The spatial coherence of noise fields evoked by continuous source distributions

In this work, analytic expressions for the spatial coherence of noise fields are derived in the modal domain with the aim of providing a sparse representation. For this purpose, the sound field in a region of interest is expressed in terms of a given pressure distribution on a virtual surrounding cylindrical or spherical surface. According to the Huygens-Fresnel principle, the sound pressure on this surface is represented by a continuous distribution of elementary line or point sources, where orthogonal basis functions characterize the spatial properties. To describe spatially windowed pressure distributions with arbitrary angular extensions, orthogonal basis functions of limited angular support are proposed. As special cases, circular and spherical pressure distributions with uncorrelated source modes of equal power are investigated. It is shown that these distributions result, respectively, in cylindrically isotropic and spherically isotropic, i.e., diffuse noise fields. The analytic expressions derived in this work allow for a prediction of the spatial coherence between arbitrary positions within the region of interest, such that no microphones need to be placed at the actual points of interest. Simulation results are presented to validate the derived relations.