# Compression Sensing for Blind Source Separation

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?

• So is there any restrictions in definition of your A? May 25, 2017 at 17:27
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 $${z}_{i} \in \mathcal{S}$$.
If the our assumption was correct, the atoms used for signals from $$\mathcal{S}_{1}$$ will be different than atoms used for signals from $$\mathcal{S}_{2}$$.