System response analysis in wavenumber domain for linear space-invariant time-varying problems

TL;DR

This paper introduces a novel method for analyzing the response of linear space-invariant systems with time-varying properties by transforming spatial fields into the wavenumber domain, enabling efficient prediction and decomposition of complex systems.

Contribution

It proposes a Fourier-based approach for system response analysis in the wavenumber domain for LSI time-varying systems, including a method for decomposing complex temporal systems into simpler subsystems.

Findings

System function in wavenumber domain can be obtained for LSI systems.

Output prediction is simplified using the system function.

Complex systems can be decomposed into sequential subsystems for easier analysis.

Abstract

Being a powerful tool for linear time-invariant (LTI) systems, system response analysis can also be applied to the so-called linear space-invariant (LSI) but time-varying systems, which is a dual of the conventional LTI problems. In this paper, we propose a system response analysis method for LSI problems by conducting Fourier transform of the field distribution on the space instead of time coordinate. Specifically, input and output signals can be expressed in the wavenumber (spatial frequency) domain. In this way, the system function in wavenumber domain can also be obtained for LSI systems. Given an arbitrary input and temporal profile of the medium, the output can be easily predicted using the system function. Moreover, for a complex temporal system, the proposed method allows for decomposing it into multiple simpler subsystems that appear in sequence in time. The system function of the whole system can be efficiently calculated by multiplying those of the individual subsystems.