Commutativity of First-order Discrete-Time Linear Time-varying Systems

TL;DR

This paper establishes explicit necessary and sufficient conditions for the commutativity of first-order discrete-time linear time-varying systems, with applications in digital system design and telecommunication.

Contribution

It provides the first comprehensive set of conditions for commutativity in first-order discrete-time systems, expanding the theoretical understanding and practical applications.

Findings

Derived explicit necessary and sufficient conditions for commutativity.

Verified results through examples including digital telecommunication applications.

Illustrated advantages of commutativity in digital system design.

Abstract

After introducing the concept of commutativity for continuous-time linear time-varying systems, the related literature and the results obtained so far are presented. For a simple introduction of the commutativity of discrete-time linear time-varying systems, the problem is formulated for first-order systems. Finally, explicit necessary and sufficient conditions for the commutativity of first-order discrete-time linear time-varying systems are derived, and their advantageous use in digital system design is illustrated; which are the main objectives of the paper. The results are verified by examples which include an application in amplitude modulation for digital telecommunication.