Sparsity-aware DOA estimation of quasi-stationary signals using nested arrays

Highlights

• The redundant components in the signal subspace can be eliminated effectively through a linear transformation.

• Formulate a sparse reconstruction problem including a reweighted ${\ell }_1$-norm minimisation subject to a weighted Frobenius norm.

• An explicit upper bound for error-suppression is provided for robust signal recovery.

• The proposed sparse-aware DOA estimation technique is extended to the wideband signal scenario.

Abstract

Direction of arrival (DOA) estimation of quasi-stationary signals (QSS) impinging on a nested array in the context of sparse representation is addressed in this paper. By exploiting the quasi-stationarity and extended virtual array structure provided inherently in the nested array, a new narrowband signal model can be obtained, achieving more degrees of freedom (DOFs) than the existing solutions. A sparsity-based recovery algorithm is proposed to fully utilise these DOFs. The suggested method is based on the sparse reconstruction for multiple measurement vector (MMV) which results from the signal subspace of the new signal model. Specifically, the notable advantages of the developed approach can be attributed to the following aspects. First, through a linear transformation, the redundant components in the signal subspace can be eliminated effectively and a covariance matrix with a reduced dimension is constructed, which saves the computational load in sparse signal reconstruction. Second, to further enhance the sparsity and fit the sampled and the actual signal subspace better, we formulate a sparse reconstruction problem that includes a reweighted ${\ell }_1$-norm minimisation subject to a weighted error-constrained Frobenius norm. Meanwhile, an explicit upper bound for error-suppression is provided for robust signal recovery. Additionally, the proposed sparsity-aware DOA estimation technique is extended to the wideband signal scenario by performing a group sparse recovery across multiple frequency bins. Last, upper bounds of the resolvable signals are derived for multiple array geometries. Extensive simulation results demonstrate the validity and efficiency of the proposed method in terms of DOA estimation accuracy and resolution over the existing techniques.

Introduction

Direction of arrival (DOA) estimation, which determines the spatial spectra of the impinging electromagnetic or acoustic waves by using sensor arrays, has been attracting considerable attention due to its application to various fields, for instance radar, sonar, and microphone array systems, to name a few. It is well known that an M-element uniform linear array (ULA) has $M-1$ degrees of freedom (DOFs), i.e., it can resolve up to $M-1$ sources or targets by using conventional subspace-based DOA estimation methods, such as MUSIC [1] and ESPRIT [2]. On the other hand, a higher number of DOFs can be achieved to identify more sources by using either distinct characteristics embedded in the signals, like noncircularity [3] of wireless communication signals and orthogonality between transmitted signals in MIMO radar [4], or the same number of sensors but sparsely placed [5], [6].

Quasi-stationary signals (QSS) represent an important class of signals that we frequently encounter in many applications such as microphone array speech processing [7] and electroencephalogram [8]. The QSS have the statistical property that remains locally static over a short period of time but varies from one local time frame to another. DOA estimation of such audio or speech signals plays an important role, for example, in teleconference systems and automatic conference minute generators. Such real-world applications usually are challenged by scenarios where more sources than the number of sensors are present simultaneously, which turns out to the so-called underdetermined DOA estimation problem. Recently, a Khatri-Rao (KR) subspace approach has been proposed in [9] to tackle this kind of issue for QSS. The work in [9] reveals that the quasi-stationarity in the time domain is not only favourable to the enhancement of DOFs in the spatial domain, i.e., $2(M-1)$ sources can be estimated with M physical sensors, but is also robust to unknown coloured noise. This work is then extended to two-dimensional array geometry, i.e., uniform circular array (UCA), and corresponding DOFs that the subspace method is able to provide is studied in [10].

To increase the number of resolvable sources beyond the number of given physical sensors, an alternative is to leverage sparse array geometries, such as emerging nested arrays [11], which are capable of providing $\mathcal{O}\left(M^2\right)$ DOFs with $\mathcal{O}\left(M\right)$ physical sensors. The idea behind this systematic sparse array design lies in achieving the higher number of DOFs by exploiting the difference co-array whose virtual sensor positions are determined by the lag differences between the physical sensors. The nested array is considered attractive as its synthetic geometry is equivalent to a ULA with larger apertures and DOFs which have a closed form expression with respect to the given number of sensors. Inspired by [11], much effort has been devoted to DOA estimation using nested arrays. In particular, by exploiting a four-level nested array with M elements, the subspace-based method in [12] further increases the identifiability to $\mathcal{O}\left(M^4\right)$ QSS.

Since the observations from the virtual array are obtained through vectorising the covariance matrix of the nested array, which is equivalent to signals with one snapshot impinging on the synthetic array, it intuitively entails a strong motivation for utilising rank restoration techniques, including spatial smoothing [11] and matrix restoration [13], to enable subsequent subspace-based methods to work properly. In [14], a hybrid approach is presented that uses a low-rank matrix denoising algorithm followed by a MUSIC-like method for DOA estimation.

In recent years, a novel approach, referred to as sparse signal reconstruction, has been introduced for statistical signal analysis and parameter estimation, exhibiting remarkable superiority in resolution and robustness to noise. Distinct from the principle of rank restoration, sparse reconstruction determines the DOA estimates by finding the sparsest supports which index an over-complete basis with respect to angular grids, without being confined by the condition of rank which is necessary for the subspace-based methods. The sparse signal reconstruction process can be regarded as the ${\ell }_0$-norm optimisation problem. Unfortunately, it is nonconvex and NP-hard to solve [15], [16].

To overcome this difficulty, many algorithms have been proposed for convex relaxation, e.g., ${\ell }_p$-norm, 0 < p ≤ 1, optimisation. In [17], a recursive weighted least-squares algorithm called FOCUSS is addressed for achieving sparsity with a single measurement vector (SMV) in the problem of DOA estimation. In [18], the vector formed by the beamformer outputs is considered as a sparse linear combination of bases of beamformer outputs by the global matched filter (GMF). So far, the most representative sparse recovery algorithm for DOA estimation is ${\ell }_1$-SVD [19], which exploits the ${\ell }_1$-norm penalty to enforce sparsity in conjunction with singular value decomposition (SVD) for dimension reduction of the problem size. A merit of the ${\ell }_1$-SVD is that it is applicable to not only the SMV case but also the multiple measurement vector (MMV) problem. Although the ${\ell }_1$-norm minimisation is a convex problem and the global minima can be guaranteed easily, it has inherent drawbacks: one is the equal penalisation which is actually prejudiced for large supports, and the other is suboptimal regularisation between the sparsity penalty and subspace fitting error, sometimes resulting in unsatisfactory sparse recovery especially at low signal-to-noise ratio (SNR).

To alleviate this problem, the iterative reweighted ${\ell }_1$ minimisation [20] was developed, where large weights are used to punish the entries who are more likely to be zero in recovered signals, whereas small weights are used to preserve the large entries. Based on [19], [20], Xu et al. proposed the weighted ${\ell }_1$-norm penalty utilising the property of the Capon spectrum function, referred to as CSW-${\ell }_1,$ to improve the performance of ${\ell }_1$-norm minimisation, and a theoretical guidance of selecting the regularisation parameter has been offered in their work [21]. Sparse representation of array covariance vectors (SRACV) [22] also employed the ${\ell }_1$-norm penalty to reconstruct the covariance vectors, while it addressed the problem in the correlation domain instead of the raw data domain. Following the work of [22], Hu et al. attempted to obtain the DOA estimates by solving an ${\ell }_2$-norm minimisation subject to a constrained ${\ell }_1$-norm [23], but the upper bound of the sparse vector is experimentally determined and no theoretically guaranteed quantity is provided. The sparse spectrum fitting (SpSF) algorithm [24] has well elaborated on the best choice of regularisation parameter for DOA and power estimation, with a reduced computational complexity. More recently, similar work has been followed by Tian et al., and a sparse representation of second-order statistics vector and ${\ell }_0$-norm approximation (SRSSV-${\ell }_0$) [25] has been discussed. In the work, a surrogate truncated ${\ell }_1$ function was exploited to approximate the ${\ell }_0$-norm, and the nonconvex minimisation problem can be handled by the difference of convex functions decomposition. Without choosing any hyperparameters such as the methods mentioned above, a sparse iterative covariance-based estimation (SPICE) [26] approach was proposed for array signal processing by the minimisation of a covariance matrix fitting criterion. Instead of approximating the ${\ell }_0$-norm with the ${\ell }_1$-norm, Hyder and Mahata exploit a family of Gaussian functions to deal with the ${\ell }_1$-norm approximation problem, and further propose an alternative strategy named JLZA-DOA [27]. However, as the iteratively updating Gaussian variances are chosen subjectively, global minima is not guaranteed.

As it becomes clear from the aforementioned discussion, the superiority of the nested array and sparse reconstruction motivate us to incorporate them to cope with DOA estimation of QSS, with further enhanced DOFs and even larger effective array aperture. It is essential for QSS to stack the KR products of covariance matrices of local frames, to achieve the array aperture expansion, nevertheless, directly applying some sparsity-inducing algorithms, like SRACV and SRSSV-${\ell }_0,$ to these KR products, which can be regarded as multiple pseudo snapshots of the expanded virtual array, cannot achieve higher DOFs as these methods are designed for the SMV not the MMV problem in the correlation domain herein. In addition, there is no reason to believe that the pseudo signals, i.e., the covariances of the true signals in each frame, are uncorrelated to each other or the number of pseudo snapshots (frames) is sufficiently large to ensure the uncorrelation, so a transformation of the MMV to the virtual SMV problem in the sparse signal representation framework, as [22], [25] have done, is not applicable. On the other hand, though other sparse recovery methods, e.g., ${\ell }_1$-SVD, CSW-${\ell }_1,$ and SpSF, can deal with the MMV problem, they require a priori knowledge of noise statistics to determine the regularisation parameter, but this is not available in the model of pseudo snapshots generated from QSS. Consequently, by utilising a nested array in conjunction with the sparse representation framework in the case of QSS, new strategies for solving the correlation-aware support recovery problem with MMV are desired.

In this paper, the DOA estimation of QSS is performed by employing a novel sparse weighted subspace reconstruction (SWSR) with a nested array. We first introduce the sparse representation framework for MMV to the QSS case, and provide a reasonable selection of the weights for the ${\ell }_1$-norm penalty, to further enforce the sparsity and approximate the original ${\ell }_0$-norm, and the regularisation parameter for the subspace fitting error, to guarantee robust sparse recovery. Besides, we derive the analytical expression of the available number of DOFs as a function of the number of physical sensors, M, and show that the maximum identifiability of the proposed method is $\mathcal{O}\left(M^2\right)$ which is twice as many as for ordinary nested array processing as the additional quasi-stationarity is exploited herein. Second, the proposed sparsity-aware scheme is extended to the wideband signal case using ${\ell }_{2,1}$-norm sparse regularisation. The wideband extension of the narrowband algorithm is not a simple average of the results obtained from different frequency bins, and a new formulation based on the group sparsity concept has to be applied to effectively exploit the information carried across the frequency bands of wideband signals, resulting in suppression of the spatial aliasing. As we shall show in the subsequent sections, the proposed method has a better estimation accuracy, a larger identifiability, and a superior resolution compared with the existing techniques. Analytical and numerical simulation results will both be given to support each other in the ensuing development.

It should be noted that a similar research issue has been addressed in [28], where the fourth-order difference co-array is exploited to gain higher DOFs. The main distinctions between SWSR and the method in [28], SAFE-CPA, are stark, which lie in: (a) SAFE-CPA virtually expands the physical array to have a much larger aperture, implemented by vectorising covariance matrices twice¹, which highly relies on the assumption that the powers of QSS are wide-sense stationary and uncorrelated with each other. However, the prerequisite cannot always satisfy real applications when the signal powers are correlated to each other, or signal quasi-stationarity cannot last for long, say several dozens of frames, which will also incur the partially correlated power case that is particularly relevant in the frame-starved scenario. In contrast, SWSR accommodates correlation between source powers even if they are fully correlated; (b) SAFE-CPA converts DOA estimation into solving a basis pursuit denoising problem, and the key parameter therein, error bound preserving the data fidelity, is chosen through trial-and-error in every experiment, which involves extra computations and is not pragmatic in real applications, whereas SWSR not only explicitly provides such an upper bound in a mathematic expression that can guarantee the signal subspace fidelity will be satisfied with a high probability, but also enforce weights on sparse supports to make ${\ell }_1$ norm further approximate ${\ell }_0$ norm; (c) SAFE-CPA is designed for quasi-stationary signals contaminated by white Gaussian noise, whereas SWSR is robust to spatially coloured noise. These differences enable SWSR to have its own advantages on DOA estimation of QSS.

The rest of the paper is organised as follows. The DOA estimation model of QSS in a nested array is introduced in Section 2. The proposed sparsity-aware algorithm is discussed in detail first based on the narrowband signal condition then extended to the wideband scenario in Section 3, where some mathematically analysis relating to the maximum number of resolvable signals is also performed. Section 4 provides extensive numerical simulations demonstrating the advantages of our method in terms of DOFs, estimation resolution, and accuracy. Finally, some concluding remarks are drawn in Section 5.

Notations: We use lower-case (upper-case) bold characters to denote vectors (matrices). In particular, the operators (·)^T, (·)*, (·)^H, ${(·)}^{-1},$ E[·], ⊗, and ○ denote the operation of transpose, conjugate, conjugate transpose, inverse, expectation, Kronecker product, and Khatri-Rao product, respectively. The complex operator j is reserved for the imaginary unit, i.e., $j=\sqrt{-1}$. The notations Pr{·}, rank(·), and vec(·) denote the probability of an event, rank of a matrix, and the vectorisation operator that turns a matrix into a vector by stacking all columns on top of each other, respectively. ‖ · ‖₀, ‖ · ‖₁, ‖ · ‖₂, and ‖ · ‖_F denote the ${\ell }_0,$ ${\ell }_1,$ Euclidean (${\ell }_2$), and Frobenius norms, respectively. The symbol diag{z₁, z₂} represents a diagonal matrix with diagonal entries z₁, z₂ and $\text{blkdiag}\{{\mathbf{Z}}_1,{\mathbf{Z}}_2\}$ represents a block diagonal matrix with diagonal entries ${\mathbf{Z}}_1,{\mathbf{Z}}_2$. The symbol $\mathbf{Z}(a:b,c:d)$ denotes a constructed submatrix with the entries from the a to bth row and c to dth column of $\mathbf{Z},$ and the symbol $\mathbf{Z}(a,b)$ denotes the entry in the ath row and bth column of $\mathbf{Z}$.