Off-grid direction-of-arrival estimation for nested arrays via uncertainty set extraction

Here, a novel off-grid direction-of-arrival (DOA) estimation algorithm for nested arrays in the context of sparse recovery is proposed. The framework of existing off-grid approaches is to correct the on-grid pa- rameter with a grid mismatch variable. From another perspective, we treat the direction-of-arrival deviation as the steering vector deviation equivalently, and the concept of steering vector uncertainty set is intro-duced to approximate the desired steering vector. Based on this idea, an optimization problem with respect to the uncertainty set for the pur-pose of steering vector reconstruction is formulated. After that, phase- ambiguity resolving and phase-difference approximation are orderly performed on the optimized uncertainty set to extract the off-grid direc- tion of arrival. Even though a coarse grid is given, simulation results are presented to demonstrate the performance advantage of the proposed direction-of-arrival estimation algorithm compared with several existing approaches.

✉ E-mail: chongyifan@nudt.edu.cn Here, a novel off-grid direction-of-arrival (DOA) estimation algorithm for nested arrays in the context of sparse recovery is proposed. The framework of existing off-grid approaches is to correct the on-grid parameter with a grid mismatch variable. From another perspective, we treat the direction-of-arrival deviation as the steering vector deviation equivalently, and the concept of steering vector uncertainty set is introduced to approximate the desired steering vector. Based on this idea, an optimization problem with respect to the uncertainty set for the purpose of steering vector reconstruction is formulated. After that, phaseambiguity resolving and phase-difference approximation are orderly performed on the optimized uncertainty set to extract the off-grid direction of arrival. Even though a coarse grid is given, simulation results are presented to demonstrate the performance advantage of the proposed direction-of-arrival estimation algorithm compared with several existing approaches.
Introduction: As one of the fundamental fields of array signal processing, direction-of-arrival (DOA) estimation has widespread applications including radar, sonar and wireless communication [1]. With common array configurations like the uniform linear array (ULA), the number of detectable targets is impossible to exceed the number of sensors. To break through such a limitation, the concept of virtual arrays derived from the sparse arrays is utilized to increase degree-of-freedom (DOF) for DOA estimation. Among the sparse arrays, a recently proposed one called the nested array [2] has attracted noticeable attention due to its systematical structure which has closed-form expressions for array configurations and achievable DOF. By exploiting the difference co-array, the DOF offered by the nested array can reach up to O(N 2 ) with only O(N ) physical sensors.
To take full advantage of the DOF superiority, DOA estimation algorithms for nested arrays are usually performed on the virtual array. However, the subspace-based approaches [2,3] require adequate snapshots, high signal-to-noise ratio (SNR) and related prior information, which limits their applicability. Inspired by the sparsity of incident signals, numerous approaches retrieve DOA via the technique of sparse recovery [4,5]. The sparsity-based approaches enhance the adaptation of DOA estimators but suffer from the basis mismatch problem because targets do not always fall on the pre-defined spatial grid exactly. Densifying the grid is not advisable since it highly increases the computational complexity and might conflict with the restricted isometry property [6]. To address this issue, the so-called off-grid approaches are proposed to correct the DOA with a grid mismatch variable. Parameterized as a nonconvex Taylor approximation model, the mismatch bias can be obtained in an iterative manner [7,8] or from the perspective of sparse Bayesian learning [9,10]. Moreover, the idea of joint sparsity [11] is used to relax the non-convex model as a convex problem that can be solved efficiently.
Off-grid targets will deteriorate the performance of sparse recovery significantly. Unlike the existing approaches, we address the off-grid targets from the perspective of steering vector reconstruction in this letter. The reduction of off-grid DOA bias is equivalent to reducing the steering vector bias, and we characterize such a bias via the steering vector uncertainty set instead of the Taylor approximation model. The proposed approach is summarized as three stages. First, we pre-estimate the DOA by a sparse signal reconstruction (SSR) approach [4] with a coarse grid. The computational complexity is low due to the coarse grid but the pre-estimation result will not be accurate enough. After that, an optimization problem with respect to the uncertainty set derived from the pre-estimated DOA is formulated to reconstruct the desired steering vector. Finally, phase-ambiguity resolving and phase-difference approximation are performed on the optimized uncertainty set to extract the off-grid DOA.
Signal model: An N-element prototype nested array consists of two ULAs where the inner one has N 1 sensors with inter-element spacing d and the outer one has N 2 sensors with spacing (N 1 + 1)d, that is, the sensors are located at {0, 1, . . . , In addition, d is usually chosen to be half wavelength λ/2.
Assume that L far-field uncorrelated narrowband signals from θ = [θ 1 , θ 2 , . . . , θ L ] T impinge on the nested array. The received signal at moment t can be modelled as where A(θ) = [a(θ 1 ), a(θ 2 ), . . . , a(θ L )] ∈ C N×L represents the manifold matrix of the nested array, denotes the signal waveform vector, and n(t ) ∼ CN (0, σ 2 n I ) denotes the additive Gaussian noise. Moreover, a(θ ) is the steering vector with the nth element taken as e − j(2π/λ)un sin θ where u n is the location of the nth sensor. To explore the second-order information of the received signal, we express the covariance matrix among L sources as where p l denotes the power of the lth source. By vectorizing the covariance matrix, we can formulate the equivalent virtual signal as The sensors of the augmented virtual array are located at {u m − u n | 1 ≤ m, n ≤ N}. After removing the repeated elements, a virtual ULA ranging from [2], and its virtual signal denoted asỹ is formed by selecting the corresponding elements from y. Sinceỹ behaves in a single-snapshot manner, we arrange the elements ofỹ in a Toeplitz matrix structure as where · denotes the th elements of the inner vector. According to [12], the covariance matrix based on spatial smoothing can be formulated as R ss =R yR H y /M. Note that R ss corresponds to the reference virtual array ranging from 0 to (M − 1)d with the steering vector expressed asã Preliminary estimation: In the context of sparse recovery, the angle range of interest is discretized into By defin-ingÃ as the collection of steering vectors over θ g , we can formulate the following constrained minimization problem with respect top where is a user-specific bound. This is a typical problem in the area of compressive sensing. Herein least absolute shrinkage and selection operator (LASSO), one of the most important techniques for sparse signal recovery, is selected to solve the above-mentioned problem by substituting 1 norm to 0 norm. Moreover, by defining B = [Ã, i] and r = [p T , σ 2 n ] T , we can reformulate (6) as where λ t is a regularization parameter. The LASSO algorithm is limited to solving real value problems, whereas y and B are generally complex. Thus, by definingŷ = [real(y) T , imag(y) T ] T andB = [real(B) T , imag(B) T ] T , we can equivalently modify (7) to Such a linear programming problem can be efficiently solved by the CVX toolbox [13]. After that, the pre-estimated DOAθ = [θ 1 ,θ 2 , . . . ,θ L ] T can be empirically obtained by selecting L dominant peaks of the LASSO-based spatial spectrum.
Off-grid DOA extraction: Suppose thatθ = [θ 1 ,θ 2 , . . . ,θ L ] T is obtained under a coarse spatial grid, and thus the computational complexity is low enough but the basis mismatch problem readily occurs. Besides, the preestimated steering vectorã(θ l ) deviates from the desired value. Herein, the steering vector of the off-grid target can be theoretically modelled as the following uncertainty set [14] whereã e is a random error vector and ε is its upper bound. We intend to utilize the uncertainty set to approximate the desired steering vector and then the DOA derived from the optimized uncertainty set will approach the true parameter. To achieve this idea, we formulate a covariance matrix fitting problem to maximize the signal power of interest p l with a constraint that the residual covariance matrix is positive semi-definite, that is, Such a covariance matrix fitting problem will shorten the Euclidean distance between the desired steering vector and the uncertainty set. By introducing a new variable κ l = 1/p l along with the standard technique of Schur complements [15], we can readily reformulate the optimization problem (10) as the following semi-definite programming (SDP) problem It is too time consuming to directly solve (11) which requires O(M 6 ) floating-point operations (flops). Noticing the particular structure of (11), we replace the signal power of interest with the Capon power, that is, p l = 1/(ã H (θ l )R −1 ssã (θ l )), followed with which (11) is equivalent tõ The solution to such an optimization problem will converge to the global optimum and (12) can be efficiently solved by the CVX [13] toolbox or using the Lagrange multiplier methodology [14] in O(M 3 ) flops. Note that the DOA parameter is encoded in the phase of the steering vector. As the phase difference ϕ l of any two adjacent elements withiñ a(θ l ) keeps equal as the filled dots illustrated in Figure 1, DOA can be theoretically extracted with only a single phase difference as which is the inverse function relating to the steering vector. However, the elements of the derived uncertainty setã o (θ l ) do not fall on the unit circle exactly and the phase differences between any two adjacent points are

6:
Repeat 3 to 5 until each element ofθ is enhanced.
unequal as the star dots illustrated in Figure 1. It is attributed thatã o (θ l ) does not hold the Vandermonde structure. Since the phase sequence of a(θ l ) orã o (θ l ) keeps increased, the phase ambiguity commonly occurs when the phase exceeds 2π . We can utilize the phase function to resolve phase ambiguity in MATLAB. In this case, phase(·) works for the pre-estimated steering vector, but it fails to insure monotonicity of the uncertainty set because its elements are randomly distributed. By regardingã(θ l ) as reference, the ith element ofã(θ l ) is always adjoint with a corresponding element which belongs toã o (θ l ) according to (9). Hence, these two interrelated elements are subjected to the identical ambiguity multiple. By denotingφ l = phase(ã(θ l )) andφ o l = phase(ã o (θ l )), we can calculate the unambiguous phase sequence of the uncertainty set bỹ where sign(·) denotes the signum function and round(·) represents rounding to the nearest integer. The expression (14) is a simple value correction where the latter part on the right side of (14) is the compensation to alleviate the effect of random distribution. With the unambiguous phase sequence of the optimized uncertainty setã o (θ l ), there are M − 1 unequal phase-difference values. Notice that these values fluctuate around the theoretical phase difference ϕ l , and thus the following statistical average can be calculated with these values to approximate the theoretical phase difference where φ i l = φ l i+1 − φ l i . Finally, by substituting φ l to (13), the enhanced estimation resultθ l can be obtained. Each step of the proposed algorithm is summarized in Algorithm 1. Moreover, the spatial spectrum of the pre-estimation algorithm can be regarded as the spectrum of the proposed algorithm by calibrating the peaks to corresponding DOAs. It is worth mentioning that the proposed algorithm can be readily extended to other augmentable sparse array configurations like co-prime arrays [16]. Simulation results: We present several numerical examples to demonstrate the advantage of the proposed off-grid DOA estimation algorithm. We consider an 8-element nested array where N 1 = N 2 = 4. A spatial grid from −90 • to 90 • with step sizes 3 • is pre-defined. By crossvalidation, λ t and ε in the proposed algorithm are set to be 0.25 and 0.75, respectively.
In the first numerical example, we test the DOF characteristic and present the on-grid SSR approach [4] for comparison. The simulation parameters are set to be SNR = 0 dB and the number of snapshots K = 500, respectively. Herein we consider 13 uncorrelated narrowband sources which are located at −61 that all 13 sources can be identified by the two algorithms with only 8 sensors, which demonstrates the effectiveness of the nested array to increase DOF. Although the SSR algorithm can find out all 13 sources, it suffers from the basis mismatch problem shown as the estimated peaks which obviously deviate from the actual sources. Compared with the SSR, the peaks yielded by the proposed algorithm are much closer to the red dashed lines as shown in Figure 2b. Consequently, the proposed algorithm is effective to alleviate the basis mismatch and produces a better estimation result with the off-grid targets.
In the second example, we evaluate the estimation accuracy of the tested algorithms, and we add two off-grid approaches named joint-LASSO (JLASSO) [11] as well as off-grid sparse Bayesian inference (OGSBI) [9] for comparison. A single target located at 19.35 • is considered first and we present the estimation results of 100 Monte Carlo runs in Figure 3. The average CPU time for running SSR, JLASSO, OGSBI and the proposed algorithm are 0.398, 0.597, 0.141 and 0.667 s, respectively. Limited by the pre-defined spatial grid, SSR can only yield peaks located at 18 • . In contrast, the other three approaches can correctly detect a target near the true location in each Monte Carlo trial, especially the proposed one. The performance is further compared in terms of   Figure 4, it is indicated that the proposed approach outperforms the other three approaches in any case, especially with high SNR or a large number of snapshots. Although the proposed algorithm is a little time consuming, it is worthwhile in accuracy-demanding applications like long-distance orientation where a slight DOA deviation will result in a much large location deviation.

Conclusion:
To address the basis mismatch problem, we propose a novel off-grid DOA estimation algorithm for nested arrays in this letter. Different from the existing approaches, the proposed one copes with off-grid targets from the perspective of steering vector reconstruction instead of mismatch variable compensation. We introduce the concept of steering vector uncertainty set to parameterize the deviation of the steering vector under a coarse grid, and then we formulate an optimization problem to obtain the optimized uncertainty set from which the off-grid DOA can be extracted after phase-ambiguity resolving and phase-difference approximation. Compared with several existing algorithms, the superiority of the proposed approach is validated through simulation results.