Linearly Constrained Minimum Variance Beamformer Research Articles

On the Implementation of the Linearly Constrained Minimum Variance Beamformer

The linearly constrained minimum variance (LCMV) method, which allows multiple linear constraints, is an extension of the classical minimum variance distortionless response filter. Depending on the spatial filter length and the desired frequency grid, a direct computation of the resulting spatial beam pattern may be prohibitive. This brief exploits the rich structure of the LCMV expression to find a nonrecursive computationally efficient implementation of the LCMV beamformer with fixed constraints. The implementation is formed via the use of the matrix inversion lemma and the fast Fourier transform. Numerical simulations indicate a dramatic computational gain, especially for fine frequency grids and multiple constraints

MEG Analysis with Spatial Filtered Reconstruction

Magnetoencephalography (MEG) is a method to measure a magnetic field generated by electrical neural activity in a brain, and it plays increasingly important role in clinical diagnoses and neurophysiological studies. However, in MEG analysis, the estimation of the brain activity, of the electric current density distribution in a brain which is represented by current dipoles, is problematic. A spatial filter and subsequent reconstruction of the current density distribution estimated by the spatial filter (spatial filtered reconstruction: SFR) are proposed. The spatial filter is designed to be used without prior or temporal information. The proposed spatial filter ensures that it concentrates the current distribution around the activated sources in the conductor. The current distribution estimated by the spatial filter is reconstructed by multiple linear regression. Redundant current dipoles are eliminated, and the current distribution is optimized in the sense of the Mallows Cp statistic. Numerical studies are demonstrated and show successful estimation by SFR in multiple-dipole cases. In single-dipole cases with SNRs of 101 and more, the location of the true dipole was successfully estimated for about 80% of the simulations. The reconstruction with multiple linear regression corrected the location of the maximum current density estimated by the proposed spatial filtering. The dipole on the correct position contributes to more than 70% of the total dipoles in the estimated current distribution in those cases. These results show that the current distribution is effectively localized by SFR. We also investigate the differences among SFR, the LCMV (linearly constrained minimum variance) beamformer and the SAM (synthetic aperture magnetometry), the representatives of spatial filters in MEG analyses. It is indicated that spatial resolution is improved by avoiding dependence on temporal information.

A short proof of the equivalence of LCMV and GSC beamforming

Charles Jim (1977) showed the conditions under which a linearly constrained minimum-variance (LCMV) beamformer produces outputs equivalent to a generalized sidelobe canceller (GSC). Buckley (1986) has stated that the proof of equivalence extends to general multilinear constraints. In this article a short proof based on complementary subspaces is presented.

Transformation-based adaptive array beamforming

Recently, the generalized eigenspace-based beamformer (GEIB) has been proposed to combat the pointing errors and to enhance the convergence speed. The weight vector of the GEIB is generated by projecting the weight vector of the linearly constrained minimum variance beamformer (LCMVB) onto a modified signal subspace. Unfortunately, numerical instability and high computational complexity have prohibited the GEIB from practical applications. In the paper, we propose the transformation-based adaptive array beamforming to overcome those problems. With the introduction of the transformation matrix, we first present an equivalent structure of the LCMVB. Based on the proposed LCMVB structure, the transformation-based GEIB is further developed without computing the modified signal subspace. With the removing of the computation of the modified signal subspace, the transformation-based GEIB becomes numerically stable and computationally efficient. Computer simulations are also given to demonstrate the correctness and usefulness of the transformation-based adaptive array beamforming.

Performance of generalized eigenspace-based beamformers in the presence of pointing errors

Here we examine the performance of the generalized eigenspace-based beamformer (GEIB) with derivative constraints in the presence of pointing errors. Under the assumption that the signal scenario consists of a desired signal, an interferer and additive receiver noise, the output signal-to-interference-plus-noise ratios (SINRs) of the GEIB and the linearly constrained minimum variance beamformer (LCMVB) are evaluated. Analysis results show that the output SINR of the GEIB considerably depends on an interferer located in the sidelobe region, while that of the LCMVB does not. As the arriving angle of the interferer varies, the output SINR of the GEIB may oscillate, especially in the case that only the unit gain constraint is applied. When the input interference-to-noise ratio (INR) approaches infinite, the level of oscillation reduces to zero and the GEIB achieves almost the optimal output SNR. We compare the performances of the GEIB and the LCMVB, and find that the GEIB can achieve nearly the optimal output signal-to-noise ratios (SNRs) when the LCMVB has a loss of 10 log M dB, where M is the sensor number, due to pointing errors.

Gaitbase — A relational database for the management of gait laboratory data

Beamformers are applied for estimating spatiotemporal characteristics of neuronal sources underlying measured MEG/EEG signals. Several MEG analysis toolboxes include an implementation of a linearly constrained minimum-variance (LCMV) beamformer. However, differences in implementations and in their results complicate the selection and application of beamformers and may hinder their wider adoption in research and clinical use. Additionally, combinations of different MEG sensor types (such as magnetometers and planar gradiometers) and application of preprocessing methods for interference suppression, such as signal space separation (SSS), can affect the results in different ways for different implementations. So far, a systematic evaluation of the different implementations has not been performed. Here, we compared the localization performance of the LCMV beamformer pipelines in four widely used open-source toolboxes (MNE-Python, FieldTrip, DAiSS (SPM12), and Brainstorm) using datasets both with and without SSS interference suppression.We analyzed MEG data that were i) simulated, ii) recorded from a static and moving phantom, and iii) recorded from a healthy volunteer receiving auditory, visual, and somatosensory stimulation. We also investigated the effects of SSS and the combination of the magnetometer and gradiometer signals. We quantified how localization error and point-spread volume vary with the signal-to-noise ratio (SNR) in all four toolboxes.When applied carefully to MEG data with a typical SNR (3–15 ​dB), all four toolboxes localized the sources reliably; however, they differed in their sensitivity to preprocessing parameters. As expected, localizations were highly unreliable at very low SNR, but we found high localization error also at very high SNRs for the first three toolboxes while Brainstorm showed greater robustness but with lower spatial resolution. We also found that the SNR improvement offered by SSS led to more accurate localization.

Generalized eigenspace-based beamformers

The generalized eigenspace-based beamformer (GEIB) is presented here, which utilizes the eigenstructure of the correlation matrix to enhance the performance of the linearly constrained minimum variance beamformer (LCMVB). The weight vector of the GEIB is found by projecting the LCMVB weight vector onto a vector subspace constructed from the eigenstructure of the correlation matrix. The GEIB and the LCMVB have the same responses to the desired signal and the interferers. However, the weight vector of the GEIB has a smaller norm and generates a lower output noise power. An additional advantage of the GEIB is that the linear constraints can be treated flexibly, i.e. each linear constraint can be chosen to be preserved or not preserved. The cost of preserving a linear constraint is to get more output noise power. In addition to developing the GEIB, we discuss the effects of imposing linear constraints on the output noise powers of the GEIB and the LCMVB. Computer simulations are also presented that demonstrate the merits of the GEIB.

Sparse LCMV beamformer design for suppression of ground clutter in airborne radar

Cancellation of the ground clutter received at an airborne phased array radar is an inherently two dimensional problem. Clutter returns are Doppler shifted due to platform motion forcing the use of processors that can resolve targets in both velocity (Doppler) and azimuth. Fully adaptive processors that operate in both dimensions require prohibitively large computation so that reduced adaptive dimension, or partially adaptive processors must be considered. In conventional partially adaptive linearly constrained minimum variance (LCMV) beamformer design the approach taken has been to represent the interference subspace with some reduced set of vectors, typically the eigenvectors associated with the largest eigenvalues of the interference covariance matrix. This technique does yield good performance but will not give the optimum performance for a given partially adaptive dimension. In this paper, an off-line method for selecting the degrees of freedom to be retained in a partially adaptive design is presented. The sequential algorithm described selects those degrees of freedom that best minimize the beamformer output mean square error. This approach leads to a sparse structure for the transformation matrix, which when implemented in a generalized sidelobe canceller (GSC) structure results in a reduction in the computational load. This approach also allows a reduction in the required adaptive dimension as compared to the eigenvector based approach. Illustrative examples demonstrate the effectiveness of this method.

On the mean-square error performance of adaptive minimum variance beamformers based on the sample covariance matrix

The authors examine the mean-square error (MSE) performance of two common implementations of adaptive linearly constrained minimum variance (LCMV) beamformers that employ the sample covariance matrix. The Type I beamformer is representative of block processing methods where the same input data is used both to compute the adaptive weights and to form the beamformer output. The Type II beamformer, as in many recursive schemes, applies adaptive weights computed from previous data to the current input. Due to correlation between the adaptive weights and the input data, the Type I LCMV beamformer exhibits signal cancellation, which is shown here to cause signal estimate bias. To explicitly account for signal cancellation, the mean-square error (MSE) and output signal-to-noise ratio (SNR) measures of the bias-corrected Type I beamformer are analyzed, thus extending previous results. Further, new analytical results for these performance measures are given for the Type II LCMV beamformer. Comparison of bias-corrected Type I and Type II implementations indicate that both methods yield exactly the same MSE and output SNR performance. >