Parametric direction-of-arrival estimation with three recurrence relations of spherical harmonics

The eigenbeam estimation of signal parameters via the rotational invariance technique (EB-ESPRIT) is a well-known subspace-based beamforming algorithm for a spherical microphone array. EB-ESPRIT uses a recurrence relation to directly estimate directional parameters expressing directions-of-arrival (DOAs) of sound sources without an exhaustive grid-search. In the conventional EB-ESPRIT, the directional parameter along the elevational direction is given by a tangent function, which inevitably produces two shortcomings. First, the tangent function becomes singular for sources near the equator in spherical coordinates. Furthermore, two sources lying in exactly opposite directions in the spherical coordinates are indistinguishable and a strong ambiguity problem arises. In this work, an EB-ESPRIT technique based on generalized eigenvalue decomposition (GEVD) is proposed to resolve the singularity and ambiguity problems. The proposed technique uses three independent recurrence relations for spherical harmonics, thus the singularity problem due to the tangent function can be completely avoided. A common transformation matrix for extracting DOAs from recurrence relations are found from the GEVD, and the use of cosine and sine functions makes it possible to find DOAs without ambiguity and without extra transforms or angle-pairing processes. It is demonstrated that the proposed method not only overcomes the singularity and ambiguity problems, but also outperforms conventional techniques in terms of DOA accuracy.