Linear shift‐invariant input–output maps do not necessarily commute

Abstract

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.