Rader's FFT Algorithm

Rader's FFT Algorithm

Rader's algorithm (1968) is a fast Fourier transform (FFT) algorithm that computes the discrete Fourier transform (DFT) of prime sizes by re-expressing the DFT as a cyclic convolution. (The other algorithm for FFTs of prime sizes, Bluestein's algorithm, also works by rewriting the DFT as a convolution.)

Since Rader's algorithm only depends upon the periodicity of the DFT kernel, it is directly applicable to any other transform (of prime order) with a similar property, such as a number-theoretic transform or the discrete Hartley transform.

The algorithm can be modified to gain a factor of two savings for the case of DFTs of real data, using a slightly modified re-indexing/permutation to obtain two half-size cyclic convolutions of real data (Chu & Burrus, 1982); an alternative adaptation for DFTs of real data, using the discrete Hartley transform, was described by Johnson & Frigo (2007).

Winograd extended Rader's algorithm to include prime-power DFT sizes (Winograd 1976; Winograd 1978), and today Rader's algorithm is sometimes described as a special case of Winograd's FFT algorithm, also called the multiplicative Fourier transform algorithm (Tolimieri et al., 1997), which applies to an even larger class of sizes. However, for composite sizes such as prime powers, the Cooley–Tukey FFT algorithm is much simpler and more practical to implement, so Rader's algorithm is typically only used for large-prime base cases of Cooley–Tukey's recursive decomposition of the DFT (Frigo and Johnson, 2005).