[Jprogramming] jwiki/Essays/FFT

[henryhrich at nc.rr.com]

My roots-of-unity verb was designed to get higher accuracy than simpler approaches.  If you just take numbers from 0 to pi you in effect lose an ULP or two of significance.  So, I calculated over a smaller & more accurate interval & used reflection and sign-change.  The results of FFT using the more accurate method were noticeably better.  There is Forum traffic about it.

Henry Rich
---- Raul Miller <rauldmiller at gmail.com> wrote: 
> Here's a variation on roots of unity:
> 
> rou=: [: (* * |)&.+. [: ^ 0j2p1 * % * i.
> 
> Except, it's not compatible with the version in
> http://www.jsoftware.com/jwiki/Essays/FFT
> 
> For reasons I do not understand the essay version only gives 4 roots
> when I ask for 8.  For compatibility purposes I could divide the
> argument by 2, but I do not know why I should be doing that (and I
> have not bothered to look close enough to understand the consequences
> of this change).
> 
> -- 
> Raul
> 
> On Sat, Jul 14, 2012 at 6:52 AM, Bo Jacoby <bojacoby at yahoo.dk> wrote:
> >
> >
> > 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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