On 22-Jun-15 21:59, Sampo Syreeni wrote:
On 2015-06-22, Vadim Zavalishin wrote:
After some googling I rediscovered (I think I already found out it one
year ago and forgot) the Paley-Wiener-Schwartz theorem for tempered
distributions, which is closely related to what I was aiming at.
It'll you land right back at the extended sampling theorem I told about,
above.
Exactly (if by "extended sampling theorem" you mean the sampling theorem
for tempered distributions). And now, by dropping the polynomial growth
on the real axis restriction in PWS, I can handle any analytic signal.
And those which are not analytic are not bandlimited anyway.
So why fret about the complex extensions?
I'm not sure which specific meaning of the word "complex" you imply
here. But the main motivation for the whole stuff is applying the BLEP
method to frequency-modulated sawtooth and triangle, where the FM is
either done in the exponential scale and/or the oscillator is
self-modulated. In this case you get exponential segments (and more
complex shapes if self-modulation is done in the exp scale I believe).
This also should cover the question of applicability of BLEP to
arbitrary signal shapes more or less.
It's just that you don't need any of that machinery in order to deal
with that mode of synthesis, and you can easily see from the
distributional theory that you can't do any better.
It seems I can do better. Because my question is not whether an
infinitely-long signal, which doesn't even have Fourier transform, is
bandlimited. My question is whether a time-limited version of that
signal is "bandlimited except exactly for the discontinuities arising
from the time-limiting".
--
Vadim Zavalishin
Reaktor Application Architect | R&D
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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