question about non-uniform sampling?

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Jerry Avins
Guest

Posted: Fri Nov 18, 2005 5:16 pm Post subject: Re: question about non-uniform sampling?

Ron N. wrote:

Quote:

Jerry Avins wrote:

Again?! The second half is not sampled at all.

I didn't say that the second half was sampled (although that would
be one method of implementing the bandpass filter). I said the second
half would be input into the low pass filter before that filter settled
enough for the sampler to sample the first half.

Ummm... If the second half isn't sampled, how is it fed to the filter?

Quote:

Instead, the first half
is oversampled enough so the the total number of samples -- all taken
from the first half -- suffices for the entire composition.

I note that you said samples "from" the first half, not "during" the
first half. They're different due to the low pass filter delay.

The output of the filter may be delayed, but there's no delay at the
input. Sampled _from_ the first half are put in _during_ the first half
with at most trivial delay. When the output emerges doesn't affect the
plot of this fantasy.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

David Tweed
Guest

Posted: Sat Nov 19, 2005 9:15 am Post subject: Re: question about non-uniform sampling?

robert bristow-johnson wrote:

Quote:

David Tweed at dtweed@acm.org wrote on 11/12/2005 09:32:
lucy wrote:
Can non-uniform sampled signal be used to perfectly reconstruct the
original continuous time signal?

Yes, but it isn't easy.

What is the Nyquist sampling rate in the non-uniform case?

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.

See here (questions 2 and 3) for a little more detail:
http://www.circuitcellar.com/library/eq/136/index.asp

hi Dave,

could you take a look at the paper that Bob Adams did in 1992 that i
reference here:

http://groups.google.com/group/comp.dsp/msg/ae7fe00eb3c8622b

i haven't cracked your brief analysis, but does that accomplish what
i was hoping would be shown that if your average sample rate is more
than twice the bandwidth, then random sampling will also be sufficient
for reconstruction?

Sorry for not replying sooner -- things got kind of busy around here
this past week.

I can't easily get at the paper itself, but your description in that
message leads me to believe that it does. Obviously, the samples have
to be unique.

I just browsed down through the rest of the thread -- it's amazing
what it turned into, thanks mainly to Jerry's stubbornness about
causality and how it relates to strict band-limiting.

-- Dave Tweed

Jerry Avins
Guest

Posted: Sat Nov 19, 2005 9:15 am Post subject: Re: question about non-uniform sampling?

David Tweed wrote:

...

Quote:

I just browsed down through the rest of the thread -- it's amazing
what it turned into, thanks mainly to Jerry's stubbornness about
causality and how it relates to strict band-limiting.

I didn't inject causality into this thread. Causality has nothing to do
with the point I tried to make. I'll try again.

A telephone conversation is sampled 8000 times a second, or 480,000
times a minute. A 4-minute monolog uses 1,920,000 samples. If it is
required only that this rate be met as an average, how may the samples
be distributed in time? Will 1,920,000 samples obtained at any time
during the monolog suffice? Suppose the sampler runs for one minute,
collecting all 1,920,000 samples in that time; can the entire monolog be
reconstructed? Suppose the sampler runs at 16,000 per second in each
even second and not at all in the odd ones? How non-uniform can the
sampling be and still support reconstruction?

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Guest

Posted: Sat Nov 19, 2005 9:15 am Post subject: Re: question about non-uniform sampling?

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 read about half the posts here so I hope I am not missing something
that is already said. The original post here seems to be about
perfectly reconstructing the signal, which would imply perfect
sampling. Other posts deal with how practical it is to reconstruct the
original signal in terms of error. The problem described by these later
posts would be dependent on the type of signal. For instance someone
gave the example of trying to sample music every half hour. Another
easier problem which was not addressed but similar is only trying to
estimate the power spectrum and not the entire signal.

Posters have pointed out that the ability to cheat Nyquest depends on
how stationary the signal is. A stationary signal is a signal where the
Fourier transform is described by a sum of impulse functions (AKA delta
functions). As the signal becomes less stationary the bandwidth of
these spikes will widen. To the extreme of white noise where the power
spectrum is a constant.

Computationally, if the signal spectrum is fairly board, I would
suggest the singular value based pseudo inverse. If there is a lot of
noise in the system then this won't give good result. To see how good
a fit is there is something called the Akaike information. It is an
information quantity that helps evaluate the tradeoff between the order
of fit and the goodness of fit.

If there are only a few frequency a better approach would be to use an
order recursive method of fitting the signal. With an order recursive
approach you can obtain an order 1 fit all the way up to an order N fit
with the same amount of computation it takes to do an order N fit. You
can use singular value decomposition to see how statically significant
each parameter you add to your fit is and you can use the method of
leaps and bounds to reduce the amount of fits you need to try.

There is a technique of model identification that uses all these
techniques called CVA (canonical vector analysis) which I believe was
created by Wallace E. Larimore. He crated software the runs in MATLAB
that uses these techniques which is called ADAPTx. Sine the software is
typically applied to control systems the models identified are
typically dynamic models. I assume he published some papers about this
technique but has not yet published a book.

David Tweed
Guest

Posted: Sat Nov 19, 2005 5:15 pm Post subject: Re: question about non-uniform sampling?

Jerry Avins wrote:

Quote:

How non-uniform can the sampling be and still support reconstruction?

It doesn't matter.

I won't get drawn into this debate; you've already conceded the point.
I just don't understand why you continue to pound away at it. If you're
making a different point, you're not making yourself clear enough to
the rest of us here.

On Tuesday, you wrote:

Quote:

I'm talking about precisely what you are. A signal that is truly
bandlimited isn't time limited, and vice versa. In the real world,
signals with finite duration can bandlimited well enough so that
we can deal with them. But when one becomes pedantic about what is
theoretically possible, on must be likewise aware of what is
theoretically impossible.

But in all the subsequent posts, you never explained exactly what it
is you feel is theoretically impossible. You keep talking in allegories
about symphonies and forests.

Do you understand the dual of what you said above? It goes like this:
Signals with finite bandwidth can be time-limited well enough so that
we can deal with them. Think about it.

-- Dave Tweed

Jerry Avins
Guest

Posted: Sat Nov 19, 2005 5:15 pm Post subject: Re: question about non-uniform sampling?

David Tweed wrote:

Quote:

Jerry Avins wrote:

How non-uniform can the sampling be and still support reconstruction?

It doesn't matter.

I won't get drawn into this debate; you've already conceded the point.
I just don't understand why you continue to pound away at it. If you're
making a different point, you're not making yourself clear enough to
the rest of us here.

On Tuesday, you wrote:

I'm talking about precisely what you are. A signal that is truly
bandlimited isn't time limited, and vice versa. In the real world,
signals with finite duration can bandlimited well enough so that
we can deal with them. But when one becomes pedantic about what is
theoretically possible, on must be likewise aware of what is
theoretically impossible.

But in all the subsequent posts, you never explained exactly what it
is you feel is theoretically impossible. You keep talking in allegories
about symphonies and forests.

Do you understand the dual of what you said above? It goes like this:
Signals with finite bandwidth can be time-limited well enough so that
we can deal with them. Think about it.

I understand, Dave. I believe I made that point early on. (Read the
paragraph three up from here.) I wrote elsewhere that such
approximations are all we can do, and that they are adequate.

It is impossible to reconstruct a signal from a set of samples that have
large gaps in time, even if the total number of samples is larger than
the number of uniformly spaced samples that are enough. Although
mathematics tells us that it is possible if the signal is perfectly
bandlimited, mathematics also tells us that such perfectly bandlimited
signals don't exist. The larger the gaps in the sampling sequence, the
worse the reconstruction will be. Where the samples are matters.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Guest

Posted: Sun Nov 20, 2005 1:15 am Post subject: Re: question about non-uniform sampling?

robert bristow-johnson wrote:

Quote:

in article 4375FA3F.921CBB9A@acm.org, David Tweed at dtweed@acm.org wrote on
11/12/2005 09:32:

lucy wrote:
Can non-uniform sampled signal be used to perfectly reconstruct the
original continuous time signal?

Yes, but it isn't easy.

What is the Nyquist sampling rate in the non-uniform case?

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.

That sounds like a sensible conclusion but how do we define bandwidth?
For instance, if you have one peak in the frequency domain, you can
measure the bandwidth as the width of the peak. But what if you have
more then one peek? If they were the same amplitude I would assume you
could add the peaks up. If they were we different amplitudes I would
think you would want to weight them somehow.

Guest

Posted: Sun Nov 20, 2005 1:15 am Post subject: Re: question about non-uniform sampling?

robert bristow-johnson wrote:

Quote:

in article 1132428061.211802.186400@g49g2000cwa.googlegroups.com,
JohnCreighton_@hotmail.com at JohnCreighton_@hotmail.com wrote on 11/19/2005
14:21:

robert bristow-johnson wrote:
in article 4375FA3F.921CBB9A@acm.org, David Tweed at dtweed@acm.org wrote on
11/12/2005 09:32:

lucy wrote:
Can non-uniform sampled signal be used to perfectly reconstruct the
original continuous time signal?

Yes, but it isn't easy.

What is the Nyquist sampling rate in the non-uniform case?

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.

careful with your quoting. i don't think i wrote a single word in the above
quote ostensibly attributed to me.

To me all it looks like is you quoted someone else. And I believe that

is all I attributed to you. So I would say it is correct but perhaps
unnecessary.

Ron N.
Guest

Posted: Sun Nov 20, 2005 1:15 am Post subject: Re: question about non-uniform sampling?

Jerry Avins wrote:

Quote:

David Tweed wrote:
Jerry Avins wrote:

How non-uniform can the sampling be and still support reconstruction?
....
It is impossible to reconstruct a signal from a set of samples that have
large gaps in time, even if the total number of samples is larger than
the number of uniformly spaced samples that are enough. Although
mathematics tells us that it is possible if the signal is perfectly
bandlimited, mathematics also tells us that such perfectly bandlimited
signals don't exist. The larger the gaps in the sampling sequence, the
worse the reconstruction will be. Where the samples are matters.

We don't get mathematically perfectly bandlimited signals in the
real world. However, engineers succeed quite well at resampling
signals using imperfectly bandlimited signals. How well one can
resample depends on how close the real world bandlimiting gets to
the ideal. The better the pre-sampling bandlimit filter, the better
one can resample that "almost" bandlimited signal, including some
ability to deal with wider gaps in the uniformity of the sampling.
Higher quality bandlimiting filters which allow gap replacement
also have a longer delay (before sampling) so that causality is
not violated.

It's not a binary decision.

Your thought experiment fails because you failed to hypothesize
a "good enough" pre-sampling filter.

IMHO. YMMV.
--
rhn A.T nicholson d.O.t C-o-M

robert bristow-johnson
Guest

Posted: Sun Nov 20, 2005 1:15 am Post subject: Re: question about non-uniform sampling?

in article 1132428061.211802.186400@g49g2000cwa.googlegroups.com,
JohnCreighton_@hotmail.com at JohnCreighton_@hotmail.com wrote on 11/19/2005
14:21:

Quote:

careful with your quoting. i don't think i wrote a single word in the above
quote ostensibly attributed to me.

Quote:

--

r b-j rbj@audioimagination.com

"Imagination is more important than knowledge."

Jerry Avins
Guest

Posted: Sun Nov 20, 2005 1:15 am Post subject: Re: question about non-uniform sampling?

Ron N. wrote:

Quote:

Jerry Avins wrote:

David Tweed wrote:

Jerry Avins wrote:

How non-uniform can the sampling be and still support reconstruction?

...

It is impossible to reconstruct a signal from a set of samples that have
large gaps in time, even if the total number of samples is larger than
the number of uniformly spaced samples that are enough. Although
mathematics tells us that it is possible if the signal is perfectly
bandlimited, mathematics also tells us that such perfectly bandlimited
signals don't exist. The larger the gaps in the sampling sequence, the
worse the reconstruction will be. Where the samples are matters.

We don't get mathematically perfectly bandlimited signals in the
real world. However, engineers succeed quite well at resampling
signals using imperfectly bandlimited signals. How well one can
resample depends on how close the real world bandlimiting gets to
the ideal. The better the pre-sampling bandlimit filter, the better
one can resample that "almost" bandlimited signal, including some
ability to deal with wider gaps in the uniformity of the sampling.
Higher quality bandlimiting filters which allow gap replacement
also have a longer delay (before sampling) so that causality is
not violated.

It's not a binary decision.

Your thought experiment fails because you failed to hypothesize
a "good enough" pre-sampling filter.

OK, do it then. Sample a signal uniformly at four times the prescribed
Nyquist rate, then discard a contiguous group consisting of half of
them. There remains twice the minimum number needed for reconstruction.
Reconstruct. Display the result and the original. When I see (or hear)
it, I'll believe that the locations of the samples in time can be arbitrary.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Jerry Avins
Guest

Posted: Sun Nov 20, 2005 1:15 am Post subject: Re: question about non-uniform sampling?

JohnCreighton_@hotmail.com wrote:

Quote:

robert bristow-johnson wrote:

in article 4375FA3F.921CBB9A@acm.org, David Tweed at dtweed@acm.org wrote on
11/12/2005 09:32:

lucy wrote:

Can non-uniform sampled signal be used to perfectly reconstruct the
original continuous time signal?

Yes, but it isn't easy.

What is the Nyquist sampling rate in the non-uniform case?

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.

That sounds like a sensible conclusion but how do we define bandwidth?
For instance, if you have one peak in the frequency domain, you can
measure the bandwidth as the width of the peak. But what if you have
more then one peek? If they were the same amplitude I would assume you
could add the peaks up. If they were we different amplitudes I would
think you would want to weight them somehow.

The bandwidth if a signal is simply the frequency range needed to
encompass all of the energy in it. In practice, when we say a signal has
a certain bandwidth, we mean that the energy outside that range is small
enough not to bollux up what the signal is intended to accomplish.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Guest

Posted: Sun Nov 20, 2005 1:15 am Post subject: Re: question about non-uniform sampling?

glen herrmannsfeldt wrote:

Quote:

Jerry Avins wrote:

Steve Underwood wrote:

(snip)
There is nothing wrong with any
extreme of non-uniformity in a purely mathematical sense. That is in a
world with infinite sampling precision and no noise due to the
converter itself.

It's also a world where signals exist for all time. I doesn't matter how
precisely one can sample and how often, nothing can be known about a
speech yet to be given, even if the mathematics of nonuniform and highly
clumped sampling shows that it can.

I am not so sure what quantum mechanics says about this.

The only thing quantum mechanics says about anything
that is not quantum mechanics is the scattering strength of
100,000 GeV particles. Which is why the only people who
even use QM are mathematicians, since both Giga's and eV's
are stastical propertites, not physical properties.

Since there is no conversion from an eV to any other
observalble macroscopic property that isn't first filtered through

General Relavity, Protons, and The Eiffel Tower, rather than
sampling theory.

Much can even said about speeches yet t be given,
since the only people who even listen to them
are the speaker's speahwriter rather than the speaker.
Which is why blow-up barbie dolls still confuse
Behaviourists, more than they do Barbie.

David Tweed
Guest

Posted: Sun Nov 20, 2005 1:15 am Post subject: Re: question about non-uniform sampling?

JohnCreighton_@hotmail.com wrote:

Quote:

David Tweed at dtweed@acm.org wrote on 11/12/2005 09:32:
lucy wrote:
What is the Nyquist sampling rate in the non-uniform case?

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.

That sounds like a sensible conclusion but how do we define bandwidth?
For instance, if you have one peak in the frequency domain, you can
measure the bandwidth as the width of the peak. But what if you have
more then one peek? If they were the same amplitude I would assume you
could add the peaks up. If they were we different amplitudes I would
think you would want to weight them somehow.

The bandwidth is defined as the bandwidth of the perfect brick-wall
low-pass or band-pass antialias filter you ran the signal through
*prior* to sampling it. The impulse response of this filter is the
interpolation function used in the system of equations you have to
solve as shown in my writeup.

-- Dave Tweed

Guest

Posted: Sun Nov 20, 2005 1:16 am Post subject: Re: question about non-uniform sampling?

David Tweed wrote:

Quote:

JohnCreighton_@hotmail.com wrote:
David Tweed at dtweed@acm.org wrote on 11/12/2005 09:32:
lucy wrote:
What is the Nyquist sampling rate in the non-uniform case?

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.

That sounds like a sensible conclusion but how do we define bandwidth?
For instance, if you have one peak in the frequency domain, you can
measure the bandwidth as the width of the peak. But what if you have
more then one peek? If they were the same amplitude I would assume you
could add the peaks up. If they were we different amplitudes I would
think you would want to weight them somehow.

The bandwidth is defined as the bandwidth of the perfect brick-wall
low-pass or band-pass antialias filter you ran the signal through
*prior* to sampling it. The impulse response of this filter is the
interpolation function used in the system of equations you have to
solve as shown in my writeup.

But, what mathematicians don't seem to understand about sampling
theory is that the perfect brickwall low-pass filter isn't made of
bricks.
Its a forest made of trees, leaves, and lakes, rather than bricks
and mrrors.
Since the rocks in the forest need no interpolation,
the lake surface needs no idiiot QM chemists and thermostats,
and they automatically generate generalized sampling theory,
both non-uniform sampling theory, uniform sampling,
and non-Gaussian interpolation functions.

Quote:

-- Dave Tweed

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