On the Recovery of a Band-Limited Signal, After Instantaneous Companding and Subsequent Band Limiting

Abstract

If f(t) is a band-limited function, with band limit −Ω to Ω, the result of instantaneously companding f(t) is in general no longer band-limited. Nevertheless, it has been proved that knowledge of merely those frequencies of the compandor output which lie in the band from — Ω to Ω is sufficient to recover the original signal f(t). An iteration formula has been proposed that, in theory, performs the desired recovery. In this paper we study in detail some of the practical questions raised by that formula. We show that the successive approximations converge to the solution f(t) at a geometric rate, uniformly for all t, and that the iteration procedure is stable. We then describe a method of performing the recovery in real time and a successful simulation, of it on a general-purpose analog computer. The circuit used in the simulation serves as a first approximation to a practical realization of the recovery scheme.