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A ‐map representations and asymptotically almost periodic responses
Author(s) -
Sandberg Irwin W.
Publication year - 2001
Publication title -
international journal of circuit theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.364
H-Index - 52
eISSN - 1097-007X
pISSN - 0098-9886
DOI - 10.1002/cta.142
Subject(s) - uniqueness , mathematics , invariant (physics) , class (philosophy) , causality (physics) , discrete mathematics , set (abstract data type) , function (biology) , lti system theory , linear system , pure mathematics , mathematical analysis , computer science , physics , quantum mechanics , artificial intelligence , evolutionary biology , mathematical physics , biology , programming language
It was recently shown that for each member G of a large class of causal time‐invariant non‐linear input–output maps, with inputs and outputs defined on the non‐negative 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 is a simple linear map that does not depend on G . In this paper, we consider non‐linear maps G that have such ‘ A ‐map representations’. We observe that these G s have extensions to a domain of inputs defined on the set of all integers. We show that these extensions possess some interesting properties including a certain important uniqueness property. As an application, we show that under the (very often satisfied) conditions of time invariance, causality, and approximately finite memory, and under typically mild boundedness and continuity conditions, the response of G to a discrete‐time asymptotically almost periodic input is an output that is always an asymptotically almost periodic function, and that the almost periodic part of the output is independent of the transient part of the input. We also give corresponding results for a continuous‐time case. Copyright © 2001 John Wiley & Sons, Ltd.