2025/06/29 by Yuxin Yang, Yang, Yuxin, Hang Zhou +8 · 2 citations
Computer Science · Engineering · Mathematics · #Digital Filter Design and Implementation #FOS: Electrical engineering #Mathematical Analysis and Transform Methods #Physics and Engineering Research Articles #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2506.23242
openalex publication_date 2025/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
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.