# Sub nyquist sampling, required number of samples for time sparse grouped signals

Question: Does it make sense to perform compressed sampling if the non zero samples are grouped in time? If so, what is the minimal length of the vector x that should be acquired to allow full signal reconstruction (of at least the non zero pulse segments)?

$y=\phi x$, where y is the compressed vector and x is the signal (sampled at nyquist)

Situation: time sparse signal (+- 97 % of the samples at noise level), but the 3% non zero samples are grouped (periodic). I'm only interested in the pulse samples (so not in the noise floor samples).

Thanks a lot!

• There's something called block sparsity in compressed sensing literature which may be relevant to your question. Oct 26, 2017 at 18:10
• now is that an analog signal? or is it already sampled and you're gonna downsample it or reduce the data? Aug 25, 2020 at 1:27
• Usually for time series like signals the idea is building the efficient dictionary and the location will be determined by the vector. Wouldn't that work for you?
– Royi
Sep 3, 2022 at 7:23

In original definition of Compressed sensing [1] it does not assume any distribution of non-zero elements in the input vector (signal), so it makes no difference if your non-zero elements are near each other or located uniformly on the sampling grid. The number of measurement, as theory states for all sparse signals, depends on how many non-zero elements you have (sparsity degree), what is your noise in input and what recovery quality your need (exact or approximation) and etc. therefore,

Does it make sense to perform compressed sampling if the non zero samples are grouped in time?

Yes!

I suggest the following video lecture of Baranuik: Video Lecture On CS

Furthermore, for model-based compressive sensing you may read [3].

[1]: Donoho, David L. "Compressed sensing." IEEE Transactions on information theory 52, no. 4 (2006): 1289-1306.

[3]: Baraniuk, Richard G., Volkan Cevher, Marco F. Duarte, and Chinmay Hegde. "Model-based compressive sensing." IEEE Transactions on information theory 56, no. 4 (2010): 1982-2001.

• In many cases you can build equivalent MAP model for many CS models. So you can actually built a probabilistic interpretation.
– Royi
Aug 31, 2019 at 6:50
• Is not that called "model-based" compressive sensing?? Jun 5, 2022 at 10:19
• I am not sure what you mean by that. But for instance, if you solve the classic ${L}_{1}$ norm regularized Least Squares, which is a CS model, you actually solve a MAP problem with Laplace distribution prior. Moreover, the idea of sparsity is with relation to some basis / dictionary and not the non zero values of the signal.
– Royi
Feb 21, 2023 at 20:22
• @Royi at this point I do not even know myself what I meant back then :D, but I guess I was referring to concept that says if you basically know how the "non-zero" elements populate the search space you can converge faster. Mar 22, 2023 at 10:39