A unified approach to noniterative linear signal restoration

Abstract

The main goal of this paper is to describe a unified framework for several noniterative algorithms for signal extrapolation reported in the literature. This unification is achieved through integral equation and Hilbert space theories. The importance of this unification is that we can bring to bear the vast body of techniques in these theories to the solution of the extrapolation problem. We will show that the so-called two-step procedures for extrapolation with different underlying models can be unified by means of noniterative algorithms for solving optimization problems in Hilbert spaces. In particular, we show that two-step procedures under a discrete-continuous model [1], [2] belong to a general class of well-known algorithms for solving linear integral equations of the first kind: given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(x), x \in A</tex> find <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t), t \in \Omega</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g(x) = \int\min{\Omega} K(x,t)f(t)dt, x \in A</tex> (1) In addition, we will show that the prolate spheroidal expansion technique is also a special case of the well-known Picard's eigenfunction procedure for the general integral problem (1). This theoretical unification, together with that presented in [3] for iterative least-squares algorithms, demonstrates that most of the well-known procedures for band-limited extrapolation can be considered as special cases of standard techniques in integral equations and operator theory.