assume as example that DFTn in (2) can be factored into four matrices ... Here, x[
a:b] denotes (Matlab or FORTRAN style) the subvector of x starting at a and
ending at b. ..... Figure 7: Recursive radix-2 decimation-in-time FFT for n = 24.
TITLE Fast Fourier Transform
BYLINE Franz Franchetti and Markus P¨uschel Department of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh, PA USA {franzf, pueschel}@ece.cmu.edu
SYNONYMS FFT, fast algorithm for the discrete Fourier transform (DFT)
DEFINITION A fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) of an input vector. Efficient means that the FFT computes the DFT of an n-element vector in O(n log n) operations in contrast to the O(n2 ) operations required for computing the DFT by definition. FFTs exist for any vector length n and for real and higher-dimensional data. Parallel FFTs have been developed since the advent of parallel computing.
DISCUSSION Introduction The discrete Fourier transform (DFT) is a ubiquitous tool in science and engineering including in digital signal processing, communication, and high-performance computing. Applications include spectral analysis, image compression, interpolation, solving partial differential equations, and many other tasks. Given n real or complex inputs x0 , . . . , xn−1 , the DFT is defined as X yk = ωnkℓ xℓ , 0 ≤ k < n, (1) 0≤ℓ
