Partial differential equation-based algorithm of sound source localization with finest granularity in both time and frequency

We deal with the problem of direction and distance estimate of sound sources in 3-D space and arrive newly at an exact and direct algorithm from the finite observation both in space and time. We first derive a partial differential equation (PDE) what we call the sound source constraint (SSC). We show that the general solution of the SSC-PDE is a diverging spherical wave from a point source with arbitrary temporal waveform. The SSC enables the observer to determine the source location (distance R and direction n) from local measurements of wavefield. As the measurements of wavefield, we consider weighted temporal integrals of arrayed microphone outputs in a finite duration. We obtain exact formula for localizing a single source from single weight measurements and multiple sources from the combination of differently weighted measurements. We examine the performance by simulating non-stationary complex multi-source environments and by using real data in a reverberant environment being common in realworld applications.