On 3/27/14 2:20 PM, Doug Houghton wrote:
Some great replies, gives me a lot to think about
Terms like "well behaved" when applied to the "functon" make me wonder
what
stipulations might be implied by the language that you'd have to be a
formal
mathmatician to interpret.
i'm not so terribly worried about the existence of audio or music
signals that are not sufficiently "well behaved"
As an example, I don't even know what the
instrinsic properties of a "function" may be in this context.
the context are audio and music signals. the end receptacle of these
signals are our ears and brains. i'm pretty sure that bandwidth
restrictions apply. that *really* nails down the "well-behaved". the
are continuous-time, finite power signals that are also bandlimited.
whether it's bandlimited to 22.05 kHz or to 48 kHz or 96 kHz, doesn't
matter. that is a quantitative issue. doesn't change the validity nor
the qualitative conditions for the theorem.
Since it's an infinit series I suppose it doesn't really matter, given
enough time you could prove out any rational requirement? which is why
you
can throw math at it.
yup. and then, when you get practical about reconstruction, you realize
that the infinite series of sinc() functions will turn into a finite
approximation to the same thing. one approach, to a finite sum, is to
truncate the sinc() function to a finite length. that is the same as
applying a rectangular window (which is often the worst kind), so then
you try the sinc() function windowed by a good window function. now
that is a slightly different low-pass filter than the ideal brick-wall
filter (which has a sinc() function for its impulse response). so then
you investigate how bad is it from different points of view (usually a
spectral POV over some frequencies of interest).
If it was just a bunch of random numbers that started
somewhere and stopped somewhere, I doubt anyone would be writing
equations
that mean anything.
???
um, you can model *any* linear and time-invariant signal reconstruction
problem (or "interpolation problem") as a specific case of a string of
impulses weighted with the samples values, x[n], going into a particular
low-pass filter. you can write equations for that. in both the time
domain (you would use these equations to implement the interpolation)
and in the frequency domain: [what does this LPF do to the baseband
signal? what does it do to the images?]
so, for even polynomial interpolation (like Lagrange or Hermite), you
can model it as a convolution with an impulse response and you can
compute the Fourier transform of that continuous-time impulse response
and see how good or how bad the frequency response is. how bad does it
kill the images and how safe is it to your original signal?
you can write equations for that, and they mean something.
I'd guess we would turn to statistics at that pint to
supply some context.
but you can make some good guesses instead of doing this as a
complicated statistical process
As a broad answer to questions posted in a couple of the replies, my
interest lies in imrpoving my understanding of specifically what the SNST
proves, and the requirements for it to be valid.
take a look at that earlier wikipedia version to show you. if you
ideally uniformly sample, your spectrum is repeatedly shifted (by
integer multiples of the sampling frequency, these are called "images")
and added together. to recover the original signal, you must remove all
of the images, yet preserve the original (that's what the brick-wall LPF
does). Only if there is no overlap of the adjacent images is it
possible to recover the original spectrum. To make that happen, the
sampling frequency must exceed twice the frequency of the highest
frequency component of a bandlimited signal. if it does not do that,
images will overlap and once you add two numbers, it's pretty hard to
separate them. so the overlapped images can be thought of as
non-overlapped frequency components that "just happened" to be at those
frequencies. those frequency components are called "aliases".
take a look at
https://en.wikipedia.org/w/index.php?title=Nyquist%E2%80%93Shannon_sampling_theorem&oldid=217945915
the sampling theorem is actually quite simple to be expressed
rigorously. it is, at least, if you accept the EE notion of the Dirac
delta function and not worry so much about it "not really being a
function", which is literally what the math folks tell us.
--
r b-j [email protected]
"Imagination is more important than knowledge."
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