Sampo Syreeni писал 2015-06-28 18:39:
What makes all of that suspect is that at first it does seem to imply
that all of the interesting spectral information is in the
discontinuities.
That's until you begin considering analytic signals having infinitely
long Taylor series. Like a sine. Or an FM'd sine. Then you realize that
such signals can contain non-zero frequency partials even without
discontinuities. So there is interesting spectral information in the
waveform itself, but whether this interesting part is bandlimited or
not? That is, whether the non-bandlimited part of the spectrum resides
in the discontinuities only.
As I wrote in the previuos mail this seems to bear a strong relationship
with the convergence of the infinite sum of BLEPs at each discontinuity
points. Roughly (or maybe even strictly?) speaking, the sums converge if
and only if the derivatives fall off sufficiently fast, which seems to
be equivalent to the relaxed (exponential growth on the complex plane,
instead of exponential growth on the imaginary axis and polynomial
growth on the real axis) Paley-Wiener-Schwartz conditions.
But as we saw, there's quite a lot of matching we
have to do to get those waveforms to interpolate their analog
counterparts.
In practice we cannot sum infinitely many BLEPs. However, the error from
truncation of their sum is related to the same rolloff speed of the
discontinuities.
So, in essence it might not make a lot of sense to talk
about where the information actually resides. The picture could be
seen as the discontinuities defining what is between them, as well as
what is between determining the discontinuities, via the various
matching constraints.
I strongly believe it does make sense. Because you want to distinguish
between the waveforms which can be antialiased by BLEP (at least
theoretically) and those which can not. For those which can the
practical error of antialiasing will be determined by the truncation of
the BLEP sum and the truncation of the BLEPs themselves (windowing in
time domain).
Ethan Duni писал 2015-06-27 02:59:
I guess the kicker with
this approach is that we require knowledge of all of the signal's
derivatives on each side of every discontinuity?
Yes, but this is not necessarily a problem.
That's lower than I would expect - you get good aliasing performance
with
that short of a filter? Even at, say, a fundamental frequency in the
high
single-digit kHz?
Could you maybe say a bit about how the BLEP method compares to a more
brute-force approach like doing "naive" hard sync in a (heavily)
oversampled domain, and then downsampling? Is there mileage to be had
by
combining oversampling with BLEP?
I do not have any specific figures sorry. And as always with such stuff,
this is highly subjective, since it depends on your musical and sound
design tastes. As to the oversampling, I'm usually running my DAW at
88kHz anyway ;) Lower aliasing from nonlinear stuff, less frequency axis
warping from BLT etc.
--
Vadim Zavalishin
Reaktor Application Architect | R&D
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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