Fast Fourier transform

Fast Fourier transform is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The purpose of this task is to calculate the FFT (Fast Fourier Transform) of an input sequence. The most general case allows for complex numbers at the input and results in a sequence of equal length, again of complex numbers. If you need to restrict yourself to real numbers the output should be the magnitude (i.e. sqrt(re²+im²)) of the complex result. The classic version is the recursive Cooley–Tukey FFT. Wikipedia has pseudocode for that. Further optimizations are possible but not required.

J

From j:Essays/FFT:

<lang j>cube =: (\$~ q:@#) :. , roots =: +@]^:(0>[) (_1^2%]) ^ i.@-: NB. The elegant version, but less accurate rou =: [: (, j.) [: (, (j.~%:0.5) , |."1&.:+.@|.@}.) [: ^@o.@:j. i.@(%&8) % -: NB. roots of unity floop =: 4 : 'for_r. i.#\$x do. (y=.{."1 y) ] x=.(+/x) ,&,:"r (-/x)*y end.' fft =: ( ] floop&.cube rou@#) f. :. ifft ifft =: (# %~ ] floop&.cube +@rou@#) f. :. fft</lang>

Example:

<lang j>fmt =: [:, sprintf~&'%7.3f '@|

```  fmt"1 (,:fft) 1 o. 3r8p1 * i.16
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
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</lang>
```

Note that fmt only displays the magnitude of the result (and shows only 3 decimals of precision).

Perl 6

<lang perl6>sub fft {

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

}

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

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

Output:

```wave:   0.000   0.924   0.707  -0.383  -1.000  -0.383   0.707   0.924   0.000  -0.924  -0.707   0.383   1.000   0.383  -0.707  -0.924
fft:    0.000   0.000   0.000   8.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   0.000   8.000   0.000   0.000
```