A MUSIC-like method for estimating quadratic phase coupling

Abstract

In this paper, a two-dimensional version of the well-known MUSIC algorithm for estimating the quadratically coupled frequency pairs (QC pairs) in a noise-corrupted complex harmonic process is proposed. It is shown that the algorithm can also, with minor modifications, be used for estimating the bispectrum of a general third-order stationary harmonic process. The algorithm involves arranging the complex third-order cumulants of the noisy harmonic process in the form of a matrix having a normal structure, i.e., a matrix with an orthonormal eigenbasis. It is shown that a necessary and sufficient condition for an ordered pair (θ₁, θ₂) in the two-dimensional frequency plane to be a QC pair is that the Kronecker product between steering vectors associated with the two frequencies θ₁ and θ₂ lies in the signal subspace of this matrix. By exploiting this result and the orthogonality between the signal and noise subspaces of the above matrix, a symmetric search function of two frequency variables (termed as the MUSIC pseudo-bispectrum estimator) is constructed using the signal eigenvectors. This function is shown to peak precisely at the QC pair locations in the two-dimensional frequency plane. Simulation results are presented, and are shown to testify to the high-resolution performance of this estimator.