Extrapolation algorithms for discrete signals with application in spectral estimation

Abstract

In this paper we present some new algorithms useful for extrapolation and spectral estimation of band-limited sequences in one and two dimensions. First we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares (MNLS) extrapolation. For example, the iterative algorithms proposed in [2] and [8]-[10] are shown to be special cases of a one-step gradient algorithm which has linear convergence. Convergence and other numerical properties are improved by going to a conjugate gradient algorithm. For noisy observations, these algorithms could be extended by considering a mean-square extrapolation criterion which gives rise to a mean-square extrapolation filter and also to a recursive extrapolation filter. Examples and application of these methods are given. Extension of these algorithms is made for problems where the signal is known to be periodic. A new set of functions called the periodic-discrete prolate spheroidal sequences (P-DPSS), analogous to DPSS [21], [22], are introduced and their properties are studied. Finally, several of these algorithms are generalized to two dimensions and the relevant equations are given.