B.S.T.J. Briefs: A Note on a Signal Recovery Problem

In a recent study 1 of the recoverability of square-integrable bandlimited signals that are distorted by a frequency-selective time-varying nonlinear system and subsequently are bandlimited to the original bands, certain assumptions were made concerning three of the four functions of frequency that characterize the linear time-invariant part of the system. These assumptions, which are stated in Section 3.4 of Ref. 1, are satisfied in most, but not all, cases of engineering interest. The purpose of this note is to report on an extension of Theorem I of Ref. 1 that covers cases in which the conditions of Section 3.4 of Ref. 1 are not met. More specifically, a proof of the following result is outlined. Theorem: Let £ 2 « and be as defined in Section II of Ref. 1. Let A, B, C, D, a, p(- ), and P be as defined in Sections 3.1, 3.2 and 3.3 of Ref. 1, and, for any ,/' £ £ 2 « , let p[f] denote the function with values t ( f ( t ) , t ) (- «>