On the statistical performance of spherical harmonics MUSIC

In this paper, we present a theoretical analysis of the statistical performance of spherical harmonics multiple signal classification (SH-MUSIC), which is a representative localization algorithm for spherical microphone arrays. What we are concerned about is the influence of spherical harmonics transformation on localization performance, which is mainly manifested in the following aspects: how the spatial aliasing affects the estimation of direction of arrivals (DOAs) in the large aperture case; and how to characterize the resolution capacity of SH-MUSIC in the small aperture case. We advance the research through the detailed examination of the null spectrum of SH-MUSIC. First, by applying the Taylor series expansion of the null spectrum, we give a bias expression to expound the impact of spatial aliasing on DOA estimation. The post-processing for bias reduction is then suggested. Second, utilizing the statistical distribution of the sample covariance matrix and also using the addition theorem of spherical harmonics, we derive an expression for the threshold at which SH-MUSIC can resolve two closely spaced sources to clarify SH-MUSIC's advantages in the small aperture case. Besides, numerical simulations are provided to verify the analytical results.