On 2015-06-30, Vadim Zavalishin wrote:
And even if what we've been talking about above does go as far as I
(following Vadim) suggested, exponential segments are still out of
the picture for now.
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.
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
mean such a segment, say over [0,1], and 0 elsewhere, is going to be an
absolutely bounded function with compact support, so that it's a
distribution with an extant Fourier transform. Just cut that into a
compact support in the Fourier domain, and come back into the time one.
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. And of course
the same with the hard phase advanced sine you also worried about.
(And yes, I was reading Schwartz's Paley-Wiener theorem the wrong way
around. You are exactly right in your interpretation of it. I kept
reading F as f, so that I kept the compactness condition on the time
side of things. In the distributional framework that fucks up the
interpretation quite royally indeed.)
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. Unless Ethan's point about convergence from two
sides somehow manages to moderate the overall decay of the "series" of
bandlimited discontinuities. But let's see once again...
We wish to know if we obtain a bandlimited signal in the result.
The formal "series" we're doing does guarantee *that*. It only has
finite frequencies by definition. It's just that it doesn't guarantee
convergence...
Represent the signal as a sum of rectangular-windowed exponentials.
Why not instead try to develop that single exponential segment as a sort
of "balanced" MacLaurin series? First take away the constant term, then
the quadratic, and so on. The error terms will go to zero over the
negative zero axis, and if you develop the thing in base e, it ought to
be analytically easy to handle from 0 to 1 as well.
Windowing it from [1] is nasty. But e.g. putting on a half-Gaussian
window ought to make the analysis easier, even as you finally reflect it
onto the Fourier side.
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.
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.
If the sum converges then the obtained signal is bandlimited, right?
If it does, yes. But I don't think it does. I think you have to go about
it another way, especially at [1]. You really hit on a nasty problem wrt
the BLEP framework, and the distributional one, where the argument of
the exponential function snaps back to zero (and the same for the sine).
I'm thinking, those points might not easily fall back to the BLEP
framework. But they *might* just be handled by another additive waveform
which is *not* derivable from BLEPs. Maybe that is still derivable in
closed form, and knowable from the closed form non-BLEP-derivation of
the discontinuity?
Now the sum of bandlimited rectangular-windowed monomials converges if
and only if the sum of their BLEP residuals converges.
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.
Don't break ranks just yet. :)
The sufficient convergence condition for the latter is that the
derivatives of the exponent roll off sufficiently fast.
But they don't, do they? When you snap an exponential back to zero, you
necessarily snap back all of its derivatives. Which are equal to each
other, right onto infinite order. The BLEP framework can't deal with
that. And the same goes for sines, as the real part of a complex
exponential: they don't lead to amplitude falloff at all, but a phase
shift with unity modulus.
So, could it be that we need quadrature terms here, as well? We *do*
know how to derive those, via the Hilbert transform, and we know how to
bandlimit them as well. Might they help?
We have seen exactly this for the sine (in my paper), where the
sufficient conditions for the BLEP convergence is that the sine
frequency is below Nyquist. Notice that the Taylor series for exp is
essentially the same as for the sine, therefore they have the same
rolloff speed. So, the exponent is "bandlimited" (in the sense that
the BLEP sum converges) under the same conditions as the sine (the
"frequency" must be below Nyquist).
If so, it works. But since I haven't done the math, I'm not too certain
you can get the sine case to work either without using quadrature terms
in addition.
And in fact, intuitively speaking, if you had to use those as well,
that'd finally tie the knot with my skepticism towards your going with
the complex, holomorphic version of Fourier theory, and what you were
talking about with Paley-Wiener.
I mean, when you introduce a hard phase shift to a sine, you don't just
modulate the waveform AM-wise. You introduce a phase discontinuity. On
the left side of it the sine has one phase, and on the right side it has
another, from -inf to +inf. To me it seems rather obvious, intuitively
speaking at least, that the discontinuity doesn't have just a symmetric
part, but an asymmetrical one as well.
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. Those can all be derived from derivatives of a
Heaviside step, but their infinitesimal or bandlimited versions don't
just go up the the frequency band like you'd think.
The derivative of a step is the Dirac, the derivative of that is the
Hilbert, the derivative of that is in its amplitude behavior just what
you'd think once again, but in angle it's the opposite of the Dirac, and
at fourth order derivative it returns back to being in phase with the
Dirac.
That's because those operators are the differential ones which classify
by their eigenfunctions the shift-invariant subspaces of functions (and
by extension distributions as well) of the function space we started
with; originally L^2 of course. Because the Fourier operator is an
isometry and whatnot, here, those operators map bijectively onto their
generators/characters in the Fourier space; that is, quadrature phase
shifts plus amplitude shifts map onto complex numbers. And then the
four-fold symmetry of the Fourier transform itself maps back into the
four-fold phase symmetry of the (actually Lie ;) ) calculus of the
differential Dirac, Hilbert, whatnot, operators, we started with on the
distributional side.
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? Because it's rather different, and
might help explain a couple of things in addition to what we've talked
about inb4. :)
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