Short-time Fourier transform

Fourier analysis

Discrete cosine transform

Mathematics

Cyclotomic fast Fourier transform

Hartley transform

Fourier transform

Algorithm

Image (mathematics)

Split-radix FFT algorithm

Mathematical analysis

Cooley–Tukey FFT algorithm

Artificial intelligence

Discrete Hartley transform

Computer science

Computation

Twiddle factor

Rader's FFT algorithm

Discrete Fourier transform (general)

Fast Fourier transform

Fractional Fourier transform

Prime-factor FFT algorithm

IEEE Transactions on Acoustics Speech and Signal Processing

This tutorial paper describes the methods for constructing fast algorithms for the computation of the discrete Fourier transform (DFT) of a real-valued series. The application of these ideas to all the major fast Fourier transform (FFT) algorithms is discussed, and the various algorithms are compared. We present a new implementation of the real-valued split-radix FFT, an algorithm that uses fewer operations than any other real-valued power-of-2-length FFT. We also compare the performance of inherently real-valued transform algorithms such as the fast Hartley transform (FHT) and the fast cosine transform (FCT) to real-valued FFT algorithms for the computation of power spectra and cyclic convolutions. Comparisons of these techniques reveal that the alternative techniques always require more additions than a method based on a real-valued FFT algorithm and result in computer code of equal or greater length and complexity.

Real-valued fast Fourier transform algorithms