Signal processing in the spherical harmonic (SH) domain has the advantages of analyzing a signal on the sphere with equal resolution in the whole space and of decomposite the frequency- and location-dependent components of the signal. Therefore, it finds recent applications in signal recovery and localization. In this paper, we consider the gridless sparse signal recovery problem in the SH domain with atomic norm minimization (ANM). Due to the absence of Vandermonde structure for spherical harmonics, the Vandermonde decomposition theorem, which is the mathematic foundation of conventional ANM approaches, is not applicable in the SH domain. To address this issue, a low-dimensional semidefinite programming (SDP) method to implement the spherical harmonic atomic norm minimization (SH-ANM) approach is proposed. This method does not rely on the Vandermonde decomposition and can recover the atomic decomposition in the SH domain directly. As an application, we develop the direction-of-arrival estimation approach based on the proposed SH-ANM method, and computer simulations demonstrate that its performance is superior to the state-of-the-art counterparts. Furthermore, we validate the results in real-life acoustics scenes for multiple speakers localization using measured data in LACATA challenge.

Introduction

Sherical harmonics (SH) are a set of orthogonal polynomials as the complete basis on the sphere which can be used for approximation of the spherical manifolds. It has been studied extensively to process the signals defined on the spherical manifolds encountered in practice, such as astrophysics, medical imaging, and audio processing, among others [1]–[۳]. The signals on the spherical manifolds can be orthogonally projected onto the vector of spherical harmonics, which is known as spherical harmonic domain. Various signal processing approaches have been extended to the SH domain for signal recovery, direction-of-arrival (DOA) estimation, etc. In last decades, several classical DOA estimation methods and their applications have been proposed in literatures, such as MUSIC [4], ESPRIT [5], propagator method [6], maximum likelihood [7] and tensor approaches [8], etc. Some conventional DOA estimation methods have been proposed for SH domain in [9]–[۱۱]. Inspired by the sparse representation technique [12], [13], sparse signal processing in the SH domain has been drawn attention to in recent years. In [14], [15], sparse recovery in the SH domain for random sampling has been studied. In [16], the `1-SVD (singular value decomposition, SVD) method in the SH domain has been proposed for combating room reverberations. In [17], [18], the sparse Bayesian learning DOA estimation methods were exploited to the SH domain.