Upon a little bit more thinking I came to the conclusion that the
expressed in the earlier post (quoted below) idea should work.
Indeed, the windowed signal y(t) can be represented as a series of
windowed monomials, by simply windowing each of the terms of its Taylor
series separately. If the statement can be shown for a monomial of an
arbitrary order, then, given BLEP convergence, it will follow for their
sum (windowed Taylor series) as well (since the series converges
pointwise on a compact interval, the uniform convergence on that
interval follows, if I'm not mistaken, and one can apply Fourier
transform to each term separately).
Now, a bandlimited version of a windowed monomial can be represented as
a sum of time-shifted BLEPs of 0th and higher orders and therefore it is
a bandlimited signal in the sense of both the classical definition and
the proposed definition.
Maybe I should write a short paper with a more detailed and more clear
explanation?
Regards,
Vadim
vadim.zavalishin писал 2015-06-13 14:50:
Ethan Duni писал 2015-06-12 23:43:
However, if
I'm following this correctly, it seems to me that the problem of
multiplication of distributions means that the whole basic set-up of
the
sampling theorem needs to be reworked to make sense in this context.
I.e.,
not much point worrying about whether to call whatever exotic
combination
of derivatives of delta functions that result from polynomials as
"band
limited" or not, when we don't have a way to relate such a property
back to
sampling/reconstruction of well-tempered distributions in the first
place.
No?
Kind of. Actually, I just had an idea of a much more clear definition
of "bandlimitedness", which doesn't rely on the sampling theorem
(which is not applicable everywhere in the context of interest), or on
the weird sequences of sinc convolution which converge only in the
average (kind of Cesaro sense) at best.
The definition applies only to "real entire" functions (that is entire
functions giving real values for real argument). In the present
context we are not interested in other functions. Particularly, any
discontinuity of the function or its derivative will make the function
non-bandlimited, so we don't need to cover those.
Let x(t) be a "real entire" function (possibly not having a Fourier
transform in any sense). Let's apply some arbitrary rectangular window
to this signal: y(t)=w(t)*x(t). This creates the discontinuities of
the function and its derivatives at the window edges. The signal y(t)
is in L_2 and thus has Fourier transform. Let BL[y] be the bandlimited
(using the Fourier transform, or equivalently, sinc convolution)
version of that signal. Now instead of bandlimiting the signal y let's
apply BLEP bandlimiting to the discontinuities of y and its
derivatives, obtaining (if the ifninite sum of BLEPs converges) some
other signal y'. The signal x is called bandlimited if for any
rectangular window w(t), the signal y' exists (the BLEPs converge) and
y'=BL[y].
This definition is well-specified and directly maps to the goals of
the BLEP approach. The conjectures are
- for the signals which are in L_2 the definition is equivalent to the
usual definition of bandlimitedness.
- if y' exists (BLEPs converge), then y'=BL[y]
If the BLEP convergence is only given within some interval of the time
axis (don't know if such cases can exist), then we can speak of
signals "bandlimited on an interval".
--
Vadim Zavalishin
Reaktor Application Architect | R&D
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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