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Re: [Phys-l] some questions related to sampling



On 02/13/2009 05:19 PM, Stefan Jeglinski wrote:

There's probably a deep conceptual issue that needs to
be addressed, but I'm not seeing it at the moment, so
in the meantime some shallower technical comments:

a) it seemed like we got something for nothing with this trick in the f-domain

I don't see it that way. We got something extra in the
frequency domain, but only because we provided something
in the time domain. Specifically, we padded the input
with a huge number of zeros.

The new input leads to a new output. There's nothing
mysterious about that. OTOH if you look at it the wrong
way it might seem mysterious, because there is a very
compact representation of the padded data ... but that's
a red herring. Conceptually and formally it is different
input data.

b) the sampling theorem CAN successfully interpolate, since we are bandlimited.

The theorem only applies to things that are *strictly*
band limited. It therefore doesn't really apply to
anything in the real world, so far as I know. Real
RC filters, RLC filters, and such do only a roughly
approximate job of stopping the stop-band signals.

Conversely, a truly band-limited interpolator would
be classified as a high-Q filter, and will generally
have some nasty ringing behavior. If you find this
counterintuitive, you can easily demonstrate it by
transforming some seemingly nice signal (such as a
square wave or sawtooth wave), chopping off the high
frequency components, and transforming back.

Bottom line: "Sampling theorem" and "practical
interpolation" are not the same thing.

If this doesn't help, please re-ask the question.