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question about non-uniform sampling?

Started by lucy November 12, 2005
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

lucy wrote:
> What is the Nyquist sampling rate in the non-uniform case?
um.. i'll take the risk of trying to answer and say that it must be the same one as if you had an uniform sampling rate matching to the shortest distance between two samples in your non-uniformly sampled signal. but i wouldnt be surprised if my answer was wrong or off-topic (i'm a newbie kinda)
"Michel Rouzic" <Michel0528@yahoo.fr> wrote in message 
news:1131784415.779792.104570@g14g2000cwa.googlegroups.com...
> > lucy wrote: >> What is the Nyquist sampling rate in the non-uniform case? > > um.. i'll take the risk of trying to answer and say that it must be the > same one as if you had an uniform sampling rate matching to the > shortest distance between two samples in your non-uniformly sampled > signal. but i wouldnt be surprised if my answer was wrong or off-topic > (i'm a newbie kinda) >
"shortest distance" - do you mean "longest"?
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 -- Dave Tweed
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!
This has been the subject of a few threads in the past. You might try Google Groups for some insight. Theory says yes if the signal is stationary, but practice is difficult. Given that the signal is bandlimited, a system of samples can be solved as n simultaneous equations. Spacing the samples uniformly in time simplifies the math but other solutions are possible. If the signal isn't stationary, then the samples must be close enough -- whatever that means -- to track the changes. When the samples bunch too closely, the disturbing effects of noise and truncation become overwhelmingly prominent. So yes; in theory, there are many circumstances in which it can be done. In practice, it is difficult. Fortunately, I have not so far had a need to try. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
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. > > 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? -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
lucy wrote:

> 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?
As others have said, if the average rate is fast enough it works. It then gets back to your other question, which is end effects for a time limited signal. I had an example in the past where someone goes into a concert hall and samples the signal at twice the required rate. This, then, is theoretically enough to include the following concert, which is not sampled at all. As you previously mentioned, time limited signals can't be band limited, so this only works for non-time limited signals. That means in infinite number of sample points. It also requires that the signal not be quantized, or at least not too coarsely. (Depending on how non-uniform it is.) -- glen
A specific, and limited, example is when you do have a reference that
indicates where in its cycle the signal is, regardless of time.

For instance with rotating machinery, you might have a reference signal
that tells you each time one of the shafts rotates to a given position.
Then, you can use that to resample the (time) signals you measured, so
that they are evenly spaced with respect to the rotation (usually
within one cycle of some part of the machinery). This then lets you
enforce that your samples always happen to sample complete cycles of
the rotation, and that gives you the happy effect that the signal is
then ideally sampled. (This corresponds to a case that I describe on
our web site: http://www.bores.com/courses/intro/freq/3_exact.htm )

This technique was called 'order processing' and was developed and
publicised by Hewlett Packard some years ago - I don't know if
references are still available or if the method is widely used still.

Chris
==============================
Chris Bore
BORES Signal Processing
www.bores.com

chris_bore@yahoo.co.uk wrote:
> A specific, and limited, example is when you do have a reference that > indicates where in its cycle the signal is, regardless of time. > > For instance with rotating machinery, you might have a reference signal > that tells you each time one of the shafts rotates to a given position. > Then, you can use that to resample the (time) signals you measured, so > that they are evenly spaced with respect to the rotation (usually > within one cycle of some part of the machinery). This then lets you > enforce that your samples always happen to sample complete cycles of > the rotation, and that gives you the happy effect that the signal is > then ideally sampled. (This corresponds to a case that I describe on > our web site: http://www.bores.com/courses/intro/freq/3_exact.htm ) > > This technique was called 'order processing' and was developed and > publicised by Hewlett Packard some years ago - I don't know if > references are still available or if the method is widely used still.
That amounts to sampling that's uniform in space, rather than in time. The results have spatial significance, rather than temporal. It's so standard in image processing that we don't even think about it. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
lucy wrote:
> Can non-uniform sampled signal be used to perfectly reconstruct the > original continuous time signal?
Interesting homework question old man. The answer is probably no. All the sample points could end up at the same point in time without further specification on the non-uniformity.
> What is the Nyquist sampling rate in the non-uniform case?
DC or infinity? YMMV. -- rhn