# Fast Fourier Transform

## Overview

A Fourier Transform (FFT) is a method of calculating the frequency components in a data set - and the inverse FFT converts back from the frequency domain - 4 applications of the FFT rotates you round the complex plane and leaves you back with the original data.

A fast Fourier transform (

**FFT**) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse.By far the most common FFT is the Cooley-Tukey algorithm :

The most well-known use of the Cooley-Tukey algorithm is to divide the transform into two pieces of size

`N / 2`at each step...

## APLX FFT Code

Note that APLX is no longer under development.

This is as given in Robert J. Korsan's article in APL Congress 1973, p 259-268, with just line labels and a few comments added.

X and Z are two row matrixes representing the imput and output real and imaginary data. The data length must be

`2*N`(N integer), and the algorithm will cope with varying N, unlike many DSP versions which are for fixed N.Zero frequency is at

`Z[1;]`, maximum frequency in the middle; from there to`¯1↑[1] Z`are negative frequencies. i.e. for an input Gaussian they transform a 'bath-tub' to a 'bath-tub'.This is an elegant algorithm, and works by transforming the input data into a array of 2×2 FFT Butterflies.

Z←fft X;N;R;M;L;P;Q;S;T;O ⍝ ⍝ Apl Congress 1973, p 267. Robert J. Korsan. ⍝ ⍝ Restructure as an array of primitive 2×2 FFT Butterflies X←(2,R←(M←⌊2⍟N←¯1↑⍴X)⍴2)⍴⍉X ⍝ Build sin and cosine table : Z←R⍴⍉2 1∘.○○(-(O←?1)-⍳P)÷P←N÷2 ⍝ Q←⍳P←M-1+L←0 T←M-~O loop:→(M≤L←L+1)⍴done X←(+⌿X),[O+¯0.5+S←M-L](-/Z×-⌿X),[O+P-0.5]+/Z×⌽-⌿X Z←(((-L)⌽Q),T)⍉R⍴((1+P↑(S-1)⍴1),2)↑Z →loop done:Z←⍉(N,2)⍴(+⌿X),[O-0.5]-⌿X

## Variants

I also have this code as

**APL\11**or**aplc**plain text - contact me if you need these :`<BUBBELBLUB J.B.W.Webber AT kent NONSENSE DOT ac DOT uk>`As Lab-Tools Ltd. I can supply well-tested variants that have a time column, work with real and imaginary data, are correctly normalised in both amplitude and time, and (say) transform a centralised Gaussian to a centralised Gaussian. Also variants that transform Q to R (and R to Q) for neutron and X-Ray scattering. These have been tested with up to 100k data point (2*17) arrays :

`<VERY SPAMFREE Beau AT BLUR Lab-Tools DOT com>`