# Theory and Applications of Fourier, Laplace, and Z Transformations

## DOI:

https://doi.org/10.48149/jciees.2022.2.1.6## Keywords:

k-space, L-space, z-space, phase, shift## Abstract

This study presents the mathematics for the implementation of direct and inverse Fourier, Laplace, and Z transformations. This research is at the intersection between signal processing, applied mathematics, and software engineering, and it provides a study guide to implementers. Mathematical concepts and details necessary to transform the math into code are provided as theoretical background. Validation is conducted for the cases when the transforms do intersect, when the transforms do not intersect, and when, in Fourier and Z-transformations, the frequency domain encodes a phase shift which is reconstructed as an image space shift. Coherence between the software implementation of the three transformations is confirmed when: 1. The real component of the complex variable s = σ + i ω is σ = 0, which is the case when Fourier and Laplace transforms are the same. 2. When the magnitude of the complex variable z = r e ^{jω} is r = 1, which is the case when Fourier and Z transforms are the same. Congruency between software implementation of transformations is confirmed comparing departing image and inverse reconstructed image. The novelty of this research is the presentation style of the theory of direct and inverse Fourier, Laplace, and Z transforms. Details provided in this research make this paper a study guide that is not found elsewhere.

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## References

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