On 06-Jul-15 04:03, Sampo Syreeni wrote:
On 2015-06-30, Vadim Zavalishin wrote:
I would say the whole thread has been started mostly because of the
exponential segments. How are they out of the picture?
They are for *now* out, because I don't yet see how they could be
bandlimited systematically within the BLEP framework.
Didn't I describe this is my previous posts?
But then evidently they can be bandlimited in all: just take a segment
and bandlimit it. It's not going to be an easy math exercise, but it
*is* going to be possible even within the distributional framework.
I'd say even without one. A time-limited segment is in L2, isn't it?
I don't think I'm good enough with integration to do that one myself.
But you, Ethan and many others on this list probably are. Once you then
have the analytic solution to that problem, I'm pretty sure you can tell
from its manifest form whether the BLEP framework cut it.
That would be a nice check, but I'm not sure I'd be able to derive an
analytic closed-form expression for the related sum of the BLEPs which
is what we need to compare against. But could you spot a mistake in my
argument otherwise?
Consider a piecewise-exponential signal being bandlimited by BLEP.
That sort of implies an infinite sum of equal amplitude BLEPs, which
probably can't converge.
I think I have addressed exactly this convergence issue in my previous
posts and in my paper. Furthermore, the convergence seems to be directly
related to the bandlimitedness of the sine (see the paper). The same
conditions hold for an exponential, hence my idea to define the
"extended bandlimitedness" based on the BLEP convergence (or rather, the
rolloff of the derivatives, which defines the BLEP convergence).
Each of these exponentials can be represented as a sum of
rectangular-windowed monomials (by windowing each term of the Taylor
series separately).
They can't: they are not finite sums, but infinite series, and I don't
think we know how to handle such series right now.
I meant "inifinite series of rectangular-windowed monomials". I'm not
sure what specifically you are referring to by "we don't know how to
handle them". We are just talking about pointwise convergence of this
series.
We can apply the BLEP method to bandlimit each of these monomials and
then sum them up.
We can handle each (actually sum of them) monomial. To finite order. But
handling the whole series towards the exponential...not so much.
Again, pointwise convergence is meant.
If the sum converges then the obtained signal is bandlimited, right?
If it does, yes. But I don't think it does.
According to my paper, it does. Unless I did a mistake, the BLEP
amplitudes roll off as 1/n, so if the derivatives (which are the BLEP
gains) roll off exponentially decaying (which they do for a sine
bandlimited to 1), the sum converges. Notice that this sufficient
condition for the BLEP convergence is fulfilled if and only if the sine
is bandlimited.
I'm pretty sure you shouldn't be thinking about the bandlimited forms,
now. The whole BLIT/BLEP theory hangs on the idea that you think about
the continuous time, unlimited form first, and only then substitute --
in the very final step -- the corresponding bandlimited primitives.
So it did, but what's wrong in doing the same for the monomial series?
The sufficient convergence condition for the latter is that the
derivatives of the exponent roll off sufficiently fast.
But they don't, do they?
Of course they do
d^n exp(a*t) /dt^n = a^n * exp(a*t)
so they roll off as a^n. The same for the sine. For a sine bandlimited
to 1 we have a<1 and thus the BLEP sum converges.
> When you snap an exponential back to zero, you
necessarily snap back all of its derivatives.
I'm not sure what you are referring to as "snapping".
I mean, when you introduce a hard phase shift to a sine, you don't just
modulate the waveform AM-wise.
I do, if we consider the same in complex numbers. This is more or less
what my paper is doing in the "ring modulation" approach.
Especially when it gets bandlimited, the way you interpolate the
waveform ain't gonna have just Diracs there, but Hilberts as well, and
both of all orders...
You lost me here a little bit. What's a Hilbert? 1/t? I thought it's the
Fourier transform of a Heaviside. How is it a derivative of a Dirac?
Furthermore, if we are talking in the spectral domain, we are going to
have issues arising from the convergence of the infinite series in the
time domain (you mentioned that the set of tempered distributions is not
closed), that's why I specifically tried to stay in the time domain.
That's the whole point: the bandlimitedness can be checked in the time
domain, without even knowing the spectrum. Maybe the spectrum even
doesn't exist for the full signal (like for an exponential), but we
don't care, since our definition works only with time-limited segments
of the signal.
So to return to the discussion... Have you actually looked at how the
phase side of the picture functions? In addition to and in separation
with the amplitude/modulus side?
I'm afraid you lost me here again. What do you mean by looking at the
amplitude/phase sides? My argument works in the time domain, where there
is no amplitude or phase. If you're suggesting to analyze what's
happening in the frequency domain, that should be possible for a sine.
Take a complex sinusoidal and multiply it by a one-sided rectangular
window. The latter is a sum of a DC and a Heaviside, so the spectrum
must be something like delta(f)+1/f, which is getting frequency-shifted
by the multiplication (time-domain) by the complex sinusoidal. So the
spectrum is something like delta(f+f0)+1/(f+f0) (also add -f0 for the
negative-frequency sinusoidal part). So what does this mean?
--
Vadim Zavalishin
Reaktor Application Architect | R&D
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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