A canonical form for discrete-time systems defined over 𝒵+

It is shown that for each member G of a large class of causal time-invariant nonlinear input-output maps, with inputs and outputs defined on the nonnegative integers, there is a functional A on the input set such that (Gs)(k) has the representation A(F_ks) for all k and each input s, in which F_k is a simple linear map that does not depend on G. More specifically, this holds-with an A that is unique in a certain important sense-for any G that has approximately finite memory and meets a certain often-satisfied additional condition. Similar results are given for a corresponding continuous-time case in which inputs and outputs are defined on IR₊. An example given shows that the members of a large family of feedback systems have these “A-map” representations