FFT

The Fast Fourier Transform (FFT) is an efficient family of algorithms for computing the discrete Fourier transform (DFT) and its inverse. By exploiting mathematical symmetries in the DFT, FFT algorithms reduce the computational complexity from O(N^2) to O(N log N), enabling practical analysis of long sequences. The FFT is not a separate transform; it is a collection of methods for evaluating the DFT more quickly.

The DFT converts a sequence of N samples into N frequency components Xk, where Xk = sum from n=0 to N−1 of x_n e^(−i 2π kn / N). The inverse DFT recombines these components to recover the original sequence. FFT algorithms typically assume N is composite and use recursive or iterative decompositions, such as radix-2, radix-4, or mixed-radix schemes, in decimation-in-time or decimation-in-frequency approaches. For arbitrary lengths, methods like Bluestein’s algorithm extend applicability.

Practically, FFTs are implemented as in-place computations and often include real-input optimizations. They are central to digital signal processing, audio and image processing, communications, radar, spectroscopy, and scientific computing. FFT-based convolution, via the convolution theorem, enables fast linear filtering and polynomial multiplication. Numerical precision and rounding errors are considerations, especially in large-scale or high-dynamic-range applications.

Key considerations for use include hardware and memory architecture, software library availability, and the specific FFT variant chosen. Widely used libraries implement optimized FFT routines and support forward and inverse transforms, real-valued inputs, and multi-dimensional transforms, reflecting the FFT’s foundational role in modern signal analysis.