Nonparametric local polynomial approximation of the time-varying frequency and amplitude

The problem of estimating the time-varying frequency, phase and amplitude of a real-valued harmonic signal is considered. It is assumed that the frequency and amplitude are unspecified rapidly time-varying functions of time. The technique is based on fitting a local polynomial approximation of the phase and amplitude which implements a high-order nonlinear nonparametric estimator. The estimator is shown to be strongly consistent and Gaussian. In particular, the convergence ratesO(h^-3/2 )and O(h^-5/2 ), where $i;h$ei; is the number of observations, are obtained for the frequency estimator when the amplitude is unknown constant or linear in time respectively. The orders of the bias and Gaussian distribution are obtained for a class of the time-varying frequency and amplitude with bounded second derivatives. The a priori amplitude information about the unknown time-varying frequency and amplitude and their derivatives can be incorporated to improve the accuracy of the estimation. Simulation results are given.