Revisiting Z Transform Laplace Inversion: To Correct flaws in Signal and System Theory

TL;DR

The paper identifies a fundamental omission in standard inverse Laplace evaluation—the neglect of the infinite-arc contribution in the Bromwich integral—and shows this biases $L^{-1}$ and, by extension, the $Z$-transform in sampled-data modeling. By enforcing the full Bromwich contour, including the infinite boundary, the authors restore internal consistency between $L^{-1}$ and the $Z$-transform and reconcile sampling theory with DTFT aliasing results. The work proposes a structural revision of the $Z$-transform, $L^{-1}$, and the Heaviside function at discontinuities, with implications for modeling, simulation, and the handling of sampling delays—extending to cases where the delay equals or exceeds one sampling interval and addressing zero-order hold behavior.

Abstract

This paper revisits the classical formulation of the Z-transform and its relationship to the inverse Laplace transform (L-1), originally developed by Ragazzini in sampled-data theory. It identifies a longstanding mathematical oversight in standard derivations, which typically neglect the contribution from the infinite arc in the complex plane during inverse Laplace evaluation. This omission leads to inconsistencies, especially at discontinuities such as t = 0. By incorporating the full Bromwich contour, including all boundary contributions, we restore internal consistency between L-1 and the Z-transform, aligning the corrected L-1 with results from Discrete-Time Fourier Transform (DTFT) aliasing theory. Consequently, this necessitates a structural revision of the Z-transform, inverse Laplace transform, and the behavior of the Heaviside step function at discontinuities, providing a more accurate foundation for modeling and analysis of sampled-data systems.