# February Fourier Talks 2008

## Carol Christou

### Title:

A Method of Incorporating Finite Source Extent Into
the Sonar Beamforming Process

### Abstract:

Conventional beamforming methods are generally formulated in terms of
far-field point sources, with emitted signals following a unidirectional
path to a sensor array. This is not a valid assumption for sources with
appreciable apparent spatial extent, such as close aboard targets in the
near-field of a sonar array. In an effort to improve target localization
and tracking, a model has been developed for incorporating the uncertainty
associated with spatial target extent into the beamforming of signals
received at an array of sensors. Assuming that the amplitudes of small
volume elements of a distributed source are statistically independent, it
is shown that the beam response may be expressed as a convolution of the
beampattern of a point source and a "directivity factor" for the actual
extended source. Equivalently, this may be thought of as an average of the
beampattern weighted by a probability density function (PDF) representing
the spatial distribution of the source about its center of mass angular
coordinates. It is then straightforward to obtain the beam response using
Fourier analysis. Results are derived theoretically for the general three
dimensional array case, and then specialized to two and one dimensional
arrays. Because signals may be beamformed in azimuth, as well as
elevation, a PDF that best allows modeling of directional response
properties is optimal. One such choice for volumetric and planar arrays
is the Von Mises-Fisher distribution, a function of the angular deviations
of the source elements from the center of mass and a parameter which
incorporates the uncertainty associated with target distance, size and
aspect angle. Multiparameter PDFs, such as the Fisher-Bingham
distribution, will also be considered. Results will be presented for
volumetric, planar and linear sonar arrays.