Abstract
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.
Highlights
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
The signals on the spherical manifolds can be orthogonally projected onto the vector of spherical harmonics, which is known as spherical harmonic domain
Due to the fact that spherical harmonics are not Vandermonde, the major challenge of extending ANM approach to SH domain is to formulate convex optimization problem in absence of Vandermonde decomposition theorem. To deal with this issue, we propose a low-dimensional semidefinite programming (SDP) formulation of spherical harmonic atomic norm minimization (SH-ANM) problem, which retrieve the atomic decomposition in SH domain directly without the foundation of Vandermonde decomposition theorem
Summary
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. The ANM approach has been extended to multidimensional frequency models [22]–[24], prior kenowledge [25], multiple measurement vectors (MMV) [26] and covariance matrix cases [27]–[29] for gridless signal recovery and DOA estimation. These conventional ANM methods depend upon the Vandermonde structure of the array manifolds, and they are limited to linear or rectangular arrays.
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