tl;dr version: The justification for DSP (equi-distant samples) is the
Whittaker-Shannon interpolation formula, which follows from the Poisson
summation formula plus some hand-waving about distributions (dirac delta
theory). Am I right?
On Fri, Mar 28, 2014 at 4:50 AM, Ethan Duni
robert bristow-johnson wrote:
...
and in my opinion, a very small amount of hand-waving regarding the
Dirac delta (to get us to the same understanding one gets at the
sophomore or junior level EE) is *much* *much* easier to gain
understanding than farting around with the Dirac delta as a
rb-j, you wrote
again, all you really need is
+inf +inf
T SUM{ delta(t-nT) } = SUM{ e^(i 2 pi k/T t) }
n=-inf k=-inf
Precisely, and one way to get there is by starting from the Poisson
Summation Formula and taking f(n) = T dirac(t-nT)
On 2014-03-27, robert bristow-johnson wrote:
the *sampling* function is periodic (that's why we call it uniform
sampling), but the function being sampled, x(t), is just any
reasonably well-behaved function of t.
Ah, yes, that much is true. But in fact, if you look a bit further,
actually
On Mar 26, 2014, at 8:42 PM, Doug Houghton doug_hough...@sympatico.ca wrote:
I'm guessing this somehow scratches at the surface of what I've read about no
signal being properly band limited unless it's infinit.
Sure, in the same sense, we don’t properly sample to digital or properly
convert
On 2014-03-27, Doug Houghton wrote:
I understand the basics, my question is in the constraints that might
be imposed on the signal or functon as referenced by the theory.
The basic theory presupposes that the signal is square integrable and
bandlimited. That's pretty much it. If you want to
On 27/03/2014 3:23 PM, Doug Houghton wrote:
Is that making any sense? I'm struggling with the fine points. I bet
this is obvious if you understand the math in the proof.
I'm following along, vaguely.
My take is that this conversation is not making enough sense to give you
the certainty you
Hi Doug-
To address some of your general questions about Fourier analysis and
relationship to sampling theory:
Broadly speaking any reasonably well-behaved signal can be decomposed into
a sum of sinusoids (actually complex exponentials but don't worry about
that detail for now). There are
On Mar 26, 2014, at 10:07 PM, Doug Houghton doug_hough...@sympatico.ca
wrote:
so is there a requirement for the signal to be periodic? or can any
series of numbers be cnsidered periodic if it is bandlimited, or infinit?
Periodic is the best word I can come up with.
--
Well, no--you can
On 2014-03-27, Stefan Sullivan wrote:
Actually, yes there IS a requirement that it be periodic.
No, there is not. And the Shannon-Nyquist theorem isn't typically proven
under under any such assumption. Furthermore, it generalizes to settings
where periodicity isn't even an option.
Hi Doug-
Regarding this:
Terms like well behaived 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. As an example, I don't even know what the
instrinsic properties of a function may be in this
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
On 3/27/14 4:05 PM, Ethan Duni wrote:
Hi Doug-
Regarding this:
Terms like well behaived 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. As an example, I don't even know what the
instrinsic
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.
I may be misremembering, but can't non-standard analysis be used to make
that whole Dirac delta function approach
In the time when Einstein started to work on his theories, the main hip
and profound mathematics of the time came to be a consequence of the
important physics problems of the time, and mostly (if I'm not
forgetting some other factors) they were the higher maths, formulated as
functional
On 3/27/14 10:58 PM, Theo Verelst wrote:
I'm glad to see some influence of my repeated mention of some of my
theoretical concerns leads to thoughts getting formulated, and more,
up to quite some, precision being present.
well, Theo, i've been thinking (and writing about
Hi Robert-
i dunno what non-standard analysis you mean.
I'm referring to the stuff based on hyperreal numbers:
http://en.wikipedia.org/wiki/Hyperreal_number
These are an extension of the extended real numbers, where each hyperreal
number has a standard part (which is an extended real) and an
is your application?
From: Doug Houghton doug_hough...@sympatico.ca
To: A discussion list for music-related DSP music-dsp@music.columbia.edu
Sent: Wednesday, March 26, 2014 10:42 PM
Subject: [music-dsp] Nyquist–Shannon sampling theorem
I can't seem to get
The application is music. I understand the basics, my question is in the
constraints that might be imposed on the signal or functon as referenced
by the theory. Is it understood to be repeating? for lack of a better term,
essentually just a mash of frequencies that bever change from start to
sorry about all the attachments, didn't see that coming.
--
dupswapdrop -- the music-dsp mailing list and website:
subscription info, FAQ, source code archive, list archive, book reviews, dsp
links
http://music.columbia.edu/cmc/music-dsp
http://music.columbia.edu/mailman/listinfo/music-dsp
Hi Doug,
I think you’re overthinking this…
There is the frequency-sensitive requirement that you can’t properly sample a
signal that has frequencies higher than half the sample rate. For music, that’s
not a problem, since our ears have a significant band limitation anyway.
So, if we have a
There is the frequency-sensitive requirement that you can’t properly sample
a signal that has frequencies higher than half the sample rate. For music,
that’s not a problem, since our ears have a significant band limitation
anyway.
This is intuitive. I think perhaps what I'm asking has
It's my understanding that the fourier theory says any signal can be created
by summing various frequencies at various phases and amplitudes.
OK, now recall that the Fourier series describes a subset of “any signal” with
a subset of “various frequencies”. It’s more like one cycle of any
so is there a requirement for the signal to be periodic? or can any series
of numbers be cnsidered periodic if it is bandlimited, or infinit? Periodic
is the best word I can come up with.
--
dupswapdrop -- the music-dsp mailing list and website:
subscription info, FAQ, source code archive,
I'm guessing this somehow scratches at the surface of what I've read
about no signal being properly band limited unless it's infinit.
You're talking about Sinc filtering (ideal low pass filter), which is
essentially an IIR filter that needs infinite past and future samples.
In practice, a
On Mar 26, 2014, at 10:07 PM, Doug Houghton doug_hough...@sympatico.ca wrote:
so is there a requirement for the signal to be periodic? or can any series of
numbers be cnsidered periodic if it is bandlimited, or infinit? Periodic is
the best word I can come up with.
--
Well, no—you can
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