Discrete-time local polynomial approximation of the instantaneous frequency

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) Ω(t) of a harmonic complex-valued signal. The LPP is interpreted as a time-frequency energy distribution over the t-(Ω(t), Ω¹(t)),...,Ω^m-1(t) 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 Ω^k-1(t) have orders O(N^-(2k+1)) and O(N^-/2(m-k+1)2m+3), k=1.2,...,m, respectively, for a polynomial Ω(t) of the degree m-1 and arbitrary smooth Ω(t) with a bounded mth derivative