Fast Computation of the DFT

Abstract

In this chapter, fast algorithms for the computation of the DFT, a set of practically efficient DFT algorithms, are presented, both for computing the DFT/IDFT of real- and complex-valued data. The DFT is the workhorse in signal and system analysis due to the availability of fast algorithms for its computation. They are basically recursive algorithms mostly used with complex-valued data, irrespective of the nature of the given data. The recursive process, the algorithms depend on, is presented using the decomposition of a waveform into its even and odd half-wave symmetry components for clear understanding. There are two major types of algorithms, depending on which end of the algorithm the decomposition starts. They are called DIT and DIF algorithms. While they are inherently designed for complex-valued data, they can be modified easily for the computation of the DFT of real-valued data. The IDFT can be computed by DFT algorithms with minor modifications. Both types of algorithms and their modifications for computing the DFT/IDFT of real-valued data are presented with a number of examples.

Adapted from my book Fourier Analysis—A Signal Processing Approach, Springer, 2018 by permission.