Wavefield modeling and array processing .I. Spatial sampling

We introduce the concept of wavefield modeling and study its relationship to array professing and direction finding. We show how the output of an array can be written as the product of a wavefield independent sampling matrix and an array independent coefficient vector. Using this decomposition, we derive the conditions under which the array output may be linearly interpolated to give the array output of a different array, within a desired accuracy. We proceed to show under what conditions the array output may be linearly interpolated to give the value of the wavefield over a continuous manifold. As a result, we also derive conditions under which the array manifold may be linearly transformed between frequencies. Finally, we treat the case where the sources are known to be limited to specific, possibly disjoint angular sectors of arbitrary shape, and show how the sampling matrix can in this case be replaced by a modified sampling matrix of lower rank. This result yields a straightforward extension of the formalism presented in this paper to the Ease of limited angular sectors. In two companion papers, we use wavefield modeling to derive practical algorithms and bounds on the performance of arrays