Compression Sensing for Blind Source Separation

Question

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 at Cleve Moler - Magic Reconstruction: Compressed Sensing 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?

Answer 1

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} $.