Robust adaptive beamforming based on covariance matrix reconstruction with RCB principle

Abstract

In this paper, a novel robust adaptive beamforming (RAB) technique based on the covariance matrix reconstruction is proposed to solve the array model mismatch. The proposed technique provides an alternative method to reconstruct the interference-plus-noise covariance matrix (IPNCM) and estimate the steering vector (SV) of the desired signal. In particular, an unknown error exists in the SV of the considered array. The proposed RAB algorithm adopts the robust Capon beamformer (RCB) principle to roughly estimate the SVs of the desired and interference signals. Based on those preliminary estimated SVs, an improved Capon power spectrum is constructed. Then the interference covariance matrix is reconstructed by utilizing the modified Capon power spectrum integrated over the union of several disjoint angular sectors. Then, the reconstructed covariance matrix is further refined by exploring the low rank property. Meanwhile, the SV of the desired signal is estimated by solving a modified quadratically constrained quadratic programming (QCQP) problem. The simulation results show that the RCB principle provides an accurate IPNCM reconstruction, and the proposed RAB algorithm outperforms the existing RAB techniques over a wide range of input signal-to-noise ratio (SNR) region under various mismatch conditions.

Introduction

As a fundamental technology for array signal processing, adaptive beamforming widely applied in the fields of radar, sonar, communication, navigation, radio astronomy, and medical imaging [1], [2], [3], [4], [5], [6] in the past decades. The minimum variance distortionless response (MVDR) beamformer is known to be the optimal beamformer that maximizes the output signal-to-interference-plus-noise ratio (SINR) [7]; however, MVDR is sensitive to any steering vector (SV) mismatch, especially when the input SNR is high. This mismatch may be caused by physical and/or practical reasons, such as sensor calibration errors, signal direction errors, amplitude and phase errors, coherent/incoherent local scattering, etc. In addition, the presence of the desired signal in the training data will severely degrade the performance of the MVDR beamformer, especially in the case of a small training sample size. These flaws have limited the application of MVDR; hence many robust adaptive beamforming (RAB) techniques have been proposed in past decades to improve the robustness of MVDR against model mismatch [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41].

Diagonal loading (DL) is one of the most commonly used methods to improve robustness [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. The idea of the DL algorithm is to add a scaled identity matrix to the sample covariance matrix. The scaler for the identity matrix, which is known as the DL factor, determines the performance of the algorithm. In general, the DL algorithm is easy for implementation if the DL factor is selected empirically. However, the optimal DL factor is difficult to obtain in practice since the precise information about the interferences is unavailable. A class of beamforming algorithms aims to maximize the output SINR subjected to some uncertainty set constraint. For example, [17] considers a worst-case constraint, and [18] considers a probabilistic constraint. [19] shows that both these two methods can be interpreted as special cases of the DL method for selecting a suitable DL factor under certain constraints.

The eigenspace-based algorithm [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] is another useful tool to develop robust beamforming algorithms. For example, [28] projects the nominal SV onto the observed signal-plus-interference subspace to mitigate the effect of the SV perturbations and sample covariance matrix errors. [29] estimates the mismatched SVs by solving a quadratically constrained quadratic programming (QCQP) problem using an interference-plus-noise subspace projection matrix. [30] improves the accuracy of the covariance matrix estimation by applying a modified projection method. These eigenspace-based algorithms improve robustness of the beamforming technique; however, these methods usually suffer from severe performance degradation at a low signal-to-noise ratio (SNR), as it is difficult to effectively distinguish the signal subspace from the noise subspace in the low SNR condition.

In order to eliminate the influence of the desired signal on adaptive beamformers, a new class of RAB methods has been developed recently, which focuses on the reconstructed interference-plus-noise covariance matrix (IPNCM) [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. [31] originally proposes the idea of IPNCM reconstruction and SV estimation, in which the IPNCM is reconstructed by utilizing the Capon power spectrum integrated over a sector region separated from the direction of the desired signal, and the perturbed SV of the desired signal is estimated by a QCQP problem. This algorithm significantly improves the robustness against look direction error, but it is still sensitive to other kinds of errors such as array calibration error and random SV error. [32] proposes a low complexity algorithm that reconstructs the IPNCM by using the oracle approximating shrinkage method. [33] proposes another low complexity algorithm based on spatial power spectral sampling (SPSS). Both algorithms are computationally efficient, but their performance deteriorates to some degree. In [34], the sparsity of the desired signal and the interference signal is exploited to reconstruct the IPNCM. [35] considers the sparsity in the coprime array setting and proposes a joint covariance matrix optimization problem to improve the estimation accuracy of the reconstructed IPNCM. In [36], the IPNCM is reconstructed by utilizing a Capon power spectrum integrated over an annulus uncertainty set around the interference SVs. This algorithm promotes the robustness against the model mismatch since the actual SV is contained in the uncertainty set. However, the vectors other than the actual SV in the uncertainty set increase the computational complexity and introduce considerable redundancy. In [37], the SV of the desired signal is estimated by the intersection of two subspaces, and the IPNCM is reconstructed by suppressing the eigenvalue of the desired signal in the sample covariance matrix. This algorithm is robust against the large look direction mismatch, but its performance deteriorates when the power of the desired signal and the power of interference are close. [38] extends the subspace projection method to estimate the SVs and the power of the interference signals, by which the IPNCM is reconstructed. [39] estimates the orientation of each interference signal by the Capon power spectrum, based on which the SV and the power of the interference signal are obtained by orthogonality. The performance of this RAB algorithm degrades when the orientation of each interference cannot be accurately obtained; hence it has limited performance in the low INR region. [40] discusses the impact of interference power estimation on the performance of the RAB algorithm, and derives a simplified RAB algorithm. However, less computational complexity also decreases the performance to some degree. [41] adopts the maximum entropy power spectrum to improve the robustness on weak (interference) signals. However, this algorithm has power overestimation significantly for strong signals.

Inspired by the idea of IPNCM reconstruction, this paper proposes a novel RAB algorithm based on the robust Capon beamformer (RCB) principle. In particular, we use the RCB principle [9] to establish a second-order cone programming (SOCP) problem to construct a more accurate Capon power spectrum. Then, we estimate the interference covariance matrix by using the improved Capon power spectrum integrated over the small angular sections in which the interference signals located. This step reduces the computation amount of integrals and avoids the redundant information caused by irrelevant directions. Meanwhile, by analyzing the rank of the actual interference covariance matrix, we use the prime eigenvector to optimize the initially estimated matrix and obtain the final estimate of the IPNCM. In addition, we estimate the SV of the desired signal. The key idea of the estimation problem is to provide a more precise subspace, thus we propose modified constraints to improve the original QCQP problem. This novel approach improves the estimation accuracy for the SV of the desired signal. The contributions of this paper include:

• We introduce the RCB principle to estimate the SVs of the desired signal and the interference signals. Comparing with the nominal SV used in the conventional algorithm, the estimated SVs contain more information about the model mismatch.

• We adopt two approaches to reduce the redundant information in the reconstructed IPNCM. On the one hand, the integration region for IPNCM reconstruction is concentrated around the orientation of the signal; on the other hand, the reconstructed IPNCM is further refined by employing its prime eigenvectors to meet the rank requirement. These approaches improve the accuracy of the reconstructed IPNCM.

• We propose a modified QCQP method to estimate the SV of the desired signal. Comparing with the conventional QCQP method, covariance matrices used in the proposed method are reconstructed from the estimated SVs; hence model mismatch information is contained in the formulation.

Using these technics, simulations show that the proposed algorithm outperforms the existing RAB methods in a wide range of input SNR under various model mismatch cases.

The remainder of this paper is organized as follows. Section 2 introduces the background of adaptive beamformers and the underlying signal model. In Section 3, we present our proposed RAB algorithm. Numerical simulations are presented in Section 4. Section 5 offers conclusion remarks.

Notations: (random) vectors will be presented by bold face lower-case letters. Matrices will be presented by bold face upper-case letters. We will use ${(\cdot )}^T$ and ${(\cdot )}^H$ to denote the transpose operation and the conjugate transpose operation, respectively; Operator $E(\cdot )$ represents the statistical expectation.