Elsevier

Signal Processing

Volume 188, November 2021, 108172
Signal Processing

Robust adaptive beamforming based on virtual sensors using low-complexity spatial sampling

https://doi.org/10.1016/j.sigpro.2021.108172Get rights and content

Highlights

  • Design a robust adaptive beamforming method for uniform linear arrays based on a virtual sensor.

  • A novel approach to reconstruct the IPNC matrix using a low-complexity spatial sampling process (LCSSP).

  • The power spectrum sampling is realized by a proposed projection matrix orthogonal to the signal subspace that retains the interference-plus-noise in a higher dimension.

  • A new robust adaptive beamforming algorithm is developed.

Abstract

The performance of robust adaptive beamforming (RAB) based on interference-plus-noise covariance (IPNC) matrix reconstruction can be degraded seriously in the presence of random mismatches (look direction and array geometry), particularly when the input signal-to-noise ratio (SNR) is high. In this work, we present a RAB technique to address covariance matrix reconstruction problems. The proposed RAB technique involves IPNC matrix reconstruction using a low-complexity spatial sampling process (LCSSP) and employs a virtual received array vector. In particular, the power spectrum sampling is realized by a proposed projection matrix in a higher dimension. The essence of the proposed technique is to avoid reconstruction of the IPNC matrix by integrating over the angular sector of the interference-plus-noise region. Simulation results are presented to verify the effectiveness of the proposed RAB approach.

Introduction

In the last decades, many adaptive beamforming techniques have been developed and applied in wireless communications, sonar, and radar due to their superior interference mitigation capability [1], [2] to standard Capon beamforming (SCB) approaches. The SCB has good interference suppression capability and can yield a high output signal to interference-plus-noise ratio (SINR) when the actual steering vector of the signal of interest (SOI) is known or the sample covariance matrix (SCM) does not contain a component associated with the SOI. However, under non-ideal conditions such as finite data samples and mismatch between the presumed and true steering vector (e.g., imperfect sensor calibration, signal pointing and look direction errors, wavefront distortions, local scattering), the performance of adaptive beamformers degrades substantially.

Several robust adaptive beamforming (RAB) techniques have been proposed to enhance robustness against the aforementioned mismatches and improve the robustness of the beamformers. These techniques mainly use the linearly constrained minimum variance (LCMV) beamformer [3], diagonal loading (DL), eigenspace-based (ESB) methods, worst case-based techniques, the probabilistically constrained approach in [4] and uncertainty set-based methods. The DL method is widely applied due to its ease of application and effectiveness to overcome SOI self-elimination. A scaled identity matrix named the DL matrix is added to the SCM, to mitigate possible ill-condition of the covariance matrix as well as to eliminate SOI suppression [5], [6]. However, it is difficult to determine an appropriate DL factor [7]. The ESB method focuses on obtaining an accurate steering vector estimate for the SOI [8], [9], [10]. To this end, the nominal steering vector is projected onto the signal-plus-interference subspace. The signal-subspace-based methods have the common drawback that it is hard to separate the signal and noise subspaces at low signal-to-noise ratio (SNR). In [11] and [12], a worst-case-based technique was proposed to achieve good performance. However, with this approach, it is very difficult to obtain the steering vector mismatch and the error bound in practical applications. Moreover, at high SNRs, the performance of this method will severely degrade in the presence of array steering vector errors. The uncertainty set-based algorithms estimate the desired signal steering vector based on the elliptical and spherical uncertainty set of the SOI steering vector by solving an optimization problem [13]. However, their performance is mainly determined by the uncertainty parameter set and it is difficult to select the optimal factor in practice [14]. Hence, the development of low-complexity RAB approaches has been a very active research topic in recent years. Nevertheless, a major cause of performance degradation in adaptive beamforming is the presence of the desired signal component in the training data, especially at high SNR.

To address this issue, many works tried to remove the SOI components by reconstruction of the interference-plus-noise covariance (IPNC) matrix instead of using the sample covariance matrix. In [15], the IPNC matrix is reconstructed by integrating the nominal steering vector and the corresponding Capon spectrum over the entire angular sector except for the region near the SOI, while the desired signal steering vector is estimated by solving a quadratically constrained quadratic programming (QCQP) problem. This method shows reasonable performance, but is sensitive to large direction-of arrival (DoA) mismatches [16], [17]. Several categories of IPNC matrix-based beamformers were then proposed, such as the beamformer in [18], which relies on a correlation coefficient method, the computationally efficient algorithms via low complexity reconstruction in [19], [20] that reconstruct the IPNC matrix by subtracting the reconstructed desired signal covariance matrix from the sample covariance matrix. The algorithm in [21] jointly estimates the theoretical IPNC matrix using the eigenvalue decomposition of the received signal’ s covariance matrix and the mismatched steering vector using the output power of the beamformer. In [22] a robust beamforming algorithm has been developed based on the IPNC matrix reconstruction using spatial power spectrum sampling (SPSS). This method has lower computational complexity, but its performance is degraded as the number of sensors is decreased. The algorithm in [23] mitigates the effect of sensor position errors by compensating for the estimated sensor position errors in the IPNC matrix reconstruction using subspace-fitting based reconstruction. In [24] a procedure analogous to those of [15], [25] is used to reconstruct the IPNC matrix and the desired signal steering vector estimation. However, the accuracy of the interference steering vector estimation is related to an ad hoc parameter. The beamformer in [26] utilizes the orthogonal subspace (OS) to eliminate the component of the SOI from the angle-related bases while in [27] a beamformer based on the IPNC and desired signal covariance matrices is proposed, which estimates all interference powers as well as the desired signal power using the principle of maximum entropy power spectrum (MEPS).

In this paper, we devise a novel low-complexity RAB approach based on IPNC matrix reconstruction that achieves nearly optimal performance by addressing the inaccurate covariance matrix construction problems. The essence of the proposed idea is based on the exploitation of virtual sensors to reconstruct the IPNC matrix using a low-complexity spatial sampling process (LCSSP). The power spectrum sampling is realized by a proposed projection matrix orthogonal to the signal subspace that retains the interference-plus-noise in a higher dimension. To this end, we extend the sensor array with a number of virtual sensors. In contrast to previously reported works with IPNC construction, we avoid the reconstruction and estimation of the IPNC matrix by integrating over the angular sector of the interference-plus-noise region. Moreover, we develop a mathematical analysis of the proposed LCSSP RAB technique along with an analysis of the computational complexity of LCSSP and existing RAB techniques. The proposed algorithm requires prior information about the angular sector of SOI and the interference-plus-noise region as well as their corresponding direction of arrival, and has a computational complexity that is cubic in the number of sensor elements.

Simulation results are presented to verify the effectiveness of the proposed LCSSP RAB method while requiring less computational complexity than competing approaches. Moreover, the proposed LCSSP RAB approach is more robust against mismatches in the steering vectors, since the projection used in our algorithm only requires very rough estimates of the angular sectors in which the signal of interest and the uncertainties must lie.

The main contributions of this paper are summarized as follows:

  • We accurately estimate the IPNC matrix without resorting to the power spectrum of the interference steering vectors directly. Instead, we define a projection matrix onto an approximation to the orthogonal space of the steering vector of the SOI, taking advantage of the higher dimensions to better approximate the interference-noise-plus space.

  • The proposed LCSSP RAB beamformer reduces the computational complexity of IPNC matrix reconstruction by avoiding integrating over the angular sector on the interference-plus noise region.

  • A performance analysis is provided for the proposed approach. Moreover, numerical examples are presented to verify the superiority of the proposed approach over existing techniques.

The remainder of the paper is organized as follows. Section 2 describes the system model, while Section 3 discusses the design of the proposed LCSSP algorithm. Section 4 presents a comparison of the computational complexity for all tested methods versus the proposed method. Simulation results are described in Section 5. Finally, conclusions are drawn in Section 6.

Notation: We adopt the notation of using boldface for vectors a (lower case), and matrices A (upper case). The conjugate operator, transpose operator and the conjugate transpose operator are denoted by the symbols (·)*, (·)T and (·)H, respectively. I and 0 denote respectively the identity matrix and the matrix (or the row vector or the column vector) with zero entries (their size is determined from the context). The mathematical expectation is expressed as E{·} while the letter j represents the imaginary unit (i.e., j=1). The Euclidean norm of the vector a is given as a where the Frobenius norm of the matrix A is depicted by AF.

Section snippets

Problem background

Consider a uniform linear array with M sensors receiving P+1 far-field narrow-band uncorrelated signals composed of one SOI and P interferences. The array sample complex vector at the tth snapshot can be presented as:r(t)=as(t)s(θs)+p=1Pap(t)s(θp)+n(t),where the angles θs, θp denote the directions of the desired signal and the pth interference, respectively. as(t), ap(t) represent signal waveforms of the desired signal and interferences, respectively. The vector n(t) denotes the additive white

Proposed LCSSP algorithm

A key idea is based on the exploitation of virtual sensors to reconstruct the IPNC matrix. In this section, we develop an effective RAB approach to estimate the IPNC matrix without resorting to the power spectrum of the interference signals directly. Instead, we define a projection matrix that operates on an approximation to the orthogonal space of the steering vector of the SOI, taking advantage of the higher dimensions to obtain a better approximation.

Under the narrowband assumption, the

Analysis

In this section, we carry out a mathematical analysis of the proposed LCSSP RAB technique along with an analysis of the computational complexity of the proposed and existing techniques.

Simulations

In our simulations, we assess the performance of the proposed LCSSP method against existing techniques using a uniform linear array with M=12 omnidirectional sensors. There are three signals impinging from the directions θ¯s=5, 30 and 30. The first signal is assumed to be the desired signal and the other two signals are interferers with 30 dB interference-to-noise (INR) ratios, respectively. We assume that the interfering signals and the desired signal are independent. In terms of the SINR

Conclusion

In this paper, a novel RAB approach based on the IPNC reconstruction matrix has been introduced. In the proposed LCSSP method, we extend the array with a number of virtual sensors to reconstruct the IPNC matrix using a low-complexity spatial sampling process. The power spectrum sampling is realized by a proposed projection matrix orthogonal to the signal subspace that retains the interference-plus-noise in a higher dimension. The proposed LCSSP approach avoids estimation of the IPNC matrix by

CRediT authorship contribution statement

Saeed Mohammadzadeh: Conceptualization, Methodology, Software, Formal analysis. Vítor H. Nascimento: Writing - review & editing, Supervision. Rodrigo C. de Lamare: Writing - review & editing. Osman Kukrer: Methodology, Writing - review & editing, Validation, Formal analysis.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported in part by the So Paulo Research Foundation (FAPESP) through the ELIOT project under Grant 2018/12579-7 and Grant 2019/19387-9.

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