The Determinability of Classes of Noisy Channels
01 December 1969
A large body of literature already exists for the problem of identifying a control system or communication channel with noisy measurements. In the usual identification problems, a certain structure is assumed at the outset in order to reduce the identification problem to one of parameter estimation. The absence of such parametrization increases the difficulty of the problem substantially. It is often not clear if identification is even possible. In this paper we are concerned with the determinability (identifiability) of quite general nonlinear operators whose outputs are corrupted by additive gaussian noise. We introduce a norm on this space of nonlinear operators and define precisely what we mean by determinability. Loosely speaking, we say that we can determine an operator H if we can choose a finite observation interval [0, T], a test signal with constrained peak value over this interval, a finite set of linear measurements over [0, T], and an estimate A of H which is a continuous function of our measurements such that A is close to H in norm with high probability. The question of determinability is of course intimately related to the kind of a priori knowledge one has of the operator. We represent this a priori information by saying that the operator H belongs to a subset 3265