Sinc Non-uniform Filtering

Hi all, 

At the moment I am working on interpolation filters for bandlimited
signals. I was wondering whether anyone knows if there is a generalization
of the sinc based interpolation method for a recurrent sampling error. 
For example the sampling times could look like 

t = [0.99   2.02   3.03   3.99   5.02   6.03  6.99    ...]

It is quite easy to compute the sinc coefficients if you want to delay all
the samples by always the same fraction of the sampling period. I know how
to calculate the filter coefficients for the Lagrange interpolation (which
is some kind of sinc filter as well) which can do the job and works with
non-uniform sampling instants. 
I was wondering how that is supposed to work for a sinc interpolation (for
the purpose of comparison). 

Any help (e.g. link to a paper) would be very appreciated.

Thank you,
Michael

Michael wrote:
> Hi all,
>
> At the moment I am working on interpolation filters for bandlimited
> signals. I was wondering whether anyone knows if there is a generalization
> of the sinc based interpolation method for a recurrent sampling error.
> For example the sampling times could look like
>
> t = [0.99   2.02   3.03   3.99   5.02   6.03  6.99    ...]
>
> It is quite easy to compute the sinc coefficients if you want to delay all
> the samples by always the same fraction of the sampling period. I know how
> to calculate the filter coefficients for the Lagrange interpolation (which
> is some kind of sinc filter as well) which can do the job and works with
> non-uniform sampling instants.
> I was wondering how that is supposed to work for a sinc interpolation (for
> the purpose of comparison).
>
> Any help (e.g. link to a paper) would be very appreciated.

Hello Michael

There is an article in the IEEE Signal Processing Magazine (Volume 24,
Issue 6, November 2007) that is concerned with interpolation of
recurrent non-uniformly sampled signals.

For your sampling scheme, the ideal interpolation works as follows:
Because the sampling times are recurrent with period 3, the ideal
interpolation uses 3 kernels (instead of only one sinc-kernel for the
uniform sampling case). Each kernel is used for the subsequences you
get by subsampling by a factor of 3.

To be more specific: assume you sampled the bandlimited signal x(t) as
follows (I shifted your sampling scheme by 0.99 to start with t=0):

x[0] = x(0)
x[1] = x(1.03)
x[2] = x(2.04)
x[3] = x(3.00)
x[4] = x(4.03)
...

Subsampling by a factor of 3 gives three sequences:

first sequence: x[0], x[3], x[6], ..., x[3 n], ...
second sequence: x[1], x[4], x[7], ..., x[3 n+1], ...
third sequence: x[2], x[5], x[8], ...., x[3 n+2], ...

The reconstruction formula then is:

       infinity
x(t) = sum x[3 n] k0(t-3 n)
       n=-infinity

       infinity
      + sum x[3 n+1] k1(t-3 n)
       n=-infinity

       infinity
      + sum x[3 n+2] k2(t-3 n)
       n=-infinity

The interpolation kernels are:

k0(t) = sinc(t/3) sin(pi/3 (t-1.03))/sin(pi/3 (-1.03)) sin(pi/3
(t-2.04))/sin(pi/3 (-2.04))

k1(t) = sinc((t-1.03)/3) sin(pi/3 t)/sin(pi/3 1.03) sin(pi/3 (t-2.04))/
sin(pi/3 (1.03-2.04))

k2(t) = sinc((t-2.04)/3) sin(pi/3 t)/sin(pi/3 2.04) sin(pi/3 (t-1.03))/
sin(pi/3 (2.04-1.03))

You can download some demo Matlab code and a derivation of the
interoplation kernels are defined here:

http://apollo.ee.columbia.edu/spm/?i=external/tipsandtricks

Look for the article "Recovering Periodically-Spaced Missing Samples".
The interpolation kernels are defined in equation (3) in the PDF that
you can download from this link. Some references are at the end of
this post. Come back if you have more questions.

Regards,
Andor

[1] J. L. Yen, "On Nonuniform Sampling of Bandwidth-Limited Signals,"
IRE Trans. Circuit Theory, vol. 3, pp. 251-257, Dec. 1956.
[2] P. P. Vaidyanathan, V. C. Liu, "Efficient Reconstruction of Band-
Limited Sequences from Nonuniformly Decimated Versions by Use of
Polyphase Filter Banks," IEEE Trans. ASSP, vol. 38, pp. 1927-1936,
Nov. 1990.
[3] Y. C. Eldar, A. V. Oppenheim, "Filterbank Reconstruction of
Bandlimited Signals from Nonuniform and Generalized Samples," IEEE
Trans. Signal Processing, vol. 48, pp. 2864-2875, Oct. 2000.
[4] R. S. Prendergast, B. C. Levy, P. J. Hurst, "Reconstruction of
Band-
Limited Periodic Nonuniformly Sampled Signals Through Multirate
Filter Banks," IEEE Trans. Circuits Systems - I: Regular Papers, vol.
51, pp. 1612-1622, Aug. 2004.
[6] A. Bariska, "Recovering Periodically-Spaced Missing Samples," IEEE
Signal Process. Mag., DSP Tips and Tricks column, vol. 24, Nov. 2007.