FAST FOURIER TRANSFORM : MATHEMATICAL FOUNDATIONS, ALGORITHMS, AND APPLICATIONS

Abstract

The Fast Fourier Transform (FFT) stands as a pivotal algorithm in both theoretical and applied mathe matics, revolutionizing the way periodic and discrete signals are analyzed in the frequency domain. This paper presents a comprehensive study of the FFT with emphasis on the Cooley-Tukey algorithm and its divide-and-conquer strategy, which minimize the computational complexity of the Discrete Fourier Transform (DFT) from O(N2 ) to O(N log N). We examine the mathematical underpinnings of the algorithm, explore its generalizations, and highlight its efficiency through comparative analysis across different software implementations including FFTW, CUFFT, and Python-based libraries. Further, the study bridges the gap between theory and application by demonstrating FFT’s critical role in signal processing, image compression, and artificial intelligence. Visualization techniques such as spectro grams and frequency-domain transformations are used to showcase FFT’s capability in extractting and interpreting complex data patterns. This work aligns the interplay between mathematical theory and computational innovation, offering insights into how classical mathematics continues to shape modern technological trends.