Prime Numbers

480 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
the Cooley–Tukey form, the phrase meaning that as in the Danielson–Lanczos
splitting identity (9.22), we cut up (decimate) the time domain—the index on
the original signal. The Gentleman–Sande FFT falls into the “decimation in
frequency” class, for which a similar game is played on the k index of the
transform elements Xk.
Algorithm 9.5.5 (FFT, in-place, in-order loop forms with bit-scramble).
Given a (D =2 d )-element signal x, the functions herein perform an FFT via
nested loops. The two essential FFTs are laid out as decimation-in-time (Cooley–
Tukey) and decimation-in-frequency (Gentleman–Sande) forms. Note that these
forms can be applied symbolically, or in number-theoretical transform mode, by
identifying properly the root of unity and the ring or field operations.
1. [Cooley–Tukey, decimation-in-time FFT]
FFT(x) {
scramble(x);
n = len(x);
for(m =1; m