Reconstruction and interpolation from irregularly spaced data

Hey everybody,

I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
2T+t2,....) .
My sampling distribution is nonuniform. And I want to get my data
values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
found some literature on this but couldnt understand it well. Somebody
please explain this to me!!

Rose

rosy27@gmail.com wrote:
> Hey everybody,
> 
> I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
> 2T+t2,....) .
> My sampling distribution is nonuniform. And I want to get my data
> values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
> found some literature on this but couldnt understand it well. Somebody
> please explain this to me!!

Haven't we been here before?

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

<rosy27@gmail.com> wrote in message 
news:1141157795.777102.259760@i39g2000cwa.googlegroups.com...
> Hey everybody,
>
> I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
> 2T+t2,....) .
> My sampling distribution is nonuniform. And I want to get my data
> values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
> found some literature on this but couldnt understand it well. Somebody
> please explain this to me!!
>

It partly depends on if your objective is a computationally efficient method 
or one that is intuitively satisfying.  Here is one of the latter:

Convolve the samples with a windowed sinx/x of some frequency of the spacing 
between zeros spacing you like, probably T.
This amounts to adding a set of these sinx/x functions each timed and 
weighted by the samples you do have.  Conceptually add over all time.
Then choose the value of the sum at the times you want.
This means you only have to compute the sum at those times and not, as 
above, over all time.

This approach assumes that the samples came from a sequence that was 
bandlimited to 1/2T - which has to be a reasonable assumption since the 
sample rate is (well at least roughly) 1/T.  If it is strictly bandlimited 
to 1/2T, then the corresponding sinx/x will reconstruct it.  The error is 
generally small for shortening these basis functions with a window.

About the FIR filter:  let's see:

I like to think of the FIR filter as the "convolver" mentioned above:
So, start with a FIR filter with *lots* of little delays and coefficients 
that represent the sinx/x from above.  In fact, you need to define this 
filter so that there will be an output at each time you want an output and 
at each time that there is an input - since we're dealing with discrete 
time, right?

OK, now you pass those samples through the filter and look at the output of 
the filter at intervals of T.  You just skip a lot of the output sums that 
happen in the filter.

There's probably a better way computationally but this might give plenty of 
insight - which is what you wanted.


Fred 

Jerry Avins wrote:

>> I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
>> 2T+t2,....) .
>> My sampling distribution is nonuniform. And I want to get my data
>> values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
>> found some literature on this but couldnt understand it well. Somebody
>> please explain this to me!!
> 
> Haven't we been here before?

And see how we *could* indeed predict that it would happen again?  :-)

Carlos
--

Carlos Moreno wrote:
> Jerry Avins wrote:
> 
>>> I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
>>> 2T+t2,....) .
>>> My sampling distribution is nonuniform. And I want to get my data
>>> values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
>>> found some literature on this but couldnt understand it well. Somebody
>>> please explain this to me!!
>>
>>
>> Haven't we been here before?
> 
> 
> And see how we *could* indeed predict that it would happen again?  :-)See the thread "Unevenly spaced data interpolation" begun on 2/1/2006 
2:10 PM

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

Carlos Moreno wrote:

 > Jerry Avins wrote:
 >
 >>> I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
 >>> 2T+t2,....) .
 >>> My sampling distribution is nonuniform. And I want to get my data
 >>> values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
 >>> found some literature on this but couldnt understand it well. Somebody
 >>> please explain this to me!!
 >>
 >>
 >>
 >> Haven't we been here before?
 >
 >
 >
 > And see how we *could* indeed predict that it would happen again? :-)

See the thread "Unevenly spaced data interpolation" begun on 2/1/2006.

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

Jerry Avins wrote:
> Carlos Moreno wrote:
> 
>  > Jerry Avins wrote:
>  >
>  >>> I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
>  >>> 2T+t2,....) .
>  >>> My sampling distribution is nonuniform. And I want to get my data
>  >>> values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
>  >>> found some literature on this but couldnt understand it well. Somebody
>  >>> please explain this to me!!
>  >>
>  >>
>  >>
>  >> Haven't we been here before?
>  >
>  >
>  >
>  > And see how we *could* indeed predict that it would happen again? :-)
> 
> See the thread "Unevenly spaced data interpolation" begun on 2/1/2006.

Oh!!  I thought you were making reference to that thread in which we
ended up in a theoretical/philosophical discussion about being able
to predict an entire movement of a symphony by sampling real fast the
first movement -- hence my comment above  :-)

Carlos
--

rosy27@gmail.com wrote:
> I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
> 2T+t2,....) .
> My sampling distribution is nonuniform. And I want to get my data
> values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
> found some literature on this but couldnt understand it well. Somebody
> please explain this to me!!

Depending on the qualifications or restrictions one puts on
t1, t2, T and the function being sampled, one could make several
interesting trick exam questions out of the above.

For instance, if the function is bounded in domain and both
t1 and t2 are +-infinity or thereabouts, the problem should
start to look familiar.

& etc. ...


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M

Ron N. wrote:
> rosy27@gmail.com wrote:
> 
>>I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
>>2T+t2,....) .
>>My sampling distribution is nonuniform. And I want to get my data
>>values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
>>found some literature on this but couldnt understand it well. Somebody
>>please explain this to me!!
> 
> 
> Depending on the qualifications or restrictions one puts on
> t1, t2, T and the function being sampled, one could make several
> interesting trick exam questions out of the above.
> 
> For instance, if the function is bounded in domain and both
> t1 and t2 are +-infinity or thereabouts, the problem should
> start to look familiar.

Familiarly unsolvable, you mean?  :-)

When you say bounded in domain, you mean bounded support?  In
such case, the bandwidth would be unbounded, and the signal
would not be representable by sampling, no matter how high
the frequency.

Carlos
--

Carlos Moreno wrote:
> Ron N. wrote:
> > rosy27@gmail.com wrote:
> >
> >>I have data values at : (0, t1, t2, T, T+t1, T+t2, 2T, 2T+t1,
> >>2T+t2,....) .
> >>My sampling distribution is nonuniform. And I want to get my data
> >>values at point say: ( t1-d,T+t1-d, 2T+t1-d) using an FIR filter. I
> >>found some literature on this but couldnt understand it well. Somebody
> >>please explain this to me!!
> >
> >
> > Depending on the qualifications or restrictions one puts on
> > t1, t2, T and the function being sampled, one could make several
> > interesting trick exam questions out of the above.
> >
> > For instance, if the function is bounded in domain and both
> > t1 and t2 are +-infinity or thereabouts, the problem should
> > start to look familiar.
>
> Familiarly unsolvable, you mean?  :-)
>
> When you say bounded in domain, you mean bounded support?  In
> such case, the bandwidth would be unbounded, and the signal
> would not be representable by sampling, no matter how high
> the frequency.

Indeed, given the problem did not limit the length of the FIR filter
to a finite value.  However, note that the data values appear to
be a one-sided vector (with "..." only to the right.)  That puts
another interesting wrinkle in the problem.

What is the difference in usefulness between a finite set of
samples taken from a perfectly bandlimited-signal, and a set
of samples taken from a signal that only has support within
the limits of that sample sets domain (thus not perfectly
bandlimited), but happens to be identical to the bandlimited
signal in that range?


IMHO. YMMV.
-- 
rhn A.T nicholson d.0.t C-o-M