Robust Adaptive Beamformers Based on Worst-Case Optimization and Constraints on Magnitude Response

In this paper, novel robust adaptive beamformers are proposed with constraints on array magnitude response. With the transformation from the array output power and the magnitude response to linear functions of the autocorrelation sequence of the array weight, the optimization of an adaptive beamformer, which is often described as a quadratic optimization problem in conventional beamforming methods, is then reformulated as a linear programming (LP) problem. Unlike conventional robust beamformers, the proposed method is able to flexibly control the robust response region with specified beamwidth and response ripple. In practice, an array has many imperfections besides steering direction error. In order to make the adaptive beamformer robust against all kinds of imperfections, worst-case optimization is exploited to reconstruct the robust beamformer. By minimizing array output power with the existence of the worst-case array imperfections, the robust beamforming can be expressed as a second-order cone programming (SOCP) problem. The resultant beamformer possesses superior robustness against arbitrary array imperfections. With the proposed methods, a large robust response region and a high signal-to-interference-plus-noise ratio (SINR) enhancement can be achieved readily. Simple implementation, flexible performance control, as well as significant SINR enhancement, support the practicability of the proposed methods.     

基于指向最小误差的宽带稳健波束形成方法

宽带自适应波束形成方法较常规延时求和波束形成方法具备更好的空间分辨能力和干扰抑制能力,但在实际应用中由于阵列误差和指向误差影响导致性能下降。文中在宽带指向最小误差(STMV)方法和窄带稳健Capon(RCB)方法基础上,提出了一种基于椭圆不确定集的宽带稳健波束形成方法。通过仿真验证分析了算法在强副瓣干扰和指向误差条件下弱目标检测能力;在幅频响应误差影响条件下的保持弱信号功率输出能力;并验证算法对快拍数要求低于STMV 方法。     

Design of broadband beamformers robust against gain and phase errors in the microphone array characteristics

Fixed broadband beamformers using small-size microphone arrays are known to be highly sensitive to errors in the microphone array characteristics. The paper describes two design procedures for designing broadband beamformers with an arbitrary spatial directivity pattern, which are robust against gain and phase errors in the microphone array characteristics. The first design procedure optimizes the mean performance of the broadband beamformer and requires knowledge of the gain and the phase probability density functions, whereas the second design procedure optimizes the worst-case performance by using a minimax criterion. Simulations with a small-size microphone array show the performance improvement that can be obtained by using a robust broadband beamformer design procedure.     

一种分解迭代二阶锥规划鲁棒自适应波束形成算法

为有效克服导向矢量大失配误差对自适应波束形成器的影响,该文提出了一种最差性能最优的分解迭代鲁棒自适应波束形成算法。该算法对非凸的幅度响应约束问题进行分解处理,将问题转化为迭代的二阶锥规划问题,从而可对鲁棒响应区的波束宽度和纹波水平进行自由控制,并可得到较高的输出信干噪比。此外,与现有大部分该类鲁棒波束形成方法相比,提出的算法直接对权矢量进行优化,无需使用谱分解算法,避免了阵列结构的限制,可适用于任意阵形。仿真结果验证了算法的正确性和有效性。     

Further Study on Robust Adaptive Beamforming With Optimum Diagonal Loading

Significant effort has gone into designing robust adaptive beamforming algorithms to improve robustness against uncertainties in array manifold. These uncertainties may be caused by uncertainty in direction-of-arrival (DOA), imperfect array calibration, near-far effect, mutual coupling, and other mismatch and modeling errors. A diagonal loading technique is obligatory to fulfil the uncertainty constraint where the diagonal loading level is amended to satisfy the constrained value. The major drawback of diagonal loading techniques is that it is not clear how to get the optimum value of diagonal loading level based on the recognized level of uncertainty constraint. In this paper, an alternative realization of the robust adaptive linearly constrained minimum variance beamforming with ellipsoidal uncertainty constraint on the steering vector is developed. The diagonal loading technique is integrated into the adaptive update schemes by means of optimum variable loading technique which provides loading-on-demand mechanism rather than fixed, continuous or ad hoc loading. We additionally enrich the proposed robust adaptive beamformers by imposing a cooperative quadratic constraint on the weight vector norm to overcome noise enhancement at low SNR. Several numerical simulations with DOA mismatch, moving jamming, and mutual coupling are carried out to explore the performance of the proposed schemes and compare their performance with other traditional and robust beamformers     

A robust STAP approach for airborne FDA radar with multiple possible prior information constraints

In frequency diverse array (FDA) space–time adaptive processing (STAP) system, the spectrum of fast-moving target is non-well focused in the spatial–temporal plane, which causes a large mismatch between the actual and presumed target steering vector and dramatically degrades the performance of FDA-STAP system. In this paper, we propose a robust FDA-STAP approach based on multiple possible prior information constraints to resolve this issue and improve the performance of FDA-STAP system. In the proposed approach, multiple possible prior information, i.e., multiple large uncertainty regions, where the actual target possibly locate in spatial–temporal domains, are utilized to cover and constraint the estimated steering vector. The multiple uncertainty regions constraints make it possible to accurately estimate the fast-moving target steering vector. Moreover, to mitigate the influence of non-focused spatial–temporal spectrum of fast-moving target, a non-focused constraints is developed using the squared norm of space–time steering vector to avoid the estimated steering vector converge to the non-focused region of fast-moving target. Finally, the proposed robust FDA-STAP is converted into a non-convex quadratically constrained quadratic programming problem, and the semidefinite relaxation technique is employed to obtain the estimated target steering vector. Numerical results indicate that the proposed method has superior performance than other methods, including well-maintained main beam direction and significant performance improvement.

     

Eigenspace-based adaptive array beamforming with robustcapabilities

This paper deals with adaptive array beamforming based on eigenspace-based (ESB) techniques with robust capabilities. It has been shown that ESB adaptive beamformers demonstrate the advantages of fast convergence speed and less sensitivity to steering angle error over conventional beamformers. In conjunction with a signal subspace construction method, we present an efficient technique to achieve the advantages of ESB adaptive beamforming with less computing cost and more robust capabilities over existing ESB techniques. Several computer simulation examples are provided for illustrating the effectiveness of the proposed technique     

Optimal Design of Nearfield Wideband Beamformers Robust Against Errors in Microphone Array Characteristics

Nearfield wideband beamformers for microphone arrays have wide applications, such as hands-free telephony, hearing aids, and speech input devices to computers. The existing design approaches for nearfield wideband beamformers are highly sensitive to errors in microphone array characteristics, i.e., microphone gain, phase, and position errors, as well as sound speed errors. In this paper, a robust design approach for nearfield wideband beamformers for microphone arrays is proposed. The robust nearfield wideband beamformers are designed based on the minimax criterion with the worst case performance optimization. The design problems can be formulated as second-order cone programming and be solved efficiently using the well-established polynomial time interior-point methods. Several interesting properties of the robust nearfield wideband beamformers are derived. Numerical examples are given to demonstrate the efficacy of the proposed beamformers in the presence of errors in microphone array characteristics.     

Robust adaptive beamforming for general-rank signal models

The performance of adaptive beamforming methods is known to degrade severely in the presence of even small mismatches between the actual and presumed array responses to the desired signal. Such mismatches may frequently occur in practical situations because of violation of underlying assumptions on the environment, sources, or sensor array. This is especially true when the desired signal components are present in the beamformer "training" data snapshots because in this case, the adaptive array performance is very sensitive to array and model imperfections. The similar phenomenon of performance degradation can occur even when the array response to the desired signal is known exactly, but the training sample size is small. We propose a new powerful approach to robust adaptive beamforming in the presence of unknown arbitrary-type mismatches of the desired signal array response. Our approach is developed for the most general case of an arbitrary dimension of the desired signal subspace and is applicable to both the rank-one (point source) and higher rank (scattered source/fluctuating wavefront) desired signal models. The proposed robust adaptive beamformers are based on explicit modeling of uncertainties in the desired signal array response and data covariance matrix as well as worst-case performance optimization. Simple closed-form solutions to the considered robust adaptive beamforming problems are derived. Our new beamformers have a computational complexity comparable with that of the traditional adaptive beamforming algorithms, while, at the same time, offer a significantly improved robustness and faster convergence rates.     

Robust adaptive beamforming using worst-case performance optimization: a solution to the signal mismatch problem

Adaptive beamforming methods are known to degrade if some of underlying assumptions on the environment, sources, or sensor array become violated. In particular, if the desired signal is present in training snapshots, the adaptive array performance may be quite sensitive even to slight mismatches between the presumed and actual signal steering vectors (spatial signatures). Such mismatches can occur as a result of environmental nonstationarities, look direction errors, imperfect array calibration, distorted antenna shape, as well as distortions caused by medium inhomogeneities, near-far mismatch, source spreading, and local scattering. The similar type of performance degradation can occur when the signal steering vector is known exactly but the training sample size is small. In this paper, we develop a new approach to robust adaptive beamforming in the presence of an arbitrary unknown signal steering vector mismatch. Our approach is based on the optimization of worst-case performance. It turns out that the natural formulation of this adaptive beamforming problem involves minimization of a quadratic function subject to infinitely many nonconvex quadratic constraints. We show that this (originally intractable) problem can be reformulated in a convex form as the so-called second-order cone (SOC) program and solved efficiently (in polynomial time) using the well-established interior point method. It is also shown that the proposed technique can be interpreted in terms of diagonal loading where the optimal value of the diagonal loading factor is computed based on the known level of uncertainty of the signal steering vector. Computer simulations with several frequently encountered types of signal steering vector mismatches show better performance of our robust beamformer as compared with existing adaptive beamforming algorithms.     

A numerical pattern synthesis algorithm for arrays

A numerical technique for pattern synthesis in arrays is presented. For a given set of elements, the technique allows one to find a set of array coefficients that steer the main beam in a given direction and yield sidelobes meeting a specified criterion, if such a set of array coefficients exists. If the pattern specifications cannot be met with the given elements, the algorithm finds the best attainable pattern. The advantage of this technique is that it can be used with an arbitrary set of array elements. Different elements in the array can have different element patterns, and the array can have arbitrary nonuniform spacing between elements. The synthesis technique is based on adaptive array theory. The given array elements are assumed to be used as the elements of an adaptive array. The main beam is pointed in the proper direction by choosing the steering vector for that direction, and the sidelobes are controlled by introducing a large number of interfering signals at many angles throughout the sidelobe region. The algorithm iterates on the interference powers until a suitable pattern is obtained     

PSO Assisted DD-ESB Beamforming with Robust Capabilities

This paper deals with adaptive array beamforming based on the decision-directed eigenspace-based (DD-ESB) technique with robust capabilities. It has been shown that DD-ESB adaptive beamformer demonstrates the advantages of better output signal-to-interference plus noise ratio performance and less sensitivity to pointing error over conventional ESB beamformers without any specific training bits. In conjugation with particle swam optimization assisted scheme to obtain more correct desired user’s transmitted bits from the output of the ESB, the more correct steering vector of the desired user can be reconstructed for DD-ESB adaptive beamforming in the presence of larger pointing error and relatively low interference-to-noise ratio. Computer simulations are provided to illustrate the effectiveness of the proposed approach.     

基于改进不确定集的稳健波束形成算法

基于不确定集约束的稳健MVDR波束形成算法在一定程度上依赖于期望信号导向矢量误差的先验知识,且当导向矢量失配较严重时干扰抑制性能也有所下降。为此,提出了一种基于投影变换的改进算法。该方法将约束方向矢量向信号干扰子空间投影,并作为新的约束方向矢量,从而等效于减小了期望信号导向矢量误差。这样,误差不确定参数只需设置为一较小的实数即可在任意导向矢量失配时获得最优的输出性能。计算机仿真结果证明了所提波束形成器具有较强的稳健性能。     

Spatial blocking filter derivative constraints for the generalizedsidelobe canceller and MUSIC

The paper presents a new approach to spatial derivative constraints for the generalized sidelobe canceller (GSC). Spatial derivative constraints have been applied to linearly constrained minimum variance beamformers to reduce the sensitivity to steering error. Earlier approaches to this problem constrained derivatives of the beamformer power and phase response, leading to beamformer performance that depended on the coordinate reference location of the array. Current approaches constrain only the beamformer power response, eliminating the problem with phase dependence. However, nonlinear minimization is required in order to solve for the linear constraint equations. An alternative approach for the GSC, which is presented in the paper, is to use derivative constraints to flatten the null of the spatial blocking filter power response. Thus, for a small steering error, the desired signal is still blocked from the noise cancelling filter, and the GSC output is unaffected by the steering error. These derivative constraints can be used with wideband array calibration, leading to effective performance in the presence of array errors. This same approach to derivative constraints can be used in other applications involving spatial blocking filters, such as the constrained MUSIC direction finding algorithm to give robustness against direction error of the known signal subspace     

A Bayesian approach to robust adaptive beamforming

An adaptive beamformer that is robust to uncertainty in source direction-of-arrival (DOA) is derived using a Bayesian approach. The DOA is assumed to be a discrete random variable with a known a priori probability density function (PDF) that reflects the level of uncertainty in the source DOA. The resulting beamformer is a weighted sum of minimum variance distortionless response (MVDR) beamformers pointed at a set of candidate DOAs, where the relative contribution of each MVDR beamformer is determined from the a posteriori PDF of the DOA conditioned on previously observed data. A simple approximation to the a posteriori PDF results in a straightforward implementation. Performance of the approximate Bayesian beamformer is compared with linearly constrained minimum variance (LCMV) beamformers and data-driven approaches that attempt to estimate signal characteristics or the steering vector from the data     

Self-Calibration-Based Robust Near-Field Adaptive Beamforming for Microphone Arrays

Near-field beamforming using a microphone array has found many applications, such as sound acquisition in small rooms. However, robust near-field adaptive beamforming (NABF) against focal point errors has not been studied much in the literature until recently. In this brief, a robust near-field adaptive beamformer is proposed. The proposed method is developed by combining a new formulation of the point-constrained NABF and a self-calibration technique, in the presence of focal point uncertainties. The proposed method suffers from no loss in the degrees of freedom for interference rejection. Compared with conventional calibration-based adaptive beamformers, the proposed method has the advantage of not needing a noise-free calibration signal. Simulation results demonstrate that the performance of the proposed method is superior to that of the existing methods     

Adaptive Beamforming Using Frequency Invariant Uniform Concentric Circular Arrays

This paper proposes new adaptive beamforming algorithms for a class of uniform concentric circular arrays (UCCAs) having near-frequency invariant characteristics. The basic principle of the UCCA frequency invariant beamformer (FIB) is to transform the received signals to the phase mode representation and remove the frequency dependence of individual phase modes through the use of a digital beamforming or compensation network. As a result, the far field pattern of the array is electronic steerable and is approximately invariant over a wider range of frequencies than the uniform circular arrays (UCAs). The beampattern is governed by a small set of variable beamformer weights. Based on the minimum variance distortionless response (MVDR) and generalized sidelobe canceller (GSC) methods, new recursive adaptive beamforming algorithms for UCCA-FIB are proposed. In addition, robust versions of these adaptive beamforming algorithms for mitigating direction-of-arrival (DOA) and sensor position errors are developed. Simulation results show that the proposed adaptive UCCA-FIBs converge much faster and reach a considerable lower steady-state error than conventional broadband UCCA beamformers without using the compensation network. Since fewer variable multipliers are required in the proposed algorithms, it also leads to lower arithmetic complexity and faster tracking performance than conventional methods.     

Doubly constrained robust Capon beamformer

The standard Capon beamformer (SCB) is known to have better resolution and much better interference rejection capability than the standard data-independent beamformer when the array steering vector is accurately known. However, the major problem of the SCB is that it lacks robustness in the presence of array steering vector errors. In this paper, we will first provide a complete analysis of a norm constrained Capon beamforming (NCCB) approach, which uses a norm constraint on the weight vector to improve the robustness against array steering vector errors and noise. Our analysis of NCCB is thorough and sheds more light on the choice of the norm constraint than what was commonly known. We also provide a natural extension of the SCB, which has been obtained via covariance matrix fitting, to the case of uncertain steering vectors by enforcing a double constraint on the array steering vector, viz. a constant norm constraint and a spherical uncertainty set constraint, which we refer to as the doubly constrained robust Capon beamformer (DCRCB). NCCB and DCRCB can both be efficiently computed at a comparable cost with that of the SCB. Performance comparisons of NCCB, DCRCB, and several other adaptive beamformers via a number of numerical examples are also presented.     

A User Parameter-Free Robust Adaptive Beamformer Based on General Linear Combination in Tandem with Steering Vector Estimation

If there is a mismatch between the assumed steering vector (SV) and the real value, the performance of adaptive beamforming methods is degraded. When the signal SV is known exactly but the sample size is small, the performance degradation can also occur. The second kind of degradation is mainly due to the mismatch between the sample covariance matrix and the real one. Almost all existing robust adaptive beamformers are proposed to improve the robustness against these two types of mismatch. Indeed, most of them are user parameter dependent, and the user parameter-free robust beamformers are scarce. As one of the shrinkage methods, the general linear combination (GLC) based beamformer is a good user parameter-free robust beamformer. However, it is only suitable for the scenarios with low sample size and/or small SV mismatch. In this paper, we propose a new robust beamformer, and it is based on general linear combination in tandem with SV estimation (GLCSVE). The proposed approach is superior to GLC in two aspects. One is that the GLCSVE beamformer performs well not only with small but also with large sample size. The other is that the GLCSVE can effectively deal with a large range of SV mismatch. Moreover, the proposed GLCSVE approach is a user parameter-free robust beamformer, and is more suitable for application than the parameter dependent approaches. The idea of our method can also be used to enhance other shrinkage based beamformers.     

改进的稳健Capon 波束形成算法性能分析

常规Capon 波束形成算法能够使波束在期望信号方向形成高增益,在干扰方向形成零陷,针对该算法在期望信号导向矢量失配的情况下,出现性能下降的问题,研究了期望信号导向矢量在不确定集约束下的求解。通过分析稳健Capon波束形成算法的特点,推导出了期望信号导向矢量在球形不确定集约束下的权矢量近似闭式解,并采用图像法,找到给定条件下的最优约束参数。在指向误差和相位误差存在情况下,对算法进行了仿真分析,仿真结果验证了算法在误差存在情况下的稳健性。