Robust adaptive beamforming based on covariance matrix reconstruction with RCB principle
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:
- 1.
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.
- 2.
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.
- 3.
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.
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 and to denote the transpose operation and the conjugate transpose operation, respectively; Operator represents the statistical expectation.
Section snippets
The signal model
We consider an array of M omnidirectional sensors receiving far-field narrowband signals from multiple sources. Denote as the direction of the desired signal, and denote , , as the direction of the interference. The array observation at time instant k can be modeled as in which , and are statistically independent components of the desired signal, interferences and white noise,
Proposed algorithm
In this section, a novel RAB algorithm based on IPNCM reconstruction is proposed. We adopt the RCB principle to estimate the mismatched SVs, and then enhance the Capon power spectrum to reconstruct the IPNCM and estimate the SV of the desired signal. The details are presented in the following subsections.
Simulation results
In this section, we conduct numerical simulation to evaluate the performance of the proposed algorithm. Specifically, a uniform linear array with omnidirectional sensors spaced half a wavelength is considered. The observer receives one desired signal from and two interferences from and . Both interference signals have interference-to-noise ratio (INR) of 20 dB. The estimated directions of these three signals are , and , respectively. In the
Conclusion
In this paper, we have developed a novel RAB method based on covariance matrix reconstruction. For the case of model mismatch, we have used the RCB principle to correct the SV of the interference signals and then used a modified Capon power spectrum and the eigen-decomposition method to reconstruct the covariance matrix. The new reconstruction method has improved the accuracy of the reconstructed IPNCM. Moreover, a modified QCQP approach to estimate the SV of the desired signal has been
CRediT authorship contribution statement
Haoran Li: Conceptualization, Formal analysis, Validation, Writing – original draft, Writing – review & editing. Jun Geng: Conceptualization, Funding acquisition, Writing – review & editing. Junhao Xie: Funding acquisition, Supervision, Writing – review & editing.
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.
Acknowledgements
The authors would like to thank the anonymous peer reviewers for their valuable comments. The authors also sincerely appreciate the associate editor enthusiastic help and responsible support. The work of J. Geng was supported by the National Natural Science Foundation of China under grant 61601144 and by the Fundamental Research Funds for the Central Universities under grant 2020009.
Haoran Li was born in Shandong, China, in Dec 1994. He received the B.S. degree in electronic information engineering from the Harbin Institute of Technology, Weihai, China, in 2017, and the M.S. degree in information and communication engineering from the Harbin Institute of Technology, Harbin, China, in 2019, where he is currently pursuing a Ph.D. degree in information and communication engineering. His research interests include joint transmit and receive beamforming, robust beamforming, and
References (48)
- et al.
Eigendecomposition of the multi-channel covariance matrix with applications to SAR-GMTI
Signal Process.
(2004) - et al.
Projection-based robust adaptive beamforming with quadratic constraint
Signal Process.
(2016) - et al.
Robust adaptive beamforming based on a new steering vector estimation algorithm
Signal Process.
(2013) - et al.
Modified projection approach for robust adaptive array beamforming
Signal Process.
(2012) - et al.
Robust adaptive beamforming based on interference covariance matrix sparse reconstruction
Signal Process.
(2014) - et al.
Robust adaptive beamforming via a novel subspace method for interference covariance matrix reconstruction
Signal Process.
(2017) - et al.
Robust adaptive beamforming via subspace for interference covariance matrix reconstruction
Signal Process.
(2020) - et al.
Rapid convergence rate in adaptive arrays
IEEE Trans. Antennas Propag.
(1974) - et al.
Robust minimum variance beamforming
IEEE Trans. Signal Process.
(2005) - et al.
Experimental performance of adaptive beamforming in a sonar environment with a towed array and moving interfering sources
IEEE Trans. Signal Process.
(2000)
Robust Capon Beamforming
Detection, Estimation, and Modulation Theory, Part IV: Optimum Array Processing
High-resolution frequency-wavenumber spectrum analysis
Proc. IEEE
On robust Capon beamforming and diagonal loading
IEEE Trans. Signal Process.
Robust Capon beamforming
IEEE Signal Process. Lett.
Robust adaptive beamforming
IEEE Trans. Acoust. Speech Signal Process.
Covariance matrix estimation errors and diagonal loading in adaptive arrays
IEEE Trans. Aerosp. Electron. Syst.
Further study on robust adaptive beamforming with optimum diagonal loading
IEEE Trans. Antennas Propag.
Iterative robust minimum variance beamforming
IEEE Trans. Signal Process.
Robust adaptive beamforming based on steering vector estimation with as little as possible prior information
IEEE Trans. Signal Process.
Probabilistic QoS constrained robust downlink multiuser MIMO transceiver design with arbitrarily distributed channel uncertainty
IEEE Trans. Wirel. Commun.
Joint transmit and receive beamforming for multiuser MIMO downlinks with channel uncertainty
IEEE Trans. Veh. Technol.
Robust adaptive beamforming using worst-case performance optimization: a solution to the signal mismatch problem
IEEE Trans. Signal Process.
Robust CDMA multiuser detectors: probability-constrained versus the worst-case-based design
IEEE Signal Process. Lett.
Cited by (14)
Robust adaptive beamforming via covariance matrix reconstruction with diagonal loading on interference sources covariance matrix
2023, Digital Signal Processing: A Review JournalA novel beamforming technique using mmWave antenna arrays for 5G wireless communication networks
2023, Digital Signal Processing: A Review JournalAutoregressive Power Spectrum-Based Covariance Matrix Reconstruction for Robust Adaptive Beamforming
2024, Circuits, Systems, and Signal ProcessingJammer Tracking Based on Efficient Covariance Matrix Reconstruction with Iterative Spatial Spectrum Sampling
2023, IEEE Transactions on Aerospace and Electronic SystemsRobust Adaptive Beamforming for Interference Suppression Based on SNR
2023, Electronics (Switzerland)
Haoran Li was born in Shandong, China, in Dec 1994. He received the B.S. degree in electronic information engineering from the Harbin Institute of Technology, Weihai, China, in 2017, and the M.S. degree in information and communication engineering from the Harbin Institute of Technology, Harbin, China, in 2019, where he is currently pursuing a Ph.D. degree in information and communication engineering. His research interests include joint transmit and receive beamforming, robust beamforming, and optimization theory.
Jun Geng received the B. E. and M. E. degrees from Harbin Institute of Technology, Harbin, China in 2007 and 2009 respectively, and the Ph.D. degree from Worcester Polytechnic Institute, MA, United States in 2015. Since June 2015, he has been an associate professor at Harbin Institute of Technology. Dr. Geng's research interests include stochastic signal processing, wireless communications and other related areas.
Junhao Xie received the B.S. degree in electronic engineering from Harbin Institute of Shipbuilding Engineering, the M.S. degree in signal and information processing from Harbin Engineering University and the Ph.D. degree in communication and information system from Harbin Institute of Technology, in 1992, 1995 and 2001 respectively. From 2004 to 2006, he was a visiting scholar at Curtin University of Technology, Perth, W.A., Australia. He is a Professor in the Department of Electronic Engineering, Harbin Institute of Technology. His research interests involve radar system analysis and modeling, array signal processing, target detection and estimation.