A new form of the Fourier transform for time-varying frequency estimation

The local polynomial time-frequency transform (LPTFT) and the local polynomial periodogram (LPP) are proposed in order to estimate a rapidly time-varying frequency omega(t) of a harmonic signal m(t) = A exp(j omega(t)t). The LPTFT gives a time-frequency energy distribution over the t - (omega(t), d omega(t)/dt,...,dm(-1)omega(t)/dt(m-1)) space, where m is a degree of the LPTFT. The LPTFT enables one to estimate both the time-varying frequency and its derivatives. The technique is based on fitting a local polynomial approximation of the frequency which implements a high-order nonparametric regression. The a priori information about bounds for the frequency and its derivatives can be incorporated to improve the accuracy of the estimation. The estimator is shown to be strongly consistent and Gaussian for a polynomial frequency. The asymptotic covariance matrix and bias of the estimators of d(s) omega(t)/dt(s), s = 0, 1, 2,..., m - 1, are obtained for the frequency with bounded m-derivative. Simulation results are presented.