Asymptotic SNR of Scalar and Vector Minimum-Variance Beamformers for Neuromagnetic Source Reconstruction

TLDR

The minimum‑variance beamformer has been extended to incorporate the three‑dimensional vector nature of neuromagnetic sources, yielding scalar and vector extensions. This paper examines the asymptotic signal‑to‑noise ratio of the outputs of these scalar and vector beamformers. The authors compare the output SNR of an optimally directed beamformer with that of the conventional vector beamformer where the pointing direction is not optimized. The optimally directed beamformer achieves the same output power and SNR as the scalar and vector extensions when its direction is optimized, outperforms the conventional vector beamformer, and numerical examples confirm these analytical results.

Abstract

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.