Re: Sound Analysis Tools

["Julius O. Smith III" <jos@xxxxxxxxxxxxxxxxxx>]

 > > Since the Fourier transform is linear, the two are equivalent.
 >
 > As the Fourier transform is windowed, there is always a time/frequency
 > resolution tradeoff. So, the interpolation is not as accurate as one
 > wants.

In the time domain, band-limited interpolation is ideal. In the
frequency domain, time-limited interpolation is ideal, since
a window was used in the time domain.  In my experience, interpolation
is generally about the same amount of work in either case, per
sample.  However, since a spectral peak is more localized in
the frequency domain, it is obviously less work to interpolate
the window-transform main lobe than it is to interpolate the
whole windowed sinusoid in the time domain.  Anyway, my only
point was that in principle subtraction in the time and freq
domains are equivalent, ignoring sampling details.

 > I think that the gaussian is the easiest window to interpret as its
 > transform is a... gaussian.

We also like the Gaussian window because its transform is a parabola
on a dB scale and so it is exactly interpolated by quadratic
interpolation in the frequency domain. However, the Gaussian
window must itself be windowed, so results are not exact.  What
window do you use on the Gaussian window?  Have you quantified the
resulting interpolation error?

Thanks,
Julius