Taylor series expansion and modified extended Prony analysis for localization

In the multiple source localization problem, many inverse routines use a rooting of a polynomial to determine the source locations. We present a rooting algorithm for locating an unknown number of three-dimensional, near-field, static sources from measurements at an arbitrarily spaced three-dimensional array. Since the sources are near-field and static, the spatial covariance matrix is always rank one, and spatial smoothing approaches are inappropriate due to the spatial diversity. We approach the solution through spherical harmonics, essentially replacing the point source function with its Taylor series expansion. We then perform a modified extended Prony analysis of the expansion coefficients to determine the number and location of the sources. The full inverse method is typically ill-conditioned, but a portion of the algorithm is suitable for synthesis analysis. We present a simulation for simplifying point charges limited to a spherical region, using an array of voltage potential measurements made outside the region