# Fast Fourier transform

The **fast Fourier transform** (**FFT**) is an algorithm to compute the discrete Fourier transform of a vector in time , where a naive implementation achieves only time. APL implementations of the fast Fourier transform began appearing as early as 1970, with an 8-line implementation by Alan R. Jones published in APL Quote-Quad.^{[1]}

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.

## Implementations

### APLX

This FFT code is implemented with the Cooley–Tukey FFT algorithm by dividing the transform into two pieces of size `N÷2`

at each step. It will run under APLX.

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 matrices representing the input 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 an 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
```

### Simple recursive implementation

```
fft←{
1>K←2÷⍨M←≢⍵:⍵
0≠1|2⍟M:'Length of the input vector must be a power of 2'
odd←∇(M⍴1 0)/⍵
even←∇(M⍴0 1)/⍵
exp←*(0J¯2×(○1)×(¯1+⍳K)÷M)
(odd+T),odd-T←even×exp
}
```

Sample usage:

fft 1 1 1 1 0 0 0 0 4 1J¯2.414213562 0 1J¯0.4142135624 0 1J0.4142135624 0 1J2.414213562

Inverse FFT can be defined for testing:

```
ifft←{(≢⍵)÷⍨+fft+⍵}
test←{⌈/(10○⊢)(⍵-ifft fft ⍵)}
test 1 1 1 1 0 0 0 0
7.850462E¯17
```

2-dimensional FFT and inverse 2D FFT:

```
fft2D←{
∨/0≠1|2⍟⍴⍵:'Matrix dimensions must be powers of 2'
⍉(fft⍤1)⍉(fft⍤1)⍵
}
ifft2D←{(≢∊⍵)÷⍨+fft2D+⍵}
```

Sample usage:

fft2D 2 2⍴⍳4 10 ¯2 ¯4 0 ifft2D fft2D 2 2⍴⍳4 1 2 3 4

### Dyalog APL

FFT appears in dfns.dws, a workspace supplied with Dyalog APL, in the context of fast multi-digit multiplication^{[2]}. Extracted from there, it is there defined as:

```
roots←{×\1,1↓(⍵÷2)⍴¯1*2÷⍵}
cube←{⍵⍴⍨2⍴⍨2⍟⍴⍵}
floop←{(⊣/⍺)∇⍣(×m)⊢(+⌿⍵),[m-0.5]⍺×[⍳m←≢⍴⍺]-⌿⍵}
FFT←{,(cube roots⍴⍵)floop cube ⍵}
```

## References

- ↑ Jones, Alan R. (IBM). "Fast Fourier transform". APL Quote-Quad Volume 1, Number 4. 1970-01.
- ↑ dfns.dws: xtimes — Fast multi-digit product using FFT