On 09-Jun-15 19:23, Ethan Duni wrote:
Could you give a little bit more of a clarification here? So the
finite-order polynomials are not bandlimited, except the DC? Any hints
to what their spectra look like? How a bandlimited polynomial would look
like?
Any hints how the spectrum of an exponential function looks like? How
does a bandlimited exponential look like? I hope we are talking about
one and the same real exponential exp(at) on (-infty,+infty) and not
about exp(-at) on [0,+infty) or exp(|a|t).
The Fourier transform does not exist for functions that blow up to +-
infinity like that.
I understood from Sampo Syreeni's answer, that Fourier transform does
exist for those functions. And that's exactly the reason for me asking
the above question.
To do frequency domain analysis of those kinds of
signals, you need to use the Laplace and/or Z transforms. Equivalently, you
can think of doing a regular Fourier transform after applying a suitable
exponential damping to the signal of interest. This will handle signals
that blow up in one direction (like the exponential), but signals that blow
up in both directions (like polynomials) remain problematic.
Not good enough. If we're talking about unilateral Laplace transform,
then it introduces a discontinuity at t=0, which immediately introduces
further non-bandlimited partials into the spectrum. I'm not sure how you
suppose to answer the question of the original signal being bandlimited
in this case. 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. 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.
That said, I'm not sure why this is relevant? Seems like you aren't so much
interested in complete exponential/polynomial functions over their entire
domain, but rather windowed versions that are restricted to some small time
region?
I am specifically interested in the functions on the entire real axis.
Further in my original email there is an explanation of the reasons.
Regards,
Vadim
--
Vadim Zavalishin
Reaktor Application Architect | R&D
Native Instruments GmbH
+49-30-611035-0
www.native-instruments.com
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