Fast Fourier transform
From Rosetta Code
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.
Contents |
[edit] Ada
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
new Ada.Numerics.Generic_Complex_Arrays (<>);
use Complex_Arrays;
function Generic_FFT (X : Complex_Vector) return Complex_Vector;
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;
Example:
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;
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)
[edit] ALGOL 68
Note: This specimen retains the original Python coding style. File: Template.Fast_Fourier_transform.a68PRIO DICE = 9; # ideally = 11 #File: test.Fast_Fourier_transform.a68
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
);
#!/usr/local/bin/a68g --script #Output:
# -*- 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, $l$
))
)
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,
[edit] 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
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
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
[edit] 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>Output
#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);
}
int main()
{
PI = atan2(1, 1) * 4;
cplx buf[] = {1, 1, 1, 1, 0, 0, 0, 0};
void show(char * s) {
printf(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;
}
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)
[edit] 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)
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;
}
}
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);
fft(data);
for (int i = 0; i < 8; ++i)
{
std::cout << data[i] << "\n";
}
return 0;
}
Output:
(4,0) (1,-2.41421) (0,0) (1,-0.414214) (0,0) (1,0.414214) (0,0) (1,2.41421)
[edit] D
import std.stdio, std.math, std.algorithm, std.range;Output:
/*auto*/ creal[] fft(/*in*/ creal[] x) /*pure nothrow*/ {
int N = x.length;
if (N <= 1) return x;
auto ev = stride(x, 2).array().fft();
auto od = stride(x[1 .. $], 2).array().fft();
auto l = map!(k => ev[k] + expi(-2*PI*k/N) * od[k])(iota(N / 2));
auto r = map!(k => ev[k] - expi(-2*PI*k/N) * od[k])(iota(N / 2));
return l.chain(r).array();
}
void main() {
writeln(fft([1.0L+0i, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0]));
}
[4+0i, 1+-2.41421i, 0+0i, 1+-0.414214i, 0+0i, 1+0.414214i, 0+0i, 1+2.41421i]
But in D2 built-in complex numbers will be deprecated.
[edit] 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 ]
[edit] 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)
[edit] 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]
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
[edit] 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:
require'printf'
fmt =: [:, sprintf~&'%7.3f'"0
('wave:',:'fft:'),.fmt"1 (,: |@fft) 1 o. 2p1*3r16 * i.16
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
Note that sprintf does not support complex arguments, so we only display the magnitude of the fft here.
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 floating point values are being calculated to the full 53 bits of mantissa precision of 64 bit floating point numbers, but only the first six digits are displayed, by default. Finally, 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?).
[edit] 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
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
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
[edit] 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.
Output:
Fourier[{1,1,1,1,0,0,0,0}, FourierParameters->{1,-1}]
{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}
[edit] 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
[edit] 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)
[edit] 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]
[edit] Perl 6
sub fft {
return @_ if @_ == 1;
my @evn = fft( @_[0,2...^* >= @_] );
my @odd = fft( @_[1,3...^* >= @_] );
my $twd = 2i * pi / @_; # twiddle factor
@odd »*=« (^@odd).map(* * $twd)».exp;
return @evn »+« @odd, @evn »-« @odd;
}
my @seq = ^16;
my $cycles = 3;
my @wave = (@seq »*» (2*pi / @seq * $cycles))».sin;
say "wave: ", @wave.fmt("%7.3f");
say "fft: ", fft(@wave)».abs.fmt("%7.3f");
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
[edit] 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) ) ) )
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))
(round (cadr 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
[edit] 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])
Output:
[(4+0j), (1-2.4142135623730949j), 0j, (1-0.41421356237309492j), 0j, (0.99999999999999989+0.41421356237309492j), 0j, (0.99999999999999967+2.4142135623730949j)]
[edit] Using module numpy
>>> 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
[edit] 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
[edit] 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.
/*REXX pgm does a fast Fourier transform (FFT) on a set of complex nums.*/
numeric digits 80
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)
do sig=0 until 2**sig>=size; end /*# of 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 hd /*┌─────────────────────────────┐*/
/*│ 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 /*└─────────────────────────────┘*/
say "FFT in " center(j+1,3) ' ' nice(#.1.j) nice(#.2.j,'i')
end
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
doubler=doubler*2; pointer=halfpointer
end
do j=0 to 2**sig%4; cmp.j=cos(j*tran); _m=hsig-j; _p=hsig+j
cmp._m=-cmp.j; cmp._p=-cmp.j
end
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 /*(above) switch 2 sets of values*/
end
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
end
double=double*2
end
call hd; do i=0 for size
say "FFT out" center(i+1,3) ' ' nice(#.1.i) nice(#.2.i,'j')
end
exit
/*─────────────────────────────────────subroutines──────────────────────*/
hd:_='numbers # real-part imaginary-part';say _;say translate(_," "copies('═',255),xrange());return
pi: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862;return pi
r2r: return arg(1)//(2*pi()) /*reduce radians to unit circle. */
cos: procedure; 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
/*subroutine to make complex numbers look nicer for output.*/
nice: procedure; parse arg x,j; numeric digits digits()%10; nz='1e-'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)
Output when using the default input of: 1 1 1 1 0
numbers # 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 numbers # 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
[edit] 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
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
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
[edit] 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)
}
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)
[edit] Tcl
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
[edit] 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>)
