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Compressed sensing approach to blind source separation of far field sources

Compressed sensing framework for blind source separation have been considered in recent years because of the inherent sparsity in the assumptions used for blind source separation. Consider a noise free instantaneous mixture signals described below:

\begin{bmatrix} x_1(t) \\ x_2(t) \\ \vdots \\ x_N(t)\end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1M}\\a_{21} & a_{22} & \cdots & a_{2M}\\\vdots & \vdots & \vdots & \vdots\\a_{N1} & a_{N2} & \cdots & a_{NM}\end{bmatrix} \begin{bmatrix} s_1(t) \\ s_2(t) \\ \vdots \\ s_M(t)\end{bmatrix}

where there are M sources and N observations. Under the assumption that only one source is active, which implies the signal space is sparse, for each time, we define the following vectors and matrix.define the vector b as:

b = \begin{bmatrix} x_1(t) & x_1(t+1) & \cdots & x_1(t+T-1) & x_2(t) & \cdots & x_N(t+T-1)\end{bmatrix} ^T

where ^T denotes the transpose. Also denote the vector s as

s = \begin{bmatrix} s_1(t) & s_1(t+1) & \cdots & s_1(t+T-1) & s_2(t) & \cdots & s_N(t+T-1)\end{bmatrix} ^T

Finally, denote the matrix A as

\begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1M}\\a_{21} & a_{22} & \cdots & A_{2M}\\\vdots & \vdots & \vdots & \vdots\\A_{N1} & A_{N2} & \cdots & A_{NM}\end{bmatrix}

where

A_{ij} = \begin{bmatrix} a_{11} &0 & \cdots &0\\0 & a_{11} & \cdots & 0\\\vdots & & \ddots & \vdots\\0 &\cdots &0& a_{11}\end{bmatrix}

The problem can then be posed as solving the equation

b = A s

With s being sparse by assumption, this is the classical compressed sensing signal recovery problem using

min||s||_0 ~~ s.t. ~~ b = A s

The signal space can be further given a sparser representation by using a dictionary D such that s = \hat{A} \hat{x} with the l_0 minimization done on \hat{x}.

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