Extrapolation and Spectral Estimation Techniques for Discrete Time Signals.

This report considers spectral estimation and extrapolation techniques for discrete time, band limited signals, i.e., signals whose bandwidth is less than 1T cycles if T sec. is the sampling interval which are observable only for a finite duration. The objective is to determine the spectrum or power spectrumof these signals. It is shown that the estimated spectrum can be improved considerably over a periodgram or Maximum entropy spectrum by first extrapolating the given observations beyond the observation interval. Also, we consider the problem of extrapolation of signal in the presence of noise or other interfering signals. Several new results and algorithms are presented. First, it is shown that some of the existing extrapolation methods for continuous signals when extended to sampled data do not converge to the exact original time-unlimited signal.Rather, one only expects to get a minimum norm least squares estimate. And, we find that Papoulis 8 iterative extrapolation algorithm is a special case of a gradient algorithm with linear convergence. It is shown that an infinite extrapolation matrix introduced in 10 does not exist and is ill conditioned at best when approximated to a finite matrix. The new extrapolation algorithms include a discrete prolate spheroidal wave function PSWF expansion, a conjugate gradient iterative algorithm, a mean square extrapolation filter and recursive Kalman filter type extrapolator.