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To be noted that I'm very new to this topic, I would like to understand the fundamentals of how to get Super Resolution in Frequency Domain estimation using the Compressed Sensing Model.

I am also looking for some references and Python/Matlab code that can help me.

Thanks a lot in advantage and happy new year, Luca

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  • $\begingroup$ Are you asking for Super Resolution in Frequency Domain? $\endgroup$
    – Royi
    Commented Jan 6, 2021 at 19:33
  • $\begingroup$ yes, the final aim is to get a super-resolution in the frequency domain. $\endgroup$
    – Luca Romano
    Commented Jan 6, 2021 at 19:39
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    $\begingroup$ So please change the question and just ask for how to get Super Resolution in for Frequency Domain estimation using Compressed Sensing Model. $\endgroup$
    – Royi
    Commented Jan 6, 2021 at 21:54
  • $\begingroup$ Could you review my answer? $\endgroup$
    – Royi
    Commented Feb 12, 2021 at 14:14

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You can employ Compressed Sensing / Sparse Representation for Super Resolution in Frequency Domain.

One way to do so is solving the problem:

$$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| F \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \lambda {\left\| \boldsymbol{x} \right\|}_{1} $$

Where the $ {L}_{1} $ norm is sparsity inducing regularization and $ F $ is the inverse DFT matrix.

Solving this, quite simple, optimization problem will yield Super Resolution of the DFT.
Super Resolution means, in that context, being able to resolve frequencies which are closer than what the observation time suggests:

enter image description here

In the above you can see the DFT of a sum of 2 sines with the given frequencies. The Gaussian model is using $ {L}_{2} $ for regularization (Which is basically damped zero padding).

You may see that the $ {L}_{1} $ could resolve the 2 sines even when they are only 0.5 [Hz] apart with an observation windows of 1 [Sec].

This is pretty nice...

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  • $\begingroup$ very interesting, could you link some refs? for code in particular $\endgroup$
    – I.M.
    Commented Jan 20, 2021 at 2:44
  • $\begingroup$ @I.M., Thank you. Feel free to +1 if you found it interesting. There are actually much better methods as well. $\endgroup$
    – Royi
    Commented Jan 20, 2021 at 10:51
  • $\begingroup$ sure, already +1, could you add code? $\endgroup$
    – I.M.
    Commented Jan 20, 2021 at 15:49
  • $\begingroup$ @Royi, thanks for the good overview. I was interested in some more technical details, in particular on 1) how to treat complex values, and 2) the relationship between dictionary size and resolution in the spare domain (dsp.stackexchange.com/questions/72346/…) $\endgroup$
    – Luca Romano
    Commented Jan 22, 2021 at 11:50
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    $\begingroup$ What issues do you see with Complex Numbers in the formulation above? By design $ F $ is a complex matrix. In the above the dictionary is fixed, not learned. It sets the grid hence the maximum potential resolution. $\endgroup$
    – Royi
    Commented Feb 1, 2021 at 16:06

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