Study of Robust Direction Finding Based on Joint Sparse Representation

Direction of Arrival (DOA) estimation stands as one of the pivotal subjects in array signal processing, playing a key role in fields such as radar, sonar and wireless communications [1]. Over the past few decades, DOA estimation techniques have been introduced and extensively studied in the academic literature, such as classical subspace-based approaches (e.g.,MUSIC,ESPRIT) [2, 3, 4, 5, 6], which rely on the eigenvalue decomposition of the sample covariance matrix. However, they require a substantial number of snapshots to obtain accurate estimates and their performance degrades significantly in the presence of non-Gaussian noise, such as impulsive noise. Variants of subspace-based methods have been proposed to deal with impulsive noise, such as ROC-MUSIC [7], FLOM-MUSIC [8]. These methods utilize fractional lower-order statistics instead of second-order statistics to tackle impulsive noise, but still require a significant number of samples. Inspired by robust statistics, the $l_{p}$-MUSIC method has been introduced [9], replacing the traditional Frobenius norm with the $l_{p}$-norm (1 $\leq$ $p$ $<$ 2) to minimize residuals, which is less sensitive to outliers than the former. Based on the work of [9], [10] introduced a generalized maximum complex correntropy criterion for suppressing outliers in impulse noise.

Recently, the development of compressive sensing theory has led to the emergence of numerous DOA estimation methods based on sparse representation [11, 12]. These methods encompass greedy algorithms [13], convex optimization [14], and sparse Bayesian learning [15]. Unlike subspace-based methods, sparse signal recovery methods [16, 17, 18] based on compressive sensing theory offer advantages in low signal-to-noise ratios, fewer snapshots, and highly correlated sources. In practice, the DOAs may not be aligned precisely with the predefined grid, leading to certain angle deviations. To address this issue, two methods are commonly employed. One approach involves increasing the number of predefined grid points; however, as the grid density increases, the correlation between adjacent atoms in the dictionary grows significantly. This contradicts the Restricted Isometry Property (RIP) condition of compressive sensing and comes with a high computational cost [19]. The other approach is to adopt an off-grid strategy, where the grid spacing is treated as a parameter. It is estimated by first or second-order Taylor expansions, effectively approximating it [20, 21]. In these approaches, employing the $l_{2}$-norm as the residual term is inefficient, as it is equally sensitive to impulse noise, making it less robust in the presence of outliers.

Robust sparse recovery methods have been proposed to mitigate the impact of outliers in [22, 23]. Different norms are used in [22, 23] as residual fitting terms, which are less sensitive to large outliers as compared to least-squares. This can, to some extent, reduce the influence of impulsive noise or outliers on DOA estimation. In [24], a novel sparse robust signal recovery framework was introduced. This method exploits the sparsity of outliers and the statistical properties of Gaussian noise, making it highly robust against outliers. In [25], a sparse Bayesian learning algorithm for DOA estimation under impulsive noise conditions was introduced, which exhibits high resolution and accuracy.

In this work, we introduce a novel DOA estimation technique for impulsive noise scenarios, where the impulsive noise is modeled as a combination of Gaussian noise and outliers and an objective function is constructed to exploit the sparsity of outliers and the statistical properties of Gaussian noises. This objective function incorporates the Minimax Logarithmic Concave (MLC) Function [26] as a sparsity-inducing term for outliers. The proposed design significantly enhances the algorithm’s robustness, particularly when dealing with severe impulsive noise conditions. Furthermore, we extend this approach to Multiple Measurement Vectors (MMV) and introduce an alternating optimization algorithm to tackle the grid mismatch problem, which employs rough estimates of the on-grid angles and the outlier matrix to obtain off-grid DOA estimates.

Consider $K$ far-field narrowband sources $s_{k}(t),(k=1,2,...,K)$ impinging on a uniform linear array (ULA) with $M$ omnidirectional elements from directions of $\bm{\theta}=[\theta_{1},...,\theta_{K}]^{T}$. The array output at snapshot $t$ is given by

where $\mathbf{y}(t)=[y_{1}(t),...,y_{M}(t)]^{T}$ is the array output, $\mathbf{A}(\bm{\theta})=[\mathbf{a}(\theta_{1}),...,\mathbf{a}(\theta_{K})]$ is the array manifold matrix, $\mathbf{a}(\theta_{k})=[1,...,e^{-j2\pi(M-1)d/\lambda_{s}\sin({\theta_{k}})}]^{T}$ is the time delay of the $k$th source at each array element, $\mathbf{s}(t)=[s_{1}(t),...,s_{K}(t)]^{T}$ are the source waveforms, $\lambda_{s}$ and $d$ denote the wavelength of sources and the distance between adjacent array elements, respectively, $d=\lambda_{s}/2$, and $\mathbf{n}(t)$ is the unknown noise. The model in (1) after $T$ snapshots can be rewritten in matrix form as

where $\mathbf{S}=[\mathbf{s}(1),...,\mathbf{s}(T)]$ and $\mathbf{N}=[\mathbf{n}(1),...,\mathbf{n}(T)]$.

Given $\mathbf{A}$ and $\mathbf{Y}$, the DOA estimation problem can be transformed into a sparse signal recovery problem. Exploiting the sparsity of the number of sources in the spatial domain, the angular space $[-\pi/2,\pi/2]$ is divided into $N$ equidistant grids $\overline{\bm{\theta}}=[\overline{\theta}_{1},...,\overline{\theta}_{N}]^{T}$, satisfying the condition $N\gg K$ . The sparse representation of (2) can be rewritten as

where $\mathbf{A(\overline{\bm{\theta}})}=[\mathbf{a}(\overline{\theta}_{1}),...,\mathbf{a}(\overline{\theta}_{N})]$ is an overcomplete dictionary and $\mathbf{X}=[\mathbf{x}(1),...,\mathbf{x}(T)]\in\mathbf{C}^{N\times T}$ is a row sparse matrix with $K$ nonzero rows. The indices of rows in $\mathbf{X}$ correspond to the positions of DOAs on the grid. If the $n$th row of $\mathbf{X}$ is nonzero, it represents the presence of a source in direction $\overline{\theta}_{n}$. Therefore, the objective of DOA estimation is to compute $\mathbf{X}$.

In this section, we detail the proposed DOA estimation algorithm based on the MLC function and an on-grid model.

In this section, we present several examples of the proposed algorithm. We consider a ULA with 10 elements. Two narrowband sources with equal power impinge on the ULA. Specifically, the uncorrelated source waveforms are generated as $s_{k}=e^{j\varphi(s_{k})}$, and the phase $\varphi(s_{k})$ follows a uniform distribution in the range $[0,2\pi]$. Unless otherwise specified, these two sources come uniformly from the interval $[-10^{\circ},0^{\circ}]$ and $[20^{\circ},30^{\circ}]$. $\mathbf{W}$ is an independent and identically distributed complex Gaussian noise matrix with mean zero and variance of $\sigma_{1}^{2}$. $\mathbf{O}=\mathbf{E}\odot\mathbf{N}$ is used to model outliers, where $\mathbf{E}$ is Bernoulli distributed with $P\left\{\mathbf{E}_{m,t}=1\right\}=p$, $p$ is the probability of outliers occurring, and $\mathbf{N}$ is a complex Gaussian noise matrix with mean zero and variance $\sigma_{2}^{2}$. The signal-to-noise ratio (SNR) and the signal-to-outlier ratio (SOR) are defined by $\sigma_{s}^{2}/\sigma_{1}^{2}$ and $\sigma_{s}^{2}/\sigma_{2}^{2}$, respectively, where $\sigma_{s}^{2}$ is variance of the source. In each simulation, 1000 Monte Carlo experiments are conducted and SOR = -20dB. The average RMSE is given by

where $N_{mc}$ is the number of Monte Carlo experiments, $\tilde{\theta}_{j,k}$ is the value estimated for the $k$th source in the $j$th Monte Carlo experiment, and ${\theta}_{k}$ is true DOA for the $k$th source. Given that some algorithms may not perform well in extreme cases, leading to a very large RMSE, we include cases where the DOA estimation error is less than $3^{\circ}$ in the RMSE. $l_{p}$-MUSIC [9], Bayes-optimal [25], SOMP-LS [29], GMCCC-MUSIC [10], and SBL-Tyler [30] are used for comparison. The grid interval is selected as $r=2^{\circ}$, and $p=1.1$ for $l_{p}$-MUSIC. For SBL-Tyler, as recommended, the dictionary size $N=18001$. We tested the relationship between SNR and DOA estimation accuracy. According to the results, when the SNR is greater than $5$dB, we use $\lambda_{1}=7,\lambda_{2}=1.4$; otherwise, we use $\lambda_{1}=4,\lambda_{2}=4$.

In the first example, the performance of various algorithms is tested under different SNRs. The number of snapshots is $T$ = 30 and $p$ is set to 0.1. From the left side of Fig. 2, it can be observed that SOMP-LS exhibits the poorest performance, mainly due to its high sensitivity to outliers. As the SNR increases, $l_{p}$-MUSIC and GMCCC-MUSIC show a reduced rate of performance improvement due to the limitation of snapshots. In contrast, methods based on sparse representation do not have strict requirements in this regard. Compared to Bayes-optimal, our proposed algorithm can more accurately mitigate the influence of outliers, leading to improved performance.

In the second example, we investigate the impact of the number of snapshots on DOA estimation performance. The SNR and $p$ are fixed at 10dB and 0.1 respectively. As demonstrated in the right side of Fig. 2, the proposed algorithm exhibits greater robustness. Interestingly, as the number of snapshots increases, GMCCC-MUSIC outperforms Bayes-optimal. This is because subspace-based algorithms heavily rely on large samples.

The left side of Fig. 3 presents the variation in algorithm performance with changes in the probability of outlier occurrences. The proposed algorithm and Bayes-optimal demonstrate stable performance, which is attributed to the separate handling of outliers in the presence of impulse noise.

In DOA estimation, the ability to distinguish between two closely located sources is one of the important criteria for assessing performance. In the fourth example, $p$ are set to 0.1. The two uncorrelated sources are considered with DOAs $\theta_{1}=-10.8^{\circ}$ and $\theta_{2}=-10.8^{\circ}+\Delta\theta$, where $\Delta\theta$ varies from $0.5^{\circ}$ to $10^{\circ}$. We consider the two sources distinguishable if $\max\limits_{k=1,2}\lvert\tilde{\theta}_{k}-\theta_{k}\lvert$ is less than $\lvert\theta_{1}-\theta_{2}\lvert/2$. From the right side of Fig. 3, we notice that our algorithm outperforms other methods.

In the last example, the performance of the algorithms under coherent (fully correlated) sources is investigated, where the parameters are the same as in example 1. From Fig. 4, we observe that the proposed algorithm has good performance in this environment, while the performance of the $l_{p}$-MUSIC and GMCCC-MUSIC algorithms significantly degrades.

We have introduced a novel DOA estimation algorithm with robustness against impulsive noise. We model impulsive noise as a combination of outliers and Gaussian noise, exploiting the statistical differences between them. By employing the ADM to solve the optimization problem, we obtain outliers matrix and the on-grid DOAs. To address the grid mismatch, we use an alternating approach with the obtained outlier matrix and on-grid DOAs to obtain the final DOAs. Simulation results demonstrate that this method exhibits excellent resilience against impulsive noise.