Unified Array Manifold Decomposition Based on Spherical Harmonics and 2-D Fourier Basis

In this paper, we derive a unified framework for orthonormal decomposition of the array manifold on scalar fields. Such fields are encountered in cases where polarization is not considered, e.g., single polarized radio waves and acoustic pressure. The results generalize and unify different decompositions of the array manifold, found in recent literature, to arrays of arbitrary geometry including conformal arrays. The concept of equivalence matrix is introduced, establishing a link between the spherical harmonics and 2-D Fourier basis functions. Under some mild assumptions that typically hold in practice, a one-to-one relationship between spherical harmonic spectra and 2-D Fourier spectra may be established. Additionally, it is shown that the rows of the equivalence matrix and the 2-D Fourier spectra of the array manifold span the same subspace. With such results the spherical harmonic and 2-D Fourier decompositions of the array manifold vector, i.e., Wavefield Modeling and 2-D Effective Aperture Distribution Function (EADF) are shown to be equivalent. Results on the modeling capabilities of both orthonormal decompositions are obtained. Moreover, the equivalence matrix is shown to facilitate noise attenuation. A fast spherical harmonic transform with complexity O(Q logQ) can be obtained by exploiting the equivalence matrix, where Q represents the total number of points on the sphere. Finally, the equivalence matrix allows to gain more insight into the relation between rotating a function on the sphere and on the torus. These contributions facilitate high-resolution array processing both in elevation and azimuth irrespective of the array structure or imperfections.