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Linear shift‐invariant input–output maps do not necessarily commute
Author(s) -
Sandberg Irwin W.
Publication year - 2000
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 - 10.1002/1097-007x(200009/10)28:5<513::aid-cta119>3.0.co;2-#
Subject(s) - commutative property , invariant (physics) , mathematics , pure mathematics , set (abstract data type) , linear operators , discrete mathematics , control theory (sociology) , computer science , mathematical analysis , mathematical physics , control (management) , artificial intelligence , bounded function , programming language
An Erratum has been published for this article in International Journal of Circuit Theory and Applications, 2001, 29:433. It is part of the engineering folklore that linear shift‐invariant input–output operators that take a set of functions (closed under translation) into itself commute in the sense that H 1 H 2 = H 2 H 1 for any two such operators H 1 and H 2 . The main purpose of this paper is to record theorems to the effect that, in a certain very reasonable discrete‐space setting, it is not true that shift‐invariant operators commute, even though H 1 H 2 = H 2 H 1 holds on certain interesting subsets of the set of inputs. A result showing the lack of commutativity for continuous‐space systems is also given. Copyright © 2000 John Wiley & Sons, Ltd.