Robust Adaptive Beamforming Research Articles

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

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.

Simultaneous Interference Localization and Array Calibration for Robust Adaptive Beamforming With Partly Calibrated Arrays

In this article, we address the problem of robust adaptive beamforming in the presence of array sensor miscalibration. We consider the use of partly calibrated linear arrays, where only a small portion of sensors have been gain-phase aligned. Our solution is based on the interference-plus-noise covariance matrix (INCM) reconstruction principle. In our solution, the INCM is reconstructed by performing simultaneous interference localization and array calibration (SILAC). Toward this end, a novel virtual baseline extension technique is presented for high-accuracy SILAC. After SILAC, the interference and noise powers are estimated, and the INCM is reconstructed subsequently. No computations of integration/summation and nonlinear optimization are involved in our beamformer, which is termed as “INCM-SILAC” beamformer. Numerical examples are offered to validate the performance of the INCM-SILAC beamformer. A MATLAB code for reproducing the results of radar application example is available at https://github.com/jinhesjtu/SILAC.git

Steering vector optimization using subspace-based constraints for robust adaptive beamforming

To address the issue of steering vector mismatch, a robust adaptive beamforming design via steering vector optimization is proposed in this paper. Different from conventional studies, this paper resolves the exact desired signal (DS) steering vector through formulating an array output power maximization problem subjected to noise subspace (NS) based and interference subspace (IS) based constraints. Under the condition that the NS is ready to be attained while the IS is hard to be got, two efficient interference-plus-noise covariance matrix (INCM) reconstruction means, i.e. direct DS matrix removal from sample covariance matrix and indirect DS blocking from training data and matrix transition, are derived to estimate the IS with high accuracy. Herein, after resolving DS steering vector, the weight vectors are thereby extracted with orthogonal projection (OP) criterion. Numerical simulations verify that the devised methods can outperform the existing ones and obtain almost optimal performance across a wide range of DS power.

Robust wideband adaptive beamforming based on covariance matrix reconstruction in the spatial‐frequency domain

Without the response variation constraint, the main-lobe responses of wideband beamforming usually vary with the frequencies of interest, and the beam pointing may deviate from the incident angle of the desired signal. If the interference-plus-noise covariance matrix (INCM) is directly replaced by the sample covariance matrix, the desired signal in the sample data may result in self-null problem, which can lead to the degradation of the performance of wideband beamformers. Through frequency division, a novel wideband adaptive beamforming is proposed by reconstructing the INCM and solving the convex optimisation problem. Using fast Fourier transform (FFT), the time-domain samples are transformed into the frequency field, and the covariance matrix of each subband can be obtained from the frequency-domain data. In the spatial-frequency domain, the INCM and signal-plus-noise covariance matrix can be respectively reconstructed by integrating the stacked steering vector and the Capon spatial spectrum over the corresponding angular region. Then, we establish the optimisation criteria, that is, minimising the output interference-plus-noise power, minimising the main-lobe spatial response variation, and constraining the response deviation of the desired signal. A novel convex optimisation model can be established, and then the weight coefficients for time-domain wideband beamforming can be obtained by solving the optimisation problem. The simulation and experimental results demonstrate that the proposed beamformer can provide better performance compared with other tested beamformers and maintain good robustness in various applications.

Fast and accurate covariance matrix reconstruction for adaptive beamforming using Gauss-Legendre quadrature

Most of the reconstruction-based robust adaptive beamforming (RAB) algorithms require the covariance matrix reconstruction (CMR) by high-complexity integral computation. A Gauss-Legendre quadrature (GLQ) method with the highest algebraic precision in the interpolation-type quadrature is proposed to reduce the complexity. The interference angular sector in RAB is regarded as the GLQ integral range, and the zeros of the three-order Legendre orthogonal polynomial is selected as the GLQ nodes. Consequently, the CMR can be efficiently obtained by simple summation with respect to the three GLQ nodes without integral. The new method has significantly reduced the complexity as compared to most state-of-the-art reconstruction-based RAB techniques, and it is able to provide the similar performance close to the optimal. These advantages are verified by numerical simulations.

Robust Adaptive Beamforming Based on Low-Complexity Discrete Fourier Transform Spatial Sampling

In this paper, a novel and robust algorithm is proposed for adaptive beamforming based on the idea of reconstructing the autocorrelation sequence (ACS) of a random process from a set of measured data. This is obtained from the first column and the first row of the sample covariance matrix (SCM) after averaging along its diagonals. Then, the power spectrum of the correlation sequence is estimated using the discrete Fourier transform (DFT). The DFT coefficients corresponding to the angles within the noise-plus-interference region are used to reconstruct the noise-plus-interference covariance matrix (NPICM), while the desired signal covariance matrix (DSCM) is estimated by identifying and removing the noise-plus-interference component from the SCM. In particular, the spatial power spectrum of the estimated received signal is utilized to compute the correlation sequence corresponding to the noise-plus-interference in which the dominant DFT coefficient of the noise-plus-interference is captured. A key advantage of the proposed adaptive beamforming is that only little prior information is required. Specifically, an imprecise knowledge of the array geometry and of the angular sectors in which the interferences are located is needed. Simulation results demonstrate that compared with previous reconstruction-based beamformers, the proposed approach can achieve better overall performance in the case of multiple mismatches over a very large range of input signal-to-noise ratios.

High-Precision Mutual Coupling Coefficient Estimation for Adaptive Beamforming

Here, a high-precision mutual coupling coefficient estimation method is proposed that is more suitable for adaptive beamforming than traditional algorithms. According to the relationship between the designed transition matrix and the signal, the proposed algorithm selects the transition matrix corresponding to the high-power signal. The high-precision estimation of the mutual coupling coefficient is obtained by using the selected transition matrix estimation, which yields relatively good estimation accuracy for the mutual coupling coefficient when the desired signal-to-noise ratio (SNR) is low and relatively robust adaptive beamforming with unknown mutual coupling. Simulation results demonstrate the validity of the proposed method.

Robust adaptive beamforming based on a method for steering vector estimation and interference covariance matrix reconstruction

Robustness is an important factor in adaptive beamforming because various mismatches that exist in reality will lead to considerable performance degradation. In this paper, a robust adaptive beamforming (RAB) algorithm is proposed based on a novel method for estimating the steering vectors (SVs). The interference-plus-noise covariance matrix (INCM) is then reconstructed by the SVs and their corresponding power estimates. As we know, the Capon power spectrum is actually a function of the SV defined in a high-dimensional domain, in which the actual SVs correspond to the highest peaks. The nominal SVs may lead to relatively lower power amplitude points around the peaks when mismatches exist, and these peaks are located in the directions of gradient vectors at the lower power points obtained by the nominal SVs. Therefore, to obtain the actual SVs, we first construct a subspace for each nominal SV in a small neighborhood of angles. Then we get the gradient vector, which is orthogonal to the corresponding nominal SV neighborhood, using a subspace-based method. Finally, we search along the gradient vector to obtain the adjusted SV that generates the highest Capon power amplitude. The interference covariance matrix (ICM) is reconstructed by the adjusted interference SVs and corresponding Capon power amplitudes. The actual SV of the signal of interest (SOI) is estimated as the adjusted SV of the SOI. Simulation results demonstrate that the proposed method is robust against various types of mismatch and is superior to other existing reconstruction-based beamforming algorithms.

Weighted data spaces for correlation-based array imaging in experimental aeroacoustics

This article discusses aeroacoustic imaging methods based on correlation measurements in the frequency domain. Standard methods in this field assume that the estimated correlation matrix is superimposed with additive white noise. In this paper we present a mathematical model for the measurement process covering arbitrarily correlated noise. The covariance matrix of correlation data is given in terms of fourth order moments. The aim of this paper is to explore the use of such additional information on the measurement data in imaging methods. For this purpose a class of weighted data spaces is introduced, where each data space naturally defines an associated beamforming method with a corresponding point spread function. This generic class of beamformers contains many well-known methods such as Conventional Beamforming, (Robust) Adaptive Beamforming or beamforming with shading. This article examines in particular weightings that depend on the noise (co)variances. In a theoretical analysis we prove that the beamformer, weighted by the full noise covariance matrix, has minimal variance among all beamformers from the described class. Application of the (co)variance weighted methods on synthetic and experimental data show that the resolution of the results is improved and noise effects are reduced.

Novel Robust Adaptive Beamformer in the Presence of Gain-Phase Errors

In this paper, we consider the problem of robust adaptive beamforming (RAB) for a linear array in the presence of gain-phase errors. By setting a few calibrated auxiliary elements on one side of a linear array, we propose a novel gain error and phase error estimation algorithm to calibrate gain-phase errors. According to the proposed estimation algorithm, the unknown gain error and phase error are jointly estimated by calculating the eigenvector corresponding to the minimum eigenvalue of the available transitional matrices. Furthermore, we calibrate the received data and compensate for the nonwhite noise caused by the calibration process. Then, the interference-plus-noise covariance matrix (INCM) is reconstructed based on the calibrated steering vector. By incorporating the reconstructed INCM with the minimum variance distortionless response principle, a novel robust adaptive beamformer for a linear array with gain-phase errors is designed. The proposed beamformer can obviously improve RAB performance in a linear array with gain-phase errors. The robustness and superiority of the designed beamformer are demonstrated in simulations.

Robust Adaptive Beamforming Based on Covariance Matrix Reconstruction With Annular Uncertainty Set and Vector Space Projection

The performance of adaptive beamforming will degrade dramatically in practical application due to system errors including signal direction error, array geometry error, gain, and phase errors. Besides, the performance will further deteriorate when the desired signal is contained in the sample data. Therefore, a robust adaptive beamforming method based on the covariance matrix reconstruction with annular uncertainty set (AUS) and vector space projection (VSP) is proposed in this letter. By integrating the corresponding Capon spectrum over the surface of AUS, the interference-plus-noise covariance matrix (INCM) and the desired signal covariance matrix are reconstructed, respectively. Because the steering vector (SV) of desired signal lies in the intersection of the signal subspaces of the sample covariance matrix and the reconstructed desired signal covariance matrix, it can be estimated by the VSP method. Finally, the adaptive weight vector is calculated based on the reconstructed INCM and the estimated SV. Simulation and experiment results show that the proposed method can effectively suppress the interference and achieve excellent performance under system errors.

Robust Wideband Adaptive Beamforming Based on Focusing Transformation and Steering Vector Compensation

The wideband adaptive beamforming method would deteriorate severely when the desired signal is contained in the training snapshots, especially the mismatched signal steering vector existed simultaneously. Therefore, a robust wideband adaptive beamforming method is proposed based on focusing transformation and steering vector compensation. In proposed method, the received wideband signals are first decomposed into narrow sub-bands. Then, based on the direction of arrivals of signals, the interference plus noise covariance matrix of each sub-band is reconstructed to remove the desired signal and transformed into the reference frequency by the focusing transformation. Subsequently, the mismatched signal steering vector is compensated based on the error vector decomposition and the optimization of output power of adaptive beamformer. Finally, the adaptive weight vector of robust wideband beamforming is calculated based on the minimum variance distortionless response criterion. The effectiveness of proposed method is verified by numerical simulations.

Signal-to-interference plus noise ratio loss constrained robust adaptive beamformer inpsired by random matrix theory

Signal-to-interference plus noise ratio loss constrained robust adaptive beamformer inpsired by random matrix theory

An Improved Sub-Array Adaptive Beamforming Technique Based on Multiple Sources of Errors

In this paper, a new robust adaptive beamforming method is proposed in order to improve the robustness against steering vector (SV) mismatches that arise from multiple types of array errors. First, the sub-array technique is applied in order to obtain the decoupled sample covariance matrix (DSCM), in which the auxiliary sensors are selected to decouple the array. The decoupled interference-plus-noise covariance matrix (DINCM) is reconstructed with the estimated interference SV and maximum eigenvalue of the DSCM. Furthermore, the desired signal SV is estimated as the corresponding eigenvector determined by the correlation coefficients of the assumed SV and eigenvectors. Finally, the optimal weighting vector is obtained by combining the reconstructed DINCM and the estimated desired signal SV. Our simulation results show significant signal-to-interference-plus-noise ratio (SINR) enhancement of the proposed method over existing methods under multiple types of array errors.

Robust adaptive beamforming for coprime array with steering vector estimation and covariance matrix reconstruction

Coprime array exhibits many advantages over the uniform linear array (ULA) with the same number of physical sensors in resolution performance and interference suppression capability. In this study, the authors take the advantages of coprime array to improve the robustness of adaptive beamformer. In the coprime virtual ULA (CV-ULA), they prove that a constructed Toeplitz matrix can be taken as the sample covariance matrix from the perspective of virtual signal characteristics. The CV-ULA Capon spectrum estimator is modified to obtain the directions and powers of all impinging signals. Since the real directions of all impinging signals are located at different angular sectors, they form independent signal subspace for each impinging signal. They also assign independent steering vector mismatches for different impinging signals to obtain their real steering vectors. The steering vector mismatch of each impinging signal is independently obtained by solving its own convex optimisation problem. They reconstruct the interference-plus-noise covariance matrix (INCM) with precise steering vectors and powers of interference signals. The proposed weight vector is computed by combining the desired signal steering vector and the reconstructed INCM. Extensive simulations show that the proposed algorithm provides robustness against many types of model mismatches.

A Robust Beamforming Algorithm Based on Covariance Matrix Reconstruction and Steering Vector Estimation

Adaptive beamforming algorithms often suffer from performance degradation due to the presence of the desired signal component in the interference-plus-noise covariance matrix (IPNCM) and mismatch of steering vector. In order to improve the robustness of the beamforming algorithm performance, a new robust adaptive beamforming algorithm is proposed in this paper. In the proposed algorithm, Capon spatial spectrum is first used to reconstruct the IPNCM. Then, based on the robust Capon beamforming algorithm based on the steering vector uncertain set, new constraints are added so that the steering vector does not converge to the interference direction. The proposed beamforming algorithm is a non-convex quadratic constraint quadratic programming (QCQP) problem that is solved using semidefinite programming (SDP) relaxation. Simulation results show that the proposed algorithm is more robust than some other existing algorithms and is closest to the optimal solution.

An uncertainty-set-shrinkage-based covariance matrix reconstruction algorithm for robust adaptive beamforming

This paper presents an uncertainty-set-shrinkage (USS) algorithm that aims to reconstruct a precise interference-plus-noise covariance matrix (INCM) and improve the performance of adaptive beamformers when steering vector (SV) mismatch exists. Both of the interference covariance matrix (ICM) and the desired signal covariance matrix (DSCM) can be divided into two parts, namely the nominal matrix reconstructed using the nominal SVs and the error matrix consisting of the residual component of the covariance matrix. By using a two-step uncertainty set shrinkage method, the proposed beamformer constructs the error matrices by integrating the estimated spatial spectrum over the shrinked uncertainty set at a cost of low computational complexity. After extracting the principal component of the reconstructed ICM and DSCM, the INCM and the SV of the source of interest (SOI) can be estimated without solving any optimization problem. Both of numerical simulations and experimental results demonstrate that the performance of the proposed algorithm is robust with several categories of SV mismatches.

Experimental study of robust acoustic beamforming for speech acquisition in reverberant and noisy environments

The performance of adaptive beamformers may suffer from significant degradation in the presence of steering vector errors, statistics estimation errors, and reverberation. To address this issue, robust beamforming methods, which were originally studied in the narrow-band cases, are studied and compared in this paper for processing acoustic and speech signals, which are broadband in nature. We study two types of methods. In the first type, the robustness of the beamformer is improved by adding a norm constraint and/or a steering vector uncertainty constraint to the optimization problem. It is worth noticing that the norm constraint also helps to control the sidelobes of the beampattern, which makes the beamformers able to suppress interference and multipath effect, thereby improving the robustness of the beamformers with respect to reverberation. Another type of methods improve the robustness using the spatial smoothing technique in which the noise covariance matrix is implicitly estimated first by subtracting an estimate of the desired signal covariance matrix with a delay-and-sum beamformer from the observation signal covariance matrix. Simulations and experiments are performed to investigate the performance of the studied robust adaptive beamformers in acoustic environments. The results show that the robust beamformers outperform their non-robust counterparts in terms of: (1) better performance in reverberation and different noise levels; (2) resilience against steering vector and noisy signal covariance matrix estimation errors; and (3) better predicted speech quality and intelligibility measured using the PESQ and STOI measures.

Accessibility and Convergence Analysis of the Beamformer Design Based on Fibonacci Branch Search

An approach toward beamforming for uniform linear array (ULA) based on a novel optimization algorithm, designated as Fibonacci branch search (FBS), is presented in this paper. The proposed FBS search strategy inspired from Fibonacci sequence principle used fundamental branch structure and interactive searching rules to obtain the global optimal solution in the search space. The structure of FBS is established by two types of multidimensional points on the basis of shortening fraction formed by the Fibonacci sequence; in this mode, interactive global searching and local optimization rules are implemented alternately to reach global optima, avoiding stagnating in local optimum. At the same time, the rigorous mathematical proof for the accessibility and convergence of FBS toward the global optimum is presented to further verify the validity of our theory and support our claim. Taking advantage of the global search ability and high convergence rate of this technique, a robust adaptive beamformer technique is also constructed here by FBS as a real-time implementation to improve the beamforming performance by preventing loss of optimal trajectory. The performance of the FBS is compared with the five typical heuristic optimization algorithms, and the reported simulation results demonstrate the superior of the proposed FBS algorithm in locating the optimal solution with higher precision and reveal the further improvement in the adaptive beamforming performance.

Flexible robust adaptive beamforming method with multiple separately widened nulls

With the development of the array signal processing techniques, more and more robust methods are proposed for non-stationary interference cancellation at the beamforming stage. Among these methods, null widening methods or well-known covariance matrix taper (CMT) methods are the research hotspots. However, the typical CMT-class methods modified the whole covariance matrix with the same taper matrix, which means that all the nulls produced by the covariance matrix will be widened to the same width in cosine angular domain. This will make a waste of degrees of freedom (DOFs) in the scene that different interference sources have a different degree of non-stationarity. In this Letter, also adopting CMT structure, a flexible widening method is discussed, which utilises the relationship between each eigenvector and the steering vector of each interference source. The proposed method can widen different nulls with different width, which can save the consumption of DOFs as many as possible according to the demand. Besides, as a subspace-based method, it can preserve the minimal sample support property. The simulation results show the effectiveness of the proposed method.