Implementation and Performance Testing of FFT and DFT Algorithms

In this document, we will implement FFT (Fast Fourier Transform) and DFT (Discrete Fourier Transform) algorithms using MATLAB and benchmark their execution speeds. First, let's briefly review the background and principles of these algorithms. FFT is an efficient computational algorithm that transforms signals from the time domain to the frequency domain, widely used in signal processing and spectral analysis due to its computational efficiency. DFT serves as the discrete version of Fourier transform for processing discrete signals. MATLAB provides built-in functions like fft() for FFT implementation and allows custom DFT coding using matrix operations or nested loops. The implementation typically requires only a few lines of code - for DFT: using vectorized operations with complex exponentials; for FFT: leveraging MATLAB's optimized built-in function. We will compare execution times using tic/toc commands for different signal lengths, analyzing computational complexity differences (DFT: O(N²) vs FFT: O(N log N)). This experiment helps understand the algorithmic principles and practical implementations in MATLAB, particularly for large-scale data processing applications where computational efficiency becomes critical. Through performance comparison, we can evaluate the trade-offs between implementation complexity and computational speed for real-world applications.