# Super Resolution in Frequency Domain Using Compressed Sensing

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

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

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:

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...

• very interesting, could you link some refs? for code in particular
– I.M.
Jan 20 at 2:44
• @I.M., Thank you. Feel free to +1 if you found it interesting. There are actually much better methods as well.
– Royi
Jan 20 at 10:51
• 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.