Foundational Correction of Z-Transform Theory: Restoring Mathematical Completeness in Sampled-Data Systems
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914323684655104 |
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| author | Yang, Yuxin Zhou, Hang Li, Chaojie Li, Xin Yan, Yingyi Zheng, Mingyang |
| author_facet | Yang, Yuxin Zhou, Hang Li, Chaojie Li, Xin Yan, Yingyi Zheng, Mingyang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23242 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Foundational Correction of Z-Transform Theory: Restoring Mathematical Completeness in Sampled-Data Systems Yang, Yuxin Zhou, Hang Li, Chaojie Li, Xin Yan, Yingyi Zheng, Mingyang Systems and Control 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. |
| title | Foundational Correction of Z-Transform Theory: Restoring Mathematical Completeness in Sampled-Data Systems |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2506.23242 |