Theory and Applications of Fourier, Laplace, and Z Transformations

This book explores the mathematical and practical aspects of implementing and testing direct and inverse Fourier, Laplace, and Z transformations in software, specifically using C++. The study evaluates the coherence of these transformations under specific conditions: when Fourier and Laplace transforms align (σ = 0) and when Fourier and Z transforms align (r = 1). Testing involves two-dimensional images, comparing original and reconstructed images to verify the accuracy of the transformations. The book also examines differences in the frequency domain for non-aligned cases (σ ≠ 0, r ≠ 1) and demonstrates the effects of phase shifts on image space. A key innovation lies in its detailed presentation of theoretical and implementation aspects, offering a practical guide for accurate software development of these mathematical transformations.