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 invarian t nonlinear input-output maps, with inputs and outputs defined on the nonne gative integers, there is a functional A on the input set such that (Gs)(k) has the representation A(F(k)s) for all k and each input s, in which F-k i s a simple linear map that does not depend on G. More specifically, this ho lds-with an A that is unique in a certain important sense-for any G that ha s approximately finite memory and meets a certain often-satisfied additiona l condition. Similar results are given for a corresponding continuous-time case in which inputs and outputs are defined on R+. An example shows that t he members of a large family of feedback systems have these "A-map" represe ntations.