I comment on http://www.jsoftware.com/jwiki/Essays/FFT. FFT means Fast Fourier
Transform and IFFT is the Inverse Fast Fourier Transform.
ifft fft i.16
0 1j_2.22045e_16 2 3j2.22045e_16 4 5j_2.22045e_16 6 7j2.22045e_16 8
9j2.22045e_16 10 11j_2.22045e_16 12
13j2.22045e_16 14 15j_2.22045e_16
1. The Cliff Reiter rounding (**|)&.+. removes the ugly deviations from exact
zero.
(**|)&.+. ifft fft i.16 NB. test
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
So the verb 'roundimag' from jwiki/Essays/FFT is unnecessary.
2. jwiki/Essays/FFT defines two versions of the verb 'roots', but none of them
are used later.
3. In jwiki/Essays/FFT the 'root of unity' verb rou is complicated for reasons
which I don't quite understand. It only works if the argument is divisible by
8. It may be defined using a 'Power of Unity' verb. The simple
definitionPoU=:13 : '1^y' always produces 1 for real arguments, but'_1^+:y'
works. So does '^0j2p1*y' and 'r.2p1*y', but I prefer'_1^+:y'.
4. Rather than complex conjugation of rou in the ifft verb you may conjugate
the input, both in fft and ifft.
5. The division by '#' can be put inside rconvolve. Then fft and ifft becomes
the same.
Then the program becomes:
convolve =: +//.@:(*/) NB. slow convolution
extend =: >.&.(2&^.)@<:@+&#{."_1,: NB. join and extend with zeros
rd =: (**|)&.+. NB. Cliff Reiter rounding
PoU =: _1^+: NB. Power of Unity
rou =: 13 :'PoU(i.y%2)%y' NB. roots of unity
floop =: 4 :'for_r. i.#$x do. (y=.{."1 y) ] x=.(+/x) ,&,:"r (-/x)*y end.'
cube =: ($~ q:@#) :. ,
fft =: (+floop&.cube rou@#) f. :. fft
rconvolve =: [:rd(%#)@:*&.fft/@extend NB. fast convolution
1 2 3 4 convolve 2 3 4 5 NB. test
2 7 16 30 34 31 20
1 2 3 4 rconvolve 2 3 4 5 NB. test
2 7 16 30 34 31 20 0
- Bo
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