Error due to non-uniform sampling

Hello,

I have a problem that is quite simple to state, but not sure how hard
it is to answer:

I have a rectangular pulse of width N, consisting of data that is
sampled uniformly at rate Fs. Hence, its Fourier transform is a sinc
function with
the frequency zeros at integral multiples of 2*pi/N.

But I later find that the data is not really uniformly sampled. So my
question: is it possible to quantify the perturbation in the ideal
sinc function due to the non-uniform sampling? Are there any practical
solutions to accounting for non-uniform sampling? Resampling is one
solution as some have mentioned in earlier threads, but any thoughts
will be helpful.

Thanks.
Dinkar

dbhat2@yahoo.com (dnb) wrote 

> I have a problem that is quite simple to state, but not sure how hard
> it is to answer:
> 
> I have a rectangular pulse of width N, consisting of data that is
> sampled uniformly at rate Fs. Hence, its Fourier transform is a sinc
> function with the frequency zeros at integral multiples of 2*pi/N.
> 
> But I later find that the data is not really uniformly sampled. So my
> question: is it possible to quantify the perturbation in the ideal
> sinc function due to the non-uniform sampling? Are there any practical
> solutions to accounting for non-uniform sampling? Resampling is one
> solution as some have mentioned in earlier threads, but any thoughts
> will be helpful.

Q: How do you _know_ it's not uniformly sampled?

The reason I ask is that if it _is_ only a rectangular pulse (0 -> 1
-> 0), then sampling it uniformly or close-to-uniformly won't show up
a big difference, will it?

The only thing you would notice is if the transitions were sampled on
one side (uniform) and then the other (non-uniform).  All the other
samples would be the same.

Q: Do you know what the non-uniformity is? Is it just phase-jitter
(random distribution around the uniform sampling instants)?  Is it
systematic?

Both can probably be catered for, but you need to be able to
characterise it.

I'm not sure if that helps; more information would probably improve
the answer. :-)

Ciao,

Peter K.

p.kootsookos@iolfree.ie (Peter Kootsookos) wrote in message news:<3fca8095.0412010654.792ade60@posting.google.com>...
> dbhat2@yahoo.com (dnb) wrote 
> 
> > I have a problem that is quite simple to state, but not sure how hard
> > it is to answer:
> > 
> > I have a rectangular pulse of width N, consisting of data that is
> > sampled uniformly at rate Fs. Hence, its Fourier transform is a sinc
> > function with the frequency zeros at integral multiples of 2*pi/N.
> > 
> > But I later find that the data is not really uniformly sampled. So my
> > question: is it possible to quantify the perturbation in the ideal
> > sinc function due to the non-uniform sampling? Are there any practical
> > solutions to accounting for non-uniform sampling? Resampling is one
> > solution as some have mentioned in earlier threads, but any thoughts
> > will be helpful.
> 
> Q: How do you _know_ it's not uniformly sampled?
> 
> The reason I ask is that if it _is_ only a rectangular pulse (0 -> 1
> -> 0), then sampling it uniformly or close-to-uniformly won't show up
> a big difference, will it?
> 
> The only thing you would notice is if the transitions were sampled on
> one side (uniform) and then the other (non-uniform).  All the other
> samples would be the same.
> 
> Q: Do you know what the non-uniformity is? Is it just phase-jitter
> (random distribution around the uniform sampling instants)?  Is it
> systematic?
> 
> Both can probably be catered for, but you need to be able to
> characterise it.
> 
> I'm not sure if that helps; more information would probably improve
> the answer. :-)
> 
> Ciao,
> 
> Peter K.

Peter,

Thanks. Well, I know the sampling is not randomly non-uniform (in
certain cases). They are deliberately spaced in the fashion. I know
that the maximum time distance between successive samples cannot
exceed Tmax. Usually, the samples are uniformly spaced, but as I said
in certain cases, the samples are deliberately spaced non-uniformly.
There is no periodic pattern too, in the sense, if I look at the time
spacing between samples over a long time, no immediate pattern
emerges. But, I know the logic as to why the samples are spaced as
they are.

Dinkar

dnb wrote:

(snip)

> Thanks. Well, I know the sampling is not randomly non-uniform (in
> certain cases). They are deliberately spaced in the fashion. I know
> that the maximum time distance between successive samples cannot
> exceed Tmax. Usually, the samples are uniformly spaced, but as I said
> in certain cases, the samples are deliberately spaced non-uniformly.
> There is no periodic pattern too, in the sense, if I look at the time
> spacing between samples over a long time, no immediate pattern
> emerges. But, I know the logic as to why the samples are spaced as
> they are.

If the samples are close to uniform spacing you might be able
to use a first order correction to the result.  The correction
would be proportional to the derivative at the desired point,
and the deviation from uniform.

-- glen

glen herrmannsfeldt wrote:

> 
> 
> dnb wrote:
> 
> (snip)
> 
>> Thanks. Well, I know the sampling is not randomly non-uniform (in
>> certain cases). They are deliberately spaced in the fashion. I know
>> that the maximum time distance between successive samples cannot
>> exceed Tmax. Usually, the samples are uniformly spaced, but as I said
>> in certain cases, the samples are deliberately spaced non-uniformly.
>> There is no periodic pattern too, in the sense, if I look at the time
>> spacing between samples over a long time, no immediate pattern
>> emerges. But, I know the logic as to why the samples are spaced as
>> they are.
> 
> 
> If the samples are close to uniform spacing you might be able
> to use a first order correction to the result.  The correction
> would be proportional to the derivative at the desired point,
> and the deviation from uniform.
> 
> -- glen

But the pulse being rectangular, the derivative is zero. As I understand
the conditions, sampling nonuniformity could only affect the measured width.

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

"dnb" <dbhat2@yahoo.com> wrote in message
news:fa0deac8.0412011109.7e7be640@posting.google.com...
> p.kootsookos@iolfree.ie (Peter Kootsookos) wrote in message
news:<3fca8095.0412010654.792ade60@posting.google.com>...
> > dbhat2@yahoo.com (dnb) wrote
> >
> > > I have a problem that is quite simple to state, but not sure how hard
> > > it is to answer:
> > >
> > > I have a rectangular pulse of width N, consisting of data that is
> > > sampled uniformly at rate Fs. Hence, its Fourier transform is a sinc
> > > function with the frequency zeros at integral multiples of 2*pi/N.
> > >
> > > But I later find that the data is not really uniformly sampled. So my
> > > question: is it possible to quantify the perturbation in the ideal
> > > sinc function due to the non-uniform sampling? Are there any practical
> > > solutions to accounting for non-uniform sampling? Resampling is one
> > > solution as some have mentioned in earlier threads, but any thoughts
> > > will be helpful.
> >
> > Q: How do you _know_ it's not uniformly sampled?
> >
> > The reason I ask is that if it _is_ only a rectangular pulse (0 -> 1
> > -> 0), then sampling it uniformly or close-to-uniformly won't show up
> > a big difference, will it?
> >
> > The only thing you would notice is if the transitions were sampled on
> > one side (uniform) and then the other (non-uniform).  All the other
> > samples would be the same.
> >
> > Q: Do you know what the non-uniformity is? Is it just phase-jitter
> > (random distribution around the uniform sampling instants)?  Is it
> > systematic?
> >
> > Both can probably be catered for, but you need to be able to
> > characterise it.
> >
> > I'm not sure if that helps; more information would probably improve
> > the answer. :-)
> >
> > Ciao,
> >
> > Peter K.
>
> Peter,
>
> Thanks. Well, I know the sampling is not randomly non-uniform (in
> certain cases). They are deliberately spaced in the fashion. I know
> that the maximum time distance between successive samples cannot
> exceed Tmax. Usually, the samples are uniformly spaced, but as I said
> in certain cases, the samples are deliberately spaced non-uniformly.
> There is no periodic pattern too, in the sense, if I look at the time
> spacing between samples over a long time, no immediate pattern
> emerges. But, I know the logic as to why the samples are spaced as
> they are.
>
> Dinkar
Hi Dinkar,
    Maybe Peter will be able to work out what you are trying to do but I'm
afraid I'm still confused.

    Are you looking at sampled data containing a single rectangular pulse or
several pulses which are assumed to be regularly spaced?

   Do you know that the pulse(s) are N sample periods in duration at a
notional average sample rate or are you inferring this from the sample
values that you have?

   What are you trying to find out about the pulse(s)? e.g. start time, stop
time , duration (amplitude/energy even)?  Would you be happy to be able to
characterise the probability that some parameter lies in a particular
interval?


Best of Luck - Mike

"Mike Yarwood" <mpyarwood@btopenworld.com> wrote in message news:<comunf$a3u$1@titan.btinternet.com>...
> "dnb" <dbhat2@yahoo.com> wrote in message
> news:fa0deac8.0412011109.7e7be640@posting.google.com...
> > p.kootsookos@iolfree.ie (Peter Kootsookos) wrote in message
>  news:<3fca8095.0412010654.792ade60@posting.google.com>...
> > > dbhat2@yahoo.com (dnb) wrote
> > >
> > > > I have a problem that is quite simple to state, but not sure how hard
> > > > it is to answer:
> > > >
> > > > I have a rectangular pulse of width N, consisting of data that is
> > > > sampled uniformly at rate Fs. Hence, its Fourier transform is a sinc
> > > > function with the frequency zeros at integral multiples of 2*pi/N.
> > > >
> > > > But I later find that the data is not really uniformly sampled. So my
> > > > question: is it possible to quantify the perturbation in the ideal
> > > > sinc function due to the non-uniform sampling? Are there any practical
> > > > solutions to accounting for non-uniform sampling? Resampling is one
> > > > solution as some have mentioned in earlier threads, but any thoughts
> > > > will be helpful.
> > >
> > > Q: How do you _know_ it's not uniformly sampled?
> > >
> > > The reason I ask is that if it _is_ only a rectangular pulse (0 -> 1
> > > -> 0), then sampling it uniformly or close-to-uniformly won't show up
> > > a big difference, will it?
> > >
> > > The only thing you would notice is if the transitions were sampled on
> > > one side (uniform) and then the other (non-uniform).  All the other
> > > samples would be the same.
> > >
> > > Q: Do you know what the non-uniformity is? Is it just phase-jitter
> > > (random distribution around the uniform sampling instants)?  Is it
> > > systematic?
> > >
> > > Both can probably be catered for, but you need to be able to
> > > characterise it.
> > >
> > > I'm not sure if that helps; more information would probably improve
> > > the answer. :-)
> > >
> > > Ciao,
> > >
> > > Peter K.
> >
> > Peter,
> >
> > Thanks. Well, I know the sampling is not randomly non-uniform (in
> > certain cases). They are deliberately spaced in the fashion. I know
> > that the maximum time distance between successive samples cannot
> > exceed Tmax. Usually, the samples are uniformly spaced, but as I said
> > in certain cases, the samples are deliberately spaced non-uniformly.
> > There is no periodic pattern too, in the sense, if I look at the time
> > spacing between samples over a long time, no immediate pattern
> > emerges. But, I know the logic as to why the samples are spaced as
> > they are.
> >
> > Dinkar
> Hi Dinkar,
>     Maybe Peter will be able to work out what you are trying to do but I'm
> afraid I'm still confused.
> 
>     Are you looking at sampled data containing a single rectangular pulse or
> several pulses which are assumed to be regularly spaced?
> 
>    Do you know that the pulse(s) are N sample periods in duration at a
> notional average sample rate or are you inferring this from the sample
> values that you have?
> 
>    What are you trying to find out about the pulse(s)? e.g. start time, stop
> time , duration (amplitude/energy even)?  Would you be happy to be able to
> characterise the probability that some parameter lies in a particular
> interval?
> 
> 
> Best of Luck - Mike


Hi Mike,

Looks like I am confusing people again. I am looking at several pulses
which are rectangular. In each pulse, there are N sampled values. The
inter-sample distance is not fixed (non-uniformly sampled) in some of
the pulses. But I know the inter-sample distance cannot exceed Tmax.

If I take the DFT of a uniformly sampled (assuming Nyquist fy
criterion is met) data, then I should get a sinc function. But if I
take the DFT of non-uniformly sampled data, but I assume it to be
uniformly sampled, then will I still get a sinc function in fy domain
(and if not, how does the perturbation relate to the non-uniform
sampling parameter)? I am interested in knowing the frequency at which
the magnitude falls to 1/2 the peak value. In the case of the sinc
function, it is easy.

If this question is fomulated erroneously or is trivial, please let me
know.

Thanks
Dinkar

Jerry Avins wrote:
> glen herrmannsfeldt wrote:
> 
> 
>>
>>dnb wrote:
>>
>>(snip)
>>
>>
>>>Thanks. Well, I know the sampling is not randomly non-uniform (in
>>>certain cases). They are deliberately spaced in the fashion. I know
>>>that the maximum time distance between successive samples cannot
>>>exceed Tmax. Usually, the samples are uniformly spaced, but as I said
>>>in certain cases, the samples are deliberately spaced non-uniformly.
>>>There is no periodic pattern too, in the sense, if I look at the time
>>>spacing between samples over a long time, no immediate pattern
>>>emerges. But, I know the logic as to why the samples are spaced as
>>>they are.
>>
>>
>>If the samples are close to uniform spacing you might be able
>>to use a first order correction to the result.  The correction
>>would be proportional to the derivative at the desired point,
>>and the deviation from uniform.
>>
>>-- glen
> 
> 
> But the pulse being rectangular, the derivative is zero. As I understand
> the conditions, sampling nonuniformity could only affect the measured width.
> 
> Jerry

A Real rectangular pulse would have infinite bandwidth, i.e. the instant 
change from any value to zero - so any type of sampling is going to have 
  problems.

Glen is suggesting linear interpolation.

Cheers,
David

David Kirkland wrote:

(snip, I wrote)

>>> If the samples are close to uniform spacing you might be able
>>> to use a first order correction to the result.  The correction
>>> would be proportional to the derivative at the desired point,
>>> and the deviation from uniform.

(snip)

> A Real rectangular pulse would have infinite bandwidth, i.e. the instant 
> change from any value to zero - so any type of sampling is going to have 
>  problems.

> Glen is suggesting linear interpolation.


I thought the question was how to find the Fourier coefficients
of a signal with slightly non-uniform spacing.  I suppose that
is related to linear interpolation, but that isn't how I was
thinking about it.  For each Fourier coefficient at the sampling
point one can determine the derivative, and that will indicate
the change in that component due to a small shift of the
sampling point.  As long as higher order terms are sufficiently
small it should work.

-- glen

glen herrmannsfeldt wrote:
> 
> 
> David Kirkland wrote:
> 
> (snip, I wrote)
> 
>>>> If the samples are close to uniform spacing you might be able
>>>> to use a first order correction to the result.  The correction
>>>> would be proportional to the derivative at the desired point,
>>>> and the deviation from uniform.
> 
> 
> (snip)
> 
>> A Real rectangular pulse would have infinite bandwidth, i.e. the 
>> instant change from any value to zero - so any type of sampling is 
>> going to have  problems.
> 
> 
>> Glen is suggesting linear interpolation.
> 
> 
> 
> I thought the question was how to find the Fourier coefficients
> of a signal with slightly non-uniform spacing.  I suppose that
> is related to linear interpolation, but that isn't how I was
> thinking about it.  For each Fourier coefficient at the sampling
> point one can determine the derivative, and that will indicate
> the change in that component due to a small shift of the
> sampling point.  As long as higher order terms are sufficiently
> small it should work.
> 
> -- glen
> 

Sorry Glen, I thought you were talking about linear interpolation in the 
  Time domain and then doing the FFT on uniformly sampled data. That's 
the  approach I had in mind.

I don't know about approaching it in the frequency domain - I'd have to 
ponder that a bit more.

Cheers,
David