FFT Fast Fourier Transform and Inverse Transform Functions

In this document, I would like to provide an extended discussion about MATLAB-implemented FFT (Fast Fourier Transform) and IFFT (Inverse Fast Fourier Transform) functions. These functions are written in M-language and can be executed on computing systems. The implementation typically utilizes MATLAB's built-in fft() and ifft() functions, which employ the Cooley-Tukey algorithm for efficient computation of discrete Fourier transforms with O(n log n) complexity. Through these functions, we can perform spectral analysis on signals to better understand their characteristics and properties. Additionally, we can explore practical applications of these functions for processing real-world data, along with optimization techniques such as zero-padding for improved frequency resolution and windowing functions to reduce spectral leakage. The code structure generally includes parameter validation, signal preprocessing, FFT/IFFT computation, and result normalization. These functions provide researchers and engineers with powerful tools for signal processing and data analysis applications, enabling efficient frequency domain manipulation and time-frequency transformation capabilities.