Fast Fourier Transform (FFT)

In this document, I would like to share knowledge about the Fast Fourier Transform (FFT). The FFT is a fascinating and powerful mathematical tool widely used in signal processing, image analysis, and various scientific domains. By converting signals or functions from the time domain to frequency domain representation, we can extract valuable information about signal characteristics and perform sophisticated analysis operations. From an implementation perspective, the FFT algorithm efficiently computes the Discrete Fourier Transform (DFT) using divide-and-conquer techniques, significantly reducing computational complexity from O(N²) to O(N log N). Common implementations include the Cooley-Tukey algorithm which recursively breaks down DFTs into smaller transforms. For those interested in mathematics and signal processing, I recommend downloading this document which provides practical code examples using libraries like numpy.fft in Python or FFTW in C++, demonstrating key functions such as fft(), ifft() for transformation and spectral analysis techniques that will offer valuable insights and new perspectives.