Non uniform sampling

Hi all,

Suppose a signal is known a prior. Time domain as well frequency
domain characteristic is pretty defined. Say fs is the sampling
frequency as per
Nyquist criteria. Number of samples in 1 sec is N. 

Let us think two cases 

1. Uniform sampling i.e sample interval is constant, I guess this is
the
usual case in most of the applications.

2. We keep N same in I sec but a variable sample interval i.e
nonuniform.
Steep signal envelope is sliced densely than flat signal envelope.

Will the second case has advantage or better performance  when it is
exposed in communication channel(assume noise and impairments) and
later recovered by the same receiver.

This could be a stupid question to an expert but it just came into my
mind.

Regards,
Santosh

"santosh nath" <santosh.nath@ntlworld.com> wrote in message
news:6afd943a.0307090755.37798597@posting.google.com...
> Hi all,
>
> Suppose a signal is known a prior. Time domain as well frequency
> domain characteristic is pretty defined. Say fs is the sampling
> frequency as per
> Nyquist criteria. Number of samples in 1 sec is N.
>
> Let us think two cases
>
> 1. Uniform sampling i.e sample interval is constant,
> I guess this is the
> usual case in most of the applications.
>
> 2. We keep N same in I sec but a variable sample
> interval i.e nonuniform. Steep signal envelope is sliced
> densely than flat signal envelope.

> Will the second case has advantage or better performance  when it is
> exposed in communication channel(assume noise and impairments) and
> later recovered by the same receiver.

There may be some kinds of signals that would make sense for.  Consider,
though, that if you do non-uniform sampling you must supply the sampling
times along with the sampled signal.

Consider, though, the MPEG digital video formats.  While they do use a
uniform frame rate they vary the amount of information stored (bits used)
for each frame.   In some sense they use a non-uniform spatial (the x and y
axes of the frame) sampling, concentrating where the picture is more
complex.

Audio compression systems usually break up the spectrum into frequency bands
and apply compression algorithms to those bands, after the signal was
uniformly sampled to begin with.

Both systems have the effect of reduced sampling of some parts of the
signal, though not quite the same as a change in sampling rate.

-- glen

santosh nath wrote:
> 
> Hi all,
> 
> Suppose a signal is known a prior. Time domain as well frequency
> domain characteristic is pretty defined. Say fs is the sampling
> frequency as per
> Nyquist criteria. Number of samples in 1 sec is N.
> 
> Let us think two cases
> 
> 1. Uniform sampling i.e sample interval is constant, I guess this is
> the
> usual case in most of the applications.
> 
> 2. We keep N same in I sec but a variable sample interval i.e
> nonuniform.
> Steep signal envelope is sliced densely than flat signal envelope.
> 
> Will the second case has advantage or better performance  when it is
> exposed in communication channel(assume noise and impairments) and
> later recovered by the same receiver.
> 
> This could be a stupid question to an expert but it just came into my
> mind.
> 
> Regards,
> Santosh

Uniform spacing is an advantage for analysis. Samples so close that that
they aren't independent count for little. Aside from that, given that
uniform sampling sufficiently often completely specifies the signal, and
that fewer than that can't be guaranteed to do that, what do you mean by
"advantage"?

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

"Jerry Avins" <jya@ieee.org> wrote in message
news:3F0C6A31.2710FD98@ieee.org...


(snip on  non-uniform sample spacing question)

> Uniform spacing is an advantage for analysis. Samples so close that that
> they aren't independent count for little. Aside from that, given that
> uniform sampling sufficiently often completely specifies the signal, and
> that fewer than that can't be guaranteed to do that, what do you mean by
> "advantage"?

Consider a digitized telephone call, where the person talks some and listens
some.  Why keep sampling and transmitting data when the person isn't
talking?   Maybe that isn't exactly the original question, but in many cases
there are parts of a signal that need more samples than others.

-- glen

In article 6afd943a.0307090755.37798597@posting.google.com, santosh nath at
santosh.nath@ntlworld.com wrote on 07/09/2003 11:55:

> Suppose a signal is known a prior. Time domain as well frequency
> domain characteristic is pretty defined. Say fs is the sampling
> frequency as per Nyquist criteria. Number of samples in 1 sec is N.
> 
> Let us think two cases
> 
> 1. Uniform sampling i.e sample interval is constant, I guess this is the
> usual case in most of the applications.
> 
> 2. We keep N same in 1 sec but a variable sample interval i.e nonuniform.
> Steep signal envelope is sliced densely than flat signal envelope.
> 
> Will the second case has advantage or better performance  when it is
> exposed in communication channel(assume noise and impairments) and
> later recovered by the same receiver.

why stop at reduced bits for constant signal values?  why not code any
constant slope, even those that are not zero-slope, with a low rate of
samples?

In article 0e0Pa.21386$H17.4975@sccrnsc02, Glen Herrmannsfeldt at
gah@ugcs.caltech.edu wrote on 07/09/2003 17:58:

> "Jerry Avins" <jya@ieee.org> wrote in message
> news:3F0C6A31.2710FD98@ieee.org...
...
>> Uniform spacing is an advantage for analysis. Samples so close that that
>> they aren't independent count for little. Aside from that, given that
>> uniform sampling sufficiently often completely specifies the signal, and
>> that fewer than that can't be guaranteed to do that, what do you mean by
>> "advantage"?
> 
> Consider a digitized telephone call, where the person talks some and listens
> some.  Why keep sampling and transmitting data when the person isn't
> talking?   Maybe that isn't exactly the original question, but in many cases
> there are parts of a signal that need more samples than others.

this is the kind of thing we audio guys like to categorize as a "lossless
compression algorithm" kinda issue.  essentially, what santosh nath is
proposing is to sample at a rate that is proportional to the abs() of the
signal passed through a differentiator.  for a given word size (maybe that
can be variable, too, as in a Huffman coder or relative) and a given or
constant signal power, the error or noise of quantization will greater for
signals of higher frequency *and* the coding bit rate is large.  so this
would be a good quantizer for low-pass signals, which are common, but it
would not be so good for high-pass signals, all other parameters kept equal.

the generalization, in my opinion, of santosh nath's question would be the
application of linear predictive coding (LPC) to the signal that "is known a
prior".  then any spectrum of signal that is known a priori can be optimally
accomodated.  for further compression of these slowy "quasi-stationary
processes", stuff like run-length encoding and huffman coding can be tossed
in to help reduce the bit rate in a lossless manner.  i s'pose other things
can be thought of.  (perhaps even those ZeoSync guys might have had a
legitimate idea, but they sure did try to sell it illegitimately - remember
that sorta fly-by-night?)

of course, data storage and data transmission is not the same as analysis,
and Jerry is right about "Uniform spacing is an advantage for analysis."  i
would probably never try to compute a spectrum of a non-uniform set of
samples without first interpolating it, in the sense it was meant to be, and
resampling uniformly.

r b-j

santosh.nath@ntlworld.com (santosh nath) wrote in message news:<6afd943a.0307090755.37798597@posting.google.com>...
> Hi all,
> 
> Suppose a signal is known a prior. Time domain as well frequency
> domain characteristic is pretty defined. Say fs is the sampling
> frequency as per
> Nyquist criteria. Number of samples in 1 sec is N. 
> 
> Let us think two cases 
> 
> 1. Uniform sampling i.e sample interval is constant, I guess this is
> the
> usual case in most of the applications.
> 
> 2. We keep N same in I sec but a variable sample interval i.e
> nonuniform.
> Steep signal envelope is sliced densely than flat signal envelope.
> 
> Will the second case has advantage or better performance  when it is
> exposed in communication channel(assume noise and impairments) and
> later recovered by the same receiver.
> 
> This could be a stupid question to an expert but it just came into my
> mind.

This not at all stupid. The very argument you make is indeed used at 
array processing facilities like seismological observatories. Since the 
array geometry is fixed once and for all at construction time, the 
sensor pattern (spatial non-uniform sampling parameters) is known and
can be incorporated in analysis. A site map of such an installation
would typically show a large array where the sensors are sparse on the 
largest scale, but tend to gradually cluster in some areas. The analyst
chooses subapertures depending on the wavelengths he or she wants to study.

Rune

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0307092144.2d6b6ba6@posting.google.com>...
> santosh.nath@ntlworld.com (santosh nath) wrote in message news:<6afd943a.0307090755.37798597@posting.google.com>...
> > Hi all,
> > 
> > Suppose a signal is known a prior. Time domain as well frequency
> > domain characteristic is pretty defined. Say fs is the sampling
> > frequency as per
> > Nyquist criteria. Number of samples in 1 sec is N. 
> > 
> > Let us think two cases 
> > 
> > 1. Uniform sampling i.e sample interval is constant, I guess this is
> > the
> > usual case in most of the applications.
> > 
> > 2. We keep N same in I sec but a variable sample interval i.e
> > nonuniform.
> > Steep signal envelope is sliced densely than flat signal envelope.
> > 
> > Will the second case has advantage or better performance  when it is
> > exposed in communication channel(assume noise and impairments) and
> > later recovered by the same receiver.
> > 
> > This could be a stupid question to an expert but it just came into my
> > mind.
> 
> This not at all stupid. The very argument you make is indeed used at 
> array processing facilities like seismological observatories. Since the 
> array geometry is fixed once and for all at construction time, the 
> sensor pattern (spatial non-uniform sampling parameters) is known and
> can be incorporated in analysis. A site map of such an installation
> would typically show a large array where the sensors are sparse on the 
> largest scale, but tend to gradually cluster in some areas. The analyst
> chooses subapertures depending on the wavelengths he or she wants to study.
> 
> Rune

IMHO, Sampling theorem is mostly used only in a approx. sense in
practice. To be strictly BW limited, signal should of infinite
length...

For finite duration signals, one can approximate
it by a wavenumber-limited (or could be exact but ) finite-power

"Seung" <kim.seung@sbcglobal.net> wrote in message
news:fdf92243.0307102021.4dfb953@posting.google.com...

(snip)

> IMHO, Sampling theorem is mostly used only in a approx. sense in
> practice. To be strictly BW limited, signal should of infinite
> length...
>
> For finite duration signals, one can approximate
> it by a wavenumber-limited (or could be exact but ) finite-power

Well, it is used in the approximate sense in many ways.  Ideally the sample
time should be infinitely narrow, and there should be no quantization error.
Though uniform spacing minimizes the effect of quantization error.

Also, human made low pass filters aren't perfect.   Though if you get the
error, including the infinite tail that a band limited signal should have,
below the quantization noise it is close enough for any practical problem.

Anyway, periodic signal can also be band limited, though.   As in previous
discussions I prefer to consider time limited signal as periodic with a
period greater than or equal to the signal length.

-- glen

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0307092144.2d6b6ba6@posting.google.com>...
> santosh.nath@ntlworld.com (santosh nath) wrote in message news:<6afd943a.0307090755.37798597@posting.google.com>...
> > Hi all,
> > 
> > Suppose a signal is known a prior. Time domain as well frequency
> > domain characteristic is pretty defined. Say fs is the sampling
> > frequency as per
> > Nyquist criteria. Number of samples in 1 sec is N. 
> > 
> > Let us think two cases 
> > 
> > 1. Uniform sampling i.e sample interval is constant, I guess this is
> > the
> > usual case in most of the applications.
> > 
> > 2. We keep N same in I sec but a variable sample interval i.e
> > nonuniform.
> > Steep signal envelope is sliced densely than flat signal envelope.
> > 
> > Will the second case has advantage or better performance  when it is
> > exposed in communication channel(assume noise and impairments) and
> > later recovered by the same receiver.
> > 
> > This could be a stupid question to an expert but it just came into my
> > mind.
> 
> This not at all stupid. The very argument you make is indeed used at 
> array processing facilities like seismological observatories. Since the 
> array geometry is fixed once and for all at construction time, the 
> sensor pattern (spatial non-uniform sampling parameters) is known and
> can be incorporated in analysis. A site map of such an installation
> would typically show a large array where the sensors are sparse on the 
> largest scale, but tend to gradually cluster in some areas. The analyst
> chooses subapertures depending on the wavelengths he or she wants to study.
> 
> Rune




The following paper may be helpful( 1-D as well as 2-D).

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.

Theoretically, it is possible to reconstruct a BW-limited signal from
an arbitrary placed N sample data where N is the number of samples
satisfying
Sampling theorem (numerical precision issues if not well distributed).
It is a further generalization of "bunched non-uniform-sampling" by
Papoulis mentioned above and an extension to 2-D. It has been
developed from quite different perspective though.

BTW, Sampling Theorem is mostly used in approximate sense (BW
limitedness = -x dB below peak, for example). A true BW limited signal
is of infinite length!


Seung

Glen Herrmannsfeldt wrote:
> 
>   ... uniform spacing minimizes the effect of quantization error.
> 
  ...

Interesting! Will you explain that please?

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