Rosenblatt’s Contributions to Deconvolution

Abstract

A simple model of deconvolution can be described as observing {x(t)} which is a convolution of a signal {s(t)} with a filter {f(j)},x = s*f. More specifically, we haveThe problem of deconvolution is to recover {s(t)} based on the output process {x(t)}. If the filter {f(j)} is known then the problem is fairly straightforward. The blind deconvolution, in signal processing terminology, is to recover {s(t)} based solely on {x(t)} without knowing {/(j)}. Statisticians may be more interested in the estimation of {f(j)} under certain conditions on {s(t)} and {/(j)}-This problem and its many variations have very broad applications in signal processing, image restoration, geo-exploration, seismology, radio astronomy among others [11,33,38,39].Assume that the signal random variables {s(t)} are independent and identically distributed with mean 0 and variance 1. Let the filter {f(j)} be a sequence of real constants such that oo