# Sinc Non-uniform Filtering

Started by January 28, 2008
```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

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
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.
```