On the Properties of Some Systems that Distort Signals - II

In this paper we study the recoverabilit.y of square-integrable bandlimited signals (with arbitrary frequency bands) that are distorted by a frequency-selective time-variable nonlinear operator and subsequently are bandlimited to the original bands. The distortion operator characterizes a very general class of systems containing linear time-invariant elements and a single time-variable nonlinear element. The subsequent bandlimiting of the system's output signals can be thought of as being due to transmission through a channel that performs filtering. Our principal result asserts that, under certain conditions that are satisfied by many realistic systems, it is possible to uniquely determine the bandlimited input to the system from a knowledge of the bandlimited version of the output, in spite of the intermediate distortion 91 92 T H E B E L L SYSTEM T E C H N I C A L J O U R N A L , J A N U A R Y ! 9 f S 4 which generally produces signals that are not bandlimited to the original frequency bands. Of course the distortion operator is assumed to be known. We show that the input signal can be determined by a stable iteration procedure in which the approximating functions converge to their limit at a rate that is at least geometric. When the physical system consists of only a single nonlinear element, our result reduces to that of Landau and Miranker, 1 and Zames.2 In the electronic circuitry of a communication system, it is often the case that an ideally linear amplifier is supplied with an approximately bandlimited input signal and that the circuitry subsequent to the amplifier introduces approximate bandlimiting.