Foundational Correction of Z-Transform Theory: Restoring Mathematical Completeness in Sampled-Data Systems

TL;DR

This paper corrects foundational issues in Z-transform theory by including previously neglected boundary contributions in inverse Laplace evaluations, leading to a more consistent and complete mathematical framework for sampled-data systems.

Contribution

It identifies and rectifies a longstanding mathematical oversight in Z-transform derivations by incorporating full Bromwich contour contributions, restoring consistency with inverse Laplace and Fourier theories.

Findings

Restores mathematical consistency between Z-transform and inverse Laplace transform.

Revises the theoretical foundation of sampled-data system analysis.

Provides a more accurate modeling framework for discontinuities in sampled-data systems.

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.