On 10-Jun-15 21:26, Ethan Duni wrote:
With bilateral Laplace transform it's also complicated, because the
damping doesn't work there, except possibly at one specific damping
setting (for an exponent, where for polynomials it doesn't work at
all), yielding a DC
Why isn't that sufficient? Do you need a bigger region of convergence for
something? Note that the region of convergence for a DC signal is also
limited to the real line/unit circle (for Laplace/Z respectively). I'm
unclear on exactly what you're trying to do with these quantities.
I'm interested in the bandlimitedness of the signal. I'm not aware of
how I can judge the bandlimitedness, if I don't know Laplace transform
on the imaginary axis.
I'm not fully sure, how to analytically extend this result to the entire
complex plane and whether this will make sense in regards to the
bandlimiting question.
I'm not sure why you want to do that extension? But, again, note that you
have the same issue extending the transform of a regular DC signal to the
entire complex plane - maybe it would be enlightening to walk through what
you do in that case?
See above and below ;)
Alright, I'll try to reiterate my previous year's ideas in here.
I'm interested in a firm (well, reasonably firm, whatever that means)
foundation of the BLEP approach. Intuitive description of the BLEP
approach is: the discontinuities of the signal and its derivatives are
the only source of nonbandlimitedness, so if we replace these
discontinuities with their bandlimited versions, we effectively
bandlimit the signal.
Now, how far can this statement be taken? This depends on the following
two issues:
- Which infinitely differentiable signals are bandlimited and which
aren't. Here come the polynomials and the exponentials, among other
practically interesting signals. One could be tempted to intuitively
think that polynomials, being integrals of a bandlimited DC are also
bandlimited and, taking the limit, so should be infinite polynomials,
particularly Taylor series, so any signal representable by its Taylor
series (particularly, any analytic signal) should be bandlimited.
However, this clearly contradicts the knowledge that FM'd sine is not
bandlimited. This leads to the second question:
- Given a point where a signal and its derivatives are discontinuous,
will the sum of the respective BLEPs converge?
In regards to the first question, we can notice any bandlimited (in the
Fourier transform sense) signal is necessarily entire in the complex
plane (if I'm not mistaken, this can be derived from the Laplace
transform properties). Also, pretty much any practically interesting
infinitely differentiable signal is also entire. So we can replace the
infinite real differentiability requirement here with a stronger
requirement of the signal being entire in the complex domain.
However, we also need to extend the definition of bandlimitedness. Since
polynomials and exponentials have no Fourier transform (let's believe
so, until Sampo Syreeni or someone else gives further clarifications
otherwise), we can't say whether they are bandlimited or not. More
precisely, the samping theorem cannot give any answer for these signals.
But "any answer" to what exactly?
The real question (why we are talking about bandlimitedness and the
sampling theorem in the first place) is not the bandlimitedness itself.
Rather we want to know, what is going to be the result of a restoration
of the discrete-time signal by the DAC. So the extended (more practical)
definition of the bandlimitedness should be something like follows:
Suppose we are given a continuous-time signal, which is then naively
sampled and further restored by a high-quality DAC. If the restored
signal is "reasonably identical" to the original signal, the signal is
called "bandlimited".
It is probably reasonable to simplify the above, and replace the "high
quality DAC" with a sequence of windowed sinc restoration filters, where
the window length is approaching infinity.
"Reasonably identical" means the following:
- the higher the quality of the DAC, the closer is the signal to the
original one
- we probably can allow a discrepancy at the DC. At least this
discrepancy seems to appear in windowed sinc filters for polynomials.
Then, the BLEP approach applicability condition is the following. Given
a bounded signal representable as a sum of an entire function and the
derivative discontinuity functions, can we bandlimit it by simply
bandlimiting the discontinuities? Apparently, we can do so if the entire
function is bandlimited and the sum of the bandlimited derivatives
converges. The conjecture is (please refer to my previuos year's posts)
that the condition (at least a sufficient one) for the entire function
being bandlimited and for the BLEP sum to converge is one and the same
and has to do with the rolloff speed of the function's derivatives as
the derivative order increases.
--
Vadim Zavalishin
Reaktor Application Architect | R&D
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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