Line Spectral Estimation with Palindromic Kernels

Abstract

Estimation of line spectra is a classical problem in signal processing and arises in many applications. The problem is to estimate the frequencies and corresponding amplitudes of a sum of sinusoidal components from noisy measurements. It can be solved with maximum likelihood methods or with suboptimal subspace methods. The constraint that the model does not have damping is difficult to impose in subspace methods. We develop an equivalent formulation as a structured low-rank approximation problem and present a necessary condition for the model to be undamped. The condition is that a vector in the kernel of a Hankel matrix of observations has palindromic structure and it leads to a linear equality constraint which is easily incorporated into a numerical algorithm. Simulations show that even for relatively high noise-to-signal ratios, the necessary condition is in practice also sufficient, i.e., the identified model does not have damping.