Computing Fast Fourier Transform (FFT) with MATLAB Implementation

This document presents a MATLAB program implementation for computing the Fast Fourier Transform (FFT). The Fast Fourier Transform (FFT) is an efficient algorithm for converting time-domain signals into frequency-domain representations. Widely used in signal processing and image analysis, FFT enables rapid and accurate analysis of spectral characteristics, providing insights into signal structure and properties. When implementing FFT in MATLAB, developers can leverage built-in functions from the Signal Processing Toolbox. MATLAB's fft() function handles complex DFT calculations with optimized performance. For basic implementation, the code structure typically involves: - Pre-processing: Signal normalization and zero-padding for power-of-two length optimization - Core computation: Using fft(x) for direct transform or implementing Cooley-Tukey algorithm for educational purposes - Post-processing: Magnitude calculation (abs(fft_result)) and frequency axis scaling The FFT algorithm follows a divide-and-conquer approach by separating input signals into even and odd-indexed components. These subsets undergo recursive FFT computations before being combined through butterfly operations. The recursive structure reduces computational complexity from O(N²) to O(N log N), making FFT ideal for large datasets and real-time signal processing applications. MATLAB's implementation advantages include: - Built-in vectorization for handling array inputs efficiently - Automatic memory management for large data sets - Integrated visualization tools (fftshift, plot) for spectrum analysis - Support for both 1D and multi-dimensional FFT computations In summary, developing an FFT program in MATLAB facilitates accurate spectral analysis while leveraging the platform's comprehensive signal processing capabilities. The combination of mathematical rigor and computational efficiency makes MATLAB an excellent environment for implementing and experimenting with Fourier transform applications.