Sampling with non-equidistant sampling period

Started by Jo Steinmann October 11, 2003
Hi,

I'm interested in sampling a signal with a non equidistant sampling 
schema, e.g.:
T1, T2, T1, T2, T1, T2, ...

It means, sampling at times:
t1 = 0
t2 = t1 + T1
t3 = t2 + T2
t4 = t3 + T1
t5 = t4 + T2
etc.

- Who knows how to analyse the spectrum of such a data sequence?

- What are the key-words used by the "community" for sampling with 
non-equidistant sampling period?

- What are the keywords for sampling with random sampling time sequence?

- What is the Nyquist frequency in such a case?

- Who has some links concering this topic?

Thanks,
Jo.

Jo Steinmann wrote:
> I'm interested in sampling a signal with a non equidistant sampling > schema, e.g.: > T1, T2, T1, T2, T1, T2, ... > > It means, sampling at times: > t1 = 0 > t2 = t1 + T1 > t3 = t2 + T2 > t4 = t3 + T1 > t5 = t4 + T2 > etc. > > - Who knows how to analyse the spectrum of such a data sequence?
I do. :-) Basically, you can re-sample the data so that the sampling periods are uniform, and then apply all of the usual tools.
> - What are the key-words used by the "community" for sampling with > non-equidistant sampling period? > > - What are the keywords for sampling with random sampling time sequence?
nonuniform sampling
> - What is the Nyquist frequency in such a case?
the number of samples, divided by the period over which they're taken, divided by 2
> - Who has some links concering this topic?
http://www.circuitcellar.com/library/eq/136/index.asp See problems 2 and 3. -- Dave Tweed
Note that interpolating regularly spaced samples with sincs make sense when
you consider those samples to be a part of an infinite sequence, because of
all the functions that are non-zero at sample time t=0, and zero at all the
other integral sample times, the sinc is the one with the narrowest
spectrum -- the only one that satisfies the Nyquist criterion.  In other
words, sinc interpolation is the only kind of linear interpolation that is
orthogonal, regardless of the number of samples, and satisfies the Nyquist
criterion.

If your samples are randomly spaced, then there is no rationale I can see
for choosing one family of bandlimited functions over another, because there
is no you can't ensure orthogonality with other samples you might get in the
future.

If your sample periods follow a periodic pattern, however, then you can and
probably should choose a basis that preservers orthogonality for any number
of samples, like the sinc does for uniform sampling.  There will be exactly
one such basis that perserves the Nyquist criterion at average_fs/2.

In the example the poster gave, we have samples at t=0, U, T, T+U, 2T, 2T+U,
etc.  The average sampling frequency is 2/T, and the whole pattern repeats
in time T.  It is easy to construct the appropriate interpolations functions
as follows:

To make the interpolation function for samples at t=kT, start with the
sinusoid that is 0 at all kT+U, then multiply it by the sinc that is 1 at
t=0 and zero at all other kT.  The result is non-zero at 0 and zero at all
other kT or kT+U.

To make the interpolation function for the samples at t=kT+U, start with the
sinusoid that is zero at all kT, then multiply it by the sinc that is 1 at
t=U and zero at all the other kT+U.  The result is non-zero at U and zero at
all other kT or kT+U.

Finally, note that in both cases, we form the interpolation function by
multiplying two other functions with spectrums that range from -1/2T to
1/2T.  Since pointwise multiplication in the time domain is convolution in
the frequency domain, we know the spectrums range from -1/T to 1/T, which
is -fs/2 to fs/2, so the Nyquist criterion is preserved.

You can use similar methods to interpolate from any periodic pattern of
samples.  If there are N sampling instants in the pattern, each basis
function is constructed from N-1 sinusoids and 1 sinc, to place the zeros
where they need to be.


"Matt Timmermans" <mt0000@sympatico.nospam-remove.ca> wrote in message
news:<h71ib.8177$G_.636289@news20.bellglobal.com>...
> Note that interpolating regularly spaced samples with sincs make sense when > you consider those samples to be a part of an infinite sequence, because of > all the functions that are non-zero at sample time t=0, and zero at all the > other integral sample times, the sinc is the one with the narrowest > spectrum -- the only one that satisfies the Nyquist criterion. In other > words, sinc interpolation is the only kind of linear interpolation that is > orthogonal, regardless of the number of samples, and satisfies the Nyquist > criterion. > > If your samples are randomly spaced, then there is no rationale I can see > for choosing one family of bandlimited functions over another, because there > is no you can't ensure orthogonality with other samples you might get in the > future. > > If your sample periods follow a periodic pattern, however, then you can and > probably should choose a basis that preservers orthogonality for any number > of samples, like the sinc does for uniform sampling. There will be exactly > one such basis that perserves the Nyquist criterion at average_fs/2. > > In the example the poster gave, we have samples at t=0, U, T, T+U, 2T, 2T+U, > etc. The average sampling frequency is 2/T, and the whole pattern repeats > in time T. It is easy to construct the appropriate interpolations functions > as follows: > > To make the interpolation function for samples at t=kT, start with the > sinusoid that is 0 at all kT+U, then multiply it by the sinc that is 1 at > t=0 and zero at all other kT. The result is non-zero at 0 and zero at all > other kT or kT+U. > > To make the interpolation function for the samples at t=kT+U, start with the > sinusoid that is zero at all kT, then multiply it by the sinc that is 1 at > t=U and zero at all the other kT+U. The result is non-zero at U and zero at > all other kT or kT+U. > > Finally, note that in both cases, we form the interpolation function by > multiplying two other functions with spectrums that range from -1/2T to > 1/2T. Since pointwise multiplication in the time domain is convolution in > the frequency domain, we know the spectrums range from -1/T to 1/T, which > is -fs/2 to fs/2, so the Nyquist criterion is preserved. > > You can use similar methods to interpolate from any periodic pattern of > samples. If there are N sampling instants in the pattern, each basis > function is constructed from N-1 sinusoids and 1 sinc, to place the zeros > where they need to be.
more general approach: This is a special case of the following non-uniform sampling. If randomly distributed N samples for the interval[0, NT) (and periodic extension there of if necessary) satisfies Nyquist sampling rate, one can consider the following thought experiment. If only one sample in the interval [0,NT) is taken at t_k, its FT will be aliased N times or spectrum is N-folded. If signal values outside [0,NT) is negligible (finite support signal), the FT will be C*f(t_k) C being a constant such as T/2pi*exp(j*t_k*omega) (see Fourier Shift Theorem). Otherwise, it will be C*F~(0), F~(0) meaning aliased spectrum at DC. In either case, these values are known. Hence if you have N samples, you can set up a system of equations (finite support case) f(t_0) | phase row vector at omega=0 | | F(0) | f(t_1) | phase row vector at omega=2pi/T | | F(2pi/T) | f(t_2) = | phase row vector at omega=2*2pi/T | | F(2*2pi/T)| . . . . . . . . . f(t_N-1) |phase row vector at omega=(N-1)*2pi/T | | F((N-1)*2pi/T)| where F(omega) <-> f(t). Now all you need is a matrix inverse to get a spectum at uniform interval. If all the samles are at uniform intervals, the matrix becomes a DFT matrix. This concept can be extended to 2-D case, but the matrix becomes singular for particular sampling patterns even though the number of samples per unit area satisfies Nyquist rate. Ref: S.P. Kim, N.K. Bose,"Reconstruction of 2-D Discrete Signals from Nonuniformly Spaced Samples,", IEE Proceedings - F, Radar and Signal Processing, Vol. 137, Pt F, No.3, pp, 197-204, June, 1990. seung kim
"Seung" <kim.seung@sbcglobal.net> wrote in message
news:fdf92243.0310122348.1fab632b@posting.google.com...
> more general approach:
No, it's not the same thing at all. There are infinitely many bind-limited signals that will match N samples over [0,NT). You are using a Fourier basis to pick one... but why *that* one? Dave uses a sinc basis to pick a different one. Neither of you will pick the one I do, so your method is not a generalization. If the sampling pattern (not the signal) is periodic but still nonuniform, you can use the technique I outlined to make a more refined choice that extends to any number of samples and lets you linearly combine output from disjoint sample sets, just like sinc interpolation does. Essentially, you solve the matching problem for infinite support, and then select the basis vectors for the samples you actually have.
If you are looking for an estimator for a smooth, continuous underlying
spectrum, I'm not sure what to do.

If you are looking to detect or measure periodic signals in unevenly
spaced data, look for "Lomb-Scargle periodogram" (or just "Lomb periodogram"),
and check out Larry Bretthorst's papers on a Bayesian look at this question:

http://bayes.wustl.edu/glb/bib.html

Look at #24-26.

-Tom Loredo

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