Signals Impervious to Time Limited Noise

Abstract

these properties can sometimes be used to separate them from it. Ways of doing this have been intensively studied by many workers, and the mathematical literature on the recovery problem, dating mostly from the pioneering work of Kolmogorov [2’] and Wiener [13], is already abundant. A lucid introduction to the whole subject, as well as a comprehensive bibliography of Russian and English articles published during the past twenty years, may be found in the exemplary book by Yaglom [14]. The main object of this paper is to exhibit some Hilbert spaces of signals having infinite past history which are completely impervious to time limited noise. Such signals must be deterministic in the sense that their values on the interval (-, -r) uniquely determine their values on the interval (-r, 0) for all r > 0. The underlying theory of these spaces, as we develop it here, relies heavily on two representation theorems proved by A. Beurling [1] and P. D. Lax [3]. Roughly speaking, these theorems state that certain translation invariant spaces can be characterized by bounded analytic functions. For such spaces it is possible to relate the signal recovery problem to the problem of determining whether a translation operator acting on the space has a bounded iverse. We first establish necessary and sufScient conditions for the existence of this inverse by investigating properties of the analytic function characterizing the operator’s domain, and then go on to derive a bound for the norm of the inverse whenever it exists. Our analysis yields an error estimate for the recovered signal which is applicable even when the noise is not assumed to be time limited.