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I am new to Signal Processing, and am interested in compression sensing for audio files. CS is based on the algorithm that, given some sampling of a signal $f$ in order to obtain a smaller (compressed) signal $b$ such that:

$$b = \phi f$$ such that $$f = \Psi c$$

and $f$ can be expressed by a linear combination of basis functions $\Psi$ and its coefficients $c$ by doing a nonlinear constrained optimization on the L1 norm of such that :

$$\phi f = b$$

I understand how $f$ can be extracted from a sample without reaching the Nyquist limit, but I'm interested in the application of Compressed Sensing to blind source separation. With blind source seapration, to use standard notation

$$A = \phi\Psi , c = x $$

Thus, L1 norm minization can be represented as

$$Ax = b$$ I have looked here for an example How can we adapt compressed sensing when we don't know A (since the whole point of BSS is that you do not know the mixing matrix) and we have to reconstruct more than one signal from b? Thank you for your help!

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  • $\begingroup$ So is there any restrictions in definition of your A? $\endgroup$ – MimSaad May 25 '17 at 17:27
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I will sketch an idea how to use Sparse Represenattion (Dictionary Learning) for BSS.

Let's say we have $ \mathcal{S} = \mathcal{S}_{1} \cup \mathcal{S}_{2} $ where $ \mathcal{S}_{1} = \left\{ {x}_{i} \right\}_{i = 1}^{m} $ and $ \mathcal{S}_{2} = \left\{ {y}_{i} \right\}_{i = 1}^{n} $.

So if we assume different sources for $ \mathcal{S}_{1} $ and $ \mathcal{S}_{2} $ then what we can do is learn a dictionary (For example, using K-SVD) for $ {z}_{i} \in \mathcal{S} $.

After we learned this Dictionary we can look at the representations of each signal.
If the our assumption was correct, the atoms used for signals from $ \mathcal{S}_{1} $ will be different than atoms used for signals from $ \mathcal{S}_{2} $.

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