Simulation of the beam response of distributed signals

Conventional beamforming methods are generally formulated in terms of far-field point sources, with emitted signals following a single path to an array. This is generally not true for sources with appreciable spatial extent, such as close aboard targets in the near field of a sonar array. In an effort to make an Acoustic Submarine Sonar simulation process more realistic, a model has been developed for incorporating the uncertainty associated with spatial target extent into the beamforming module of the simulator. It is assumed that the amplitudes of small volume elements of a spread source are statistically independent and that the observation update time ΔT is short enough for the distances from the array center to the source volume elements to remain approximately constant during ΔT. It is shown that the beam response of an extended source may be expressed as a convolution of the beampattern of a point source and a "directivity factor" for the spread source. Equivalently, this may be thought of as an average of the beampattern weighted by the probability density function (PDF) representing the spatial distribution of a source about its center of mass angular coordinates. Results are derived theoretically for the general three-dimensional (3-D) array case, and then specialized to two- and one-dimensional (2-D and 1-D) arrays. Because beamforming is done in azimuth and elevation, a PDF that best allows modeling of directional response properties should be chosen. The choice in this analysis for volumetric and planar arrays is the von Mises-Fisher distribution, which is a function of the angular deviations of the source elements from the center of mass and a parameter that incorporates the uncertainty associated with target distance, size, and aspect angle. Results are presented for spherical, hull, and towed linear arrays.