Linearly Constrained Minimum Variance Beamformers Research Articles

Towards an objective evaluation of EEG/MEG source estimation methods – The linear approach

The spatial resolution of EEG/MEG source estimates, often described in terms of source leakage in the context of the inverse problem, poses constraints on the inferences that can be drawn from EEG/MEG source estimation results. Software packages for EEG/MEG data analysis offer a large choice of source estimation methods but few tools to experimental researchers for methods evaluation and comparison. Here, we describe a framework and tools for objective and intuitive resolution analysis of EEG/MEG source estimation based on linear systems analysis, and apply those to the most widely used distributed source estimation methods such as L2-minimum-norm estimation (L2-MNE) and linearly constrained minimum variance (LCMV) beamformers. Within this framework it is possible to define resolution metrics that define meaningful aspects of source estimation results (such as localization accuracy in terms of peak localization error, PLE, and spatial extent in terms of spatial deviation, SD) that are relevant to the task at hand and can easily be visualized. At the core of this framework is the resolution matrix, which describes the potential leakage from and into point sources (point-spread and cross-talk functions, or PSFs and CTFs, respectively). Importantly, for linear methods these functions allow generalizations to multiple sources or complex source distributions. This paper provides a tutorial-style introduction into linear EEG/MEG source estimation and resolution analysis aimed at experimental (rather than methods-oriented) researchers. We used this framework to demonstrate how L2-MNE-type as well as LCMV beamforming methods can be evaluated in practice using software tools that have only recently become available for routine use. Our novel methods comparison includes PLE and SD for a larger number of methods than in similar previous studies, such as unweighted, depth-weighted and normalized L2-MNE methods (including dSPM, sLORETA, eLORETA) and two LCMV beamformers. The results demonstrate that some methods can achieve low and even zero PLE for PSFs. However, their SD as well as both PLE and SD for CTFs are far less optimal for all methods, in particular for deep cortical areas. We hope that our paper will encourage EEG/MEG researchers to apply this approach to their own tasks at hand.

Sensor Selection for Relative Acoustic Transfer Function Steered Linearly-Constrained Beamformers

For multi-microphone speech enhancement, different microphones might have different contributions, assome are even marginal. This is more likely to happen in wireless acoustic sensor networks (WASNs), where somesensors might be distant. In this work, we therefore consider sensor selection for linearly-constrained beamformers. Theproposed sensor selection approach is formulated by minimizing the total output noise power and constraining thenumber of selected sensors. As the considered sensor selection problem requires the relative acoustic transfer function(RTF), the covariance whitening based RTF estimation or a direct-path RTF approximation is exploited. For a singletarget source, we can thus substitute the estimated RTF or the assumed RTF to the original problem formulation in orderto design a minimum variance distortionless response (MVDR) beamformer. Alternatively, we can integrate the two RTFsto design a linearly constrained minimum variance (LCMV) beamformer in order to alleviate the effects of RTFestimation/approximation errors. By leveraging the superiority of LCMV beamformers, the proposed approach can beapplied to the multi-source case. An evaluation using a simulated large-scale WASN demonstrates that the integration ofRTFs for the sensor selection based LCMV beamformer can be beneficial as opposed to relying on either of theindividual RTF steered sensor selection based MVDR beamformers. We conclude that the sensors that are close to thetarget source(s) and also some around the coherent interferers are more informative.

Recursive Extended Instrumental Variable Based LCMV Beamformers for Planar Radial Coprime Arrays Under Spatially Colored Noise

This article proposes a new recursive linearly constrained minimum variance (LCMV) beamformer based on the extended instrumental variable (EIV) method for planar radial coprime arrays (PRCAs) under spatially colored noise. The proposed recursive LCMV beamformer is able to deal with multiple constraints with high precision and low complexity and can be applicable to various array geometrical configurations. Taking advantage of the EIV vector, the proposed beamformer can effectively combat the additive color noise with unknown noise covariance matrix. We develop our recursive LCMV beamformer based on the square-root (SR) EIV algorithm due to its improved numerical stability than the conventional EIV-based algorithms. Furthermore, we studied a class of planar arrays called PRCAs, which consists of a set of linear coprime arrays arranged radially at various azimuth angles. The coprime array property is utilized to enlarge the array aperture leading to higher resolution and stronger interference rejection and it offers additional flexibility in the tradeoffs between array complexity and performance. Simulation results demonstrate that the proposed recursive SREIV-based LCMV beamformer outperforms the conventional QR decomposition based LCMV beamformers in the resolution and suppression of interferences under various scenarios. The PRCAs tested outperform the uniform rectangular arrays with the same number of elements. Moreover, better performance can be achieved with more linear subarrays at the expense of increased complexity.

Cognitive-Driven Binaural Beamforming Using EEG-Based Auditory Attention Decoding

Identifying the target speaker in hearing aid applications is an essential ingredient to improve speech intelligibility. Recently, a least-squares-based auditory attention decoding (AAD) method has been proposed to identify the target speaker from single-trial EEG recordings in an acoustic scenario with two competing speakers. Aiming at enhancing the target speaker and suppressing the interfering speaker and ambient noise, in this article, we propose a cognitive-driven speech enhancement system, consisting of a binaural beamformer which is steered based on AAD and estimated relative transfer function (RTF) vectors, which require estimates of the direction-of-arrivals (DOAs) of both speakers. For binaural beamforming and to generate reference signals for AAD, we consider either minimum-variance-distortionless-response (MVDR) beamformers or linearly-constrained-minimum-variance (LCMV) beamformers. Contrary to the binaural MVDR beamformer, the binaural LCMV beamformer allows to preserve the spatial impression of the acoustic scene and to control the suppression of the interfering speaker, which is important when intending to switch attention between speakers. The speech enhancement performance of the proposed system is evaluated in terms of the binaural signal-to-interference-plus-noise ratio ( $\text {SINR}$ ) improvement in anechoic and reverberant conditions. Furthermore, we investigate the impact of RTF and DOA estimation errors and AAD errors on the speech enhancement performance. The experimental results show that the proposed system using LCMV beamformers yields a larger decoding performance and binaural $\text {SINR}$ improvement compared to using MVDR beamformers.

White noise reduction for wideband linear array signal processing

The performance of wideband array signal processing algorithms is dependant on the noise level in the system. In this thesis, a method is proposed for reducing the level of white noise in wideband arrays via a judiciously designed spatial transformation followed by a bank of high-pass filters. The method is initially introduced for uniform linear arrays (ULAs) and analysed in detail. The spectrum of the signal and noise after being processed by the proposed noise reduction method is analysed, and the correlation matrix of the processed noise is derived. The reduced noise level leads to a higher signal-to-noise ratio (SNR) for the system, which can have a significant effect on the performance improvement of various beamforming methods and other array signal processing applications such as direction of arrival (DOA) estimation. The performance of two well-known beamformers, the reference signal based (RSB) beamformer and the linearly constrained minimum variance (LCMV) beamformer is reviewed. Then, the theoretical effect of applying the proposed noise reduction method as a pre-processing step on the performance enhancement of RSB and LCMV beamformers is studied. The theoretical results are then confirmed by simulation. As a representative example of wideband DOA estimation application, a compressive sensing-based DOA estimation method is employed to demonstrate the improved estimation by applying the pre-processing noise reduction method, which is confirmed by simulation. Next, the idea is extended to wideband non-uniform linear arrays (NLAs). Since, NLA does not have a uniform spacing, the beam response of the row vectors of the transformation is distorted. Therefore, the transformation is re-designed using the least squares method to satisfy the band-pass requirements of the transformation. Simulation results show a satisfactory improvement in the the performance of RSB and LCMV beamformers for the NLA structure. The idea is further extended to uniform rectangular arrays (URAs) and uniform circular arrays (UCAs), as two major types of the planar arrays. Two methods are proposed for reducing the effect of white noise in wideband URAs and for each one, a different transformation is designed. The first one is based on a two-dimensional (2D) transformation and the second one is an adaptation of the method developed for the ULA case. The developed method for the UCA structure is based on a one-dimensional (1D) transformation, with modified modulation for the transformation to satisfy the required band-pass characteristics of the transformation. Same as linear array structures, the RSB and LCMV beamformers are used to demonstrate the performance enhancement of the method for planar arrays.

Robust sparse array design for adaptive beamforming against DOA mismatch

The performance of an adaptive beamformer is significantly influenced by its array configuration. The problem of optimum array configuration for minimum variance distortionless response (MVDR) beamformers has been recently investigated under the assumption of accurate estimate or prior exact knowledge of the source direction of arrival (DOA). Inaccuracies in DOA can lead to significant performance degradation. Improving the robustness of MVDR beamformers has commonly been achieved by adding appropriate constraints in the determination of beamforming weights, such as the linearly constrained minimum variance (LCMV) beamformer. This work examines the sensitivity of different sparse array configurations towards uncertainty in the source DOA. It proposes enhancing system robustness through optimizing array configurations. The sparse array design problem is formulated in terms of maximizing the output signal-to-interference-plus-noise ratio (SINR) of the MVDR and LCMV beamformers. The constrained maximization problem is expressed as the fraction of matrix determinants, and a sequential convex programming algorithm is adopted for the solution of the corresponding non-convex problem. Numerical examples are presented to validate the robustness of configured sparse array MVDR and LCMV beamformers for small errors in source directional angles.

The LDA beamformer: Optimal estimation of ERP source time series using linear discriminant analysis

We introduce a novel beamforming approach for estimating event-related potential (ERP) source time series based on regularized linear discriminant analysis (LDA). The optimization problems in LDA and linearly-constrained minimum-variance (LCMV) beamformers are formally equivalent. The approaches differ in that, in LCMV beamformers, the spatial patterns are derived from a source model, whereas in an LDA beamformer the spatial patterns are derived directly from the data (i.e., the ERP peak). Using a formal proof and MEG simulations, we show that the LDA beamformer is robust to correlated sources and offers a higher signal-to-noise ratio than the LCMV beamformer and PCA. As an application, we use EEG data from an oddball experiment to show how the LDA beamformer can be harnessed to detect single-trial ERP latencies and estimate connectivity between ERP sources. Concluding, the LDA beamformer optimally reconstructs ERP sources by maximizing the ERP signal-to-noise ratio. Hence, it is a highly suited tool for analyzing ERP source time series, particularly in EEG/MEG studies wherein a source model is not available.

Multi-line transmission combined with minimum variance beamforming in medical ultrasound imaging.

Increasing medical ultrasound imaging frame rate is important in several applications such as cardiac diagnostic imaging, where it is desirable to be able to examine the temporal behavior of fast phases in the cardiac cycle. This is particularly true in 3-D imaging, where current frame rate is still much slower than standard 2-D, B-mode imaging. Recently, a method that increases frame rate, labeled multi-line transmission (MLT), was reintroduced and analyzed. In MLT scanning, the transmission is simultaneously focused at several directions. This scan mode introduces artifacts that stem from the overlaps of the receive main lobe with the transmit side lobes of additional transmit directions besides the one of interest. Similar overlaps occur between the transmit main lobe with receive side lobes. These artifacts are known in the signal processing community as cross-talk. Previous studies have concentrated on proper transmit and receive apodization, as well as transmit directions arrangement in the transmit event, to reduce the cross-talk artifacts. This study examines the possibility of using adaptive beamforming, specifically, minimum variance (MV) and linearly constrained minimum variance (LCMV) beamforming, to reduce the cross-talk artifacts, and maintain or even improve image quality characteristics. Simulation results, as well as experimental phantom and in vivo cardiac data, demonstrate the feasibility of reducing cross-talk artifacts with MV beamforming. The MV and LCMV results achieve superior spatial resolution, not only over other MLT methods with data-independent apodization, but even over that of single-line transmission (SLT) without receive apodization. The MV beamformer is shown to be less sensitive to wider transmit profiles required to reduce the transmit crosstalk artifacts. MV beamforming, combined with the wider transmit profiles, can provide a good approach for MLT scanning with reduced cross-talk artifacts, without compromising spatial resolution, and even improving it. We also demonstrate that the MV and LCMV beamformers lead to almost identical results. This is because of their very similar beampatterns, except for the sharp nullifying properties that the LCMV beamformer has around interfering beams.

Low-Complexity Addition or Removal of Sensors/Constraints in LCMV Beamformers

We address the application of the linearly constrained minimum variance (LCMV) beamformer in sensor networks. In signal processing applications, it is common to have a redundancy in the number of nodes, fully covering the area of interest. Here we consider suboptimal LCMV beamformers utilizing only a subset of the available sensors for signal enhancement applications. Multiple desired and interfering sources scenarios in multipath environments are considered. We assume that an oracle entity determines the group of sensors participating in the spatial filtering, denoted as the active sensors. The oracle is also responsible for updating the constraints set according to either sensors or sources activity or dynamics. Any update of the active sensors or of the constraints set necessitates recalculation of the beamformer and increases the power consumption. As power consumption is a most valuable resource in sensor networks, it is important to derive efficient update schemes. In this paper, we derive procedures for adding or removing either an active sensor or a constraint from an existing LCMV beamformer. Closed-form, as well as generalized sidelobe canceller (GSC)-form implementations, are derived. These procedures use the previous beamformer to save calculations in the updating process. We analyze the computational burden of the proposed procedures and show that it is much lower than the computational burden of the straightforward calculation of their corresponding beamformers.

Distributed Node-Specific LCMV Beamforming in Wireless Sensor Networks

In this paper, we consider the linearly constrained distributed adaptive node-specific signal estimation (LC-DANSE) algorithm, which generates a node-specific linearly constrained minimum variance (LCMV) beamformer, i.e., with node-specific linear constraints, at each node of a wireless sensor network. The algorithm significantly reduces the number of signals that are exchanged between nodes, and yet obtains the optimal LCMV beamformers as if each node has access to all the signals in the network. We consider the case where all the steering vectors are known, as well as the blind beamforming case where the steering vectors are not known. We formally prove convergence and optimality for both versions of the LC-DANSE algorithm. We also consider the case where nodes update their local beamformers simultaneously instead of sequentially, and we demonstrate by means of simulations that applying a relaxation is often required to obtain a converging algorithm in this case. We also provide simulation results that demonstrate the effectiveness of the algorithm in a realistic speech enhancement scenario.

A Study of the LCMV and MVDR Noise Reduction Filters

In real-world environments, the signals captured by a set of microphones in a speech communication system are mixtures of the desired signal, interference, and ambient noise. A promising solution for proper speech acquisition (with reduced noise and interference) in this context consists in using the linearly constrained minimum variance (LCMV) beamformer to reject the interference, reduce the overall mixture energy, and preserve the target signal. The minimum variance distortionless response beamformer (MVDR) is also commonly known to reduce the interference-plus-noise energy without distorting the desired signal. In either case, it is of paramount importance to accurately quantify the achieved noise and interference reduction. Indeed, it is quite reasonable to ask, for instance, about the price that has to be paid in order to achieve total removal of the interference without distorting the target signal when using the LCMV. Besides, it is fundamental to understand the effect of the MVDR on both noise and interference. In this correspondence, we investigate the performance of the MVDR and LCMV beamformers when the interference and ambient noise coexist with the target source. We demonstrate a new relationship between both filters in which the MVDR is decomposed into the LCMV and a matched filter (MVDR solution in the absence of interference). Both components are properly weighted to achieve maximum interference-plus-noise reduction. We investigate the performance of the MVDR, LCMV, and matched filters and elaborate new closed-form expressions for their output signal-to-interference ratio (SIR) and output signal-to-noise ratio (SNR). We theoretically demonstrate the tradeoff that has to be made between noise reduction and interference rejection. In fact, the total removal of the interference may severely amplify the residual ambient noise. Conversely, totally focussing on noise reduction leads to increased level of residual interference. The proposed study is finally supported by several numerical examples.

Broad-band beamforming and the generalized sidelobe canceller

This correspondence identifies general conditions under which the generalized sidelobe canceller (GSC) structure implements a linearly constrained minimum variance (LCMV) broad-band beam-former. In previously published work, conditions have been presented for the case in which the linear constraint is restricted to fixed gain in the steer direction. Here, that work is extended to include general linear constraints. A geometric interpretation is presented to provide insight into the relationship between the GSC structure and LCMV beamformers, and several recent developments in broad-band beam-forming are reviewed and related to the GSC structure.