Direct algebraic localization of a proximate source from instantaneous or short‐time Fourier transform measurements on circular array.

Abstract

The purpose of this study is to obtain an inversion formula of direction and distance of a proximate source from sound pressure distribution on a circular array. The approach will be important both in theoretical aspects and in practical usage for providing rapid initial estimates in more general multi‐source conditions. Our approach is based on the weighted integral method (WIM) [IEEE Trans. SP 57, (2009), Inverse Problems 26, 015011 (2010)] for signal/source parameter estimation. The WIM is composed of two steps: (1) describing the problem with a partial differential equation (PDE) and a measurement scheme in a finite region and (2) integrating the PDE on the measurement region with appropriate weight functions to obtain equations for solving the parameters. The integration by parts eliminates the differential‐of‐field terms in the PDE. The weight functions are chosen so that the differentials generated instead by the integration have simpler forms and also the integral boundary terms at edges of the region will vanish successfully. We begin with the location‐constraint PDE [ASA/ASJ meeting (2006)] and obtain exact formulas, an instantaneous one including time‐differentials and an STFT‐based one with frequency decomposition, for a proximate source. We show several experimental results using 16‐element circular microphone array.