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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

Royi
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Luca Romano
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1 Answers1

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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...

Royi
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