Discrete-time local polynomial approximation of the instantaneous frequency

Abstract

The local polynomial approximation (LPA) of the time-varying phase is used to develop a new form of the Fourier transform and the local polynomial periodogram (LPP) as an estimator of the instantaneous frequency (IF) Q(t) of a harmonic complex-valued signal. The LPP is interpreted as a time-frequency energy distribution over the t - (a(f),a(1)(f),...,a(m-1)(f)) space, where m is a degree of the LPA. The variance and bias of the estimate are studied for the short- and long-time asymptotic behavior of the IF estimates. In particular, it is shown that the optimal asymptotic mean squared errors of the estimates of Cpk~l'(t) have orders 0(AT-(2fc+i)) and Q(]y-2(m+31} ), k - 1,2,..., respectively, for a polynomial £l(t) of the degree m -1 and arbitrary smooth £l(t) with a bounded rath derivative. © 1998 IEEE.