Eigensubspace algorithms for estimating the polyspectral parameters of harmonic processes

Abstract

The polyspectral parameters of a harmonic process are defined by the locations and strengths of the polyspectral impulses in the higher dimensional frequency space. MUSIC and ESPRIT-like algorithms for extracting these parameters, when the signal is corrupted by coloured Gaussian noise of unknown statistics, are proposed. The MUSIC-like algorithm involves constructing cumulant matrices having Hermitian structures. A one to one correspondence between the locations of the polyspectral peaks and certain 'steering vectors' in the signal subspace of these cumulant matrices is then set up via the Kronecker product map. The construction of the MUSIC pseudo-polyspectrum is based on this correspondence and the orthogonal eigenstructure of the cumulant matrices. The ESPRIT-like algorithms exploit rotational invariance properties of 'shifted cumulant matrices' to extract the polyspectral parameters from their generalized eigenstructure. Apart from determining the locations of the polyspectral peaks from rank reducing numbers of cumulant matrix pencils, the information contained in the generalized eigenvectors is used to extract the strengths of the polyspectral impulses.