Fast Fourier transform: Difference between revisions

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=={{header|Julia}}==
=={{header|Julia}}==
Julia has a built-in FFT function
Julia has a built-in FFT function:
<lang julia>fft([1,1,1,1,0,0,0,0])</lang>
<lang julia>fft([1,1,1,1,0,0,0,0])</lang>
Output:
<lang julia>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</lang>


=={{header|Liberty BASIC}}==
=={{header|Liberty BASIC}}==

Revision as of 15:42, 6 November 2013

Task
Fast Fourier transform
You are encouraged to solve this task according to the task description, using any language you may know.

The purpose of this task is to 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 pseudocode for that. Further optimizations are possible but not required.

Ada

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

<lang Ada> with Ada.Numerics.Generic_Complex_Arrays;

generic 

  with package Complex_Arrays is
     new Ada.Numerics.Generic_Complex_Arrays (<>);
  use Complex_Arrays;

function Generic_FFT (X : Complex_Vector) return Complex_Vector; 

</lang>

<lang Ada> with Ada.Numerics; with Ada.Numerics.Generic_Complex_Elementary_Functions;

function Generic_FFT (X : Complex_Vector) return Complex_Vector is 

  package Complex_Elementary_Functions is
     new Ada.Numerics.Generic_Complex_Elementary_Functions
       (Complex_Arrays.Complex_Types);

  use Ada.Numerics;
  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; </lang>

Example:

<lang Ada> with Ada.Numerics.Complex_Arrays; use Ada.Numerics.Complex_Arrays; with Ada.Complex_Text_IO; use Ada.Complex_Text_IO; with Ada.Text_IO; use Ada.Text_IO;

with Ada.Numerics.Complex_Elementary_Functions; 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; </lang>

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<lang algol68>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

);</lang>File: test.Fast_Fourier_transform.a68<lang algol68>#!/usr/local/bin/a68g --script #

  1. -*- coding: utf-8 -*- #

MODE SCALAR = COMPL; PROC (COMPL)COMPL scalar exp = complex exp; PR READ "Template.Fast_Fourier_transform.a68" PR

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, $l$
 ))

)</lang>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,

BBC BASIC

Works with: BBC BASIC for Windows

<lang bbcbasic> @% = &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
       READ 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

</lang> 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.<lang C>#include <stdio.h>

  1. include <math.h>
  2. 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); }

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

void show(const char * s) { 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])); }

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

return 0; }</lang>Output<lang>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)</lang>

C++

<lang cpp>#include <complex>

  1. include <iostream>
  2. include <valarray>

const double PI = 3.141592653589793238460;

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

// Cooley–Tukey FFT (in-place) 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;
   }

}

// 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;

}</lang> 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)

D

Standard Version

<lang d>import std.stdio, std.numeric;

void main() { 

   [1.0, 1, 1, 1, 0, 0, 0, 0].fft.writeln;

}</lang>

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. <lang d>import std.stdio, std.algorithm, std.range, std.math;

const(creal)[] fft(in creal[] x) /*pure nothrow*/ { 

   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() { 

   [1.0L+0i, 1, 1, 1, 0, 0, 0, 0].fft.writeln;

}</lang>

Output:
[4+0i, 1+-2.41421i, 0+0i, 1+-0.414214i, 0+0i, 1+0.414214i, 0+0i, 1+2.41421i]

Phobos Complex Version

<lang d>import std.stdio, std.algorithm, std.range, std.math, std.complex;

auto fft(T)(in T[] x) /*pure nothrow*/ { 

   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] + cast(T)E(-2*PI*k/N) * od[k]);
   auto r = iota(N / 2).map!(k=> ev[k] - cast(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;

}</lang>

Output:
[4+0i, 1-2.41421i, 0+0i, 1-0.414214i, 0+0i, 1+0.414214i, 0+0i, 1+2.41421i]

Factor

<lang 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 }

} </lang>

Fortran

<lang 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</lang>

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 )

GAP

<lang gap># Here an implementation with no optimization (O(n^2)).

  1. 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]);

  1. [ 4, 1-E(8)-E(8)^2-E(8)^3, 0, 1-E(8)+E(8)^2-E(8)^3,
  2. 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, 1, 0, 0, 0, 0 ]</lang>

Go

<lang 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)
   }

}</lang> 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)

Haskell

<lang haskell>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]</lang>

And the 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

J

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

<lang j>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@#</lang>

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):

<lang j> (**+)&.+. (,: 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</lang>

Here is a representation of an example which appears in some of the other implementations, here: <lang J> 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</lang>

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?).

Julia

Julia has a built-in FFT function: <lang julia>fft([1,1,1,1,0,0,0,0])</lang> Output: <lang julia>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</lang>

Liberty BASIC

<lang lb> 

   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
       read dummy: Re( N) =dummy
       read dummy: Im( N) =dummy
   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

</lang> 

   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


Mathematica

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.

<lang Mathematica> Fourier[{1,1,1,1,0,0,0,0}, FourierParameters->{1,-1}] </lang>

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}

MATLAB / Octave

Matlab/Octave have a builtin FFT function.

<lang MATLAB> fft([1,1,1,1,0,0,0,0]') </lang> 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

<lang maxima>load(fft)$ fft([1, 2, 3, 4]); [2.5, -0.5 * %i - 0.5, -0.5, 0.5 * %i - 0.5]</lang>

OCaml

This is a simple implementation of the Cooley-Tukey pseudo-code <lang OCaml>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)</lang>

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)

PARI/GP

Naive implementation, using the same testcase as Ada: <lang parigp>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])</lang> 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]

Perl

Translation of: Perl 6

<lang Perl>use strict; use warnings; use Math::Complex;

sub fft { 

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

}


my @seq = 0 .. 15; my $cycles = 3; my @wave = map { sin( $_ * 2*pi/ @seq * $cycles ) } @seq; print "wave: ", join " ", map { sprintf "%7.3f", $_ } @wave; print "\n"; print "fft: ", join " ", map { sprintf "%7.3f", abs $_ } fft(@wave);</lang>

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

Perl 6

<lang perl6>sub fft { 

   return @_ if @_ == 1;
   my @evn = fft( @_[0,2...^* >= @_] );
   my @odd = fft( @_[1,3...^* >= @_] );
   my $twd = 2i * pi / @_; # twiddle factor
   @odd[$_] *= exp( $_ * $twd ) for ^@odd;
   return @evn »+« @odd, @evn »-« @odd;

}

my @seq = ^16; my $cycles = 3; my @wave = map { sin( 2*pi * $_ / @seq * $cycles ) }, @seq; say "wave: ", @wave.fmt("%7.3f");

say "fft: ", fft(@wave)».abs.fmt("%7.3f");</lang>

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

PicoLisp

Works with: PicoLisp version 3.1.0.3

<lang PicoLisp># 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) ) ) )</lang>

Test: <lang PicoLisp>(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))
     (round (cadr R)) ) )</lang>

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

<lang 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;</lang> 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

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

<lang prolog>:- 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), mul(Cx,O,M), add(E,M,V0), sub(E,M,V1).

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). </lang>

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

<lang python>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])
   return [even[k] + exp(-2j*pi*k/N)*odd[k] for k in xrange(N/2)] + \
          [even[k] - exp(-2j*pi*k/N)*odd[k] for k in xrange(N/2)]

print fft([1.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0])</lang>

Output: 

[(4+0j), (1-2.4142135623730949j), 0j, (1-0.41421356237309492j), 0j, (0.99999999999999989+0.41421356237309492j), 0j, (0.99999999999999967+2.4142135623730949j)]

Using module numpy

<lang python>>>> from numpy.fft import fft >>> from numpy import array >>> a = array((0.0, 0.924, 0.707, -0.383, -1.0, -0.383, 0.707, 0.924, 0.0, -0.924, -0.707, 0.383, 1.0, 0.383, -0.707, -0.924)) >>> print( ' '.join("%5.3f" % abs(f) for f in fft(a)) ) 0.000 0.001 0.000 8.001 0.000 0.001 0.000 0.001 0.000 0.001 0.000 0.001 0.000 8.001 0.000 0.001</lang>

R

The function "fft" is readily available in R <lang R>fft(c(1,1,1,1,0,0,0,0))</lang> 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>

  1. lang racket

(require math) (array-fft (array #[1. 1. 1. 1. 0. 0. 0. 0.])) </lang>

Output: <lang racket> (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])

</lang>

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, so the functions COS and R2R
(normalization of radians) have been included here, as well as the constant pi .

This 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 called zero-padding. <lang rexx>/*REXX pgm does a fast Fourier transform (FFT) on a set of complex nums.*/ numeric digits 85 arg data; if data= then data='1 1 1 1 0' /*no data? Then use default*/ data=translate(data, 'J', "I") /*allow use of i as well as j */ size=words(data); pad=left(,6) /*PAD: for indenting/padding SAYs*/ 

 do sig=0 until 2**sig>=size      ;end /* #  args exactly a power of 2?*/
 do j=size+1 to 2**sig;data=data 0;end /*add zeroes until a power of 2.*/

size=words(data); call hdr /*┌─────────────────────────────┐*/ 

                                      /*│ Numbers in  data  can be in │*/
 do j=0  for size                     /*│ 7 formats:   real           │*/
 _=word(data,j+1)                     /*│              real,imag      │*/
 parse var  _  #.1.j  ','  #.2.j      /*│                  ,imag      │*/
 if right(#.1.j,1)=='J'    then       /*│                   nnnJ      │*/
     parse var #.1.j #2.j "J" @.1.j   /*│                   nnnj      │*/
   do p=1  for 2     /*omitted?*/     /*│                   nnnI      │*/
   #.p.j=word(#.p.j 0, 1)             /*│                   nnni      │*/
   end   /*p*/                        /*└─────────────────────────────┘*/
 say pad " FFT   in "    center(j+1,7)  pad  nice(#.1.j)  nice(#.2.j,'i')
 end     /*j*/

say; say; tran=2*pi()/2**sig;  !.=0 hsig=2**sig%2; counterA=2**(sig-sig%2); pointer=counterA; doubler=1 

 do sig-sig%2;  halfpointer=pointer%2
           do i=halfpointer  by pointer  to counterA-halfpointer
           _=i-halfpointer;  !.i=!._+doubler
           end   /*i*/
 doubler=doubler*2;  pointer=halfpointer
 end   /*sig-sig%2*/

        do j=0 to 2**sig%4; cmp.j=cos(j*tran);  _m=hsig-j;  cmp._m=-cmp.j
                                                _p=hsig+j;  cmp._p=-cmp.j
        end  /*j*/

counterB=2**(sig%2) 

 do i=0  for counterA;    p=i*counterB
     do j=0 for counterB; h=p+j; _=!.j*counterB+!.i; if _<=h then iterate
     parse value  #.1._ #.1.h #.2._ #.2.h   with  #.1.h #.1._ #.2.h #.2._
     end   /*j*/                      /* [↓]  switch two sets of values*/
 end       /*i*/

double=1; do sig  ; w=2**sig%2%double 

                 do k=0  for double ;  lb=w*k           ; lh=lb+2**sig%4
                   do j=0 for w     ;  a=j*double*2+k   ; b=a+double
                   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*/
               double=double+double
               end                  /*sig*/

call hdr 

       do i=0  for size
       say pad " FFT  out " center(i+1,7) pad nice(#.1.i) nice(#.2.i,'j')
       end   /*i*/

exit /*stick a fork in it, we're done.*/ /*──────────────────────────────────HDR subroutine──────────────────────*/ hdr: _='───data─── num real-part imaginary-part'; say pad _ 

     say pad  translate(_, " "copies('═',256), " "xrange());    return

/*──────────────────────────────────PI subroutine───────────────────────────────────*/ pi: return , /*add more digs if NUMERIC DIGITS > 85. */ 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628 /*──────────────────────────────────R2R subroutine──────────────────────*/ r2r: return arg(1) // (2*pi()) /*reduce radians to unit circle. */ /*──────────────────────────────────COS subroutine──────────────────────*/ cos: procedure; parse arg x; x=r2r(x); return .sincos(1,1,-1) .sincos: parse arg z,_,i; x=x*x; p=z 

        do k=2 by 2; _=-_*x/(k*(k+i)); z=z+_; if z=p then leave; p=z; end
        return z

/*──────────────────────────────────NICE subroutine─────────────────────*/ nice: procedure; parse arg x,j /*makes complex nums look nicer. */ numeric digits digits()%10; nz='1e-'digits() /*show ≈10% of DIGITS.*/ if abs(x)<nz then x=0; x=x/1; if x=0 & j\== then return x=format(x,,digits()); if pos('.',x)\==0 then x=strip(x,'T',0) x=strip(x,,'.'); if x>=0 then x=' '||x; return left(x||j,digits()+4)</lang> output when using the default input 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           -2.4142136j
        FFT  out     3            0
        FFT  out     4            1           -0.41421356j
        FFT  out     5            0
        FFT  out     6            1            0.41421356j
        FFT  out     7            0
        FFT  out     8            1            2.4142136j

Run BASIC

<lang runbasic>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 

   read rel(i)
   read img(i)

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</lang> 

  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

Scala

Library: Scala
Works with: Scala version 2.9.1

<lang Scala>object FFT extends App { 

 import scala.math._
 
 case class Complex(re: Double, im: Double = 0.0) {
   def +(x: Complex): Complex = Complex((this.re+x.re),(this.im+x.im))
   def -(x: Complex): Complex = Complex((this.re-x.re),(this.im-x.im))
   def *(x: Complex): Complex = Complex(this.re*x.re-this.im*x.im,this.re*x.im+this.im*x.re)
 }

 def fft(f: List[Complex]): List[Complex] = {
   import Stream._
   require((f.size==0)||(from(0) map {x=>pow(2,x).toInt}).takeWhile(_<2*f.size).toList.exists(_==f.size)==true,"list size "+f.size+" not allowed!")
   f.size match {
     case 0 => Nil
     case 1 => f
     case n => {
       val cis: Double => Complex = phi => Complex(cos(phi),sin(phi))
       val e = fft(f.zipWithIndex.filter(_._2%2==0).map(_._1))
       val o  = fft(f.zipWithIndex.filter(_._2%2!=0).map(_._1))
       import scala.collection.mutable.ListBuffer
       val lb = new ListBuffer[Pair[Int, Complex]]()    
       for (k <- 0 to n/2-1) {
         lb += Pair(k,e(k)+o(k)*cis(-2*Pi*k/n))
         lb += Pair(k+n/2,e(k)-o(k)*cis(-2*Pi*k/n))
       }
       lb.toList.sortWith((x,y)=>x._1<y._1).map(_._2)
     }  
   }
 }

 fft(List(Complex(1),Complex(1),Complex(1),Complex(1),Complex(0),Complex(0),Complex(0),Complex(0))).foreach(println)

}</lang> Output: 

Complex(4.0,0.0)
Complex(1.0,-2.414213562373095)
Complex(0.0,0.0)
Complex(1.0,-0.4142135623730949)
Complex(0.0,0.0)
Complex(0.9999999999999999,0.4142135623730949)
Complex(0.0,0.0)
Complex(0.9999999999999997,2.414213562373095)

Tcl

Library: Tcllib (Package: math::constants)
Library: Tcllib (Package: math::fourier)

<lang tcl>package require math::constants package require math::fourier

math::constants::constants pi

  1. 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

  1. Uses FFT if input length is power of 2, and a less efficient algorithm otherwise

set fft [math::fourier::dft $wave]

  1. Convert to magnitudes for printing

set fft2 [waveMagnitude $fft] printwave fft2</lang> 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. <lang ursala>#import nat

  1. 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

  1. cast %jLW

t = (f,g)</lang> 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>)
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