Separation of Additive Traveling Wave Signals Using Spatiotemporal Filtering.

Abstract

Linear systems analysis is applied to the problem of separating additive combinations of traveling wave signals on the basis of propagation velocity. The nerve compound action potential and electroencephalographic waves measured from the scalp are neurobiological examples of such signals. Fourier analysis of traveling waves permits the establishment of a firm mathematical basis for the central concept of "velocity filters" and their realizations in the form of two-dimensional systems whose independent variables are space and time. A traveling wave with a specific velocity produces a uniquely oriented line mass in frequency-wave number space where the frequency dimension results from Fourier transforming along the time dimension and the wave number results from Fourier transforming along the spatial dimension. The velocity filter is a two dimensional filter in frequency-wave number terms. Several forms of velocity filters are considered. The practical constraints imposed by sampling and windowing in both space and time are examined in detail. Analytical tools are developed for the evaluation of spatiotemporal windows and are employed to illustrate the inherent velocity selectivity of a family of windows. A significant unexpected relation is uncovered among signal b and width and propagation velocity and the spatial sampling interval required to avoid aliasing. It is shown that some previous attempts to use multi-dimensional digital filters to separate neurological signals may not have adequately considered important aspects of aliasing, leakage and window effects. Correct design procedures are developed including the new concept of skew sampling. Computer simulations provide quantitative comparisons of the efficacy of a practical digital filter in separating a pair of traveling waves. Worst-case results are obtained for various spatial sample sizes and several forms of high-pass frequency filtering. Numerical examples are provided along with a discussion of hardware implementation.