On 2002-02-28 19:48 +0100, mark verbos wrote:

> Grant Richter wrote:
> 
> > >Now you lost me. What is a windowing function?
> >
> >Here's how I understand it, maybe some of the math guru's can straighten me
> >out. A continuous time Fourier transform of a sine wave has a discrete peak
> >at one point as long a the sine wave goes from negative infinity to 
> >positive infinity. If there is a discontinuity (the sine wave is not
> >infinite in duration), that discontinuity itself has harmonic content and
> >the transform is no longer a single discrete peak.
> >
> >In the digital world, using sampled data, a non-windowed function shows
> >"bleeding" (rounding at the bottom) between the frequency bins of the
> >transform. By windowing the function (multiplying by another function that
> >changes the edges) we reduce bleeding in the frequency bins and make the
> >transform more "discrete" looking.
> >
> >It seems something like that should also apply to a continuous time
> >transform. Changes in the shape of the discontinuity (edges of the window)
> >should appear in the output of the transform and change the short term
> >harmonic spectra.
> >
> >The only time I have seen this mentioned is Electronotes #45 Page 21
> >"Transform methods in musical engineering" where Bernie talks about how the
> >Fourier series fails for gated sine waves. But he does not say what does
> >happens exactly, or why fading in a sine wave vs. gating it removes the
> >clicks mathematically.
>
> wouldn't that suggest that the sine wave should be synched to the note 
> start? That way beginning of a note is always comming from a zero point. 
> Then the click that results from a short attack would be eliminated. 
> Assuming that the abrupt jump to the zero state doesn't cause a click at 
> the end of the previous note ;)

Yes, but that won't be enough. That removes the discontinuity in
the value but not the discontinuity in the derivative. In other
words, the curve does not jump from one value to another but it 
still has a sharp angle and it translates as a sort of dull click.

      2_082  2002-03-06  README
        842  2002-03-06  gatedsine.cc
      1_591  2002-03-06  gatedsine.png
     56_044  2002-03-06  gatedsine.wav