Distribution theory approach to implementing directional acoustic sensors

Abstract

The objective of directional acoustic sensors is to provide high directivity while occupying a small amount of space. An idealized point sensor achieves this objective from a knowledge of the spatial partial derivatives of acoustic pressure at a point in space. Direct measurement of these derivatives is difficult in practice. Consequently, it is expedient to come up with indirect methods. The use of pressure sensors to construct finite-difference approximations is an example of such a method. This paper utilizes the theory of distributions to derive another indirect method for estimating the various spatial partial derivatives of the pressure. This alternate method is then used to construct a multichannel filter which processes the acoustic pressure by mean of three-dimensional integral transforms throughout a 6epsilon-length cube centered at the origin. The output of the multichannel filter is a spatially and temporally filtered version of the pressure at the origin. The temporal filter is a lowpass Gaussian filter whose bandwidth is inversely proportional to epsilon. Finally, the lattice method for numerical multiple integration is utilized to develop a discrete-spatial version of the multichannel filter.