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".
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