matrix, which is used to pro- of x(t), defined as pose the STF-MUSIC algorithm for effective DOA estimation of sparsely sampled nonstationary signals. The proposed H Rxx = E[x(t)x (t)]. (3) STF-MUSIC method achieves effective source discrimination capability, leading to improved DOA estimation performance. In practice, it is commonly estimated from the measured Notations. We use lower-case (upper-case) bold charac- snapshots as −1 ters to denote vectors (matrices). Fx and Fx respectively N represent the discrete Fourier Transform (DFT) and inverse 1 X H Rˆ xx = x(t)x (t). (4) DFT (IDFT) with respect to x. In particular, IN denotes the N N × N identity matrix. (·)∗ denotes complex conjugate of t=1 a complex value, and (·)T and (·)H respectively denote the Let Gˆ denote the noise subspace of matrix Rˆ xx. The transpose and conjugate transpose of a matrix or vector. In N×K MUSIC technique estimates the DOAs by determining the addition, C denotes the complete set of N × K complex values of ϑ for which the following spatial spectrum is entries. maximized [31]: 1 2. PROBLEM FORMULATION P (ϑ) = , (5) hH(ϑ)Gˆ Gˆ Hh(ϑ) 2.1. Signal Model where h(ϑ) is the steering vector corresponding to DOA ϑ. In narrowband array processing, consider K nonstationary It is noted that missing data in each antenna will result in signals impinging on an array consisting of N sensors. The a missing entry in the estimation of Rˆ xx. N × 1 received data vector x(t) and the K × 1 source signal vector d(t) are related by 3. CS-BASED SPARSE TIME-FREQUENCY MUSIC

x(t) = y(t) + n(t) = H(θ)d(t) + n(t), (1) 3.1. Spatial Time-Frequency Distributions where t ∈ [1, ··· ,T ] is the time index of a total number of T The Cohen’s class of bilinear TFD for signals x1(t) and x2(t) N×K is given by [4] data samples, H(θ) = [h(θ1), h(θ2), ··· , h(θK )] ∈ C is the steering matrix with h(θq) denoting the steering vector Z ∞ Z ∞ −j2πθt−j2πfτ corresponding to the DOA θq of the q-th source. n(t) ∈ Dx1x2 (t, f)= Ax1x2 (θ, τ)φ(θ, τ)e dθdτ, N×1 is an additive noise vector which consists of indepen- −∞ −∞ C (6) dent and identically distributed (i.i.d.) zero-mean, white and 2 complex Gaussian distributed processes with variance σnIN . where Ax1x2 (θ, τ) is referred to as the ambiguity function, For simplicity but without loss of generality, we assume σn to which is given by be the same for all sensors, i.e., σn = σ, for n = 1, ··· ,N. The noise elements are assumed to be independent of the Z ∞ τ τ A (θ, τ) = x (t + )x∗(t − )e−j2πθtdt. (7) signals, which are assumed to be deterministic. Here, each x1x2 1 2 −∞ 2 2 T element of vector d(t) = [d1(t), d2(t), ··· , dK (t)] is a single component FM signal. Consider the thinned sampling In the above expressions, f, θ and τ, respectively, denote of the array observations with a random pattern applied to the frequency index, the frequency shift, and the time lag. each array sensor, where the number of missing samples is The kernel φ(θ, τ) characterizes the distribution and will be

Mq satisfying Mq < T for q = 1, ··· ,N. As such, for discussed later. Dx1x2 (t, f) is referred to as auto-terms when the q-th array sensor, the thinned observation, xq(t), can be x1(t) = x2(t), and as cross-terms when x1(t) 6= x2(t). expressed as the product of yq(t), expressed in (1), and the Expressions (6) can be used to defined the data STFD matrix, following observation mask, D(t, f), whose (i, k)th element is Dxixk (t, f). For the linear data model of (1), the STFD matrix can be ( decomposed as [11, 14] 1, if t ∈ Sq, bq(t) = (2) 0, if t∈ / q, S Dxx(t, f)=Dyy(t, f)+Dyn(t, f)+Dny(t, f)+Dnn(t, f). (8) where Sq ⊂ {1, ··· ,T } is the set of observed time instants and its cardinality is |Sq| = T − Mq. Note that bq(t) may or Under the uncorrelated signal and noise assumption and the may not be the same for each sensor. zero-mean noise property, the expectation of the cross-term STFD matrices between the signal and noise vectors is zero, Then, an improved kernel in the multi-sensor platform is i.e., E[Dyn(t, f)] = E[Dny(t, f)] = 0, and it follows that obtained by replacing A(r, ψ) in (10) by AΣ(r, ψ) in (11), yielding better auto-term enhancement and artifact reduction. E[Dxx(t, f)] = Dyy(t, f) + E[Dnn(t, f)] H (9) = HDdd(t, f)H + E[Dnn(t, f)], 3.3. Compressive Sensing Model where Ddd(t, f) is the source TFD matrix whose entries The auto- and cross-sensor TFDs can be computed as the are the auto- and cross-source TFDs of the source. For 2-D Fourier transform of the corresponding kernelled auto- narrowband array signal processing applications, matrix H and cross-sensor AFs, i.e., Axixk (r, ψ)Φ(r, ψ), for i, k = holds the spatial information and maps the auto- and cross- 1, ··· ,N. By converting the kernelled and averaged AFs to source TFDs into auto- and cross-sensor TFDs [14]. Ex- the rectangular coordinate system, and denoting the result as pression (9) is similar to (4) which used in DOA estimation ˜ Axixk (θ, τ), the corresponding TFDs is expressed as problems relating the signal correlation matrix to the data D (t, f) = F −1{F [A˜ (θ, τ)]}. (12) spatial correlation matrix. xixk θ τ xixk Alternatively, we can obtain the TFDs through a group 3.2. Multi-sensor Adaptive Optimal Kernel sparse reconstruction from the same kernelled and averaged AFs. While earlier sparse TFD reconstructions were based Missing measurements produce artifacts in the TFD and AF on the 2-D Fourier transform relationship between the AF domains which resemble additive noise in the sense that they and the TFD [22, 24], it is shown in [19, 25] that the 1-D spread over the entire respective domains of joint-variable Fourier transform relationship between the IAF and the TFD representations [19]. These artifacts can be mitigated by yields simpler computations and enables the exploitation of applying proper TF kernels. In practice, to simplify the selec- local sparsity in the TF domain. tion of auto-terms and suppress the cross-terms and artifacts, For each pair of AF and IAF, the 1-D inverse Fourier we apply a smoothing kernel that significantly decreases the transform of Axx(θ, τ) with respect to θ yielding the ker- contribution of the cross-terms in the TF plane. As discussed nelled IAF, in [19, 23], the well-known data-dependent adaptive optimal −1 ˜ kernel (AOK) is uesd for this purpose. AOK is obtained by Cxixk (t, τ) = Fθ [Axixk (θ, τ)]. (13) solving the following optimization problem defined in the For each pair of IAF and TFSR, we denote c[t] as a polar coordinates [6]: xixk vector that consists of all IAF entries between xi(t) and xk(t) Z 2πZ ∞ along the τ dimension corresponding to time t, and w[t] max |A(r, ψ)Φ(r, ψ)|2rdrdψ xixk Φ 0 0 as a vector contains all the corresponding TFD entries with r2 respect to the frequency for the same time t. According to the subject to Φ(r, ψ) = exp − , (10) 2σ(ψ) Fourier relationship between the IAF and the TFD, we obtain Z 2π c[t] = Φw[t] + [t] , 1 xixk xixk xixk (14) 2 σ(ψ)dψ ≤ α, 4π 0 for 1 ≤ t ≤ T and i, k = 1, ··· ,N. Note in the [t] where α ≥ 0. In this expression, A(r, ψ) is the AF, and above expression that all the wxixk entries have the sparsity Φ(r, ψ) is the kernel function, both defined in the polar support for different sensor pairs, i.e., the respective positions [t] coordinates. of the nonzero entries of wxixk are the same for different In the underlying multi-sensor system, as discussed in pairs of i and j. In addition, since the same Φ is shared [23], averaging the AF obtained from each array sensor, for all sensor pairs, the above problem is referred to as a Aq(r, ψ), q = 1, ..., N, can enhance the auto-terms and multiple measurement vector (MMV) model. Such problem reduce cross-terms. Meanwhile, the missing data samples can be solved using block OMP (BOMP) [26], group Lasso yield artifacts that randomly spread over the entire AF do- (GLasso) [27], or multi-task BCS [29, 30]. In this paper, main, and the overall variance increases with more missing the BOMP algorithm is used as it is simple and allows data samples. As different sampling patterns in each sensor specification of the sparsity, but other techniques may also generate distinct artifact distributions, averaging the AFs over be used. all sensors effectively reduces the presence of such artifacts induced from missing samples. 3.4. Sparse Time-Frequency MUSIC Let Aq(r, ψ) denote the auto-sensor AF in the qth sensor, q = 1, ..., N. The averaged AF over all sensors is given by To perform the STF-MUSIC, we form an STFD matrix corresponding to a TF region S, expressed as N 1 X ˆ (S) X A (r, ψ) = A (r, ψ). (11) D = Dxx(t, f). (15) Σ N q xx q=1 (t,f)∈S 3

0.4 0.4 2 0.3 0.3 1 0.2 0.2 Frequency Frequency 0 0.1 0.1

waveform−real part −1 0 0 20 40 60 80 100 120 20 40 60 80 100 120 Time Time −2 0 20 40 60 80 100 120 Time (a) PWVD (b) TFD from CS reconstruction

Fig. 1. Waveform with missing samples marked in red circles. Fig. 2. PWVD and CS-based TFD of the sparsely samples signals.

5 5

Depending on the separability of the TFSRs of the sources, 0 0 the TF region S may include only a single source, a subset −5 −5 of sources, or all the sources. Generally, the selection of −10 −10 ) (dB) ) (dB) θ −15 θ −15 fewer sources in the TFSRs yields better DOA estimation P( P( performance [11]. −20 −20 −25 −25 ˆ (S) ˆ (S) Let G denote the noise subspace of matrix Dxx . The −30 −30 −50 0 50 −50 0 50 proposed STF-MUSIC technique estimates the DOAs of K0 Angle θ (degree) Angle θ (degree) (where 1 ≤ K0 ≤ K) selected sources by determining (a) STF-MUSIC (b) MUSIC the values of ϑ for which the following spatial spectrum is maximized, Fig. 3. DOA estimation results. 1 P (S)(ϑ) = . (16) strong cross-terms are present between the auto-terms. As hH (ϑ)Gˆ (S)(Gˆ (S))H h(ϑ) such, it is difficult to perform instantaneous frequency estimation and source selection from this result. In comparison, When K < K, the above procedure should be repeated for 0 Fig. 2(b) shows the TFD obtained from the CS technique for other TF regions until the DOA of all sources are estimated. the first sensor, where the AOK function is obtained based It is also noted that, with the source selection capability, the on array averaged AF. The TFD shown in Fig. 2(b) clearly total number of sources, K, may exceed the total number of reveal the true instantaneous frequencies of the two impinging sensors, N, as long as K ≤ N − 1 is satisfied in each TF 0 sources for individual source selection. As a result, we can region in which the STF-MUSIC algorithm is performed. separately perform the proposed STF-MUSIC on each of the two sources. 4. SIMULATION RESULTS The estimated STF-MUSIC pseudo spectra, overlaid for separated obtained results of both sources, are shown in Fig. In this section, we consider a uniform linear array (ULA) 3(a) for three trials for each source, where the estimated of N = 4 sensors with a half-wavelength interelement spectrum peaks respectively point to each DOA. On the other spacing. Two nonstationary signals (K = 2) impinge on the hand, conventional MUSIC algorithm, which does not have array with their respective arbitrary spatial signatures. The the capability for source selection, fails to separate the two signals emitted from the two sources are polynomial-phase sources in its estimated pseudo spectra, as shown in Fig. 3(b). FM signals with closely separated signatures. Their IF laws are expressed as 5. CONCLUSION 2 2 f1(t) = 0.05 + 0.1t/T + 0.1t /T , 2 2 (17) f2(t) = 0.15 + 0.1t/T + 0.1t /T , Compressive sensing techniques, combined with proper data- dependent kernel design, enable effective time-frequency sig- where t = [1, ··· ,T ] and T = 128. The two sources imping nature reconstruction from randomly sampled nonstationary ◦ ◦ from respective angles of θ1 = −3 and θ2 = 3 . The input signals due to the sparsity of such signals in their time- signal-to-noise ratio (SNR) is 0dB for both signals, and 50% frequency signal representations (TFSRs). The group sparsity of the data samples are randomly missing in each array sensor of the TFSRs of the signals observed in different sensors with different missing patterns. The real part of the waveform allows auto- and cross-sensor time-frequency distribution re- is shown in Fig. 4. construction, which leads to the formation of the spatial time- Fig. 2(a) shows the pseudo Wigner-Ville distribution frequency distribution matrix. 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