Fast Fourier transform
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
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
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
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)
ALGOL 68
Note: This specimen retains the original Python coding style.
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;
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, $l$
))
)- 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
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
READ in{(I%)}.r#
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
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 .. $].stride(2).array.fft;
alias E = std.complex.expi;
auto l = iota(N / 2).map!(k => ev[k] + T(E(-2* PI * k/N)) * od[k]);
auto r = iota(N / 2).map!(k => ev[k] - T(E(-2* PI * k/N)) * od[k]);
return l.chain(r).array;
}
void main() {
[1.0, 1, 1, 1, 0, 0, 0, 0].map!complex.array.fft.writeln;
}
- Output:
[4+0i, 1-2.41421i, 0+0i, 1-0.414214i, 0+0i, 1+0.414214i, 0+0i, 1+2.41421i]
Delphi
program Fast_Fourier_transform;
{$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);
readln;
end.
- Output:
4 + 0i 1 - 2,41421356237309i 0 + 0i 1 - 0,414213562373095i 0 + 0i 1 + 0,414213562373095i 0 + 0i 1 + 2,41421356237309i
EasyLang
sysconf radians
func[] cmult a[] b[] .
return [ a[1] * b[1] - a[2] * b[2] a[1] * b[2] + a[2] * b[1] ]
.
func[] cadd a[] b[] .
return [ a[1] + b[1] a[2] + b[2] ]
.
func[] csub a[] b[] .
return [ a[1] - b[1] a[2] - b[2] ]
.
func[] cexp a[] .
p = pow 2.718281828459045235 a[1]
return [ p * cos a[2] p * sin a[2] ]
.
func cabs a[] .
return sqrt (a[1] * a[1] + a[2] * a[2])
.
proc fft x[] . y[][] .
n = len x[]
if n = 1
y[][] = [ [ x[1] 0 ] ]
return
.
for i = 1 step 2 to len x[]
xeven[] &= x[i]
xodd[] &= x[i + 1]
.
fft xeven[] even[][]
fft xodd[] odd[][]
y[][] = [ ]
for k to n div 2
t[][] &= cmult cexp [ 0 -2 * pi * (k - 1) / n ] odd[k][]
.
for k to n div 2
y[][] &= cadd even[k][] t[k][]
.
for k to n div 2
y[][] &= csub even[k][] t[k][]
.
.
fft [ 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 ] r[][]
for i to len r[][]
write cabs r[i][] & " "
.
- Output:
4 2.61 0 1.08 0 1.08 0 2.61
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
READ(REL[I],IMG[I])
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
'--------------------------------------
'in-place complex-to-complex FFT adapted from
'[ 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
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]
- 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
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[evenIndex] = even.add(exp);
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;
}
public Complex add(Complex b) {
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] = even[k].add(t, odd[k]);
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;
}
Complex.prototype.add = function(other, dst)
{
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;
(e cadd't),e csub't};
:[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.append {list.map Cadd :ev :t}
{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}}}} }}
{def Cadd
{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
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
endM 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
function complex.__mt.__add (c1, c2)
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
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
else: odds.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
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
-- read only attributes
::attribute r GET
::attribute i GET
::method add
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
forward message("ADD")
::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
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
https://www.freepascal.org/advantage.var
(*)
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
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) a[i] = sq_add(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) -- pad with 0's 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 = $imaginary;
}
function Add($other, $dst){
$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] = $even[$k]->Add($t, $odd[$k]);
$amplitudes[$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
# 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
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
Note: Similar algorithmically to the python example.
:- 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).
- 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)
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.*/
size=words(data); pad= left('', 5) /*PAD: for indenting and padding SAYs.*/
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
say pad " FFT out " center(z+1,7) pad fmt(#.1.z) fmt(#.2.z,'j')
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)
/*──────────────────────────────────────────────────────────────────────────────────────*/
hdr: _=pad ' data num' pad " real─part " pad pad ' imaginary─part '
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
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
endNum 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
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
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
; 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
complexAdd(top,z) ++ complexSubtract(top,z);- 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
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
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
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
// https://community.cadence.com/cadence_blogs_8/b/fv/posts/create-a-sine-wave-generator-using-systemverilog
// 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
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
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)
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
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)
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