Deepening Understanding of Fast Fourier Transform (FFT) Fundamentals

In this article, we will conduct the following discussions: 1. Develop deeper understanding of Fast Fourier Transform (FFT) basic principles through experimental implementations, including practical MATLAB/Python code examples demonstrating FFT algorithm structure and computational efficiency compared to direct DFT calculation. 2. Investigate the relationship between FFT point count and spectral resolution, with code illustrations showing how zero-padding affects frequency bin spacing. Compare computational characteristics between zero-padded sequence FFT and original sequence FFT using practical signal processing examples. Discrete Fourier Transform (DFT) and convolution constitute two of the most fundamental and frequently used operations in signal processing. They encompass broad signal processing domains including signal and system analysis/synthesis. Crucially, DFT and convolution maintain mathematical interdependence: convolution operations can be implemented through DFT algorithms using the convolution theorem (fft(conv(a,b)) ≈ fft(a)*fft(b)), while numerous other algorithms including correlation analysis, digital filtering, and power spectral estimation can be DFT-based implementations. Conversely, DFT computations can be transformed into convolution operations through mathematical manipulations, highlighting their underlying connections in computational mathematics. The above content expands the original discussion while preserving core concepts, incorporating practical implementation perspectives for signal processing practitioners.