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jim
Guest
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 Posted:
Mon Nov 21, 2005 5:16 pm Post subject:
Re: question about non-uniform sampling? |
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William Hughes wrote:
| Quote: |
The simple point is that you can only predict the entire
signal from samples taken in the first half if this
signal is band limited. And you cannot find a band
limited signal that adequately approximates your real
world signal without processing the entire signal. So
there is no causality paradox.
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Yes, but there is one thing that seems to be generally ignored. Not only
does the signal have to be bandlimited - it has to be bandlimited rather
precisely and you have to possess knowledge of exactly how it is
bandlimited.
Take the concert example: if you have a set of uniform samples of an
entire concert you then have a FFT of the entire concert as well as the
capability to produce a continuous precisely bandlimited function of the
concert. You can now re-sample that continuous function with a
non-uniform scheme where you end up with a new set of samples of the
same number as before clustered in the first half of the concert. Can
you now reconstruct the entire concert? No, you can't unless you know
the exact time interval of the whole concert. It's not enough to simply
say the samples represent a bandlimited signal, you need to know what
the actual frequency limits are.
-jim
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David Tweed
Guest
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| Posted:
Mon Nov 21, 2005 5:16 pm Post subject:
Re: question about non-uniform sampling? |
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robert bristow-johnson wrote:
| Quote: | it seems to me that you need to solve a system of an infinite
number of equations that have an infinite number of unknowns,
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No, you only have one equation for each nonuniform sample you took.
The solution is the corresponding set (same number) of uniform samples.
You have to evaluate the sinc function N^2 times to get the coefficients
for the equations.
-- Dave Tweed |
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William Hughes
Guest
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| Posted:
Mon Nov 21, 2005 5:16 pm Post subject:
Re: question about non-uniform sampling? |
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jim wrote:
| Quote: | William Hughes wrote:
The simple point is that you can only predict the entire
signal from samples taken in the first half if this
signal is band limited. And you cannot find a band
limited signal that adequately approximates your real
world signal without processing the entire signal. So
there is no causality paradox.
Yes, but there is one thing that seems to be generally ignored. Not only
does the signal have to be bandlimited - it has to be bandlimited rather
precisely and you have to possess knowledge of exactly how it is
bandlimited.
Take the concert example: if you have a set of uniform samples of an
entire concert you then have a FFT of the entire concert as well as the
capability to produce a continuous precisely bandlimited function of the
concert.
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Nit. You can only produce a bandlimited approximation (unless
the concert was bandlimited (unlikely))
| Quote: | You can now re-sample that continuous function with a
non-uniform scheme where you end up with a new set of samples of the
same number as before clustered in the first half of the concert. Can
you now reconstruct the entire concert? No, you can't unless you know
the exact time interval of the whole concert. It's not enough to simply
say the samples represent a bandlimited signal, you need to know what
the actual frequency limits are.
|
Indeed. (There is of course a similar problem in the uniform case,
but it is not as extreme as the fact that you know the samples
are uniform gives you a lot of information, so it is possible
to calculate a frequency limit below which there is no ambiguity.
In the non uniform case we cannot do this without knowing
the length of the signal to be reconstructed.)
-William Hughes |
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Rob Johnson
Guest
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| Posted:
Mon Nov 21, 2005 5:16 pm Post subject:
Re: question about non-uniform sampling? |
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In article <4381C656.BA380516@acm.org>,
David Tweed <dtweed@acm.org> wrote:
| Quote: | robert bristow-johnson wrote:
it seems to me that you need to solve a system of an infinite
number of equations that have an infinite number of unknowns,
No, you only have one equation for each nonuniform sample you took.
The solution is the corresponding set (same number) of uniform samples.
You have to evaluate the sinc function N^2 times to get the coefficients
for the equations.
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Are you saying that you can exctly reproduce any band-limited signal
with finitely many samples? That is not only amazing, but also
impossible. Perhaps I am misinterpreting your article; it is hard
to tell if you are trying to exactly reproduce the original band
limited signal. Your article says: "Once you have these coefficients,
you can plug them into the reconstruction equation given previously to
reconstruct the original signal." Since you don't say "approximately"
or "exactly", it is hard to tell the intent. If you are approximating
the original signal, it would be nice to state that and what the error
is.
Certainly it is possible to construct a band-limited function that
agrees with the sampled function at finitely many sampled points.
There are infinitely many functions with the same band limit that
agree with the sampled function at the same sample points. However,
this is different than reconstructing the sampled signal.
Rob Johnson <rob@trash.whim.org>
take out the trash before replying |
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William Hughes
Guest
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| Posted:
Mon Nov 21, 2005 11:46 pm Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | William Hughes wrote:
...
But the vaiues of the requisite number of samples cannot be determined
until after the entire signal has been processed. So reconstruction
cannot begin until after the entire signal. There is no causality
paradox.
I can't imagine how causality entered into this discussion.
|
Your "counterexamples" have been reductio ad absurdum based
on causality (you cannot reconstruct this
because you don't know what it is yet).
| Quote: | As I
understood the original assertion about nonuniform sampling,
reconstruction is possible so long as the average sampling interval (or
was it rate?) meets the Nyquist criterion. For uniform sampling of a
signal of bandwidth B Hz and duration D seconds, at least 2BD samples
are needed. The assertion stated that 2BD samples also suffices for
nonuniform sampling, and that the location of the samples within the
signal is irrelevant.
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Yes, this was the original claim. (Note that we are assuming
that all samples are at exact times and of infinite precision).
| Quote: | There are still knowledgeable people here who think I am recalcitrant
for claiming that this won't work. I can only conclude that I haven't
adequately communicated the proposed conditions. Shall I try again?
Consider a two-second telephone monolog. B = 3.5 kHz; D = 2. By sampling
at 8 KHz, there is some processing margin. With uniform sampling, there
will be 16,000 samples, and according to the assertion, 16,000 samples
taken anywhere in the monolog suffice to characterize it.
|
Not quite, the signal must be band limited. The assertion
applies only after the filtering (but see below).
| Quote: | Before sampling, the signal is passed through a very sharp lowpass
filter that cuts off at 3.5 KHz. The cutoff of the analog filter is so
sharp and deep -- its impulse response is so long -- that the delay
through it is several days, but we're patient
|
So we start sampling long after the monologue is finished.
| Quote: | At the appropriate time,
sampling begins; not at 8 KHz, but 16. After one second of sampling, all
of the necessary 16,000 samples have been recorded. Although the filter
continues to produce output, the sampler can stop.
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Correct. (Note that the values of the samples we obtain depend
on the whole signal).
| Quote: |
I claim that the 16,000 samples so obtained meet the conditions of the
assertion but do not support the reconstruction of the entire monolog.
Do think my claim is false?
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Yes.
All your arguments so far have been
of the form "we can't reconstruct the signal yet,
we don't know what it is going to be". However, in the above
case we do know what the signal is going to be and that
information was used in creating the samples. Do you have another
argument?
[assume the first second was silence. Will the samples
obtained be 0. No Are they too small for practical
work. Yes.]
- William Hughes |
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William Hughes
Guest
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| Posted:
Mon Nov 21, 2005 11:54 pm Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | William Hughes wrote:
...
Indeed. (There is of course a similar problem in the uniform case,
but it is not as extreme as the fact that you know the samples
are uniform gives you a lot of information, so it is possible
to calculate a frequency limit below which there is no ambiguity.
In the non uniform case we cannot do this without knowing
the length of the signal to be reconstructed.)
From this I conclude you mean that spacing the samples nonuniformly
reduces the information they can yield.
|
Only to the extent that given samples taken at
an aribtrary time you cannot determine the length
of the sampled signal from the number of samples.
Given uniform samples and the sampling interval
you can. The difference in information does
not depend on the extent of the nonuniformity.
I've been saying that if the
| Quote: | nonuniformity is great enough, they become essentially useless. We're on
the same track. "No problem" becomes "no way".
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No. Making the samples less uniform does not reduce the
amount of information they hold.
- William Hughes |
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William Hughes
Guest
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| Posted:
Tue Nov 22, 2005 12:02 am Post subject:
Re: question about non-uniform sampling? |
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jim wrote:
| Quote: | William Hughes wrote:
Nit. You can only produce a bandlimited approximation (unless
the concert was bandlimited (unlikely))
Whether you think it resembles "the concert" or not is not relevant.
The important thing is we would have a bandlimited function which we can
hypothetically sample for the purpose of analyzing the process of
non-uniform sampling.
You can now re-sample that continuous function with a
non-uniform scheme where you end up with a new set of samples of the
same number as before clustered in the first half of the concert. Can
you now reconstruct the entire concert? No, you can't unless you know
the exact time interval of the whole concert. It's not enough to simply
say the samples represent a bandlimited signal, you need to know what
the actual frequency limits are.
Indeed. (There is of course a similar problem in the uniform case,
but it is not as extreme as the fact that you know the samples
are uniform gives you a lot of information, so it is possible
to calculate a frequency limit below which there is no ambiguity.
In the non uniform case we cannot do this without knowing
the length of the signal to be reconstructed.)
Yes, but that's far from a trivial problem
|
Dividing the length of the signal by the number of samples
is non-trivial?
-William Hughes |
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Jerry Avins
Guest
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| Posted:
Tue Nov 22, 2005 12:30 am Post subject:
Re: question about non-uniform sampling? |
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William Hughes wrote:
...
| Quote: | No. Making the samples less uniform does not reduce the
amount of information they hold.
|
It certainly reduces the information you can extract with practical
means. Careful preparation of the signal before it's sampled is really a
red herring. The usual case is something like a log of recorded
measurements, each with time of measurement noted. Suppose them to be
the level of a river or the speed of a ship. On most days, there are two
measurements, some have three or more, and for a few days there are
none. Given these data, How well can the continuous function be deduced?
...
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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Jerry Avins
Guest
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| Posted:
Tue Nov 22, 2005 12:36 am Post subject:
Re: question about non-uniform sampling? |
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William Hughes wrote:
| Quote: | ... Do you have another argument?
[assume the first second was silence. Will the samples
obtained be 0. No Are they too small for practical
work. Yes.]
|
Another argument: you can talk about it, but you can't do it.
The answer to the question about nonuniform sampling is simple: theory
and practice diverge. If the nonuniformity is small, approximate results
are possible. Any other answer holds out false hopes.
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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William Hughes
Guest
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| Posted:
Tue Nov 22, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | William Hughes wrote:
...
In other words it is possible in theory but not in practice.
OK. We agree on that.
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Except for the fact that you indicated you didn't think it
was possible in theory either.
| Quote: |
Lucy, in beginning this thread, asked,
###################
Hi all,
Can non-uniform sampled signal be used to perfectly reconstruct the
original continuous time signal?
What is the Nyquist sampling rate in the non-uniform case?
Thanks a lot!
###################
How would you answer her?
|
With something like
Believe it or not, it's the same as the uniform case ... the number
of samples over the time interval must exceed twice the bandwidth
of the signal.
I would relegate the fact that the Nyquist sampling theorem for uniform
sampling
has immediate practical application and the analgous therorem
for non-uniform sampling does not to more detailed discussion.
-William Hughes |
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Ron N.
Guest
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| Posted:
Tue Nov 22, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | The discussion started so:
lucy wrote:
Hi all,
Can non-uniform sampled signal be used to perfectly reconstruct the
original continuous time signal?
What is the Nyquist sampling rate in the non-uniform case?
Thanks a lot!
-L
I didn't see that as purely theoretical. Do you?
|
Can one "perfectly" reconstruct anything in a non-theoretical
sense? To do so requires a signal that has been passed
through a perfect brick wall bandlimiting filter, and a sampler
with absolutely no quantization error, which, except for some
special case signals (0V DC, etc.), exist only in theory.
In the practical sense, on can ask how much non-uniformity
will so many bits of precision, and such-and-so a prefilter,
still allow reconstruction within some error bound. But that
wasn't what the OP asked (or was asked in his/her homework
assignment.)
IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M |
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David Tweed
Guest
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| Posted:
Tue Nov 22, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | Another argument: you can talk about it, but you can't do it.
The answer to the question about nonuniform sampling is simple:
theory and practice diverge. If the nonuniformity is small,
approximate results are possible. Any other answer holds out
false hopes.
|
That's the crux. This discussion started out as purely theoretical.
No one here argues that it's practical, and you were the only one
trying to bring practice into the picture. But you seemed to be
trying to say that there was something theoretically impossible
about it.
-- Dave Tweed |
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Jerry Avins
Guest
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| Posted:
Tue Nov 22, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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David Tweed wrote:
| Quote: | Jerry Avins wrote:
Another argument: you can talk about it, but you can't do it.
The answer to the question about nonuniform sampling is simple:
theory and practice diverge. If the nonuniformity is small,
approximate results are possible. Any other answer holds out
false hopes.
That's the crux. This discussion started out as purely theoretical.
No one here argues that it's practical, and you were the only one
trying to bring practice into the picture. But you seemed to be
trying to say that there was something theoretically impossible
about it.
|
The discussion started so:
lucy wrote:
| Quote: | Hi all,
Can non-uniform sampled signal be used to perfectly reconstruct the
original continuous time signal?
What is the Nyquist sampling rate in the non-uniform case?
Thanks a lot!
-L
|
I didn't see that as purely theoretical. Do you?
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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William Hughes
Guest
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| Posted:
Tue Nov 22, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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Jerry Avins wrote:
| Quote: | William Hughes wrote:
...
No. Making the samples less uniform does not reduce the
amount of information they hold.
It certainly reduces the information you can extract with practical
means. Careful preparation of the signal before it's sampled is really a
red herring. The usual case is something like a log of recorded
measurements, each with time of measurement noted. Suppose them to be
the level of a river or the speed of a ship. On most days, there are two
measurements, some have three or more, and for a few days there are
none. Given these data, How well can the continuous function be deduced?
|
Possibly quite well possibly poorly. But the relevance
of this is nill. No one besides you seems to think that
anyone in this thread is claiming practical results for arbitrarily
non-uniform sampling. |
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Jerry Avins
Guest
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| Posted:
Tue Nov 22, 2005 1:16 am Post subject:
Re: question about non-uniform sampling? |
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William Hughes wrote:
...
| Quote: | In other words it is possible in theory but not in practice.
|
OK. We agree on that.
Lucy, in beginning this thread, asked,
###################
Hi all,
Can non-uniform sampled signal be used to perfectly reconstruct the
original continuous time signal?
What is the Nyquist sampling rate in the non-uniform case?
Thanks a lot!
###################
How would you answer her?
Jerry
--
Engineering is the art of making what you want from things you can get.
ŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻŻ |
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