# Information request about sub-Nyquist and irregular sampling

Started by October 1, 2009
```Hi,

I am new to signal processing and have very simple question. Where can I
find a brief explanation about sub-Nyquist and irregular sampling. I
searched the web by almost all possible keywords, but what I found is not
so useful. It would be very nice if you give me some book titles.

```
```>Hi,
>
>I am new to signal processing and have very simple question. Where can I
>find a brief explanation about sub-Nyquist and irregular sampling. I
>searched the web by almost all possible keywords, but what I found is
not
>so useful. It would be very nice if you give me some book titles.

I don't think a brief explanation is possible, because considerable
complexity is normally needed to make these techniques workable.

Sub-Nyquist sampling is used in lots of applications, especially military
ones, but there is normally something fairly complex going on to allow such
sampling to be usable. For example, many radar and sonar systems use
sub-Nyquist sampling, and a selection of sampling rates. By correlating
across the different aliases at the various sampling rates, the true
range/velocity/whatever can be resolved. The basic idea is pretty simple,
but the details quickly get quite messy. There are books which cover these
topics.

I guess the most widely used application for irregular sampling is in
oscilloscopes. I expect you can find material on that topic.

Steve

```
```Hi,

As Steve have mentioned subNyquist sampling problems are detailed in many
textbooks, for example in R.Lyons "Understanding Digital Signal
Processing". There is also a fundamental article on sampling frequency
selection and noise issues: R.G.Vaughan, N.L.Scott and D.R.White "The
Theory of Bandpass Sampling" in IEEE trans. on Signa&#1076; Processing,
vol.39,no.9.

What do you mean by irregular sampling here? If you mean time-interleaved
sampling, which is in fact irregular because of timing mismatches between
individual channels, then you can use this article as a starting point:
http://www.analog.com/library/analogDialogue/cd/vol37n3.pdf#page=5

--
Alexander

```
```On Oct 2, 4:11&nbsp;am, "Alexander Sotnikov" <sotni...@scideco.ru> wrote:
> Hi,
>
> As Steve have mentioned subNyquist sampling problems are detailed in many
> textbooks, for example in R.Lyons "Understanding Digital Signal
> Processing". There is also a fundamental article on sampling frequency
> selection and noise issues: R.G.Vaughan, N.L.Scott and D.R.White "The
> Theory of Bandpass Sampling" in IEEE trans. on Signa&#1076; Processing,
> vol.39,no.9.
>
> What do you mean by irregular sampling here? If you mean time-interleaved
> sampling, which is in fact irregular because of timing mismatches between
> individual channels, then you can use this article as a starting point:http://www.analog.com/library/analogDialogue/cd/vol37n3.pdf#page=5
>
> --
> Alexander

I wrote a paper on a very simple technique that can be explained
without much fany math, back in 1992. It only covers the periodic
missing-sample case.

http://www.aes.org/e-lib/browse.cfm?elib=7022#

```
```>Hi,
>
>As Steve have mentioned subNyquist sampling problems are detailed in
many
>textbooks, for example in R.Lyons "Understanding Digital Signal
>Processing". There is also a fundamental article on sampling frequency
>selection and noise issues: R.G.Vaughan, N.L.Scott and D.R.White "The
>Theory of Bandpass Sampling" in IEEE trans. on Signa&#1076; Processing,
>vol.39,no.9.

What has bandpass sampling to do with sub-Nyquist sampling? Bandpass
sampling (unless it is screwed up) complies with the sampling theorem
requirements for sampling rate.

>What do you mean by irregular sampling here? If you mean
time-interleaved
>sampling, which is in fact irregular because of timing mismatches
between
>individual channels, then you can use this article as a starting point:
>http://www.analog.com/library/analogDialogue/cd/vol37n3.pdf#page=5

Steve

```
```On 3 Okt., 16:15, Robert Adams <robert.ad...@analog.com> wrote:

> I wrote a paper on a very simple technique that can be explained
> without much fany math, back in 1992. It only covers the periodic
> missing-sample case.
>
> http://www.aes.org/e-lib/browse.cfm?elib=7022#

That was an excellent article! It taught me two things:

1. From an oversampled signal, any finite number of "missing" samples
can be reconstructed (using the "eigen-filter" approach outlined in
your article). This implies that any discrete signal with only a
finite number of non-zero samples is _never_ oversampled (the discrete
version of the finite-bandwidth-infinite-time-support theorem).

2. Even an infinite amount of missing samples can be reconstructed if
the oversampling factor is sufficient.

You showed the particlar case where 1-out-of-M samples was missing. It
was a neat exercise to extend this result to the N-out-of-M case for
periodic missing samples patterns. I wrote this up in a DSP Tips and
Tricks article in the November 2007 issue of the Signal Processing
Magazine.

To the OP: your question is not simple at all. Use google books to
have a look at "Nonuniform sampling: theory and practice" from Farokh
A. Marvasti.

Regards,
Andor
```
```>What has bandpass sampling to do with sub-Nyquist sampling? Bandpass
>sampling (unless it is screwed up) complies with the sampling theorem
>requirements for sampling rate.

I was wrong about that. I confused sub-Nyquist sampling with subsampling,
the term which is sometimes used for bandpass sampling.

--
Alexander

```