Asymptotic SNR of scalar and vector minimum-variance beamformers for neuromagnetic source reconstruction

To reconstruct neuromagnetic sources, the minimum-variance beamformer has been extended to incorporate the three-dimensional vector nature of the sources, and two types of extensions-the scalar- and vector-type extensions-have been proposed. This paper discusses the asymptotic signal-to-noise ratio (SNR) of the outputs of these two types of beamformers. We first show that these two types of beamformers give exactly the same output power and output SNR if the beamformer pointing direction is optimized. We then compare the output SNR of the beamformer with optimum direction to that of the conventional vector beamformer formulation where the beamformer pointing direction is not optimized. The comparison shows that the beamformer with optimum direction gives an output SNR superior to that of the conventional vector beamformer. Numerical examples validating the results of the analysis are presented.     

Spatio-temporal reconstruction of bilateral auditory steady-state responses using MEG beamformers

A rapidly growing number of neuromagnetic studies focus on the analysis of auditory steady-state responses (ASSR) in relation to a diverse array of factors including age, selective attention, and presence of psychopathology. The objectives of these studies require accurate spatio-temporal estimation of the underlying neural generators, a challenging task due to the relatively low signal strength and high correlation between bilateral auditory cortical sources. This paper evaluates the performance of two beamforming schemes that can potentially overcome such difficulties: 1) the linearly constrained minimum variance beamformer with partial sensor coverage (LCMV-PSC), and 2) the multiple constrained minimum-variance beamformer with coherent source region suppression (MCMV-CSRS). Simulation experiments are conducted to assess the impact of source parameters on the reconstruction accuracy. The results indicate that the LCMV-PSC method is prone to localization errors that essentially occur along medio-lateral directions, increase with source depth, and are associated to amplitude and phase distortions of the estimated time courses of activity. Comparatively, the MCMV-CSRS method exhibits precise localization and minimal amplitude and phase distortion for a broad range of relative interferer's positions within the suppression region. The results from the numerical experiments are validated on real magnetoencephalographic (MEG) data collected from a 40-Hz ASSR paradigm.     

Coherent interference suppression with complementally transformedadaptive beamformer

This paper proposes a beamforming scheme for suppressing coherent interference with an array of arbitrary geometry. The scheme first uses estimates of the source directions to construct a transformation, which removes the desired signal while retaining the coherent interference. Optimum beamforming is then performed on the transformed data containing only interference and noise to produce the maximum output signal-to-interference-plus-noise ratio (SINR). Analysis and numerical results demonstrate that the proposed complementally transformed beamformer significantly outperforms the conventional multiply constrained minimum variance (MCMV) beamformers     

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     

Effect of pointing errors on the performance of the projectionbeamformer

The projection beamformer generates a steering constraint vector for beamforming by projecting the original steering vector onto the signal-plus-interference subspace of the correlation matrix. This work investigates the effect of pointing errors on the performance of the projection beamformer. A general expression for the output SINR is first derived for the projection beamformer with multiple interferers present. The expression can also be applied to the directional-constraint beamformer except that the steering constraint vector is replaced by the steering vector. Based on the expression for the output SINR, the sensitivity of the projection and the directional-constraint beamformers to pointing errors are examined for the cases of no interferer and a single interferer. Moreover, the performance of the projection beamformer with the source number overestimated is evaluated. Theoretical analysis validated by computer simulations indicates that the projection beamformer performs better than the directional-constraint beamformer, even when the source number is overestimated     

Broadband beamforming using Laguerre filters

Space-time processing is a well-substantiated method for designing broadband beamformers. In the conventional Frost space-time beamformer, tapped delay line (TDL) filters are used in each branch of the array to create a wideband response for interference suppression. In this article a new space-time beamforming method is introduced in which Laguerre filters replace the traditional TDL filters in the Frost beamformer. The Laguerre filters are fundamentally IIR filters but with only one pole in their structure. Unlike other IIR-based space-time beamforming methods, the proposed method does not need an adaptive procedure for the pole adjustment and is inherently stable. Simulation results show superior performance of the proposed method compared to the Frost beamformer and comparable results against other IIR-based beamformers with much less computational complexity and guaranteed stability.     

Novel Transmit Beamforming Schemes for Time-Selective Fading Multiantenna Systems

Transmit beamforming has been widely adopted for wireless systems with multiple transmit antennas. For a block fading channel, the Grassmannian beamformer has been shown to provide very good performance for finite rate feedback. However, the original Grassmannian beamformer does not take the time domain correlation of the channel fading into consideration. In this paper, based on a first-order autoregressive (AR1) dynamic fading model, we develop two new classes of beamforming algorithms that exploit the interframe correlations in the channel fading. We first introduce an algorithm based on a standard predictive vector quantization (PVQ) approach, and the resulting PVQ beamformer accomplishes superior power delivery at the receiver. However, the error performance of the PVQ beamformer is not satisfactory at high signal-to-noise ratios, and it also has a high implementation complexity. To resolve these issues, we then develop a novel successive beamforming (SBF) algorithm. The new SBF scheme uses the knowledge of the previous fading blocks to aid the beamforming codebook design of the current fading block. The beamforming codebook is constructed based on the successive partition of the surface of a spherical cap. The new SBF scheme accomplishes nearly the same performance as that of the PVQ beamformer, and it has a much simpler implementation. Through numerical simulations, we demonstrate that the proposed beamformers outperform the other previously proposed beamformers at various fading scenarios     

A recursive least squares implementation for LCMP beamforming underquadratic constraint

Quadratic constraints on the weight vector of an adaptive linearly constrained minimum power (LCMP) beamformer can improve robustness to pointing errors and to random perturbations in sensor parameters. We propose a technique for implementing a quadratic inequality constraint with recursive least squares (RLS) updating. A variable diagonal loading term is added at each step, where the amount of loading has a closed-form solution. Simulations under different scenarios demonstrate that this algorithm has better interference suppression than both the RLS beamformer with no quadratic constraint and the RLS beamformer using the scaled projection technique, as well as faster convergence than LMS beamformers     

Modified constant modulus algorithm for digital signal processingadaptive antennas with microwave analog beamforming

This paper proposes a novel algorithm, called modified constant modulus algorithm (M-CMA), which is able to give adaptability to microwave beamforming phased array antennas. Since microwave analog beamformers basically require much fewer RF devices than digital beamformers, microwave analog beamformers based on M-CMA, that is, adaptive microwave beamformers, can be cheaply fabricated. Therefore, they are very suitable for mobile communication systems where both miniaturization and low cost are required for the mobile terminals. M-CMA obtains a gradient vector by a combination of analytical calculation and perturbation of the microwave beamforming control voltage. Though M-CMA is implemented with a digital signal processor, M-CMA controls the microwave analog beamformer by utilizing the gradient vector. The microwave analog beamformer based on M-CMA is analyzed to have the following characteristics: (1) the beamformer can point its main beam to the desired direction in additive white Gaussian noise (AWGN) channels; (2) although the beamformer may possibly fail in ill solutions in cochannel interference (CCI) channels, M-CMA can converge to the optimum solution when the desired direction is roughly a priori known     

Derivative-constrained frequency-domain wideband DOA estimation

In this paper, we present a new approach for wideband direction of arrival (DOA) estimation incorporating derivative constraints into the optimization problem. The Taylor series expansion of steering vector with respect to frequency exhibits the dependence of sensor snapshots on the derivatives of the array manifold matrix, suppression of which reduces the data to follow the narrowband data model. Two new formulations following a beamformer framework are developed based on this observation. While the first formulation estimates accurate source directions with apriori knowledge of the signal subspace dimension, the second formulation provides DOA estimates without estimating the number of sources. The coherent averaging at the reference frequency enables the two formulations to handle coherent sources. The effectiveness of the proposed formulations is illustrated with simulations.     

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.     

基于变换数据空间的相干脑电源定位算法

不同脑区之间的相互协作对大脑完成认知任务具有重要意义.脑区电活动的相干性被认为是这种协作的表现形式.从头表脑电无创地三维定位相干源有助于了解大脑的内在机制.传统的MUSIC算法不能定位相干源.本文发展了一种在变换数据空间的MUSIC算法用于相干源定位.首先根据先验信息大致估计相干源区的范围,然后设计能压制相干源区的数据变换矩阵.最后在变换后的数据空间定位相干源.不同条件下的计算模拟实验表明,相比其它方法,这种方法具有更高的定位精度,运算速度也更快.     

一种能有效接收相干信号的波束形成技术

在假设事先估计得到不相关干扰信号方向的条件下,本文提出了一种能有效接收相干信号的波束形成方法.该方法首先利用估计得到的不相关干扰信号的方向构造变换矩阵,然后基于变换矩阵估计得到期望信号和相干干扰信号的合成导向矢量,最后利用得到的合成导向矢量和阵列相关矩阵求得波束形成的权矢量.计算机仿真结果证实本文所提方法基本上达到理论上的最优性能,且能快速收敛.     

Maximum likelihood localization of sources in noise modeled as astable process

This paper introduces a new class of robust beamformers which perform optimally over a wide range of non-Gaussian additive noise environments. The maximum likelihood approach is used to estimate the bearing of multiple sources from a set of snapshots when the additive interference is impulsive in nature. The analysis is based on the assumption that the additive noise can be modeled as a complex symmetric α-stable (SαS) process. Transform-based approximations of the likelihood estimation are used for the general SαS class of distributions while the exact probability density function is used for the Cauchy case. It is shown that the Cauchy beamformer greatly outperforms the Gaussian beamformer in a wide variety of non-Gaussian noise environments, and performs comparably to the Gaussian beamformer when the additive noise is Gaussian. The Cramer-Rao bound for the estimation error variance is derived for the Cauchy case, and the robustness of the SαS beamformers in a wide range of impulsive interference environments is demonstrated via simulation experiments     

Widely linear generalized sidelobe canceling beamforming with variable diagonal loading

We propose a robust widely linear (WL) beamformer for noncircular (NC) signals in the presence of angle of arrival (AOA) errors or array random perturbations. In our beamformer, the block conjugate structure of covariance matrix is exploited to avoid updating the full weight vector, which reduces the computational loads. We add a variable diagonal loading term in the weight vector to improve the robustness of the WL beamformer. Moreover, the orthogonality constraint of block matrix is not required to calculate the amount of diagonal loading. Computer-simulation results show that the proposed WL beamforming provides improved performance over the conventional linear beamformers for NC signals in the presence of AOA errors and array random perturbations.     

脉冲噪声环境中鲁棒的自适应波束形成方法

本文提出一种脉冲噪声环境中的自适应波束形成方法.方法假定噪声服从对称 α 稳定(S α S:Symmetric α -stable)分布,首先定义分数低阶阵列响应,然后根据最小方差无畸变响应波束形成器(MVDR)提出分数低阶最小方差无畸变响应波束形成器(FrMVDR).理论上证明了当阶数小于噪声特征指数的一半时,分数低阶阵列输出功率有界.计算机仿真实验证明了本文提出的FrMVDR波束形成器在高斯噪声和非高斯脉冲噪声环境中性能都优于MVDR和其他有关的基于分数低阶矩的波束形成器,是一种鲁棒的自适应波束形成器.     

A spatial filtering approach to robust adaptive beaming

The problem of controlling the superresolution in adaptive beamformers is treated. A straightforward method is presented that works for both narrowband and broadband arrays. The method is based on forming the blocking matrix in a general sidelobe canceller structure using a spatial FIR filter. The suppression of this spatial filter and the implicit noise of the leaky least-mean-square algorithm together determine the beamformer     

基于自适应滤波的DPC-MAB SAR方位向信号重建

偏置相位中心方位多波束(DPC-MAB)SAR系统利用方位向多个子孔径同时接收回波,在PRF较低时根据多波束回波的相关性对其进行联合处理,实现方位向信号的重构,从而在保证高方位分辨率的同时增加测绘带宽。该文针对DPC-MAB SAR系统研究了基于波束形成理论的多普勒解模糊方法。传统非自适应的波束形成算法敏感于通道幅相误差以及各类噪声,而基于Capon算法的自适应最小方差无畸变(MVDR)最优波束形成器在多个模糊分量相关性较强时性能大大衰减,针对以往方法的局限性与不足之处,该文提出了一种改进的最小输出能量(MOE)波束形成算法,该算法首先利用多通道回波数据自适应的估计统计自相关矩阵,根据子空间投影理论对理想导向矢量进行修正,然后利用多重约束条件有效抑制相关模糊分量,解模糊性能明显提升。仿真实验和实测数据处理均验证了其有效性。     

Performance of an MEG adaptive-beamformer source reconstruction technique in the presence of additive low-rank interference

The influence of external interference on neuromagnetic source reconstruction by adaptive beamformer techniques was investigated. In our analysis, we assume that the interference has the following two properties: First, it is additive and uncorrelated with brain activity. Second, its temporal behavior can be characterized by a few distinct activities, and as a result, the spatio-temporal matrix of the interference has a few distinctly large singular values. Namely, the interference can be modeled as a low-rank signal. Under these assumptions, our analysis shows that the adaptive beamformer techniques are insensitive to interference when its spatial singular vectors are so different from a lead field vector of a brain source that the generalized cosine between these two vectors is much smaller than unity. Four types of numerical examples verifying this conclusion are presented.     

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.