Linearly Constrained Minimum Variance Source Localization and Spectral Estimation

A signal's spectrum is a representation of the signal in terms of elementary basis functions which facilitates the extraction of desired information. For a temporal signal, the spectrum is one-dimensional and expresses the time-domain signal as a linear combination of sinusoidal basis functions. A space-time signal possesses a multidimensional Fourier transform known as the wavenumber-frequency spectrum, which represents the space-time signal as a weighted summation of monochromatic plane waves. The spatial and temporal frequencies are not separable, as spatial frequency is itself a function of the temporal frequency. Thus, it seems natural to analyze and estimate the spatial and temporal frequency components in tandem. It is therefore surprising that conventional spectral estimation methods focus on either the spatial or temporal dimension, without any regard for the other. Spatial spectral estimation is commonly referred to as source localization, as the direction of the wavenumber vector is indeed the direction of propagation. Conventional methods analyze a solely spatial aperture without accounting for the temporal structure of the desired signal. Conversely, temporal spectral estimation is performed using a single sensor, and thus the signal aperture is purely temporal. This paper proposes a spatiotemporal framework for spectral estimation based on the linearly constrained minimum variance (LCMV) beamforming method proposed by Frost in 1972. The aperture consists of an array of sensors, each storing a set of previous temporal samples. It is first shown that by taking into account the temporal structure of the desired signal, the ensuing source location estimate is more robust to the effects of noise and reverberation. Unlike conventional localizers, the LCMV steered beam
temporally
focuses the array onto the desired signal. The desired signal is modeled by an autoregressive (AR) process, and the resulting AR coefficients are embedded in the linear constraints. As a result, the rate of anomalous estimates is significantly reduced as compared to existing techniques. Moreover, it is then demonstrated that by employing multiple sensors and steering the array to the assumed source location, the estimate of the desired signal's temporal spectrum contains a lesser contribution from the unwanted noise and reverberation.