Time-Frequency Analysis using Particle Filtering: Closed-Form Optimal Importance Function and Sampling Procedure for a Single Time-Varying Harmonic

We consider the problem of tracking the frequency and complex amplitude of a time-varying (TV) harmonic signal using particle filtering (PF) tools. Similar to previous PF approaches to TV spectral analysis, we assume that the frequency and complex amplitude evolve according to a Gaussian AR(1) model; but we concentrate on the important special case of a single TV harmonic. For this case, we show that the optimal importance function (that minimizes the variance of the particle weights) can be computed in closed form. We also develop a suitable procedure to sample from the optimal importance function. The end result is a custom PF solution that is more efficient than generic ones, and can be used in a broad range of important applications that postulate a single TV harmonic component, e.g., TV Doppler estimation in communications and radar.