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