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

The local polynomial Fourier transform (LPFT) and the local polynomial periodogram (LPP) are proposed in order to estimate the rapidly time-varying instantaneous frequency (IF) Ω(t) of a harmonic signal. The LPFT gives the time-frequency power distribution over the t-(Ω(t), dΩ(t)/dt,...,d^m-1Ω(t)/dt ^m-1) space, where m is a degree of the LPFT. The LPFT enables one to estimate both the time-varying frequency and its derivatives. The technique is based on fitting the 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 asymptotic mean square errors, bias and covariance, of the estimators of d^sΩ(t)/dt^s, s=0, 1,2,...,m^-1, are obtained. The considered estimators are high-order generalization of the short-time Fourier transform. The comparative study of the asymptotic variance and bias of the estimates is presented