Linear Shift-Invariant Input-Output Maps Do Not Necessarily Commute

摘要

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_1H_2=H_2H_1 for any two such operators H_1 and H_2. The main purpose of this paper is to record theroems to the effect that, in a certain very reasonable discrete-space setting, it is not true that shift-invariant operators commute, even though H_1H_2=H_2H_1 holds on certain interesting subsets of the set of inputs. A result showing the lack of cummutativity for continuous-space systems is also given.