# Fast Fourier transform

(Redirected from FFT)
Fast Fourier transform
You are encouraged to solve this task according to the task description, using any language you may know.

Calculate the   FFT   (Fast Fourier Transform)   of an input sequence.

The most general case allows for complex numbers at the input and results in a sequence of equal length, again of complex numbers. If you need to restrict yourself to real numbers, the output should be the magnitude   (i.e.:   sqrt(re2 + im2))   of the complex result.

The classic version is the recursive Cooley–Tukey FFT. Wikipedia has pseudo-code for that. Further optimizations are possible but not required.

## 11l

Translation of: Python
```F fft(x)
V n = x.len
I n <= 1
R x
V even = fft(x[(0..).step(2)])
V odd  = fft(x[(1..).step(2)])
V t = (0 .< n I/ 2).map(k -> exp(-2i * math:pi * k / @n) * @odd[k])
R (0 .< n I/ 2).map(k -> @even[k] + @t[k]) [+]
(0 .< n I/ 2).map(k -> @even[k] - @t[k])

print(fft([Complex(1.0), 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0]).map(f -> ‘#1.3’.format(abs(f))).join(‘ ’))```
Output:
```4.000 2.613 0.000 1.082 0.000 1.082 0.000 2.613
```

The FFT function is defined as a generic function, instantiated upon a user instance of Ada.Numerics.Generic_Complex_Arrays.

```with Ada.Numerics.Generic_Complex_Arrays;

generic
with package Complex_Arrays is
use Complex_Arrays;
function Generic_FFT (X : Complex_Vector) return Complex_Vector;
```
```with Ada.Numerics;

function Generic_FFT (X : Complex_Vector) return Complex_Vector is

package Complex_Elementary_Functions is
(Complex_Arrays.Complex_Types);

use Complex_Elementary_Functions;
use Complex_Arrays.Complex_Types;

function FFT (X : Complex_Vector; N, S : Positive)
return Complex_Vector is
begin
if N = 1 then
return (1..1 => X (X'First));
else
declare
F : constant Complex  := exp (Pi * j / Real_Arrays.Real (N/2));
Even : Complex_Vector := FFT (X, N/2, 2*S);
Odd  : Complex_Vector := FFT (X (X'First + S..X'Last), N/2, 2*S);
begin
for K in 0..N/2 - 1 loop
declare
T : constant Complex := Odd (Odd'First + K) / F ** K;
begin
Odd  (Odd'First  + K) := Even (Even'First + K) - T;
Even (Even'First + K) := Even (Even'First + K) + T;
end;
end loop;
return Even & Odd;
end;
end if;
end FFT;
begin
return FFT (X, X'Length, 1);
end Generic_FFT;
```

Example:

```with Ada.Numerics.Complex_Arrays;  use Ada.Numerics.Complex_Arrays;

with Generic_FFT;

procedure Example is
function FFT is new Generic_FFT (Ada.Numerics.Complex_Arrays);
X : Complex_Vector := (1..4 => (1.0, 0.0), 5..8 => (0.0, 0.0));
Y : Complex_Vector := FFT (X);
begin
Put_Line ("       X              FFT X ");
for I in Y'Range loop
Put (X (I - Y'First + X'First), Aft => 3, Exp => 0);
Put (" ");
Put (Y (I), Aft => 3, Exp => 0);
New_Line;
end loop;
end;
```
Output:
```       X              FFT X
( 1.000, 0.000) ( 4.000, 0.000)
( 1.000, 0.000) ( 1.000,-2.414)
( 1.000, 0.000) ( 0.000, 0.000)
( 1.000, 0.000) ( 1.000,-0.414)
( 0.000, 0.000) ( 0.000, 0.000)
( 0.000, 0.000) ( 1.000, 0.414)
( 0.000, 0.000) ( 0.000, 0.000)
( 0.000, 0.000) ( 1.000, 2.414)
```

## ALGOL 68

Translation of: Python

Note: This specimen retains the original Python coding style.

Works with: ALGOL 68 version Revision 1 - one minor extension to language used - PRAGMA READ, similar to C's #include directive.
Works with: ALGOL 68G version Any - tested with release algol68g-2.3.5.

File: Template.Fast_Fourier_transform.a68

```PRIO DICE = 9; # ideally = 11 #

OP DICE = ([]SCALAR in, INT step)[]SCALAR: (
### Dice the array, extract array values a "step" apart ###
IF step = 1 THEN
in
ELSE
INT upb out := 0;
[(UPB in-LWB in)%step+1]SCALAR out;
FOR index FROM LWB in BY step TO UPB in DO
out[upb out+:=1] := in[index] OD;
out[@LWB in]
FI
);

PROC fft = ([]SCALAR in t)[]SCALAR: (
### The Cooley-Tukey FFT algorithm ###
IF LWB in t >= UPB in t THEN
in t[@0]
ELSE
[]SCALAR t = in t[@0];
INT n = UPB t + 1, half n = n % 2;
[LWB t:UPB t]SCALAR coef;

[]SCALAR even = fft(t    DICE 2),
odd = fft(t[1:]DICE 2);

COMPL i = 0 I 1;

REAL w =  2*pi / n;
FOR k FROM LWB t TO half n-1 DO
COMPL cis t = scalar exp(0 I (-w * k))*odd[k];
coef[k]          := even[k] + cis t;
coef[k + half n] := even[k] - cis t
OD;
coef
FI
);```

File: test.Fast_Fourier_transform.a68

```#!/usr/local/bin/a68g --script #
# -*- coding: utf-8 -*- #

MODE SCALAR = COMPL;
PROC (COMPL)COMPL scalar exp = complex exp;

FORMAT real fmt := \$g(0,3)\$;
FORMAT real array fmt := \$f(real fmt)", "\$;
FORMAT compl fmt := \$f(real fmt)"⊥"f(real fmt)\$;
FORMAT compl array fmt := \$f(compl fmt)", "\$;

test:(
[]COMPL
tooth wave ft = fft((1, 1, 1, 1, 0, 0, 0, 0)),
one and a quarter wave ft = fft((0, 0.924, 0.707,-0.383,-1,-0.383, 0.707, 0.924,
0,-0.924,-0.707, 0.383, 1, 0.383,-0.707,-0.924));
printf((
\$"Tooth wave: "\$,compl array fmt, tooth wave ft, \$l\$,
\$"1¼ cycle wave: "\$, compl array fmt, one and a quarter wave ft, \$l\$
))
)```
Output:
```Tooth wave: 4.000⊥.000, 1.000⊥-2.414, .000⊥.000, 1.000⊥-.414, .000⊥.000, 1.000⊥.414, .000⊥.000, 1.000⊥2.414,
1¼ cycle wave: .000⊥.000, .000⊥.001, .000⊥.000, .000⊥-8.001, .000⊥.000, -.000⊥-.001, .000⊥.000, .000⊥.001, .000⊥.000, .000⊥-.001, .000⊥.000, -.000⊥.001, .000⊥.000, -.000⊥8.001, .000⊥.000, -.000⊥-.001,
```

## APL

Translation of: Fortran
Works with: Dyalog APL
```fft←{
1>k←2÷⍨N←⍴⍵:⍵
0≠1|2⍟N:'Argument must be a power of 2 in length'
even←∇(N⍴0 1)/⍵
odd←∇(N⍴1 0)/⍵
T←even×*(0J¯2×(○1)×(¯1+⍳k)÷N)
(odd+T),odd-T
}
```

Example:

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

## BBC BASIC

```      @% = &60A

DIM Complex{r#, i#}
DIM in{(7)} = Complex{}, out{(7)} = Complex{}
DATA 1, 1, 1, 1, 0, 0, 0, 0

PRINT "Input (real, imag):"
FOR I% = 0 TO 7
out{(I%)}.r# = in{(I%)}.r#
PRINT in{(I%)}.r# "," in{(I%)}.i#
NEXT

PROCfft(out{()}, in{()}, 0, 1, DIM(in{()},1)+1)

PRINT "Output (real, imag):"
FOR I% = 0 TO 7
PRINT out{(I%)}.r# "," out{(I%)}.i#
NEXT
END

DEF PROCfft(b{()}, o{()}, B%, S%, N%)
LOCAL I%, t{} : DIM t{} = Complex{}
IF S% < N% THEN
PROCfft(o{()}, b{()}, B%, S%*2, N%)
PROCfft(o{()}, b{()}, B%+S%, S%*2, N%)
FOR I% = 0 TO N%-1 STEP 2*S%
t.r# = COS(-PI*I%/N%)
t.i# = SIN(-PI*I%/N%)
PROCcmul(t{}, o{(B%+I%+S%)})
b{(B%+I% DIV 2)}.r# = o{(B%+I%)}.r# + t.r#
b{(B%+I% DIV 2)}.i# = o{(B%+I%)}.i# + t.i#
b{(B%+(I%+N%) DIV 2)}.r# = o{(B%+I%)}.r# - t.r#
b{(B%+(I%+N%) DIV 2)}.i# = o{(B%+I%)}.i# - t.i#
NEXT
ENDIF
ENDPROC

DEF PROCcmul(c{},d{})
LOCAL r#, i#
r# = c.r#*d.r# - c.i#*d.i#
i# = c.r#*d.i# + c.i#*d.r#
c.r# = r#
c.i# = i#
ENDPROC
```
Output:
```Input (real, imag):
1,         0
1,         0
1,         0
1,         0
0,         0
0,         0
0,         0
0,         0
Output (real, imag):
4,         0
1,  -2.41421
0,         0
1, -0.414214
0,         0
1,  0.414214
0,         0
1,   2.41421```

## C

Inplace FFT with O(n) memory usage. Note: array size is assumed to be power of 2 and not checked by code; you can just pad it with 0 otherwise.

Also, `complex` is C99 standard.

```#include <stdio.h>
#include <math.h>
#include <complex.h>

double PI;
typedef double complex cplx;

void _fft(cplx buf[], cplx out[], int n, int step)
{
if (step < n) {
_fft(out, buf, n, step * 2);
_fft(out + step, buf + step, n, step * 2);

for (int i = 0; i < n; i += 2 * step) {
cplx t = cexp(-I * PI * i / n) * out[i + step];
buf[i / 2]     = out[i] + t;
buf[(i + n)/2] = out[i] - t;
}
}
}

void fft(cplx buf[], int n)
{
cplx out[n];
for (int i = 0; i < n; i++) out[i] = buf[i];

_fft(buf, out, n, 1);
}

void show(const char * s, cplx buf[]) {
printf("%s", s);
for (int i = 0; i < 8; i++)
if (!cimag(buf[i]))
printf("%g ", creal(buf[i]));
else
printf("(%g, %g) ", creal(buf[i]), cimag(buf[i]));
}

int main()
{
PI = atan2(1, 1) * 4;
cplx buf[] = {1, 1, 1, 1, 0, 0, 0, 0};

show("Data: ", buf);
fft(buf, 8);
show("\nFFT : ", buf);

return 0;
}
```
Output:
```Data: 1 1 1 1 0 0 0 0
FFT : 4 (1, -2.41421) 0 (1, -0.414214) 0 (1, 0.414214) 0 (1, 2.41421)```

### OS X / iOS

OS X 10.7+ / iOS 4+

```#include <stdio.h>
#include <Accelerate/Accelerate.h>

void fft(DSPComplex buf[], int n) {
float inputMemory[2*n];
float outputMemory[2*n];
// half for real and half for complex
DSPSplitComplex inputSplit = {inputMemory, inputMemory + n};
DSPSplitComplex outputSplit = {outputMemory, outputMemory + n};

vDSP_ctoz(buf, 2, &inputSplit, 1, n);

vDSP_DFT_Setup setup = vDSP_DFT_zop_CreateSetup(NULL, n, vDSP_DFT_FORWARD);

vDSP_DFT_Execute(setup,
inputSplit.realp, inputSplit.imagp,
outputSplit.realp, outputSplit.imagp);

vDSP_ztoc(&outputSplit, 1, buf, 2, n);
}

void show(const char *s, DSPComplex buf[], int n) {
printf("%s", s);
for (int i = 0; i < n; i++)
if (!buf[i].imag)
printf("%g ", buf[i].real);
else
printf("(%g, %g) ", buf[i].real, buf[i].imag);
printf("\n");
}

int main() {
DSPComplex buf[] = {{1,0}, {1,0}, {1,0}, {1,0}, {0,0}, {0,0}, {0,0}, {0,0}};

show("Data: ", buf, 8);
fft(buf, 8);
show("FFT : ", buf, 8);

return 0;
}
```
Output:
```Data: 1 1 1 1 0 0 0 0
FFT : 4 (1, -2.41421) 0 (1, -0.414214) 0 (1, 0.414214) 0 (1, 2.41421) ```

## C#

```using System;
using System.Numerics;
using System.Linq;
using System.Diagnostics;

// Fast Fourier Transform in C#
public class Program {

/* Performs a Bit Reversal Algorithm on a postive integer
* for given number of bits
* e.g. 011 with 3 bits is reversed to 110 */
public static int BitReverse(int n, int bits) {
int reversedN = n;
int count = bits - 1;

n >>= 1;
while (n > 0) {
reversedN = (reversedN << 1) | (n & 1);
count--;
n >>= 1;
}

return ((reversedN << count) & ((1 << bits) - 1));
}

/* Uses Cooley-Tukey iterative in-place algorithm with radix-2 DIT case
* assumes no of points provided are a power of 2 */
public static void FFT(Complex[] buffer) {
#if false
int bits = (int)Math.Log(buffer.Length, 2);
for (int j = 1; j < buffer.Length / 2; j++) {

int swapPos = BitReverse(j, bits);
var temp = buffer[j];
buffer[j] = buffer[swapPos];
buffer[swapPos] = temp;
}
// Said Zandian
// The above section of the code is incorrect and does not work correctly and has two bugs.
// BUG 1
// The bug is that when you reach and index that was swapped previously it does swap it again
// Ex. binary value n = 0010 and Bits = 4 as input to BitReverse routine and  returns 4. The code section above //     swaps it. Cells 2 and 4 are swapped. just fine.
//     now binary value n = 0010 and Bits = 4 as input to BitReverse routine and returns 2. The code Section
//     swap it. Cells 4 and 2 are swapped.     WROOOOONG
//
// Bug 2
// The code works on the half section of the cells. In the case of Bits = 4 it means that we are having 16 cells
// The code works on half the cells        for (int j = 1; j < buffer.Length / 2; j++) buffer.Length returns 16
// and divide by 2 makes 8, so j goes from 1 to 7. This covers almost everything but what happened to 1011 value
// which must be swap with 1101. and this is the second bug.
//
// use the following corrected section of the code. I have seen this bug in other languages that uses bit
// reversal routine.

#else
for (int j = 1; j < buffer.Length; j++)
{
int swapPos = BitReverse(j, bits);
if (swapPos <= j)
{
continue;
}
var temp = buffer[j];
buffer[j] = buffer[swapPos];
buffer[swapPos] = temp;
}

// First the full length is used and 1011 value is swapped with 1101. Second if new swapPos is less than j
// then it means that swap was happen when j was the swapPos.

#endif

for (int N = 2; N <= buffer.Length; N <<= 1) {
for (int i = 0; i < buffer.Length; i += N) {
for (int k = 0; k < N / 2; k++) {

int evenIndex = i + k;
int oddIndex = i + k + (N / 2);
var even = buffer[evenIndex];
var odd = buffer[oddIndex];

double term = -2 * Math.PI * k / (double)N;
Complex exp = new Complex(Math.Cos(term), Math.Sin(term)) * odd;

buffer[evenIndex] = even + exp;
buffer[oddIndex] = even - exp;

}
}
}
}

public static void Main(string[] args) {
Complex[] input = {1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0};

FFT(input);

Console.WriteLine("Results:");
foreach (Complex c in input) {
Console.WriteLine(c);
}
}
}
```
Output:
```Results:
(4, 0)
(1, -2.41421356237309)
(0, 0)
(1, -0.414213562373095)
(0, 0)
(1, 0.414213562373095)
(0, 0)
(1, 2.41421356237309)
```

## C++

```#include <complex>
#include <iostream>
#include <valarray>

const double PI = 3.141592653589793238460;

typedef std::complex<double> Complex;
typedef std::valarray<Complex> CArray;

// Cooley–Tukey FFT (in-place, divide-and-conquer)
// Higher memory requirements and redundancy although more intuitive
void fft(CArray& x)
{
const size_t N = x.size();
if (N <= 1) return;

// divide
CArray even = x[std::slice(0, N/2, 2)];
CArray  odd = x[std::slice(1, N/2, 2)];

// conquer
fft(even);
fft(odd);

// combine
for (size_t k = 0; k < N/2; ++k)
{
Complex t = std::polar(1.0, -2 * PI * k / N) * odd[k];
x[k    ] = even[k] + t;
x[k+N/2] = even[k] - t;
}
}

// Cooley-Tukey FFT (in-place, breadth-first, decimation-in-frequency)
// Better optimized but less intuitive
// !!! Warning : in some cases this code make result different from not optimased version above (need to fix bug)
// The bug is now fixed @2017/05/30
void fft(CArray &x)
{
// DFT
unsigned int N = x.size(), k = N, n;
double thetaT = 3.14159265358979323846264338328L / N;
Complex phiT = Complex(cos(thetaT), -sin(thetaT)), T;
while (k > 1)
{
n = k;
k >>= 1;
phiT = phiT * phiT;
T = 1.0L;
for (unsigned int l = 0; l < k; l++)
{
for (unsigned int a = l; a < N; a += n)
{
unsigned int b = a + k;
Complex t = x[a] - x[b];
x[a] += x[b];
x[b] = t * T;
}
T *= phiT;
}
}
// Decimate
unsigned int m = (unsigned int)log2(N);
for (unsigned int a = 0; a < N; a++)
{
unsigned int b = a;
// Reverse bits
b = (((b & 0xaaaaaaaa) >> 1) | ((b & 0x55555555) << 1));
b = (((b & 0xcccccccc) >> 2) | ((b & 0x33333333) << 2));
b = (((b & 0xf0f0f0f0) >> 4) | ((b & 0x0f0f0f0f) << 4));
b = (((b & 0xff00ff00) >> 8) | ((b & 0x00ff00ff) << 8));
b = ((b >> 16) | (b << 16)) >> (32 - m);
if (b > a)
{
Complex t = x[a];
x[a] = x[b];
x[b] = t;
}
}
//// Normalize (This section make it not working correctly)
//Complex f = 1.0 / sqrt(N);
//for (unsigned int i = 0; i < N; i++)
//	x[i] *= f;
}

// inverse fft (in-place)
void ifft(CArray& x)
{
// conjugate the complex numbers
x = x.apply(std::conj);

// forward fft
fft( x );

// conjugate the complex numbers again
x = x.apply(std::conj);

// scale the numbers
x /= x.size();
}

int main()
{
const Complex test[] = { 1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0 };
CArray data(test, 8);

// forward fft
fft(data);

std::cout << "fft" << std::endl;
for (int i = 0; i < 8; ++i)
{
std::cout << data[i] << std::endl;
}

// inverse fft
ifft(data);

std::cout << std::endl << "ifft" << std::endl;
for (int i = 0; i < 8; ++i)
{
std::cout << data[i] << std::endl;
}
return 0;
}
```
Output:
```fft
(4,0)
(1,-2.41421)
(0,0)
(1,-0.414214)
(0,0)
(1,0.414214)
(0,0)
(1,2.41421)

ifft
(1,-0)
(1,-5.55112e-017)
(1,2.4895e-017)
(1,-5.55112e-017)
(5.55112e-017,0)
(5.55112e-017,5.55112e-017)
(0,-2.4895e-017)
(5.55112e-017,5.55112e-017)
```

## Common Lisp

As the longer standing solution below didn't work out for me and I don't find it very nice, I want to give another one, that's not just a plain translation. Of course it could be optimized in several ways. The function uses some non ASCII symbols for better readability and condenses also the inverse part, by a keyword.

```(defun fft (a &key (inverse nil) &aux (n (length a)))
"Perform the FFT recursively on input vector A.
Vector A must have length N of power of 2."
(declare (type boolean inverse)
(type (integer 1) n))
(if (= n 1)
a
(let* ((n/2 (/ n 2))
(2iπ/n (complex 0 (/ (* 2 pi) n (if inverse -1 1))))
(⍵_n (exp 2iπ/n))
(⍵ #c(1.0d0 0.0d0))
(a0 (make-array n/2))
(a1 (make-array n/2)))
(declare (type (integer 1) n/2)
(type (complex double-float) ⍵ ⍵_n))
(symbol-macrolet ((a0[j]  (svref a0 j))
(a1[j]  (svref a1 j))
(a[i]   (svref a i))
(a[i+1] (svref a (1+ i))))
(loop :for i :below (1- n) :by 2
:for j :from 0
:do (setf a0[j] a[i]
a1[j] a[i+1])))
(let ((â0 (fft a0 :inverse inverse))
(â1 (fft a1 :inverse inverse))
(â (make-array n)))
(symbol-macrolet ((â[k]     (svref â k))
(â[k+n/2] (svref â (+ k n/2)))
(â0[k]    (svref â0 k))
(â1[k]    (svref â1 k)))
(loop :for k :below n/2
:do (setf â[k]     (+ â0[k] (* ⍵ â1[k]))
â[k+n/2] (- â0[k] (* ⍵ â1[k])))
:when inverse
:do (setf â[k]     (/ â[k] 2)
â[k+n/2] (/ â[k+n/2] 2))
:do (setq ⍵ (* ⍵ ⍵_n))
:finally (return â)))))))
```

From here on the old solution.

```;;; This is adapted from the Python sample; it uses lists for simplicity.
;;; Production code would use complex arrays (for compiler optimization).
;;; This version exhibits LOOP features, closing with compositional golf.
(defun fft (x &aux (length (length x)))
;; base case: return the list as-is
(if (<= length 1) x
;; collect alternating elements into separate lists...
(loop for (a b) on x by #'cddr collect a into as collect b into bs finally
;; ... and take the FFT of both;
(let* ((ffta (fft as)) (fftb (fft bs))
;; incrementally phase shift each element of the 2nd list
(aux (loop for b in fftb and k from 0 by (/ pi length -1/2)
collect (* b (cis k)))))
;; finally, concatenate the sum and difference of the lists
(return (mapcan #'mapcar '(+ -) `(,ffta ,ffta) `(,aux ,aux)))))))
```
Output:
```;;; Demonstrates printing an FFT in both rectangular and polar form:
CL-USER> (mapc (lambda (c) (format t "~&~6F~6@Fi = ~6Fe^~6@Fipi"
(realpart c) (imagpart c) (abs c) (/ (phase c) pi)))
(fft '(1 1 1 1 0 0 0 0)))

4.0  +0.0i =    4.0e^  +0.0ipi
1.0-2.414i = 2.6131e^-0.375ipi
0.0  +0.0i =    0.0e^  +0.0ipi
1.0-0.414i = 1.0824e^-0.125ipi
0.0  +0.0i =    0.0e^  +0.0ipi
1.0+0.414i = 1.0824e^+0.125ipi
0.0  +0.0i =    0.0e^  +0.0ipi
1.0+2.414i = 2.6131e^+0.375ipi
;;; MAPC also returns the FFT data, which looks like this:
(#C(4.0 0.0) #C(1.0D0 -2.414213562373095D0) #C(0.0D0 0.0D0)
#C(1.0D0 -0.4142135623730949D0) #C(0.0 0.0)
#C(0.9999999999999999D0 0.4142135623730949D0) #C(0.0D0 0.0D0)
#C(0.9999999999999997D0 2.414213562373095D0))
```

## Crystal

Translation of: Ruby
```require "complex"

def fft(x : Array(Int32 | Float64)) #: Array(Complex)
return [x[0].to_c] if x.size <= 1
even = fft(Array.new(x.size // 2) { |k| x[2 * k] })
odd  = fft(Array.new(x.size // 2) { |k| x[2 * k + 1] })
c = Array.new(x.size // 2) { |k| Math.exp((-2 * Math::PI * k / x.size).i) }
codd = Array.new(x.size // 2) { |k| c[k] * odd[k] }
return Array.new(x.size // 2) { |k| even[k] + codd[k] } + Array.new(x.size // 2) { |k| even[k] - codd[k] }
end

fft([1,1,1,1,0,0,0,0]).each{ |c| puts c }
```
Output:
```4.0 + 0.0i
1.0 - 2.414213562373095i
0.0 + 0.0i
1.0 - 0.4142135623730949i
0.0 + 0.0i
0.9999999999999999 + 0.4142135623730949i
0.0 + 0.0i
0.9999999999999997 + 2.414213562373095i
```

## D

### Standard Version

```void main() {
import std.stdio, std.numeric;

[1.0, 1, 1, 1, 0, 0, 0, 0].fft.writeln;
}
```
Output:
`[4+0i, 1-2.41421i, 0-0i, 1-0.414214i, 0+0i, 1+0.414214i, 0+0i, 1+2.41421i]`

### creals Version

Built-in complex numbers will be deprecated.

```import std.stdio, std.algorithm, std.range, std.math;

const(creal)[] fft(in creal[] x) pure /*nothrow*/ @safe {
immutable N = x.length;
if (N <= 1) return x;
const ev = x.stride(2).array.fft;
const od = x[1 .. \$].stride(2).array.fft;
auto l = iota(N / 2).map!(k => ev[k] + expi(-2*PI * k/N) * od[k]);
auto r = iota(N / 2).map!(k => ev[k] - expi(-2*PI * k/N) * od[k]);
return l.chain(r).array;
}

void main() @safe {
[1.0L+0i, 1, 1, 1, 0, 0, 0, 0].fft.writeln;
}
```
Output:
`[4+0i, 1+-2.41421i, 0+0i, 1+-0.414214i, 0+0i, 1+0.414214i, 0+0i, 1+2.41421i]`

### Phobos Complex Version

```import std.stdio, std.algorithm, std.range, std.math, std.complex;

auto fft(T)(in T[] x) pure /*nothrow @safe*/ {
immutable N = x.length;
if (N <= 1) return x;
const ev = x.stride(2).array.fft;
const od = x[1 .. \$].stride(2).array.fft;
alias E = std.complex.expi;
auto l = iota(N / 2).map!(k => ev[k] + T(E(-2* PI * k/N)) * od[k]);
auto r = iota(N / 2).map!(k => ev[k] - T(E(-2* PI * k/N)) * od[k]);
return l.chain(r).array;
}

void main() {
[1.0, 1, 1, 1, 0, 0, 0, 0].map!complex.array.fft.writeln;
}
```
Output:
`[4+0i, 1-2.41421i, 0+0i, 1-0.414214i, 0+0i, 1+0.414214i, 0+0i, 1+2.41421i]`

## Delphi

Library: System.Math
Translation of: C#
```program Fast_Fourier_transform;

{\$APPTYPE CONSOLE}

uses
System.SysUtils,
System.VarCmplx,
System.Math;

function BitReverse(n: UInt64; bits: Integer): UInt64;
var
count, reversedN: UInt64;
begin
reversedN := n;
count := bits - 1;

n := n shr 1;

while n > 0 do
begin
reversedN := (reversedN shl 1) or (n and 1);
dec(count);
n := n shr 1;
end;

Result := ((reversedN shl count) and ((1 shl bits) - 1));
end;

procedure FFT(var buffer: TArray<Variant>);
var
j, bits: Integer;
tmp: Variant;
begin
bits := Trunc(Log2(length(buffer)));

for j := 1 to High(buffer) do
begin
var swapPos := BitReverse(j, bits);
if swapPos <= j then
Continue;

tmp := buffer[j];
buffer[j] := buffer[swapPos];
buffer[swapPos] := tmp;
end;

var N := 2;
while N <= Length(buffer) do
begin
var i := 0;
while i < Length(buffer) do
begin
for var k := 0 to N div 2 - 1 do
begin
var evenIndex := i + k;
var oddIndex := i + k + (N div 2);
var _even := buffer[evenIndex];
var _odd := buffer[oddIndex];
var term := -2 * PI * k / N;
var _exp := VarComplexCreate(Cos(term), Sin(term)) * _odd;

buffer[evenIndex] := _even + _exp;
buffer[oddIndex] := _even - _exp;
end;
i := i + N;
end;
N := N shl 1;
end;

end;

const
input: array of Double = [1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0];

var
inputc: TArray<Variant>;

begin
SetLength(inputc, length(input));
for var i := 0 to High(input) do
inputc[i] := VarComplexCreate(input[i]);

FFT(inputc);

for var c in inputc do
writeln(c);
end.
```
Output:
```4 + 0i
1 - 2,41421356237309i
0 + 0i
1 - 0,414213562373095i
0 + 0i
1 + 0,414213562373095i
0 + 0i
1 + 2,41421356237309i```

## EchoLisp

```(define -∏*2 (complex 0 (* -2 PI)))

(define (fft xs N)
(if (<= N 1) xs
(let* [
(N/2 (/ N 2))
(even (fft (for/vector ([i (in-range 0 N 2)]) [xs i]) N/2))
(odd  (fft (for/vector ([i (in-range 1 N 2)]) [xs i]) N/2))
]
(for ((k N/2)) (vector*= odd k  (exp (/ (* -∏*2 k) N ))))
(vector-append (vector-map + even odd) (vector-map - even odd)))))

(define data #( 1 1 1 1  0 0 0 0 ))

(fft data 8)
→ #( 4+0i 1-2.414213562373095i 0+0i 1-0.4142135623730949i
0+0i 1+0.4142135623730949i 0+0i 1+2.414213562373095i)
```

## ERRE

```PROGRAM FFT

CONST CNT=8

!\$DYNAMIC
DIM REL[0],IMG[0],CMP[0],V[0]

BEGIN
SIG=INT(LOG(CNT)/LOG(2)+0.9999)
REAL1=2^SIG

REAL=REAL1-1
REAL2=INT(REAL1/2)
REAL4=INT(REAL1/4)
REAL3=REAL4+REAL2

!\$DIM REL[REAL1],IMG[REAL1],CMP[REAL3]

FOR I=0 TO CNT-1 DO
END FOR

DATA(1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0)

SIG2=INT(SIG/2)
SIG1=SIG-SIG2
CNT1=2^SIG1
CNT2=2^SIG2

!\$DIM V[CNT1-1]
V[0]=0
DV=1
PTR=CNT1

FOR J=1 TO SIG1 DO
HLFPTR=INT(PTR/2)
PT=CNT1-HLFPTR
FOR I=HLFPTR TO PT STEP PTR DO
V[I]=V[I-HLFPTR]+DV
END FOR
DV=2*DV
PTR=HLFPTR
END FOR

K=2*π/REAL1

FOR X=0 TO REAL4 DO
CMP[X]=COS(K*X)
CMP[REAL2-X]=-CMP[X]
CMP[REAL2+X]=-CMP[X]
END FOR

PRINT("FFT: BIT REVERSAL")

FOR I=0 TO CNT1-1 DO
IP=I*CNT2
FOR J=0 TO CNT2-1 DO
H=IP+J
G=V[J]*CNT2+V[I]
IF G>H THEN
SWAP(REL[G],REL[H])
SWAP(IMG[G],IMG[H])
END IF
END FOR
END FOR

T=1
FOR STAGE=1 TO SIG DO
PRINT("STAGE:";STAGE)
D=INT(REAL2/T)
FOR II=0 TO T-1 DO
L=D*II
LS=L+REAL4
FOR I=0 TO D-1 DO
A=2*I*T+II
B=A+T
F1=REL[A]
F2=IMG[A]
CNT1=CMP[L]*REL[B]
CNT2=CMP[LS]*IMG[B]
CNT3=CMP[LS]*REL[B]
CNT4=CMP[L]*IMG[B]
REL[A]=F1+CNT1-CNT2
IMG[A]=F2+CNT3+CNT4
REL[B]=F1-CNT1+CNT2
IMG[B]=F2-CNT3-CNT4
END FOR
END FOR
T=2*T
END FOR

PRINT("NUM REAL     IMAG")
FOR I=0 TO REAL DO
IF ABS(REL[I])<1E-5 THEN REL[I]=0 END IF
IF ABS(IMG[I])<1E-5 THEN IMG[I]=0 END IF
PRINT(I;"";)
WRITE("##.###### ##.######";REL[I];IMG[I])
END FOR
END PROGRAM```
Output:
```FFT: BIT REVERSAL
STAGE: 1
STAGE: 2
STAGE: 3
NUM REAL     IMAG
0  4.000000  0.000000
1  1.000000 -2.414214
2  0.000000  0.000000
3  1.000000 -0.414214
4  0.000000  0.000000
5  1.000000  0.414214
6  0.000000  0.000000
7  1.000000  2.414214
```

## Factor

```IN: USE math.transforms.fft
IN: { 1 1 1 1 0 0 0 0 } fft .
{
C{ 4.0 0.0 }
C{ 1.0 -2.414213562373095 }
C{ 0.0 0.0 }
C{ 1.0 -0.4142135623730949 }
C{ 0.0 0.0 }
C{ 0.9999999999999999 0.4142135623730949 }
C{ 0.0 0.0 }
C{ 0.9999999999999997 2.414213562373095 }
}
```

## Fortran

```module fft_mod
implicit none
integer,       parameter :: dp=selected_real_kind(15,300)
real(kind=dp), parameter :: pi=3.141592653589793238460_dp
contains

! In place Cooley-Tukey FFT
recursive subroutine fft(x)
complex(kind=dp), dimension(:), intent(inout)  :: x
complex(kind=dp)                               :: t
integer                                        :: N
integer                                        :: i
complex(kind=dp), dimension(:), allocatable    :: even, odd

N=size(x)

if(N .le. 1) return

allocate(odd((N+1)/2))
allocate(even(N/2))

! divide
odd =x(1:N:2)
even=x(2:N:2)

! conquer
call fft(odd)
call fft(even)

! combine
do i=1,N/2
t=exp(cmplx(0.0_dp,-2.0_dp*pi*real(i-1,dp)/real(N,dp),kind=dp))*even(i)
x(i)     = odd(i) + t
x(i+N/2) = odd(i) - t
end do

deallocate(odd)
deallocate(even)

end subroutine fft

end module fft_mod

program test
use fft_mod
implicit none
complex(kind=dp), dimension(8) :: data = (/1.0, 1.0, 1.0, 1.0, 0.0,

0.0, 0.0, 0.0/)
integer :: i

call fft(data)

do i=1,8
write(*,'("(", F20.15, ",", F20.15, "i )")') data(i)
end do

end program test
```
Output:
```(   4.000000000000000,   0.000000000000000i )
(   1.000000000000000,  -2.414213562373095i )
(   0.000000000000000,   0.000000000000000i )
(   1.000000000000000,  -0.414213562373095i )
(   0.000000000000000,   0.000000000000000i )
(   1.000000000000000,   0.414213562373095i )
(   0.000000000000000,   0.000000000000000i )
(   1.000000000000000,   2.414213562373095i )```

## FreeBASIC

```'Graphic fast Fourier transform demo,
'press any key for the next image.
'131072 samples: the FFT is fast indeed.

'screen resolution
const dW = 800, dH = 600
'--------------------------------------
type samples
declare constructor (byval p as integer)

'sw = 0 forward transform
'sw = 1 reverse transform
declare sub FFT (byval sw as integer)

'draw mythical birds
declare sub oiseau ()

'plot frequency and amplitude
declare sub famp ()

'plot transformed samples
declare sub bird ()

as double x(any), y(any)
as integer fl, m, n, n2
end type

constructor samples (byval p as integer)
m = p
'number of points
n = 1 shl p
n2 = n shr 1
'real and complex values
redim x(n - 1), y(n - 1)
end constructor

'--------------------------------------
'[ http://paulbourke.net/miscellaneous/dft/ ]

sub samples.FFT (byval sw as integer)
dim as double c1, c2, t1, t2, u1, u2, v
dim as integer i, j = 0, k, L, l1, l2

'bit reversal sorting
for i = 0 to n - 2
if i < j then
swap x(i), x(j)
swap y(i), y(j)
end if

k = n2
while k <= j
j -= k: k shr= 1
wend
j += k
next i

'initial cosine & sine
c1 = -1.0
c2 = 0.0
'loop for each stage
l2 = 1
for L = 1 to m
l1 = l2: l2 shl= 1

'initial vertex
u1 = 1.0
u2 = 0.0
'loop for each sub DFT
for k = 1 to l1
'butterfly dance
for i = k - 1 to n - 1 step l2
j = i + l1
t1 = u1 * x(j) - u2 * y(j)
t2 = u1 * y(j) + u2 * x(j)
x(j) = x(i) - t1
y(j) = y(i) - t2
x(i) += t1
y(i) += t2
next i

'next polygon vertex
v =  u1 * c1 - u2 * c2
u2 = u1 * c2 + u2 * c1
u1 = v
next k

'half-angle sine
c2 = sqr((1.0 - c1) * .5)
if sw = 0 then c2 = -c2
'half-angle cosine
c1 = sqr((1.0 + c1) * .5)
next L

'scaling for reverse transform
if sw then
for i = 0 to n - 1
x(i) /= n
y(i) /= n
next i
end if
end sub

'--------------------------------------
'Gumowski-Mira attractors "Oiseaux mythiques"
'[ http://www.atomosyd.net/spip.php?article98 ]

sub samples.oiseau
dim as double a, b, c, t, u, v, w
dim as integer dx, y0, dy, i, k

'bounded non-linearity
if fl then
a = -0.801
dx = 20: y0 =-1: dy = 12
else
a = -0.492
dx = 17: y0 =-3: dy = 14
end if
window (-dx, y0-dy)-(dx, y0+dy)

'dissipative coefficient
b = 0.967
c = 2 - 2 * a

u = 1: v = 0.517: w = 1

for i = 0 to n - 1
t = u
u = b * v + w
w = a * u + c * u * u / (1 + u * u)
v = w - t

'remove bias
t = u - 1.830
x(i) = t
y(i) = v
k = 5 + point(t, v)
pset (t, v), 1 + k mod 14
next i
sleep
end sub

'--------------------------------------
sub samples.famp
dim as double a, s, f = n / dW
dim as integer i, k
window

k = iif(fl, dW / 5, dW / 3)
for i = k to dW step k
line (i, 0)-(i, dH), 1
next i

a = 0
k = 0: s = f - 1
for i = 0 to n - 1
a += x(i) * x(i) + y(i) * y(i)

if i > s then
a = log(1 + a / f) * 0.045
if k then
line -(k, (1 - a) * dH), 15
else
pset(0, (1 - a) * dH), 15
end if

a = 0
k += 1: s += f
end if
next i
sleep
end sub

sub samples.bird
dim as integer dx, y0, dy, i, k

if fl then
dx = 20: y0 =-1: dy = 12
else
dx = 17: y0 =-3: dy = 14
end if
window (-dx, y0-dy)-(dx, y0+dy)

for i = 0 to n - 1
k = 2 + point(x(i), y(i))
pset (x(i), y(i)), 1 + k mod 14
next i
sleep
end sub

'main
'--------------------------------------
dim as integer i, p = 17
'n = 2 ^ p
dim as samples z = p

screenres dW, dH, 4, 1

for i = 0 to 1
z.fl = i
z.oiseau

'forward
z.FFT(0)

'amplitude plot with peaks at the
'± winding numbers of the orbits.
z.famp

'reverse
z.FFT(1)

z.bird
cls
next i
end
```

(Images only)

## Frink

Frink has a built-in FFT function that can produce results based on different conventions. The following is not the default convention, but matches many of the other results in this page.

```a = FFT[[1,1,1,1,0,0,0,0], 1, -1]
println[joinln[format[a, 1, 5]]]```
Output:
```4.00000
( 1.00000 - 2.41421 i )
0.00000
( 1.00000 - 0.41421 i )
0.00000
( 1.00000 + 0.41421 i )
0.00000
( 1.00000 + 2.41421 i )
```

## GAP

```# Here an implementation with no optimization (O(n^2)).
# In GAP, E(n) = exp(2*i*pi/n), a primitive root of the unity.

Fourier := function(a)
local n, z;
n := Size(a);
z := E(n);
return List([0 .. n - 1], k -> Sum([0 .. n - 1], j -> a[j + 1]*z^(-k*j)));
end;

InverseFourier := function(a)
local n, z;
n := Size(a);
z := E(n);
return List([0 .. n - 1], k -> Sum([0 .. n - 1], j -> a[j + 1]*z^(k*j)))/n;
end;

Fourier([1, 1, 1, 1, 0, 0, 0, 0]);
# [ 4, 1-E(8)-E(8)^2-E(8)^3, 0, 1-E(8)+E(8)^2-E(8)^3,
#   0, 1+E(8)-E(8)^2+E(8)^3, 0, 1+E(8)+E(8)^2+E(8)^3 ]

InverseFourier(last);
# [ 1, 1, 1, 1, 0, 0, 0, 0 ]
```

## Go

```package main

import (
"fmt"
"math"
"math/cmplx"
)

func ditfft2(x []float64, y []complex128, n, s int) {
if n == 1 {
y[0] = complex(x[0], 0)
return
}
ditfft2(x, y, n/2, 2*s)
ditfft2(x[s:], y[n/2:], n/2, 2*s)
for k := 0; k < n/2; k++ {
tf := cmplx.Rect(1, -2*math.Pi*float64(k)/float64(n)) * y[k+n/2]
y[k], y[k+n/2] = y[k]+tf, y[k]-tf
}
}

func main() {
x := []float64{1, 1, 1, 1, 0, 0, 0, 0}
y := make([]complex128, len(x))
ditfft2(x, y, len(x), 1)
for _, c := range y {
fmt.Printf("%8.4f\n", c)
}
}
```
Output:
```(  4.0000 +0.0000i)
(  1.0000 -2.4142i)
(  0.0000 +0.0000i)
(  1.0000 -0.4142i)
(  0.0000 +0.0000i)
(  1.0000 +0.4142i)
(  0.0000 +0.0000i)
(  1.0000 +2.4142i)
```

## Golfscript

```#Cooley-Tukey

{.,.({[\.2%fft\(;2%fft@-1?-1\?-2?:w;.,,{w\?}%[\]zip{{*}*}%]zip.{{+}*}%\{{-}*}%+}{;}if}:fft;

[1 1 1 1 0 0 0 0]fft n*```
Output:
```4+0i
1.0000000000000002-2.414213562373095i
0.0+0.0i
0.9999999999999996-0.4142135623730949i
0+0i
1.0000000000000002+0.41421356237309515i
0.0+0.0i
1.0+2.414213562373095i
```

```import Data.Complex

-- Cooley-Tukey
fft [] = []
fft [x] = [x]
fft xs = zipWith (+) ys ts ++ zipWith (-) ys ts
where n = length xs
ys = fft evens
zs = fft odds
(evens, odds) = split xs
split [] = ([], [])
split [x] = ([x], [])
split (x:y:xs) = (x:xt, y:yt) where (xt, yt) = split xs
ts = zipWith (\z k -> exp' k n * z) zs [0..]
exp' k n = cis \$ -2 * pi * (fromIntegral k) / (fromIntegral n)

main = mapM_ print \$ fft [1,1,1,1,0,0,0,0]
```
Output:
```4.0 :+ 0.0
1.0 :+ (-2.414213562373095)
0.0 :+ 0.0
1.0 :+ (-0.4142135623730949)
0.0 :+ 0.0
0.9999999999999999 :+ 0.4142135623730949
0.0 :+ 0.0
0.9999999999999997 :+ 2.414213562373095
```

## Idris

```module Main

import Data.Complex

concatPair : List (a, a) -> List (a)
concatPair xs with (unzip xs)
| (xs1, xs2) = xs1 ++ xs2

fft' : List (Complex Double) -> Nat -> Nat -> List (Complex Double)
fft' (x::xs) (S Z) _ = [x]
fft' xs n s = concatPair \$ map (\(x1,x2,k) =>
let eTerm = ((cis (-2 * pi * ((cast k) - 1) / (cast n))) * x2) in
(x1 + eTerm, x1 - eTerm)) \$ zip3 left right [1..n `div` 2]

where
left : List (Complex Double)
right : List (Complex Double)
left  = fft' (xs) (n `div` 2) (2 * s)
right = fft' (drop s xs) (n `div` 2) (2 * s)

-- Recursive Cooley-Tukey with radix-2 DIT case
-- assumes no of points provided are a power of 2
fft : List (Complex Double) -> List (Complex Double)
fft [] = []
fft xs = fft' xs (length xs) 1

main : IO()
main = traverse_ printLn \$ fft [1,1,1,1,0,0,0,0]
```
Output:
```4 :+ 0
1 :+ -2.414213562373095
0 :+ 0
1 :+ -0.4142135623730949
0 :+ 0
0.9999999999999999 :+ 0.4142135623730949
0 :+ 0
0.9999999999999997 :+ 2.414213562373095
```

## J

Based on j:Essays/FFT, with some simplifications -- sacrificing accuracy, optimizations and convenience which are not relevant to the task requirements, for clarity:

```cube  =: (\$~ q:@#) :. ,
rou   =: ^@j.@o.@(% #)@i.@-:  NB. roots of unity
floop =: 4 : 'for_r. i.#\$x do. (y=.{."1 y) ] x=.(+/x) ,&,:"r (-/x)*y end.'
fft   =: ] floop&.cube rou@#
```

Example (first row of result is sine, second row of result is fft of the first row, (**+)&.+. cleans an irrelevant least significant bit of precision from the result so that it displays nicely):

```   (**+)&.+. (,: fft) 1 o. 2p1*3r16 * i.16
0 0.92388 0.707107 0.382683 1 0.382683 0.707107 0.92388 0 0.92388 0.707107 0.382683 1 0.382683 0.707107 0.92388
0       0        0      0j8 0        0        0       0 0       0        0        0 0      0j8        0       0
```

Here is a representation of an example which appears in some of the other implementations, here:

```   Re=: {.@+.@fft
Im=: {:@+.@fft
M=: 4#1 0
M
1 1 1 1 0 0 0 0
Re M
4 1 0 1 0 1 0 1
Im M
0 2.41421 0 0.414214 0 _0.414214 0 _2.41421
```

Note that Re and Im are not functions of 1 and 0 but are functions of the complete sequence.

Also note that J uses a different character for negative sign than for subtraction, to eliminate ambiguity (is this a single list of numbers or are lists being subtracted?).

## Java

Translation of: C sharp
```import static java.lang.Math.*;

public class FastFourierTransform {

public static int bitReverse(int n, int bits) {
int reversedN = n;
int count = bits - 1;

n >>= 1;
while (n > 0) {
reversedN = (reversedN << 1) | (n & 1);
count--;
n >>= 1;
}

return ((reversedN << count) & ((1 << bits) - 1));
}

static void fft(Complex[] buffer) {

int bits = (int) (log(buffer.length) / log(2));
for (int j = 1; j < buffer.length / 2; j++) {

int swapPos = bitReverse(j, bits);
Complex temp = buffer[j];
buffer[j] = buffer[swapPos];
buffer[swapPos] = temp;
}

for (int N = 2; N <= buffer.length; N <<= 1) {
for (int i = 0; i < buffer.length; i += N) {
for (int k = 0; k < N / 2; k++) {

int evenIndex = i + k;
int oddIndex = i + k + (N / 2);
Complex even = buffer[evenIndex];
Complex odd = buffer[oddIndex];

double term = (-2 * PI * k) / (double) N;
Complex exp = (new Complex(cos(term), sin(term)).mult(odd));

buffer[oddIndex] = even.sub(exp);
}
}
}
}

public static void main(String[] args) {
double[] input = {1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0};

Complex[] cinput = new Complex[input.length];
for (int i = 0; i < input.length; i++)
cinput[i] = new Complex(input[i], 0.0);

fft(cinput);

System.out.println("Results:");
for (Complex c : cinput) {
System.out.println(c);
}
}
}

class Complex {
public final double re;
public final double im;

public Complex() {
this(0, 0);
}

public Complex(double r, double i) {
re = r;
im = i;
}

return new Complex(this.re + b.re, this.im + b.im);
}

public Complex sub(Complex b) {
return new Complex(this.re - b.re, this.im - b.im);
}

public Complex mult(Complex b) {
return new Complex(this.re * b.re - this.im * b.im,
this.re * b.im + this.im * b.re);
}

@Override
public String toString() {
return String.format("(%f,%f)", re, im);
}
}
```
```Results:
(4,000000 + 0,000000 i)
(1,000000 + -2,414214 i)
(0,000000 + 0,000000 i)
(1,000000 + -0,414214 i)
(0,000000 + 0,000000 i)
(1,000000 + 0,414214 i)
(0,000000 + 0,000000 i)
(1,000000 + 2,414214 i)```

## JavaScript

Complex fourier transform & it's inverse reimplemented from the C++ & Python variants on this page.

```/*
complex fast fourier transform and inverse from
http://rosettacode.org/wiki/Fast_Fourier_transform#C.2B.2B
*/
function icfft(amplitudes)
{
var N = amplitudes.length;
var iN = 1 / N;

//conjugate if imaginary part is not 0
for(var i = 0 ; i < N; ++i)
if(amplitudes[i] instanceof Complex)
amplitudes[i].im = -amplitudes[i].im;

//apply fourier transform
amplitudes = cfft(amplitudes)

for(var i = 0 ; i < N; ++i)
{
//conjugate again
amplitudes[i].im = -amplitudes[i].im;
//scale
amplitudes[i].re *= iN;
amplitudes[i].im *= iN;
}
return amplitudes;
}

function cfft(amplitudes)
{
var N = amplitudes.length;
if( N <= 1 )
return amplitudes;

var hN = N / 2;
var even = [];
var odd = [];
even.length = hN;
odd.length = hN;
for(var i = 0; i < hN; ++i)
{
even[i] = amplitudes[i*2];
odd[i] = amplitudes[i*2+1];
}
even = cfft(even);
odd = cfft(odd);

var a = -2*Math.PI;
for(var k = 0; k < hN; ++k)
{
if(!(even[k] instanceof Complex))
even[k] = new Complex(even[k], 0);
if(!(odd[k] instanceof Complex))
odd[k] = new Complex(odd[k], 0);
var p = k/N;
var t = new Complex(0, a * p);
t.cexp(t).mul(odd[k], t);
amplitudes[k + hN] = even[k].sub(t, even[k]);
}
return amplitudes;
}

//test code
//console.log( cfft([1,1,1,1,0,0,0,0]) );
//console.log( icfft(cfft([1,1,1,1,0,0,0,0])) );
```

Very very basic Complex number that provides only the components required by the code above.

```/*
basic complex number arithmetic from
http://rosettacode.org/wiki/Fast_Fourier_transform#Scala
*/
function Complex(re, im)
{
this.re = re;
this.im = im || 0.0;
}
{
dst.re = this.re + other.re;
dst.im = this.im + other.im;
return dst;
}
Complex.prototype.sub = function(other, dst)
{
dst.re = this.re - other.re;
dst.im = this.im - other.im;
return dst;
}
Complex.prototype.mul = function(other, dst)
{
//cache re in case dst === this
var r = this.re * other.re - this.im * other.im;
dst.im = this.re * other.im + this.im * other.re;
dst.re = r;
return dst;
}
Complex.prototype.cexp = function(dst)
{
var er = Math.exp(this.re);
dst.re = er * Math.cos(this.im);
dst.im = er * Math.sin(this.im);
return dst;
}
Complex.prototype.log = function()
{
/*
although 'It's just a matter of separating out the real and imaginary parts of jw.' is not a helpful quote
the actual formula I found here and the rest was just fiddling / testing and comparing with correct results.
http://cboard.cprogramming.com/c-programming/89116-how-implement-complex-exponential-functions-c.html#post637921
*/
if( !this.re )
console.log(this.im.toString()+'j');
else if( this.im < 0 )
console.log(this.re.toString()+this.im.toString()+'j');
else
console.log(this.re.toString()+'+'+this.im.toString()+'j');
}
```

## jq

Currently jq has no support for complex numbers, so the following implementation uses [x,y] to represent the complex number x+iy.

#### Complex number arithmetic

```# multiplication of real or complex numbers
def cmult(x; y):
if (x|type) == "number" then
if  (y|type) == "number" then [ x*y, 0 ]
else [x * y[0], x * y[1]]
end
elif (y|type) == "number" then cmult(y;x)
else [ x[0] * y[0] - x[1] * y[1],  x[0] * y[1] + x[1] * y[0]]
end;

def cplus(x; y):
if (x|type) == "number" then
if  (y|type) == "number" then [ x+y, 0 ]
else [ x + y[0], y[1]]
end
elif (y|type) == "number" then cplus(y;x)
else [ x[0] + y[0], x[1] + y[1] ]
end;

def cminus(x; y): cplus(x; cmult(-1; y));

# e(ix) = cos(x) + i sin(x)
def expi(x): [ (x|cos), (x|sin) ];```

#### FFT

```def fft:
length as \$N
| if \$N <= 1 then .
else   ( [ .[ range(0; \$N; 2) ] ] | fft) as \$even
| ( [ .[ range(1; \$N; 2) ] ] | fft) as \$odd
| (1|atan * 4) as \$pi
| [ range(0; \$N/2) | cplus(\$even[.];  cmult( expi(-2*\$pi*./\$N); \$odd[.] )) ] +
[ range(0; \$N/2) | cminus(\$even[.]; cmult( expi(-2*\$pi*./\$N); \$odd[.] )) ]
end;```

Example:

`[1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0] | fft`
Output:
```[[4,-0],[1,-2.414213562373095],
[0,0],[1,-0.4142135623730949],
[0,0],[0.9999999999999999,0.4142135623730949],
[0,0],[0.9999999999999997,2.414213562373095]]
```

## Julia

```using FFTW # or using DSP

fft([1,1,1,1,0,0,0,0])
```
Output:
```8-element Array{Complex{Float64},1}:
4.0+0.0im
1.0-2.41421im
0.0+0.0im
1.0-0.414214im
0.0+0.0im
1.0+0.414214im
0.0+0.0im
1.0+2.41421im
```

An implementation of the radix-2 algorithm, which works for any vector for length that is a power of 2:

```function fft(a)
y1 = Any[]; y2 = Any[]
n = length(a)
if n ==1 return a end
wn(n) = exp(-2*π*im/n)
y_even = fft(a[1:2:end])
y_odd = fft(a[2:2:end])
w = 1
for k in 1:Int(n/2)
push!(y1, y_even[k] + w*y_odd[k])
push!(y2, y_even[k] - w*y_odd[k])
w = w*wn(n)
end
return vcat(y1,y2)
end
```

## Klong

```fft::{ff2::{[n e o p t k];n::#x;
f::{p::2:#x;e::ff2(*'p);o::ff2({x@1}'p);k::-1;
t::{k::k+1;cmul(cexp(cdiv(cmul([0 -2];(k*pi),0);n,0));x)}'o;
:[n<2;x;f(x)]};
n::#x;k::{(2^x)<n}{1+x}:~1;n#ff2({x,0}'x,&(2^k)-n)}
```

Example (rounding to 4 decimal digits):

```        all(rndn(;4);fft([1 1 1 1 0 0 0 0]))
```
Output:
```[[4.0 0.0]
[1.0 -2.4142]
[0.0 0.0]
[1.0 -0.4142]
[0.0 0.0]
[1.0 0.4142]
[0.0 0.0]
[1.0 2.4142]]```

## Kotlin

From Scala.

```import java.lang.Math.*

class Complex(val re: Double, val im: Double) {
operator infix fun plus(x: Complex) = Complex(re + x.re, im + x.im)
operator infix fun minus(x: Complex) = Complex(re - x.re, im - x.im)
operator infix fun times(x: Double) = Complex(re * x, im * x)
operator infix fun times(x: Complex) = Complex(re * x.re - im * x.im, re * x.im + im * x.re)
operator infix fun div(x: Double) = Complex(re / x, im / x)
val exp: Complex by lazy { Complex(cos(im), sin(im)) * (cosh(re) + sinh(re)) }

override fun toString() = when {
b == "0.000" -> a
a == "0.000" -> b + 'i'
im > 0 -> a + " + " + b + 'i'
else -> a + " - " + b + 'i'
}

private val a = "%1.3f".format(re)
private val b = "%1.3f".format(abs(im))
}
```
```object FFT {
fun fft(a: Array<Complex>) = _fft(a, Complex(0.0, 2.0), 1.0)
fun rfft(a: Array<Complex>) = _fft(a, Complex(0.0, -2.0), 2.0)

private fun _fft(a: Array<Complex>, direction: Complex, scalar: Double): Array<Complex> =
if (a.size == 1)
a
else {
val n = a.size
require(n % 2 == 0, { "The Cooley-Tukey FFT algorithm only works when the length of the input is even." })

var (evens, odds) = Pair(emptyArray<Complex>(), emptyArray<Complex>())
for (i in a.indices)
if (i % 2 == 0) evens += a[i]
else odds += a[i]
evens = _fft(evens, direction, scalar)
odds = _fft(odds, direction, scalar)

val pairs = (0 until n / 2).map {
val offset = (direction * (java.lang.Math.PI * it / n)).exp * odds[it] / scalar
val base = evens[it] / scalar
Pair(base + offset, base - offset)
}
var (left, right) = Pair(emptyArray<Complex>(), emptyArray<Complex>())
for ((l, r) in pairs) { left += l; right += r }
left + right
}
}
```
```fun Array<*>.println() = println(joinToString(prefix = "[", postfix = "]"))

fun main(args: Array<String>) {
val data = arrayOf(Complex(1.0, 0.0), Complex(1.0, 0.0), Complex(1.0, 0.0), Complex(1.0, 0.0),
Complex(0.0, 0.0), Complex(0.0, 2.0), Complex(0.0, 0.0), Complex(0.0, 0.0))

val a = FFT.fft(data)
a.println()
FFT.rfft(a).println()
}
```
Output:
```[4.000 + 2.000i, 2.414 + 1.000i, -2.000, 2.414 + 1.828i, 2.000i, -0.414 + 1.000i, 2.000, -0.414 - 3.828i]
[1.000, 1.000, 1.000, 1.000, 0.000, 2.000i, 0.000, 0.000]```

## Lambdatalk

```1) the function fft

{def fft
{lambda {:s :x}
{if {= {list.length :x} 1}
then :x
else {let { {:s :s}
{:ev {fft :s {evens :x}} }
{:od {fft :s {odds  :x}} } }
{let { {:ev :ev} {:t {rotate :s :od 0 {list.length :od}}} }
{list.map Csub :ev :t}} }}}}}

{def rotate
{lambda {:s :f :k :N}
{if {list.null? :f}
then nil
else {cons {Cmul {car :f} {Cexp {Cnew 0 {/ {* :s {PI} :k} :N}}}}
{rotate :s {cdr :f} {+ :k 1} :N}}}}}

2) functions for lists

We add to the existing {lambda talk}'s list primitives a small set of functions required by the function fft.

{def evens
{lambda {:l}
{if {list.null? :l}
then nil
else {cons {car :l} {evens {cdr {cdr :l}}}}}}}

{def odds
{lambda {:l}
{if {list.null? {cdr :l}}
then nil
else {cons {car {cdr :l}} {odds {cdr {cdr :l}}}}}}}

{def list.map
{def list.map.r
{lambda {:f :a :b :c}
{if {list.null? :a}
then :c
else {list.map.r :f {cdr :a} {cdr :b}
{cons {:f {car :a} {car :b}} :c}} }}}
{lambda {:f :a :b}
{list.map.r :f {list.reverse :a} {list.reverse :b} nil}}}

{def list.append
{def list.append.r
{lambda {:a :b}
{if {list.null? :b}
then :a
else {list.append.r {cons {car :b} :a} {cdr :b}}}}}
{lambda {:a :b}
{list.append.r :b {list.reverse :a}} }}

3) functions for Cnumbers

{lambda talk} has no primitive functions working on complex numbers. We add the minimal set required by the function fft.

{def Cnew
{lambda {:x :y}
{cons :x :y} }}

{def Cnorm
{lambda {:c}
{sqrt {+ {* {car :c} {car :c}}
{* {cdr :c} {cdr :c}}}} }}

{lambda {:x :y}
{cons {+ {car :x} {car :y}}
{+ {cdr :x} {cdr :y}}} }}

{def Csub
{lambda {:x :y}
{cons {- {car :x} {car :y}}
{- {cdr :x} {cdr :y}}} }}

{def Cmul
{lambda {:x :y}
{cons {- {* {car :x} {car :y}} {* {cdr :x} {cdr :y}}}
{+ {* {car :x} {cdr :y}} {* {cdr :x} {car :y}}}} }}

{def Cexp
{lambda {:x}
{cons {* {exp {car :x}} {cos {cdr :x}}}
{* {exp {car :x}} {sin {cdr :x}}}} }}

{def Clist
{lambda {:s}
{list.new {map {lambda {:i} {cons :i 0}} :s}}}}

4) testing

Applying the fft function on such a sample (1 1 1 1 0 0 0 0) where numbers have been promoted as complex

{list.disp {fft -1 {Clist 1 1 1 1 0 0 0 0}}} ->

(4 0)
(1 -2.414213562373095)
(0 0)
(1 -0.4142135623730949)
(0 0)
(0.9999999999999999 0.4142135623730949)
(0 0)
(0.9999999999999997 2.414213562373095)

A more usefull example can be seen in http://lambdaway.free.fr/lambdaspeech/?view=zorg
```

## Liberty BASIC

```    P =8
S  =int( log( P) /log( 2) +0.9999)

Pi =3.14159265
R1 =2^S

R =R1 -1
R2 =div( R1,  2)
R4 =div( R1,  4)
R3 =R4 +R2

Dim Re( R1), Im( R1), Co( R3)

for N =0 to P -1
next N

data    1, 0,      1, 0,      1, 0,      1, 0,      0, 0,     0, 0,      0, 0,       0, 0

S2 =div( S, 2)
S1 =S -S2
P1 =2^S1
P2 =2^S2

dim V( P1 -1)
V( 0) =0
DV =1
DP =P1

for J =1 to S1
HA =div( DP, 2)
PT =P1 -HA
for I =HA to PT step DP
V( I) =V( I -HA) +DV
next I
DV =DV +DV
DP =HA
next J

K =2 *Pi /R1

for X =0 to R4
COX =cos( K *X)
Co( X) =COX
Co( R2 -X) =0 -COX
Co( R2 +X) =0 -COX
next X

print "FFT: bit reversal"

for I =0 to P1 -1
IP =I *P2
for J =0 to P2 -1
H =IP +J
G =V( J) *P2 +V( I)
if G >H then temp =Re( G): Re( G) =Re( H): Re( H) =temp
if G >H then temp =Im( G): Im( G) =Im( H): Im( H) =temp
next J
next I

T =1

for stage =0 to S -1
print "  Stage:- "; stage
D =div( R2, T)
for Z =0 to T -1
L   =D *Z
LS  =L +R4
for I =0 to D -1
A      =2 *I *T +Z
B      =A +T
F1     =Re( A)
F2     =Im( A)
P1     =Co( L)  *Re( B)
P2     =Co( LS) *Im( B)
P3     =Co( LS) *Re( B)
P4     =Co( L)  *Im( B)
Re( A) =F1 +P1 -P2
Im( A) =F2 +P3 +P4
Re( B) =F1 -P1 +P2
Im( B) =F2 -P3 -P4
next I
next Z
T =T +T
next stage

print "   M          Re( M)       Im( M)"

for M =0 to R
if abs( Re( M)) <10^-5 then Re( M) =0
if abs( Im( M)) <10^-5 then Im( M) =0
print "   "; M, Re( M), Im( M)
next M

end

wait

function div( a, b)
div =int( a /b)
end function

end```
```   M          Re( M)       Im( M)
0          4             0
1          1.0           -2.41421356
2          0             0
3          1.0           -0.41421356
4          0             0
5          1.0           0.41421356
6          0             0
7          1.0           2.41421356
```

## Lua

```-- operations on complex number
complex = {__mt={} }

function complex.new (r, i)
local new={r=r, i=i or 0}
setmetatable(new,complex.__mt)
return new
end

return complex.new(c1.r + c2.r, c1.i + c2.i)
end

function complex.__mt.__sub (c1, c2)
return complex.new(c1.r - c2.r, c1.i - c2.i)
end

function complex.__mt.__mul (c1, c2)
return complex.new(c1.r*c2.r - c1.i*c2.i,
c1.r*c2.i + c1.i*c2.r)
end

function complex.expi (i)
return complex.new(math.cos(i),math.sin(i))
end

function complex.__mt.__tostring(c)
return "("..c.r..","..c.i..")"
end

-- Cooley–Tukey FFT (in-place, divide-and-conquer)
-- Higher memory requirements and redundancy although more intuitive
function fft(vect)
local n=#vect
if n<=1 then return vect end
-- divide
local odd,even={},{}
for i=1,n,2 do
odd[#odd+1]=vect[i]
even[#even+1]=vect[i+1]
end
-- conquer
fft(even);
fft(odd);
-- combine
for k=1,n/2 do
local t=even[k] * complex.expi(-2*math.pi*(k-1)/n)
vect[k] = odd[k] + t;
vect[k+n/2] = odd[k] - t;
end
return vect
end

function toComplex(vectr)
vect={}
for i,r in ipairs(vectr) do
vect[i]=complex.new(r)
end
return vect
end

-- test
data = toComplex{1, 1, 1, 1, 0, 0, 0, 0};

-- this works for old lua versions & luaJIT (depends on version!)
-- print("orig:", unpack(data))
-- print("fft:", unpack(fft(data)))

-- Beginning with Lua 5.2 you have to write
print("orig:", table.unpack(data))
print("fft:", table.unpack(fft(data)))
```

## Maple

Maple has a built-in package DiscreteTransforms, and FourierTransform and InverseFourierTransform are in the commands available from that package. The FourierTransform command offers an FFT method by default.

```with( DiscreteTransforms ):

FourierTransform( <1,1,1,1,0,0,0,0>, normalization=none );```
```                         [       4. + 0. I        ]
[                        ]
[1. - 2.41421356237309 I ]
[                        ]
[       0. + 0. I        ]
[                        ]
[1. - 0.414213562373095 I]
[                        ]
[       0. + 0. I        ]
[                        ]
[1. + 0.414213562373095 I]
[                        ]
[       0. + 0. I        ]
[                        ]
[1. + 2.41421356237309 I ]
```

Optionally, the FFT may be performed inplace on a Vector of hardware double-precision complex floats.

```v := Vector( [1,1,1,1,0,0,0,0], datatype=complex[8] ):

FourierTransform( v, normalization=none, inplace ):

v;```
```                         [       4. + 0. I        ]
[                        ]
[1. - 2.41421356237309 I ]
[                        ]
[       0. + 0. I        ]
[                        ]
[1. - 0.414213562373095 I]
[                        ]
[       0. + 0. I        ]
[                        ]
[1. + 0.414213562373095 I]
[                        ]
[       0. + 0. I        ]
[                        ]
[1. + 2.41421356237309 I ]
```
```InverseFourierTransform( v, normalization=full, inplace ):

v;```
```                       [          1. + 0. I          ]
[                             ]
[          1. + 0. I          ]
[                             ]
[          1. + 0. I          ]
[                             ]
[          1. + 0. I          ]
[                             ]
[          0. + 0. I          ]
[                             ]
[          0. + 0. I          ]
[                             ]
[                   -17       ]
[5.55111512312578 10    + 0. I]
[                             ]
[          0. + 0. I          ]
```

## Mathematica / Wolfram Language

Mathematica has a built-in FFT function which uses a proprietary algorithm developed at Wolfram Research. It also has an option to tune the algorithm for specific applications. The options shown below, while not default, produce output that is consistent with most other FFT routines.

```Fourier[{1,1,1,1,0,0,0,0}, FourierParameters->{1,-1}]
```
Output:
`{4. + 0. I, 1. - 2.4142136 I, 0. + 0. I, 1. - 0.41421356 I, 0. + 0. I, 1. + 0.41421356 I, 0. + 0. I, 1. + 2.4142136 I}`

Here is a user-space definition for good measure.

```fft[{x_}] := {N@x}
fft[l__] :=
Join[#, #] &@fft@l[[1 ;; ;; 2]] +
Exp[(-2 \[Pi] I)/Length@l (Range@Length@l - 1)] (Join[#, #] &@
fft[l[[2 ;; ;; 2]]])

fft[{1, 1, 1, 1, 0, 0, 0, 0}] // Column
```
Output:
```4.
1. -2.41421 I
0. +0. I
1. -0.414214 I
0.
1. +0.414214 I
0. +0. I
1. +2.41421 I
```

## MATLAB / Octave

Matlab/Octave have a builtin FFT function.

``` fft([1,1,1,1,0,0,0,0]')
```
Output:
```ans =

4.00000 + 0.00000i
1.00000 - 2.41421i
0.00000 + 0.00000i
1.00000 - 0.41421i
0.00000 + 0.00000i
1.00000 + 0.41421i
0.00000 - 0.00000i
1.00000 + 2.41421i```

## Maxima

```load(fft)\$
fft([1, 2, 3, 4]);
[2.5, -0.5 * %i - 0.5, -0.5, 0.5 * %i - 0.5]
```

## Nim

Translation of: Python
```import math, complex, strutils

# Works with floats and complex numbers as input
proc fft[T: float | Complex[float]](x: openarray[T]): seq[Complex[float]] =
let n = x.len
if n == 0: return

result.newSeq(n)

if n == 1:
result[0] = (when T is float: complex(x[0]) else: x[0])
return

var evens, odds = newSeq[T]()
for i, v in x:
if i mod 2 == 0: evens.add v
var (even, odd) = (fft(evens), fft(odds))

let halfn = n div 2

for k in 0 ..< halfn:
let a = exp(complex(0.0, -2 * Pi * float(k) / float(n))) * odd[k]
result[k] = even[k] + a
result[k + halfn] = even[k] - a

for i in fft(@[1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0]):
echo formatFloat(abs(i), ffDecimal, 3)
```
Output:
```4.000
2.613
0.000
1.082
0.000
1.082
0.000
2.613```

## OCaml

This is a simple implementation of the Cooley-Tukey pseudo-code

```open Complex

let fac k n =
let m2pi = -4.0 *. acos 0.0 in
polar 1.0 (m2pi*.(float k)/.(float n))

let merge l r n =
let f (k,t) x = (succ k, (mul (fac k n) x) :: t) in
let z = List.rev (snd (List.fold_left f (0,[]) r)) in
(List.map2 add l z) @ (List.map2 sub l z)

let fft lst =
let rec ditfft2 a n s =
if n = 1 then [List.nth lst a] else
let odd = ditfft2 a (n/2) (2*s) in
let even = ditfft2 (a+s) (n/2) (2*s) in
merge odd even n in
ditfft2 0 (List.length lst) 1;;

let show l =
let pr x = Printf.printf "(%f %f) " x.re x.im in
(List.iter pr l; print_newline ()) in
let indata = [one;one;one;one;zero;zero;zero;zero] in
show indata;
show (fft indata)
```
Output:
```(1.000000 0.000000) (1.000000 0.000000) (1.000000 0.000000) (1.000000 0.000000) (0.000000 0.000000) (0.000000 0.000000) (0.000000 0.000000) (0.000000 0.000000)
(4.000000 0.000000) (1.000000 -2.414214) (0.000000 0.000000) (1.000000 -0.414214) (0.000000 0.000000) (1.000000 0.414214) (0.000000 0.000000) (1.000000 2.414214)
```

## ooRexx

Translation of: PL/I

Output as shown in REXX

```Numeric Digits 16
list='1 1 1 1 0 0 0 0'
n=words(list)
x=.array~new(n)
Do i=1 To n
x[i]=.complex~new(word(list,i),0)
End
Call show 'FFT  in',x
call fft x
Call show 'FFT out',x
Exit

show: Procedure
Use Arg data,x
Say '---data---   num       real-part   imaginary-part'
Say '----------   ---       ---------   --------------'
Do i=1 To x~size
say data right(i,7)'       ' x[i]~string
End
Return

fft: Procedure
Use Arg in
Numeric Digits 16
n=in~size
If n=1 Then Return
odd=.array~new(n/2)
even=.array~new(n/2)
Do j=1 To n By 2; odd[(j+1)/2]=in[j]; End
Do j=2 To n By 2; even[j/2]=in[j]; End
Call fft odd
Call fft even
pi=3.14159265358979323E0
n_2=n/2
Do i=1 To n_2
w=-2*pi*(i-1)/N
t=.complex~new(rxCalcCos(w,,'R'),rxCalcSin(w,,'R'))*even[i]
in[i]=odd[i]+t
in[i+n_2]=odd[i]-t
End
Return

::class complex
::method init
expose r i
use strict arg r, i = 0

-- complex instances are immutable, so these are
::attribute r GET
::attribute i GET

expose r i
Numeric Digits 16
use strict arg other
if other~isa(.complex) then
return self~class~new(r + other~r, i + other~i)
else return self~class~new(r + other, i)

::method subtract
expose r i
Numeric Digits 16
use strict arg other
if other~isa(.complex) then
return self~class~new(r - other~r, i - other~i)
else return self~class~new(r - other, i)

::method "+"
Numeric Digits 16
-- need to check if this is a prefix plus or an addition
if arg() == 0 then
return self  -- we can return this copy since it is immutable
else

::method "-"
Numeric Digits 16
-- need to check if this is a prefix minus or a subtract
if arg() == 0 then
forward message("NEGATIVE")
else
forward message("SUBTRACT")

::method times
expose r i
Numeric Digits 16
use strict arg other
if other~isa(.complex) then
return self~class~new(r * other~r - i * other~i, r * other~i + i * other~r)
else return self~class~new(r * other, i * other)

::method "*"
Numeric Digits 16
forward message("TIMES")

::method string
expose r i
Numeric Digits 12
Select
When i=0 Then
If r=0 Then
Return '0'
Else
Return format(r,1,9)
When i>0 Then
Return format(r,1,9)' +'format(i,1,9)'i'
Otherwise
Return format(r,1,9)' -'format(abs(i),1,9)'i'
End

::method formatnumber private
use arg value
Numeric Digits 16
if value > 0 then return "+" value
else return "-" value~abs

::requires rxMath library
```
Output:
```---data---   num       real-part   imaginary-part
----------   ---       ---------   --------------
FFT  in       1        1.000000000
FFT  in       2        1.000000000
FFT  in       3        1.000000000
FFT  in       4        1.000000000
FFT  in       5        0
FFT  in       6        0
FFT  in       7        0
FFT  in       8        0
---data---   num       real-part   imaginary-part
----------   ---       ---------   --------------
FFT out       1        4.000000000
FFT out       2        1.000000000 -2.414213562i
FFT out       3        0
FFT out       4        1.000000000 -0.414213562i
FFT out       5        0
FFT out       6        1.000000000 +0.414213562i
FFT out       7        0
FFT out       8        1.000000000 +2.414213562i```

## PARI/GP

Naive implementation, using the same testcase as Ada:

```FFT(v)=my(t=-2*Pi*I/#v,tt);vector(#v,k,tt=t*(k-1);sum(n=0,#v-1,v[n+1]*exp(tt*n)));
FFT([1,1,1,1,0,0,0,0])```
Output:
`[4.0000000000000000000000000000000000000, 1.0000000000000000000000000000000000000 - 2.4142135623730950488016887242096980786*I, 0.E-37 + 0.E-38*I, 1.0000000000000000000000000000000000000 - 0.41421356237309504880168872420969807856*I, 0.E-38 + 0.E-37*I, 0.99999999999999999999999999999999999997 + 0.41421356237309504880168872420969807860*I, 4.701977403289150032 E-38 + 0.E-38*I, 0.99999999999999999999999999999999999991 + 2.4142135623730950488016887242096980785*I]`

differently, and even with "graphics"

```install( FFTinit, Lp );
install( FFT, GG );
k = 7; N = 2 ^ k;
CIRC = FFTinit(k);

v = vector( N, i, 3 * sin( 1 * i*2*Pi/N) + sin( 33 *i*2*Pi/N) );
w = FFT(v, CIRC);
\\print("Signal");
\\plot( i = 1, N, v[ floor(i) ] );
print("Spectrum");
plot( i = 1, N / 2 , abs( w[floor(i)] ) * 2 / N );```
Output:
```Spectrum
3 |"'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''|
|:                                                             |
|:                                                             |
|:                                                             |
|:                                                             |
|:                                                             |
|:                                                             |
|:                                                             |
|:                                                             |
|:                                                             |
|:                                                             |
: :                                                            |
: :                                                            |
: :                                                            |
: :                              x                             |
: :                              :                             |
: :                              :                             |
: :                              :                             |
: :                             : :                            |
: :                             : :                            |
: :                             : :                            |
0 _,_______________________________,______________________________
1                                                             64```

## Pascal

### Recursive

Works with: Free Pascal version 3.2.0
```PROGRAM RDFT;

(*)

Free Pascal Compiler version 3.2.0 [2020/06/14] for x86_64
The free and readable alternative at C/C++ speeds
compiles natively to almost any platform, including raspberry PI *
Can run independently from DELPHI / Lazarus

For debian Linux: apt -y install fpc
It contains a text IDE called fp

(*)

USES

crt,
math,
sysutils,
ucomplex;

{\$WARN 6058 off : Call to subroutine "\$1" marked as inline is not inlined}
(*) Use for variants and ucomplex (*)

TYPE

table = array  of complex;

PROCEDURE Split ( T: table ; EVENS: table; ODDS:table ) ;

VAR

k:  integer ;

BEGIN

FOR k := 0 to Length ( T ) - 1 DO

IF Odd ( k ) THEN

ODDS [ k DIV 2 ]    := T [ k ]

ELSE

EVENS [ k DIV 2 ]   := T [ k ]

END;

PROCEDURE WriteCTable ( L: table ) ;

VAR

x   :integer ;

BEGIN

FOR x := 0  to length ( L ) - 1 DO

BEGIN

Write   ( Format ('%3.3g ' , [ L [ x ].re ] ) ) ;

IF ( L [ x ].im >= 0.0 ) THEN Write ( '+' ) ;

WriteLn ( Format ('%3.5gi' , [ L [ x ].im ] ) ) ;

END ;

END;

FUNCTION FFT ( L : table ): table ;

VAR

k       :   integer ;
N       :   integer ;
halfN   :   integer ;
E       :   table   ;
Even    :   table   ;
O       :   table   ;
Odds    :   table   ;
T       :   complex ;

BEGIN

N   :=  length ( L )        ;

IF N < 2 THEN

EXIT ( L )  ;

halfN := ( N DIV 2 )        ;

SetLength ( E,    halfN )   ;

SetLength ( O,    halfN )   ;

Split     ( L, E, O )       ;

SetLength ( L, 0 )   	    ;

SetLength ( Even, halfN )   ;

Even :=     FFT ( E )       ;

SetLength ( E   , 0 )       ;

SetLength ( Odds, halfN )   ;

Odds :=     FFT ( O )       ;

SetLength ( O   , 0 )       ;

SetLength ( L,    N )       ;

FOR k := 0 to halfN - 1 DO

BEGIN

T               := Cexp ( -2 * i * pi * k / N ) * Odds [ k ];

L [ k ]         := Even [ k ] + T                      	    ;

L [ k + halfN ] := Even [ k ] - T                      	    ;

END ;

SetLength ( Even, 0 )   ;

SetLength ( Odds, 0 )   ;

FFT :=  L               ;

END ;

VAR

Ar  :   array of complex ;

x   :   integer ;

BEGIN

SetLength ( Ar, 8 ) ;

FOR x := 0 TO 3 DO

BEGIN

Ar [ x ]        :=  1.0 ;

Ar [ x + 4 ]    :=  0.0 ;

END;

WriteCTable ( FFT ( Ar ) )  ;

SetLength ( Ar, 0 ) ;

END.
(*)
Output:

4 +  0i
1 -2.4142i
0 +  0i
1 -0.41421i
0 +  0i
1 +0.41421i
0 +  0i
1 +2.4142i
```

JPD 2021/12/26

## Perl

Translation of: Raku
```use strict;
use warnings;
use Math::Complex;

sub fft {
return @_ if @_ == 1;
my @evn = fft(@_[grep { not \$_ % 2 } 0 .. \$#_ ]);
my @odd = fft(@_[grep { \$_ % 2 } 1 .. \$#_ ]);
my \$twd = 2*i* pi / @_;
\$odd[\$_] *= exp( \$_ * -\$twd ) for 0 .. \$#odd;
return
(map { \$evn[\$_] + \$odd[\$_] } 0 .. \$#evn ),
(map { \$evn[\$_] - \$odd[\$_] } 0 .. \$#evn );
}

print "\$_\n" for fft qw(1 1 1 1 0 0 0 0);
```
Output:
```4
1-2.41421356237309i
0
1-0.414213562373095i
0
1+0.414213562373095i
0
1+2.41421356237309i```

## Phix

```--
-- demo\rosetta\FastFourierTransform.exw
-- =====================================
--
--  Originally written by Robert Craig and posted to EuForum Dec 13, 2001
--

constant REAL = 1, IMAG = 2

type complex(sequence x)
return length(x)=2 and atom(x[REAL]) and atom(x[IMAG])
end type

function p2round(integer x)
-- rounds x up to a power of two
integer p = 1
while p<x do
p += p
end while
return p
end function

function log_2(atom x)
-- return log2 of x, or -1 if x is not a power of 2
if x>0 then
integer p = -1
while floor(x)=x do
x /= 2
p += 1
end while
if x=0.5 then
return p
end if
end if
return -1
end function

function bitrev(sequence a)
-- bitrev an array of complex numbers
integer j=1, n = length(a)
a = deep_copy(a)
for i=1 to n-1 do
if i<j then
{a[i],a[j]} = {a[j],a[i]}
end if
integer k = n/2
while k<j do
j -= k
k /= 2
end while
j = j+k
end for
return a
end function

function cmult(complex arg1, complex arg2)
-- complex multiply
return {arg1[REAL]*arg2[REAL]-arg1[IMAG]*arg2[IMAG],
arg1[REAL]*arg2[IMAG]+arg1[IMAG]*arg2[REAL]}
end function

function ip_fft(sequence a)
-- perform an in-place fft on an array of complex numbers
-- that has already been bit reversed
integer n = length(a)
integer ip, le, le1
complex u, w, t

for l=1 to log_2(n) do
le = power(2, l)
le1 = le/2
u = {1, 0}
w = {cos(PI/le1), sin(PI/le1)}
for j=1 to le1 do
for i=j to n by le do
ip = i+le1
t = cmult(a[ip], u)
a[ip] = sq_sub(a[i],t)
end for
u = cmult(u, w)
end for
end for
return a
end function

function fft(sequence a)
integer n = length(a)
if log_2(n)=-1 then
puts(1, "input vector length is not a power of two, padded with 0's\n\n")
n = p2round(n)
for j=length(a)+1 to n do
a = append(a,{0, 0})
end for
end if
a = ip_fft(bitrev(a))
-- reverse output from fft to switch +ve and -ve frequencies
for i=2 to n/2 do
integer j = n+2-i
{a[i],a[j]} = {a[j],a[i]}
end for
return a
end function

function ifft(sequence a)
integer n = length(a)
if log_2(n)=-1 then ?9/0 end if -- (or as above?)
a = ip_fft(bitrev(a))
-- modifies results to get inverse fft
for i=1 to n do
a[i] = sq_div(a[i],n)
end for
return a
end function

constant a = {{1, 0},
{1, 0},
{1, 0},
{1, 0},
{0, 0},
{0, 0},
{0, 0},
{0, 0}}

printf(1, "Results of %d-point fft:\n\n", length(a))
ppOpt({pp_Nest,1,pp_IntFmt,"%10.6f",pp_FltFmt,"%10.6f"})
pp(fft(a))
printf(1, "\nResults of %d-point inverse fft (rounded to 6 d.p.):\n\n", length(a))
pp(ifft(fft(a)))
```
Output:
```Results of 8-point fft:

{{  4.000000,  0.000000},
{  1.000000, -2.414214},
{  0.000000,  0.000000},
{  1.000000, -0.414214},
{  0.000000,  0.000000},
{  1.000000,  0.414214},
{  0.000000,  0.000000},
{  1.000000,  2.414214}}

Results of 8-point inverse fft (rounded to 6 d.p.):

{{  1.000000,  0.000000},
{  1.000000, -0.000000},
{  1.000000, -0.000000},
{  1.000000, -0.000000},
{  0.000000,  0.000000},
{  0.000000,  0.000000},
{  0.000000,  0.000000},
{  0.000000,  0.000000}}
```

## PHP

Complex Fourier transform the inverse reimplemented from the C++, Python & JavaScript variants on this page.

Complex Class File:

```<?php

class Complex
{
public \$real;
public \$imaginary;

function __construct(\$real, \$imaginary){
\$this->real = \$real;
\$this->imaginary = \$imaginary;
}

\$dst->real = \$this->real + \$other->real;
\$dst->imaginary = \$this->imaginary + \$other->imaginary;
return \$dst;
}

function Subtract(\$other, \$dst){

\$dst->real = \$this->real - \$other->real;
\$dst->imaginary = \$this->imaginary - \$other->imaginary;
return \$dst;
}

function Multiply(\$other, \$dst){
//cache real in case dst === this
\$r = \$this->real * \$other->real - \$this->imaginary * \$other->imaginary;
\$dst->imaginary = \$this->real * \$other->imaginary + \$this->imaginary * \$other->real;
\$dst->real = \$r;
return \$dst;
}

function ComplexExponential(\$dst){
\$er = exp(\$this->real);
\$dst->real = \$er * cos(\$this->imaginary);
\$dst->imaginary = \$er * sin(\$this->imaginary);
return \$dst;
}
}
```

Example:

```<?php

include 'complex.class.php';

function IFFT(\$amplitudes)
{
\$N = count(\$amplitudes);
\$iN = 1 / \$N;

// Conjugate if imaginary part is not 0
for(\$i = 0; \$i < \$N; ++\$i){
if(\$amplitudes[\$i] instanceof Complex){
\$amplitudes[\$i]->imaginary = -\$amplitudes[\$i]->imaginary;
}
}

// Apply Fourier Transform
\$amplitudes = FFT(\$amplitudes);

for(\$i = 0; \$i < \$N; ++\$i){
//Conjugate again
\$amplitudes[\$i]->imaginary = -\$amplitudes[\$i]->imaginary;
// Scale
\$amplitudes[\$i]->real *= \$iN;
\$amplitudes[\$i]->imaginary *= \$iN;
}
return \$amplitudes;
}

function FFT(\$amplitudes)
{
\$N = count(\$amplitudes);
if(\$N <= 1){
return \$amplitudes;
}

\$hN = \$N / 2;

\$even =  array_pad(array() , \$hN, 0);
\$odd =  array_pad(array() , \$hN, 0);
for(\$i = 0; \$i < \$hN; ++\$i){
\$even[\$i] = \$amplitudes[\$i*2];
\$odd[\$i] = \$amplitudes[\$i*2+1];
}
\$even = FFT(\$even);
\$odd = FFT(\$odd);

\$a = -2*PI();
for(\$k = 0; \$k < \$hN; ++\$k){
if(!(\$even[\$k] instanceof Complex)){
\$even[\$k] = new Complex(\$even[\$k], 0);
}

if(!(\$odd[\$k] instanceof Complex)){
\$odd[\$k] = new Complex(\$odd[\$k], 0);
}
\$p = \$k/\$N;
\$t = new Complex(0, \$a * \$p);

\$t->ComplexExponential(\$t);
\$t->Multiply(\$odd[\$k], \$t);

\$amplitudes[\$k + \$hN] = \$even[\$k]->Subtract(\$t, \$even[\$k]);
}
return \$amplitudes;
}

function EchoSamples(&\$samples){
echo "Index\tReal\t\t\t\tImaginary" . PHP_EOL;
foreach(\$samples as \$key=>&\$sample){
echo  "\$key\t" . number_format(\$sample->real, 13) . "\t\t\t\t" . number_format(\$sample->imaginary, 13) . PHP_EOL;
}
}

// Input Amplitudes
\$time_amplitude_samples = array(1,1,1,1,0,0,0,0);

// echo input for reference
echo 'Input '. PHP_EOL;
echo "Index\tReal" . PHP_EOL;
foreach(\$time_amplitude_samples as \$key=>&\$sample){
echo  "\$key\t" . number_format(\$sample, 13) . PHP_EOL;
}
echo PHP_EOL;

// Do FFT and echo results
echo 'FFT '. PHP_EOL;
\$frequency_amplitude_samples = FFT(\$time_amplitude_samples);
EchoSamples(\$frequency_amplitude_samples);
echo PHP_EOL;

// Do inverse FFT and echo results
echo 'Inverse FFT '. PHP_EOL;
\$frequency_back_to_time_amplitude_samples = IFFT(\$frequency_amplitude_samples);
EchoSamples(\$frequency_back_to_time_amplitude_samples);
echo PHP_EOL;
```

Output:
```Input
Index   Real
0       1.0000000000000
1       1.0000000000000
2       1.0000000000000
3       1.0000000000000
4       0.0000000000000
5       0.0000000000000
6       0.0000000000000
7       0.0000000000000

FFT
Index   Real                            Imaginary
0       4.0000000000000                         0.0000000000000
1       1.0000000000000                         -2.4142135623731
2       0.0000000000000                         0.0000000000000
3       1.0000000000000                         -0.4142135623731
4       0.0000000000000                         0.0000000000000
5       1.0000000000000                         0.4142135623731
6       0.0000000000000                         0.0000000000000
7       1.0000000000000                         2.4142135623731

Inverse FFT
Index   Real                            Imaginary
0       1.0000000000000                         0.0000000000000
1       1.0000000000000                         0.0000000000000
2       1.0000000000000                         0.0000000000000
3       1.0000000000000                         0.0000000000000
4       0.0000000000000                         0.0000000000000
5       0.0000000000000                         0.0000000000000
6       0.0000000000000                         0.0000000000000
7       0.0000000000000                         0.0000000000000
```

## PicoLisp

Works with: PicoLisp version 3.1.0.3
```# apt-get install libfftw3-dev

(scl 4)

(de FFTW_FORWARD . -1)
(de FFTW_ESTIMATE . 64)

(de fft (Lst)
(let
(Len (length Lst)
In (native "libfftw3.so" "fftw_malloc" 'N (* Len 16))
Out (native "libfftw3.so" "fftw_malloc" 'N (* Len 16))
P (native "libfftw3.so" "fftw_plan_dft_1d" 'N
Len In Out FFTW_FORWARD FFTW_ESTIMATE ) )
(struct In NIL (cons 1.0 (apply append Lst)))
(native "libfftw3.so" "fftw_execute" NIL P)
(prog1 (struct Out (make (do Len (link (1.0 . 2)))))
(native "libfftw3.so" "fftw_destroy_plan" NIL P)
(native "libfftw3.so" "fftw_free" NIL Out)
(native "libfftw3.so" "fftw_free" NIL In) ) ) )```

Test:

```(for R (fft '((1.0 0) (1.0 0) (1.0 0) (1.0 0) (0 0) (0 0) (0 0) (0 0)))
(tab (6 8)
(round (car R))
Output:
``` 4.000   0.000
1.000  -2.414
0.000   0.000
1.000  -0.414
0.000   0.000
1.000   0.414
0.000   0.000
1.000   2.414```

## PL/I

```test: PROCEDURE OPTIONS (MAIN, REORDER); /* Derived from Fortran Rosetta Code */

/* In-place Cooley-Tukey FFT */
FFT: PROCEDURE (x) RECURSIVE;
DECLARE  x(*) COMPLEX FLOAT (18);
DECLARE  t    COMPLEX FLOAT (18);
DECLARE ( N, Half_N ) FIXED BINARY (31);
DECLARE ( i, j ) FIXED BINARY (31);
DECLARE (even(*), odd(*)) CONTROLLED COMPLEX FLOAT (18);
DECLARE pi FLOAT (18) STATIC INITIAL ( 3.14159265358979323E0);

N = HBOUND(x);

if N <= 1 THEN return;

allocate odd((N+1)/2), even(N/2);

/* divide */
do j = 1 to N by 2; odd((j+1)/2) = x(j); end;
do j = 2 to N by 2; even(j/2)    = x(j); end;

/* conquer */
call fft(odd);
call fft(even);

/* combine */
half_N = N/2;
do i=1 TO half_N;
t = exp(COMPLEX(0, -2*pi*(i-1)/N))*even(i);
x(i)        = odd(i) + t;
x(i+half_N) = odd(i) - t;
end;

FREE odd, even;

END fft;

DECLARE data(8)  COMPLEX FLOAT (18) STATIC INITIAL (
1, 1, 1, 1, 0, 0, 0, 0);
DECLARE ( i ) FIXED BINARY (31);

call fft(data);

do i=1 TO 8;
PUT SKIP LIST ( fixed(data(i), 25, 12) );
end;

END test;```
Output:
```    4.000000000000+0.000000000000I
1.000000000000-2.414213562373I
0.000000000000+0.000000000000I
1.000000000000-0.414213562373I
0.000000000000+0.000000000000I
0.999999999999+0.414213562373I
0.000000000000+0.000000000000I
0.999999999999+2.414213562373I```

## POV-Ray

```//cmd: +w0 +h0 -F -D
//Stockham algorithm
//Inspiration: http://wwwa.pikara.ne.jp/okojisan/otfft-en/optimization1.html

#version 3.7;
global_settings{ assumed_gamma 1.0 }
#default{ finish{ ambient 1 diffuse 0 emission 0}}

#macro Cstr(Comp)
concat("<",vstr(2, Comp,", ",0,-1),"j>")
#end

#macro CdebugArr(data)
#for(i,0, dimension_size(data, 1)-1)
#debug concat(Cstr(data[i]), "\n")
#end
#end

#macro R2C(Real) <Real, 0> #end

#macro CmultC(C1, C2) <C1.x * C2.x - C1.y * C2.y, C1.y * C2.x + C1.x * C2.y>#end

#macro Conjugate(Comp) <Comp.x, -Comp.y> #end

#macro IsPowOf2(X)
bitwise_and((X > 0), (bitwise_and(X, (X - 1)) = 0))
#end

#macro _FFT0(X, Y, N, Stride, EO)
#local M = div(N, 2);
#local Theta = 2 * pi / N;
#if(N = 1)
#if(EO)
#for(Q, 0, Stride-1)
#local Y[Q] = X[Q];
#end
#end
#else
#for(P, 0, M-1)
#local Fp = P * Theta;
#local Wp = <cos(Fp), -sin(Fp)>;
#for(Q, 0, Stride-1)
#local A = X[Q + Stride * (P + 0)];
#local B = X[Q + Stride * (P + M)];
#local Y[Q + Stride * (2 * P + 0)] = A + B;
#local Y[Q + Stride * (2 * P + 1)] = CmultC((A-B), Wp);
#end
#end
_FFT0(Y, X, div(N, 2), 2 * Stride, !EO)
#end
#end

#macro FFT(X)
#local N = dimension_size(X, 1);
#if(IsPowOf2(N)=0)
#error "length of input is not a power of two"
#end
#local Y = array[N];
_FFT0(X, Y, N, 1, false)
#undef Y
#end

#macro IFFT(X)
#local N = dimension_size(X,1);
#local Fn = R2C(1/N);
#for(P, 0, N-1)
#local X[P] = Conjugate(CmultC(X[P],Fn));
#end
#local Y = array[N];
_FFT0(X, Y, N, 1, false)
#undef Y
#for(P, 0, N-1)
#local X[P] = Conjugate(X[P]);
#end
#end

#declare data = array[8]{1.0,1.0,1.0,1.0,0.0,0.0,0.0,0.0};
#declare cdata = array[8];
#debug "\n\nData\n"
#for(i,0,dimension_size(data,1)-1)
#declare cdata[i] = R2C(data[i]);
#debug concat(Cstr(cdata[i]), "\n")
#end

#debug "\n\nFFT\n"
FFT(cdata)
CdebugArr(cdata)

#debug "\nPower\n"
#for(i,0,dimension_size(cdata,1)-1)
#debug concat(str(cdata[i].x * cdata[i].x + cdata[i].y * cdata[i].y, 0, -1), "\n")
#end

#debug "\nIFFT\n"
IFFT(cdata)
CdebugArr(cdata)
#debug "\n"
```
Output:
```Data
<1.000000, 0.000000j>
<1.000000, 0.000000j>
<1.000000, 0.000000j>
<1.000000, 0.000000j>
<0.000000, 0.000000j>
<0.000000, 0.000000j>
<0.000000, 0.000000j>
<0.000000, 0.000000j>

FFT
<4.000000, 0.000000j>
<1.000000, -2.414214j>
<0.000000, 0.000000j>
<1.000000, -0.414214j>
<0.000000, 0.000000j>
<1.000000, 0.414214j>
<0.000000, 0.000000j>
<1.000000, 2.414214j>

Power
16.000000
6.828427
0.000000
1.171573
0.000000
1.171573
0.000000
6.828427

IFFT
<1.000000, 0.000000j>
<1.000000, -0.000000j>
<1.000000, -0.000000j>
<1.000000, -0.000000j>
<0.000000, -0.000000j>
<0.000000, 0.000000j>
<0.000000, 0.000000j>
<0.000000, 0.000000j>
```

## PowerShell

```Function FFT(\$Arr){
\$Len = \$Arr.Count

If(\$Len -le 1){Return \$Arr}

\$Len_Over_2 = [Math]::Floor((\$Len/2))

\$Output  = New-Object System.Numerics.Complex[] \$Len

\$EvenArr = @()
\$OddArr  = @()

For(\$i = 0; \$i -lt \$Len; \$i++){
If(\$i % 2){
\$OddArr+=\$Arr[\$i]
}Else{
\$EvenArr+=\$Arr[\$i]
}
}

\$Even = FFT(\$EvenArr)
\$Odd  = FFT(\$OddArr)

For(\$i = 0; \$i -lt \$Len_Over_2; \$i++){
\$Twiddle = [System.Numerics.Complex]::Exp([System.Numerics.Complex]::ImaginaryOne*[Math]::Pi*(\$i*-2/\$Len))*\$Odd[\$i]

\$Output[\$i]             = \$Even[\$i] + \$Twiddle
\$Output[\$i+\$Len_Over_2] = \$Even[\$i] - \$Twiddle
}

Return \$Output
}
```
Output:
```PS C:\> FFT((1,1,1,1,0,0,0,0))

Real          Imaginary        Magnitude              Phase
----          ---------        ---------              -----
4                  0                4                  0
1  -2.41421356237309 2.61312592975275  -1.17809724509617
0                  0                0                  0
1 -0.414213562373095 1.08239220029239 -0.392699081698724
0                  0                0                  0
1  0.414213562373095 1.08239220029239  0.392699081698724
0                  0                0                  0
1   2.41421356237309 2.61312592975275   1.17809724509617```

## Prolog

Translation of: Python

Note: Similar algorithmically to the python example.

Works with: SWI Prolog version Version 6.2.6 by Jan Wielemaker, University of Amsterdam
```:- dynamic twiddles/2.
%_______________________________________________________________
% Arithemetic for complex numbers; only the needed rules
add(cx(R1,I1),cx(R2,I2),cx(R,I)) :- R is R1+R2, I is I1+I2.
sub(cx(R1,I1),cx(R2,I2),cx(R,I)) :- R is R1-R2, I is I1-I2.
mul(cx(R1,I1),cx(R2,I2),cx(R,I)) :- R is R1*R2-I1*I2, I is R1*I2+R2*I1.
polar_cx(Mag, Theta, cx(R, I)) :-     % Euler
R is Mag * cos(Theta), I is Mag * sin(Theta).
%___________________________________________________
% FFT Implementation. Note: K rdiv N is a rational number,
% making the lookup in dynamic database predicate twiddles/2 very
% efficient.  Also, polar_cx/2 gets called only when necessary- in
% this case (N=8), exactly 3 times: (where Tf=1/4, 1/8, or 3/8).
tw(0,cx(1,0)) :- !.                    % Calculate e^(-2*pi*k/N)
tw(Tf, Cx) :- twiddles(Tf, Cx), !.     % dynamic match?
tw(Tf, Cx) :- polar_cx(1.0, -2*pi*Tf, Cx), assert(twiddles(Tf, Cx)).

fftVals(N, Even, Odd, V0, V1) :-       % solves all V0,V1 for N,Even,Odd
nth0(K,Even,E), nth0(K,Odd,O), Tf is K rdiv N, tw(Tf,Cx),

split([],[],[]). % split [[a0,b0],[a1,b1],...] into [a0,a1,...] and [b0,b1,...]
split([[V0,V1]|T], [V0|T0], [V1|T1]) :- !, split(T, T0, T1).

fft([H], [H]).
fft([H|T], List) :-
length([H|T],N),
findall(Ve, (nth0(I,[H|T],Ve),I mod 2 =:= 0), EL), !, fft(EL, Even),
findall(Vo, (nth0(I,T,Vo),I mod 2 =:= 0),OL), !, fft(OL, Odd),
findall([V0,V1],fftVals(N,Even,Odd,V0,V1),FFTVals),    % calc FFT
split(FFTVals,L0,L1), append(L0,L1,List).
%___________________________________________________
test :- D=[cx(1,0),cx(1,0),cx(1,0),cx(1,0),cx(0,0),cx(0,0),cx(0,0),cx(0,0)],
time(fft(D,DRes)), writef('fft=['), P is 10^3, !,
(member(cx(Ri,Ii), DRes), R is integer(Ri*P)/P, I is integer(Ii*P)/P,
write(R), (I>=0, write('+'),fail;write(I)), write('j, '),
fail; write(']'), nl).
```
Output:
``` test.
% 681 inferences, 0.000 CPU in 0.001 seconds (0% CPU, Infinite Lips)
fft=[4+0j, 1-2.414j, 0+0j, 1-0.414j, 0+0j, 1+0.414j, 0+0j, 1+2.414j, ]
true.```

## Python

### Python: Recursive

```from cmath import exp, pi

def fft(x):
N = len(x)
if N <= 1: return x
even = fft(x[0::2])
odd =  fft(x[1::2])
T= [exp(-2j*pi*k/N)*odd[k] for k in range(N//2)]
return [even[k] + T[k] for k in range(N//2)] + \
[even[k] - T[k] for k in range(N//2)]

print( ' '.join("%5.3f" % abs(f)
for f in fft([1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0])) )
```
Output:
`4.000 2.613 0.000 1.082 0.000 1.082 0.000 2.613`

### Python: Using module numpy

```>>> from numpy.fft import fft
>>> from numpy import array
>>> a = array([1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0])
>>> print( ' '.join("%5.3f" % abs(f) for f in fft(a)) )
4.000 2.613 0.000 1.082 0.000 1.082 0.000 2.613
```

## R

The function "fft" is readily available in R

```fft(c(1,1,1,1,0,0,0,0))
```
Output:
`4+0.000000i 1-2.414214i 0+0.000000i 1-0.414214i 0+0.000000i 1+0.414214i 0+0.000000i 1+2.414214i`

## Racket

```#lang racket
(require math)
(array-fft (array #[1. 1. 1. 1. 0. 0. 0. 0.]))
```
Output:
```(fcarray
#[4.0+0.0i
1.0-2.414213562373095i
0.0+0.0i
1.0-0.4142135623730949i
0.0+0.0i
0.9999999999999999+0.4142135623730949i
0.0+0.0i
0.9999999999999997+2.414213562373095i])
```

## Raku

(formerly Perl 6)

Works with: rakudo 2022-07
 This example is in need of improvement: Not numerically accurate
```sub fft {
return @_ if @_ == 1;
my @evn = fft( @_[0, 2 ... *] );
my @odd = fft( @_[1, 3 ... *] ) Z*
map &cis, (0, -tau / @_ ... *);
return flat @evn »+« @odd, @evn »-« @odd;
}

.say for fft <1 1 1 1 0 0 0 0>;
```
Output:
```4+0i
1-2.414213562373095i
0+0i
1-0.4142135623730949i
0+0i
0.9999999999999999+0.4142135623730949i
0+0i
0.9999999999999997+2.414213562373095i
```

For the fun of it, here is a purely functional version:

```sub fft {
@_ == 1 ?? @_ !!
fft(@_[0,2...*]) «+«
fft(@_[1,3...*]) «*« map &cis, (0,-τ/@_...^-τ)
}
```

## REXX

This REXX program is modeled after the   Run BASIC   version and is a   radix-2 DIC   (decimation-in-time)
form of the   Cooley-Turkey FFT   algorithm,   and as such, this simplified form assumes that the number of
data points is equal to an exact power of two.

Note that the REXX language doesn't have any higher math functions, such as the functions   COS   and   R2R
(cosine   and   reduce radians to a unit circle).

A normalization of radians function   (r2r)   has been included here, as well as the constant   pi.

This REXX program also adds zero values   if   the number of data points in the list doesn't exactly equal to a
power of two.   This is known as   zero-padding.

```/*REXX program performs a  fast Fourier transform  (FFT)  on a set of  complex numbers. */
numeric digits length( pi() )   -  length(.)     /*limited by the  PI  function result. */
arg data                                         /*ARG verb uppercases the DATA from CL.*/
if data=''  then data= 1 1 1 1 0                 /*Not specified?  Then use the default.*/
do p=0  until  2**p>=size         ;   end      /*number of args exactly a power of 2? */
do j=size+1 to 2**p;  data= data 0;   end      /*add zeroes to DATA 'til a power of 2.*/
size= words(data);      ph= p % 2   ;   call hdr         /*╔═══════════════════════════╗*/
/* [↓] TRANSLATE allows I & J*/  /*║ Numbers in data can be in ║*/
do j=0  for size                                /*║ seven formats:  real      ║*/
_= translate( word(data, j+1), 'J', "I")        /*║                 real,imag ║*/
parse  var  _    #.1.j  ''  \$  1     "," #.2.j  /*║                     ,imag ║*/
if \$=='J'  then parse var #.1.j #2.j "J" #.1.j  /*║                      nnnJ ║*/
/*║                      nnnj ║*/
do m=1  for  2;      #.m.j= word(#.m.j 0, 1)  /*║                      nnnI ║*/
end   /*m*/          /*omitted part?  [↑] */  /*║                      nnni ║*/
/*╚═══════════════════════════╝*/
say pad ' FFT   in '     center(j+1, 7)     pad    fmt(#.1.j)     fmt(#.2.j, "i")
end     /*j*/
say
tran= pi()*2 / 2**p;     !.=0;    hp= 2**p %2;       A= 2**(p-ph);      ptr= A;     dbl= 1
say
do p-ph;        halfPtr=ptr % 2
do i=halfPtr  by ptr  to A-halfPtr;  _= i - halfPtr;   !.i= !._ + dbl
end   /*i*/
ptr= halfPtr;                     dbl= dbl + dbl
end   /*p-ph*/

do j=0  to 2**p%4;  cmp.j= cos(j*tran);      _= hp - j;            cmp._= -cmp.j
_= hp + j;            cmp._= -cmp.j
end  /*j*/
B= 2**ph
do i=0      for A;            q= i * B
do j=0  for B;   h=q+j;   _= !.j*B+!.i;    if _<=h  then iterate
parse value  #.1._  #.1.h  #.2._  #.2.h    with    #.1.h  #.1._  #.2.h  #.2._
end   /*j*/                              /* [↑]  swap  two sets of values. */
end       /*i*/
dbl= 1
do p                    ;       w= hp % dbl
do k=0   for dbl      ;      Lb= w * k            ;          Lh= Lb + 2**p % 4
do j=0 for w        ;       a= j * dbl * 2 + k  ;           b=  a + dbl
r= #.1.a;  i= #.2.a ;      c1= cmp.Lb * #.1.b   ;          c4= cmp.Lb * #.2.b
c2= cmp.Lh * #.2.b   ;          c3= cmp.Lh * #.1.b
#.1.a= r + c1 - c2      ;       #.2.a= i + c3 + c4
#.1.b= r - c1 + c2      ;       #.2.b= i - c3 - c4
end     /*j*/
end       /*k*/
dbl= dbl + dbl
end         /*p*/
call hdr
do z=0  for size
end   /*z*/                             /*[↑] #s are shown with ≈20 dec. digits*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
cos: procedure; parse arg x;  q= r2r(x)**2;      z=1;    _=1;   p=1   /*bare bones COS. */
do k=2  by 2;  _=-_*q/(k*(k-1));  z=z+_;  if z=p  then return z;   p=z;  end  /*k*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
fmt: procedure; parse arg y,j;          y= y/1   /*prettifies complex numbers for output*/
if abs(y) < '1e-'digits() %4  then y= 0;    if y=0 & j\==''  then return ''
dp= digits()%3;  y= format(y, dp%6+1, dp);  if pos(.,y)\==0  then y= strip(y, 'T', 0)
y=  strip(y, 'T', .);                       return left(y || j, dp)
/*──────────────────────────────────────────────────────────────────────────────────────*/
say _;   say translate(_,  " "copies('═', 256),  " "xrange());                 return
/*──────────────────────────────────────────────────────────────────────────────────────*/
pi:  return 3.1415926535897932384626433832795028841971693993751058209749445923078164062862
r2r: return arg(1)  //  ( pi() * 2 )             /*reduce the radians to a unit circle. */
```

Programming note:   the numeric precision (decimal digits) is only restricted by the number of decimal digits in the
pi   variable   (which is defined in the penultimate assignment statement in the REXX program.

output   when using the default inputs of:     1   1   1   1   0
```         data      num         real─part                        imaginary─part
══════════   ═══       ═════════════               ════════════════════════════
FFT   in     1              1
FFT   in     2              1
FFT   in     3              1
FFT   in     4              1
FFT   in     5              0
FFT   in     6              0
FFT   in     7              0
FFT   in     8              0

data      num         real─part                        imaginary─part
══════════   ═══       ═════════════               ════════════════════════════
FFT  out     1              4
FFT  out     2              1                        -2.4142135623730950488
FFT  out     3              0
FFT  out     4              1                        -0.4142135623730950488
FFT  out     5              0
FFT  out     6              1                         0.4142135623730950488
FFT  out     7              0
FFT  out     8              1                         2.4142135623730950488
```

## Ruby

```def fft(vec)
return vec if vec.size <= 1
evens_odds = vec.partition.with_index{|_,i| i.even?}
evens, odds = evens_odds.map{|even_odd| fft(even_odd)*2}
evens.zip(odds).map.with_index do |(even, odd),i|
even + odd * Math::E ** Complex(0, -2 * Math::PI * i / vec.size)
end
end

fft([1,1,1,1,0,0,0,0]).each{|c| puts "%9.6f %+9.6fi" % c.rect}
```
Output:
``` 4.000000 +0.000000i
1.000000 -2.414214i
-0.000000 -0.000000i
1.000000 -0.414214i
0.000000 -0.000000i
1.000000 +0.414214i
0.000000 -0.000000i
1.000000 +2.414214i
```

## Run BASIC

```cnt  = 8
sig  = int(log(cnt) /log(2) +0.9999)

pi    = 3.14159265
real1 = 2^sig

real  = real1 -1
real2 = int(real1 /  2)
real4 = int(real1 /  4)
real3 = real4 +real2

dim rel(real1)
dim img(real1)
dim cmp(real3)

for i = 0 to cnt -1
next i

data    1,0, 1,0, 1,0, 1,0, 0,0, 0,0, 0,0, 0,0

sig2 = int(sig / 2)
sig1 = sig -sig2
cnt1 = 2^sig1
cnt2 = 2^sig2

dim v(cnt1 -1)
v(0) = 0
dv   = 1
ptr  = cnt1

for j = 1 to sig1
hlfPtr = int(ptr / 2)
pt     = cnt1 - hlfPtr
for i = hlfPtr to pt step ptr
v(i) = v(i -hlfPtr) + dv
next i
dv = dv + dv
ptr = hlfPtr
next j

k = 2 *pi /real1

for x = 0 to real4
cmp(x)         = cos(k *x)
cmp(real2 - x) = 0 - cmp(x)
cmp(real2 + x) = 0 - cmp(x)
next x

print "fft: bit reversal"

for i = 0 to cnt1 -1
ip = i *cnt2
for j = 0 to cnt2 -1
h = ip +j
g = v(j) *cnt2 +v(i)
if g >h then
temp   = rel(g)
rel(g) = rel(h)
rel(h) = temp
temp   = img(g)
img(g) = img(h)
img(h) = temp
end if
next j
next i

t = 1
for stage = 1 to sig
print "  stage:- "; stage
d = int(real2 / t)
for ii = 0 to t -1
l   = d *ii
ls  = l +real4
for i = 0 to d -1
a      = 2 *i *t +ii
b      = a +t
f1     = rel(a)
f2     = img(a)
cnt1   = cmp(l)  *rel(b)
cnt2   = cmp(ls) *img(b)
cnt3   = cmp(ls) *rel(b)
cnt4   = cmp(l)  *img(b)
rel(a) = f1 + cnt1 - cnt2
img(a) = f2 + cnt3 + cnt4
rel(b) = f1 - cnt1 + cnt2
img(b) = f2 - cnt3 - cnt4
next i
next ii
t = t +t
next stage

print "  Num   real   imag"
for i = 0 to real
if abs(rel(i)) <10^-5 then rel(i) = 0
if abs(img(i)) <10^-5 then img(i) = 0
print "   "; i;"   ";using("##.#",rel(i));"    ";img(i)
next i
end```
```  Num   real   imag
0    4.0    0
1    1.0    -2.41421356
2    0.0    0
3    1.0    -0.414213565
4    0.0    0
5    1.0    0.414213562
6    0.0    0
7    1.0    2.41421356```

## Rust

Translation of: C
```extern crate num;
use num::complex::Complex;
use std::f64::consts::PI;

const I: Complex<f64> = Complex { re: 0.0, im: 1.0 };

pub fn fft(input: &[Complex<f64>]) -> Vec<Complex<f64>> {
fn fft_inner(
buf_a: &mut [Complex<f64>],
buf_b: &mut [Complex<f64>],
n: usize,    // total length of the input array
step: usize, // precalculated values for t
) {
if step >= n {
return;
}

fft_inner(buf_b, buf_a, n, step * 2);
fft_inner(&mut buf_b[step..], &mut buf_a[step..], n, step * 2);
// create a slice for each half of buf_a:
let (left, right) = buf_a.split_at_mut(n / 2);

for i in (0..n).step_by(step * 2) {
let t = (-I * PI * (i as f64) / (n as f64)).exp() * buf_b[i + step];
left[i / 2] = buf_b[i] + t;
right[i / 2] = buf_b[i] - t;
}
}

// round n (length) up to a power of 2:
let n_orig = input.len();
let n = n_orig.next_power_of_two();
// copy the input into a buffer:
let mut buf_a = input.to_vec();
// right pad with zeros to a power of two:
buf_a.append(&mut vec![Complex { re: 0.0, im: 0.0 }; n - n_orig]);
// alternate between buf_a and buf_b to avoid allocating a new vector each time:
let mut buf_b = buf_a.clone();
fft_inner(&mut buf_a, &mut buf_b, n, 1);
buf_a
}

fn show(label: &str, buf: &[Complex<f64>]) {
println!("{}", label);
let string = buf
.into_iter()
.map(|x| format!("{:.4}{:+.4}i", x.re, x.im))
.collect::<Vec<_>>()
.join(", ");
println!("{}", string);
}

fn main() {
let input: Vec<_> = [1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0]
.into_iter()
.map(|x| Complex::from(x))
.collect();
show("input:", &input);
let output = fft(&input);
show("output:", &output);
}
```
Output:
```input:
1.0000+0.0000i, 1.0000+0.0000i, 1.0000+0.0000i, 1.0000+0.0000i, 0.0000+0.0000i, 0.0000+0.0000i, 0.0000+0.0000i, 0.0000+0.0000i
output:
4.0000+0.0000i, 1.0000-2.4142i, 0.0000+0.0000i, 1.0000-0.4142i, 0.0000+0.0000i, 1.0000+0.4142i, 0.0000+0.0000i, 1.0000+2.4142i
```

## Scala

Library: Scala
Works with: Scala version 2.10.4

Imports and Complex arithmetic:

```import scala.math.{ Pi, cos, sin, cosh, sinh, abs }

case class Complex(re: Double, im: Double) {
def +(x: Complex): Complex = Complex(re + x.re, im + x.im)
def -(x: Complex): Complex = Complex(re - x.re, im - x.im)
def *(x: Double):  Complex = Complex(re * x, im * x)
def *(x: Complex): Complex = Complex(re * x.re - im * x.im, re * x.im + im * x.re)
def /(x: Double):  Complex = Complex(re / x, im / x)

override def toString(): String = {
val a = "%1.3f" format re
val b = "%1.3f" format abs(im)
(a,b) match {
case (_, "0.000") => a
case ("0.000", _) => b + "i"
case (_, _) if im > 0 => a + " + " + b + "i"
case (_, _) => a + " - " + b + "i"
}
}
}

def exp(c: Complex) : Complex = {
val r = (cosh(c.re) + sinh(c.re))
Complex(cos(c.im), sin(c.im)) * r
}
```

The FFT definition itself:

```def _fft(cSeq: Seq[Complex], direction: Complex, scalar: Int): Seq[Complex] = {
if (cSeq.length == 1) {
return cSeq
}
val n = cSeq.length
assume(n % 2 == 0, "The Cooley-Tukey FFT algorithm only works when the length of the input is even.")

val evenOddPairs = cSeq.grouped(2).toSeq
val evens = _fft(evenOddPairs map (_(0)), direction, scalar)
val odds  = _fft(evenOddPairs map (_(1)), direction, scalar)

def leftRightPair(k: Int): Pair[Complex, Complex] = {
val base = evens(k) / scalar
val offset = exp(direction * (Pi * k / n)) * odds(k) / scalar
(base + offset, base - offset)
}

val pairs = (0 until n/2) map leftRightPair
val left  = pairs map (_._1)
val right = pairs map (_._2)
left ++ right
}

def  fft(cSeq: Seq[Complex]): Seq[Complex] = _fft(cSeq, Complex(0,  2), 1)
def rfft(cSeq: Seq[Complex]): Seq[Complex] = _fft(cSeq, Complex(0, -2), 2)
```

Usage:

```val data = Seq(Complex(1,0), Complex(1,0), Complex(1,0), Complex(1,0),
Complex(0,0), Complex(0,2), Complex(0,0), Complex(0,0))

println(fft(data))
println(rfft(fft(data)))
```
Output:
```Vector(4.000 + 2.000i, 2.414 + 1.000i, -2.000, 2.414 + 1.828i, 2.000i, -0.414 + 1.000i, 2.000, -0.414 - 3.828i)
Vector(1.000, 1.000, 1.000, 1.000, 0.000, 2.000i, 0.000, 0.000)```

## Scheme

Works with: Chez Scheme
```; Compute and return the FFT of the given input vector using the Cooley-Tukey Radix-2
; Decimation-in-Time (DIT) algorithm.  The input is assumed to be a vector of complex
; numbers that is a power of two in length greater than zero.

(define fft-r2dit
(lambda (in-vec)
; The constant ( -2 * pi * i ).
(define -2*pi*i (* -2.0i (atan 0 -1)))
; The Cooley-Tukey Radix-2 Decimation-in-Time (DIT) procedure.
(define fft-r2dit-aux
(lambda (vec start leng stride)
(if (= leng 1)
(vector (vector-ref vec start))
(let* ((leng/2 (truncate (/ leng 2)))
(evns (fft-r2dit-aux vec 0 leng/2 (* stride 2)))
(odds (fft-r2dit-aux vec stride leng/2 (* stride 2)))
(dft (make-vector leng)))
(do ((inx 0 (1+ inx)))
((>= inx leng/2) dft)
(let ((e (vector-ref evns inx))
(o (* (vector-ref odds inx) (exp (* inx (/ -2*pi*i leng))))))
(vector-set! dft inx (+ e o))
(vector-set! dft (+ inx leng/2) (- e o))))))))
; Call the Cooley-Tukey Radix-2 Decimation-in-Time (DIT) procedure w/ appropriate
; arguments as derived from the argument to the fft-r2dit procedure.
(fft-r2dit-aux in-vec 0 (vector-length in-vec) 1)))

; Test using a simple pulse.

(let* ((inp (vector 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0))
(dft (fft-r2dit inp)))
(printf "In:  ~a~%" inp)
(printf "DFT: ~a~%" dft))
```
Output:
```In:  #(1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0)
DFT: #(4.0 1.0-2.414213562373095i 0.0-0.0i 1.0-0.4142135623730949i 0.0 1.0+0.41421356237309515i 0.0+0.0i 0.9999999999999997+2.414213562373095i)
```

## Scilab

Scilab has a builtin FFT function.

```fft([1,1,1,1,0,0,0,0]')
```

## SequenceL

```import <Utilities/Complex.sl>;
import <Utilities/Math.sl>;
import <Utilities/Sequence.sl>;

fft(x(1)) :=
let
n := size(x);

top := fft(x[range(1,n-1,2)]);
bottom := fft(x[range(2,n,2)]);

d[i] := makeComplex(cos(2.0*pi*i/n), -sin(2.0*pi*i/n)) foreach i within 0...(n / 2 - 1);

z := complexMultiply(d, bottom);
in
x when n <= 1
else
Output:
```cmd:>fft(makeComplex([1,1,1,1,0,0,0,0],0))
[(Imaginary:0.00000000,Real:4.00000000),(Imaginary:-2.41421356,Real:1.00000000),(Imaginary:0.00000000,Real:0.00000000),(Imaginary:-0.41421356,Real:1.00000000),(Imaginary:0.00000000,Real:0.00000000),(Imaginary:0.41421356,Real:1.00000000),(Imaginary:0.00000000,Real:0.00000000),(Imaginary:2.41421356,Real:1.00000000)]
```

## Sidef

Translation of: Perl
```func fft(arr) {
arr.len == 1 && return arr

var evn = fft([arr[^arr -> grep { .is_even }]])
var odd = fft([arr[^arr -> grep { .is_odd  }]])
var twd = (Num.tau.i / arr.len)

^odd -> map {|n| odd[n] *= ::exp(twd * n)}
(evn »+« odd) + (evn »-« odd)
}

var cycles = 3
var sequence = 0..15
var wave = sequence.map {|n| ::sin(n * Num.tau / sequence.len * cycles) }
say "wave:#{wave.map{|w| '%6.3f' % w }.join(' ')}"
say "fft: #{fft(wave).map { '%6.3f' % .abs }.join(' ')}"
```
Output:
```wave: 0.000  0.924  0.707 -0.383 -1.000 -0.383  0.707  0.924  0.000 -0.924 -0.707  0.383  1.000  0.383 -0.707 -0.924
fft:  0.000  0.000  0.000  8.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  8.000  0.000  0.000
```

## Stata

### Mata

See the fft function in Mata help, and in the FAQ: How can I calculate the Fourier coefficients of a discretely sampled function in Stata?.

```. mata
: a=1,2,3,4
: fft(a)
1         2         3         4
+-----------------------------------------+
1 |       10   -2 - 2i        -2   -2 + 2i  |
+-----------------------------------------+
: end
```

### fft command

Stata can also compute FFT using the undocumented fft command. Here is an example showing its syntax. A time variable must have been set prior to calling this command. Notice that in order to get the same result as Mata's fft() function, in both the input and the output variables the imaginary part must be passed first.

```clear
set obs 4
gen t=_n
gen x=_n
gen y=0
tsset t
fft y x, gen(v u)
list u v, noobs
```

Output

```  +-----------------+
|  u            v |
|-----------------|
| 10            0 |
| -2           -2 |
| -2   -2.449e-16 |
| -2            2 |
+-----------------+```

## Swift

Translation of: Kotlin
```import Foundation
import Numerics

typealias Complex = Numerics.Complex<Double>

extension Complex {
var exp: Complex {
Complex(cos(imaginary), sin(imaginary)) * Complex(cosh(real), sinh(real))
}

var pretty: String {
let fmt = { String(format: "%1.3f", \$0) }
let re = fmt(real)
let im = fmt(abs(imaginary))

if im == "0.000" {
return re
} else if re == "0.000" {
return im
} else if imaginary > 0 {
return re + " + " + im + "i"
} else {
return re + " - " + im +  "i"
}
}
}

func fft(_ array: [Complex]) -> [Complex] { _fft(array, direction: Complex(0.0, 2.0), scalar: 1) }
func rfft(_ array: [Complex]) -> [Complex] { _fft(array, direction: Complex(0.0, -2.0), scalar: 2) }

private func _fft(_ arr: [Complex], direction: Complex, scalar: Double) -> [Complex] {
guard arr.count > 1 else {
return arr
}

let n = arr.count
let cScalar = Complex(scalar, 0)

precondition(n % 2 == 0, "The Cooley-Tukey FFT algorithm only works when the length of the input is even.")

var (evens, odds) = arr.lazy.enumerated().reduce(into: ([Complex](), [Complex]()), {res, cur in
if cur.offset & 1 == 0 {
res.0.append(cur.element)
} else {
res.1.append(cur.element)
}
})

evens = _fft(evens, direction: direction, scalar: scalar)
odds = _fft(odds, direction: direction, scalar: scalar)

let (left, right) = (0 ..< n / 2).map({i -> (Complex, Complex) in
let offset = (direction * Complex((.pi * Double(i) / Double(n)), 0)).exp * odds[i] / cScalar
let base = evens[i] / cScalar

return (base + offset, base - offset)
}).reduce(into: ([Complex](), [Complex]()), {res, cur in
res.0.append(cur.0)
res.1.append(cur.1)
})

return left + right
}

let dat = [Complex(1.0, 0.0), Complex(1.0, 0.0), Complex(1.0, 0.0), Complex(1.0, 0.0),
Complex(0.0, 0.0), Complex(0.0, 2.0), Complex(0.0, 0.0), Complex(0.0, 0.0)]

print(fft(dat).map({ \$0.pretty }))
print(rfft(f).map({ \$0.pretty }))
```
Output:
```["4.000 + 2.000i", "2.414 + 1.000i", "-2.000", "2.414 + 1.828i", "2.000", "-0.414 + 1.000i", "2.000", "-0.414 - 3.828i"]
["1.000", "1.000", "1.000", "1.000", "0.000", "2.000", "0.000", "0.000"]```

## SystemVerilog

Translation of: Java

Differently from the java implementation I have not implemented a complex type. I think it would worth only if the simulators supported operator overloading, since it is not the case I prefer to expand the complex operations, that are trivial for any electrical engineer to understand :D

I could have written a more beautiful code by using non-blocking assignments in the bit_reverse_order function, but it could not be coded in a function, so FFT could not be implemented as a function as well.

```package math_pkg;
// Inspired by the post
// import functions directly from C library
//import dpi task      C Name = SV function name
import "DPI" pure function real cos (input real rTheta);
import "DPI" pure function real sin(input real y);
import "DPI" pure function real atan2(input real y, input real x);
endpackage : math_pkg

// Encapsulates the functions in a parameterized class
// The FFT is implemented using floating point arithmetic (systemverilog real)
// Complex values are represented as a real vector [1:0], the index 0 is the real part
// and the index 1 is the imaginary part.
class fft_fp #(
parameter LOG2_NS = 7,
parameter NS = 1<<LOG2_NS
);

static function void bit_reverse_order(input real buffer_in[0:NS-1][1:0], output real buffer[0:NS-1][1:0]);
begin
for(reg [LOG2_NS:0] j = 0; j < NS; j = j + 1) begin
reg [LOG2_NS-1:0] ij;
ij = {<<{j[LOG2_NS-1:0]}}; // Right to left streaming
buffer[j][0] = buffer_in[ij][0];
buffer[j][1] = buffer_in[ij][1];
end
end
endfunction
// SystemVerilog FFT implementation translated from Java
static function void transform(input real buffer_in[0:NS-1][1:0], output real buffer[0:NS-1][1:0]);
begin
static real pi = math_pkg::atan2(0.0, -1.0);
bit_reverse_order(buffer_in, buffer);
for(int N = 2; N <= NS; N = N << 1) begin
for(int i = 0; i < NS; i = i + N) begin
for(int k =0; k < N/2; k = k + 1) begin
int evenIndex;
int oddIndex;
real theta;
real wr, wi;
real zr, zi;
evenIndex = i + k;
oddIndex  = i + k + (N/2);
theta     = (-2.0*pi*k/real'(N));
// Call to the DPI C functions
// (it could be memorized to save some calls but I dont think it worthes)
// w = exp(-2j*pi*k/N);
wr = math_pkg::cos(theta);
wi = math_pkg::sin(theta);
// x = w * buffer[oddIndex]
zr = buffer[oddIndex][0] * wr - buffer[oddIndex][1] * wi;
zi = buffer[oddIndex][0] * wi + buffer[oddIndex][1] * wr;
// update oddIndex before evenIndex
buffer[ oddIndex][0] = buffer[evenIndex][0] - zr;
buffer[ oddIndex][1] = buffer[evenIndex][1] - zi;
// because evenIndex is in the rhs
buffer[evenIndex][0] = buffer[evenIndex][0] + zr;
buffer[evenIndex][1] = buffer[evenIndex][1] + zi;
end
end
end
end
endfunction
// Implements the inverse FFT using the following identity
// ifft(x) = conj(fft(conj(x))/NS;
static function void invert(input real buffer_in[0:NS-1][1:0], output real buffer[0:NS-1][1:0]);
real tmp[0:NS-1][1:0];
begin
// Conjugates the input
for(int i = 0; i < NS; i = i + 1) begin
tmp[i][0] = buffer_in[i][0];
tmp[i][1] = -buffer_in[i][1];
end
transform(tmp, buffer);
// Conjugate and scale the output
for(int i = 0; i < NS; i = i + 1) begin
buffer[i][0] = buffer[i][0]/NS;
buffer[i][1] = -buffer[i][1]/NS;
end
end
endfunction

endclass
```

Now let's perform the standard test

```/// @Author: Alexandre Felipe (o.alexandre.felipe@gmail.com)
/// @Date: 2018-Jan-25
///
module fft_model_sanity;
initial begin
real x[0:7][1:0]; // input data
real X[0:7][1:0]; // transformed data
real y[0:7][1:0]; // inverted data
for(int i = 0; i < 8; i = i + 1)x[i][0] = 0.0;
for(int i = 4; i < 8; i = i + 1)x[i][1] = 0.0;
for(int i = 0; i < 4; i = i + 1)x[i][0] = 1.0;
fft_fp #(.LOG2_NS(3), .NS(8))::transform(x, X);
\$display("Direct FFT");
for(int i = 0; i < 8; i = i + 1) begin
\$display("(%f, %f)", X[i][0], X[i][1]);
end
\$display("Inverse FFT");
fft_fp #(.LOG2_NS(3), .NS(8))::invert(X, y);
for(int i = 0; i < 8; i = i + 1) begin
\$display("(%f, %f)", y[i][0], y[i][1]);
end
end
endmodule
```

By running the sanity test it outputs the following

```Direct FFT
(4.000000, 0.000000)
(1.000000, -2.414214)
(0.000000, 0.000000)
(1.000000, -0.414214)
(0.000000, 0.000000)
(1.000000, 0.414214)
(0.000000, 0.000000)
(1.000000, 2.414214)
Inverse FFT
(1.000000, 0.000000)
(1.000000, -0.000000)
(1.000000, 0.000000)
(1.000000, -0.000000)
(0.000000, 0.000000)
(0.000000, 0.000000)
(0.000000, -0.000000)
(0.000000, 0.000000)
```

Giving some indication that the test is correct.

A more reliable test is to implement the Discrete Fourier Transform by its definition and compare the results obtained by FFT and by definition evaluation. For that let's create a class with a random data vector, and each time the vector is randomized the FFT is calculated and the output is compared by the result obtained by the definition.

```/// @Author: Alexandre Felipe (o.alexandre.felipe@gmail.com)
/// @Date: 2018-Jan-25
///
class fft_definition_checker #(
parameter LOG2_NS = 3,
parameter NS = 1<<LOG2_NS,
parameter NB = 10);
rand logic [NB:0] x_bits[0:NS-1][1:0];
static real TWO_PI = 2.0*math_pkg::atan2(0.0, -1.0);
real w[0:NS-1][1:0];
function new;
foreach(w[i]) begin
w[i][0] = math_pkg::cos(TWO_PI * i / real'(NS));
w[i][1] =-math_pkg::sin(TWO_PI * i / real'(NS));
end
endfunction
function void post_randomize;
real x[0:NS-1][1:0];
real X[0:NS-1][1:0];
real X_ref[0:NS-1][1:0];
real errorEnergy;
begin
// Convert randomized binary numbers to real (floating point)
foreach(x_bits[i]) begin
x[i][0] = x_bits[i][0];
x[i][1] = x_bits[i][1];
end

////               START THE MAGIC HERE           ////
fft_fp #(.LOG2_NS(LOG2_NS), .NS(NS))::transform(x, X);
////                 END OF THE MAGIC            ////

/// Calculate X_ref, the discrete Fourier transform by the definition ///
foreach(X_ref[k]) begin
X_ref[k] = '{0.0, 0.0};
foreach(x[i]) begin
X_ref[k][0] = X_ref[k][0] + x[i][0] * w[(i*k) % NS][0] - x[i][1] * w[(i*k) % NS][1];
X_ref[k][1] = X_ref[k][1] + x[i][0] * w[(i*k) % NS][1] + x[i][1] * w[(i*k) % NS][0];
end
end

// Measure the error
errorEnergy = 0.0;
foreach(X[k]) begin
errorEnergy = errorEnergy + (X_ref[k][0] - X[k][0]) * (X_ref[k][0] - X[k][0]);
errorEnergy = errorEnergy + (X_ref[k][1] - X[k][1]) * (X_ref[k][1] - X[k][1]);
end
\$display("FFT of %d integers %d bits (error @ %g)", NS, NB, errorEnergy / real'(NS));
end
endfunction
endclass
```

Now let's create a code that tests the FFT with random inputs for different sizes. Uses a generate block since the number of samples is a parameter and must be defined at compile time.

```/// @Author: Alexandre Felipe (o.alexandre.felipe@gmail.com)
/// @Date: 2018-Jan-25
///
module fft_test_by_definition;
genvar LOG2_NS;
generate for(LOG2_NS = 3; LOG2_NS < 7; LOG2_NS = LOG2_NS + 1) begin
initial begin
fft_definition_checker #(.NB(10), .LOG2_NS(LOG2_NS), .NS(1<<LOG2_NS)) chkInst;
chkInst = new;
repeat(5) assert(chkInst.randomize()); // randomize and check the outputs
end
end
endgenerate
endmodule
```

Simulating the fft_test_by_definition we get the following output:

```FFT of           8 integers          10 bits (error @ 3.11808e-25)
FFT of           8 integers          10 bits (error @ 7.86791e-25)
FFT of           8 integers          10 bits (error @ 7.26776e-25)
FFT of           8 integers          10 bits (error @ 2.75458e-25)
FFT of           8 integers          10 bits (error @ 4.83061e-25)
FFT of          16 integers          10 bits (error @ 1.73615e-24)
FFT of          16 integers          10 bits (error @ 3.00742e-24)
FFT of          16 integers          10 bits (error @ 1.70818e-24)
FFT of          16 integers          10 bits (error @ 2.47367e-24)
FFT of          16 integers          10 bits (error @ 2.13661e-24)
FFT of          32 integers          10 bits (error @ 9.52803e-24)
FFT of          32 integers          10 bits (error @ 1.19533e-23)
FFT of          32 integers          10 bits (error @ 6.50223e-24)
FFT of          32 integers          10 bits (error @ 8.05807e-24)
FFT of          32 integers          10 bits (error @ 7.07355e-24)
FFT of          64 integers          10 bits (error @ 3.54266e-23)
FFT of          64 integers          10 bits (error @ 2.952e-23)
FFT of          64 integers          10 bits (error @ 3.41618e-23)
FFT of          64 integers          10 bits (error @ 3.66977e-23)
FFT of          64 integers          10 bits (error @ 3.4069e-23)
```

As expected the error is small and it increases with the number of terms in the FFT.

## Tcl

Library: Tcllib (Package: math::constants)
Library: Tcllib (Package: math::fourier)
```package require math::constants
package require math::fourier

math::constants::constants pi
# Helper functions
proc wave {samples cycles} {
global pi
set wave {}
set factor [expr {2*\$pi * \$cycles / \$samples}]
for {set i 0} {\$i < \$samples} {incr i} {
lappend wave [expr {sin(\$factor * \$i)}]
}
return \$wave
}
proc printwave {waveName {format "%7.3f"}} {
upvar 1 \$waveName wave
set out [format "%-6s" \${waveName}:]
foreach value \$wave {
append out [format \$format \$value]
}
puts \$out
}
proc waveMagnitude {wave} {
set out {}
foreach value \$wave {
lassign \$value re im
lappend out [expr {hypot(\$re, \$im)}]
}
return \$out
}

set wave [wave 16 3]
printwave wave
# Uses FFT if input length is power of 2, and a less efficient algorithm otherwise
set fft [math::fourier::dft \$wave]
# Convert to magnitudes for printing
set fft2 [waveMagnitude \$fft]
printwave fft2
```
Output:
```wave:   0.000  0.924  0.707 -0.383 -1.000 -0.383  0.707  0.924  0.000 -0.924 -0.707  0.383  1.000  0.383 -0.707 -0.924
fft2:   0.000  0.000  0.000  8.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  0.000  8.000  0.000  0.000
```

## Ursala

The `fftw` library is callable from Ursala using the syntax `..u_fw_dft` for a one dimensional forward discrete Fourier transform operating on a list of complex numbers. Ordinarily the results are scaled so that the forward and reverse transforms are inverses of each other, but additional scaling can be performed as shown below to conform to convention.

```#import nat
#import flo

f = <1+0j,1+0j,1+0j,1+0j,0+0j,0+0j,0+0j,0+0j>    # complex sequence of 4 1's and 4 0's

g = c..mul^*D(sqrt+ float+ length,..u_fw_dft) f  # its fft

#cast %jLW

t = (f,g)```
Output:
```(
<
1.000e+00+0.000e+00j,
1.000e+00+0.000e+00j,
1.000e+00+0.000e+00j,
1.000e+00+0.000e+00j,
0.000e+00+0.000e+00j,
0.000e+00+0.000e+00j,
0.000e+00+0.000e+00j,
0.000e+00+0.000e+00j>,
<
4.000e+00+0.000e+00j,
1.000e+00-2.414e+00j,
0.000e+00+0.000e+00j,
1.000e+00-4.142e-01j,
0.000e+00+0.000e+00j,
1.000e+00+4.142e-01j,
0.000e+00+0.000e+00j,
1.000e+00+2.414e+00j>)```

## VBA

Works with: VBA
Translation of: BBC_BASIC

Written and tested in Microsoft Visual Basic for Applications 7.1 under Office 365 Excel; but is probably useable under any recent version of VBA.

```Option Base 0

Private Type Complex
re As Double
im As Double
End Type

Private Function cmul(c1 As Complex, c2 As Complex) As Complex
Dim ret As Complex
ret.re = c1.re * c2.re - c1.im * c2.im
ret.im = c1.re * c2.im + c1.im * c2.re
cmul = ret
End Function

Public Sub FFT(buf() As Complex, out() As Complex, begin As Integer, step As Integer, N As Integer)
Dim i As Integer, t As Complex, c As Complex, v As Complex
If step < N Then
FFT out, buf, begin, 2 * step, N
FFT out, buf, begin + step, 2 * step, N

i = 0
While i < N
t.re = Cos(-WorksheetFunction.Pi() * i / N)
t.im = Sin(-WorksheetFunction.Pi() * i / N)
c = cmul(t, out(begin + i + step))
buf(begin + (i \ 2)).re = out(begin + i).re + c.re
buf(begin + (i \ 2)).im = out(begin + i).im + c.im
buf(begin + ((i + N) \ 2)).re = out(begin + i).re - c.re
buf(begin + ((i + N) \ 2)).im = out(begin + i).im - c.im
i = i + 2 * step
Wend
End If
End Sub

' --- test routines:

Private Sub show(r As Long, txt As String, buf() As Complex)
Dim i As Integer
r = r + 1
Cells(r, 1) = txt
For i = LBound(buf) To UBound(buf)
r = r + 1
Cells(r, 1) = buf(i).re: Cells(r, 2) = buf(i).im
Next
End Sub

Sub testFFT()
Dim buf(7) As Complex, out(7) As Complex
Dim r As Long, i As Integer
buf(0).re = 1: buf(1).re = 1: buf(2).re = 1: buf(3).re = 1

r = 0
show r, "Input (real, imag):", buf
FFT out, buf, 0, 1, 8
r = r + 1
show r, "Output (real, imag):", out
End Sub```
Output:
```Input (real, imag):
1    0
1    0
1    0
1    0
0    0
0    0
0    0
0    0

Output (real, imag):
4    0
1   -2.414213562
0    0
1   -0.414213562
0    0
1    0.414213562
0    0
1    2.414213562```

## V (Vlang)

Translation of: Go
```import math.complex
import math
fn ditfft2(x []f64, mut y []Complex, n int, s int) {
if n == 1 {
y[0] = complex(x[0], 0)
return
}
ditfft2(x, mut y, n/2, 2*s)
ditfft2(x[s..], mut y[n/2..], n/2, 2*s)
for k := 0; k < n/2; k++ {
tf := cmplx.Rect(1, -2*math.pi*f64(k)/f64(n)) * y[k+n/2]
y[k], y[k+n/2] = y[k]+tf, y[k]-tf
}
}

fn main() {
x := [f64(1), 1, 1, 1, 0, 0, 0, 0]
mut y := []Complex{len: x.len}
ditfft2(x, mut y, x.len, 1)
for c in y {
println("\${c:8.4f}")
}
}```
Output:
``` i       d
2    3.21851142
3    4.38567760
4    4.60094928
5    4.65513050
6    4.66611195
7    4.66854858
8    4.66906066
9    4.66917155
10    4.66919515
11    4.66920026
12    4.66920098
13    4.66920537
```

## Wren

Translation of: Go
Library: Wren-complex
Library: Wren-fmt
```import "./complex" for Complex
import "./fmt" for Fmt

var ditfft2 // recursive
ditfft2 = Fn.new {|x, y, n, s|
if (n == 1) {
y[0] = Complex.new(x[0], 0)
return
}
var hn = (n/2).floor
ditfft2.call(x, y, hn, 2*s)
var z = y[hn..-1]
ditfft2.call(x[s..-1], z, hn, 2*s)
for (i in hn...y.count) y[i] = z[i-hn]
for (k in 0...hn) {
var tf = Complex.fromPolar(1, -2 * Num.pi * k / n) * y[k + hn]
var t = y[k]
y[k] = y[k] + tf
y[k + hn] = t - tf
}
}

var x = [1, 1, 1, 1, 0, 0, 0, 0]
var y = List.filled(x.count, null)
for (i in 0...y.count) y[i] = Complex.zero
ditfft2.call(x, y, x.count, 1)
for (c in y) Fmt.print("\$6.4z", c)
```
Output:
```4.0000 + 0.0000i
1.0000 - 2.4142i
0.0000 + 0.0000i
1.0000 - 0.4142i
0.0000 + 0.0000i
1.0000 + 0.4142i
0.0000 + 0.0000i
1.0000 + 2.4142i
```

## zkl

```var [const] GSL=Import("zklGSL");	// libGSL (GNU Scientific Library)
v:=GSL.ZVector(8).set(1,1,1,1);
GSL.FFT(v).toList().concat("\n").println();  // in place```
Output:
```(4.00+0.00i)
(1.00-2.41i)
(0.00+0.00i)
(1.00-0.41i)
(0.00+0.00i)
(1.00+0.41i)
(0.00+0.00i)
(1.00+2.41i)
```