A direct algebraic method of instantaneous wave source localization

We deal with the problem of direction and distance estimate of emission sources in 3-D space. We first introduce a partial differential equation (PDE) what we call the location constraint PDE (LC-PDE), which provides a necessary and sufficient description of wavefield generated by a source at direction n and distance R. To remove differentials, we integrate the LC-PDE in a finite rectangular area with complex sinusoidal weight functions. By using a well-known class of window function to eliminate the integral boundary terms, we show an exact finite set of algebraic equations can be obtained including n, R as the unknown variables and small number of 2-D discrete Fourier transform (DFT) components of wavefield as the measurements. Owing to frequency domain separation by DFT, it can be applied to multiple sources with sufficient angle difference. We examine this method with numerical simulation assuming noisy measurements and multiple source conditions.