Compressed Sensing Construction for Underdetermined Source Separation

Abstract

Underdetermined blind source separation based on compressed sensing (CS) has already been proven to be an effective mechanism from an experimental viewpoint. In this study, we develop a theoretical result and show that, under a certain sparsity constraint for the restricted isometry property, the accuracy of CS when retrieving sources is guaranteed. This theoretical result can be regarded as a generalization of the blocked polynomial deterministic matrix theory and has been confirmed using numerical examples.

Appendix: Proof of Theorem 2

From Eq. (5), the length and sparsity of signal $\varvec{\uptheta }$ are NL and K, respectively. $\Gamma \in \{1,\;2,\;\ldots ,\;NL\}$ and ${\#}(\Gamma )=K$. Without loss of generality, we assume that the source signals $\mathbf{s}$ are sparse in the time domain, then the sparsity of $\varvec{\uptheta }_i$ is $K_i$. Then, the Gramian matrix can be written as follows:

$$\begin{aligned} \mathbf{D}_\Gamma =\varvec{\Pi }_\Gamma ^\mathrm {T} \varvec{\Pi }_\Gamma= & {} \left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\varvec{\Pi }_{\Gamma _1 } }&{} \mathbf{0}&{} \cdots &{} \mathbf{0} \\ \mathbf{0}&{} {\varvec{\Pi }_{\Gamma _2 } }&{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{0}&{} \mathbf{0}&{} \cdots &{} {\varvec{\Pi }_{\Gamma _L } } \\ \end{array} }} \right) _{K\times ML}^{\mathrm{T}} \left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\varvec{\Pi }_{\Gamma _1 } }&{} \mathbf{0}&{} \cdots &{} \mathbf{0} \\ \mathbf{0}&{} {\varvec{\Pi }_{\Gamma _2 } }&{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{0}&{} \mathbf{0}&{} \cdots &{} {\varvec{\Pi }_{\Gamma _L } } \\ \end{array} }} \right) _{ML\times K}\nonumber \\ \qquad= & {} \left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\varvec{\Pi }_{\Gamma _1 }^\mathrm {T} \varvec{\Pi }_{\Gamma _1 } }&{} 0&{} \cdots &{} \mathbf{0} \\ \mathbf{0}&{} {\varvec{\Pi }_{\Gamma _2 }^\mathrm {T} \varvec{\Pi }_{\Gamma _2 } }&{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{0}&{} \mathbf{0}&{} \cdots &{} {\varvec{\Pi }_{\Gamma _L }^\mathrm {T} \varvec{\Pi }_{\Gamma _L } } \\ \end{array} }} \right) _{K\times K}\nonumber \\ \qquad\doteq & {} \left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {\mathbf{D}_{\Gamma _1 } }&{} \mathbf{0}&{} \cdots &{} \mathbf{0} \\ \mathbf{0}&{} {\mathbf{D}_{\Gamma _2 } }&{} \cdots &{} \mathbf{0} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \mathbf{0}&{} \mathbf{0}&{} \cdots &{} {\mathbf{D}_{\Gamma _L } } \\ \end{array} }} \right) , \end{aligned}$$

(12)

where $\mathbf{D}_{\Gamma _i } =\varvec{\Pi }_{\Gamma _i }^\mathrm {T} \varvec{\Pi }_{\Gamma _i }$; That is,

$$\begin{aligned} \mathbf{D}_\Gamma (k,\;h)=\langle \varvec{\Pi }_\Gamma (\cdot ,\;k),\varvec{\Pi }_\Gamma (\cdot ,\;h)\rangle =\sum _{j=1}^{ML} {\varvec{\Pi }_\Gamma (j,\;k)} \varvec{\Pi }_\Gamma (j,\;h). \end{aligned}$$

(13)

From Eq. (12), we have

• (i) when $k,\;h\in \Gamma _i$

  • (a) if $k=h$, then $\mathbf{D}_\Gamma (k,\;k)=1$;

  • (b) if $k\ne h,\mathbf{D}_\Gamma (k,\;h)=\sum \limits _{j=1}^{ML} {\varvec{\Pi }_\Gamma (j,\;k)} \varvec{\Pi }_\Gamma (j,\;h)=\langle \mathbf{a}_{{k}'} ,\mathbf{a}_{{h}'} \rangle$.

• (ii) when $k\in \Gamma _i ,\;h\in \Gamma _j ,\;i\ne j,\;\mathbf{D}_\Gamma (k,\;h)=0$.

Because any off-diagonal entry of $\mathbf{D}_\Gamma$ is less than $\mathop {\max }\limits _{k\ne h} \{\mathbf{D}_\Gamma (k,\;h)\}$, the off-diagonal entries in any row or column of $\mathbf{D}_\Gamma$ have sum

$$\begin{aligned} \delta _i \doteq (K_i -1)\mathop {\max }\limits _{k, h\in \Gamma _i , k\ne h} \{\mathbf{D}_\Gamma (k,\;h)\}. \quad \hbox {If }~\delta _i <1 \end{aligned}$$

(14)

then

$$\begin{aligned} K_i =\frac{\delta _i }{\mathop {\max }\limits _{k, h\in \Gamma _i , k\ne h} \{\mathbf{D}_\Gamma (k,\;h)\}}+1. \end{aligned}$$

(15)

Let $\delta =\mathop {\max }\limits _{i=1, 2, \ldots , L} \delta _i <1$, then

$$\begin{aligned} K= & {} \sum _{i=1}^L {K_i } =\sum _{i=1}^L {\left( {\frac{\delta _i }{\mathop {\max }\limits _{k, h\in \Gamma _i , k\ne h} \{\mathbf{D}_\Gamma (k,\;h)\}}+1} \right) } \nonumber \\\le & {} \sum _{i=1}^L {\left( {\frac{\delta }{\mathop {\max }\limits _{k, h\in \Gamma _i , k\ne h} \{\mathbf{D}_\Gamma (k,\;h)\}}+1} \right) } . \end{aligned}$$

(16)

Hence, we write

$$\begin{aligned} \mathbf{D}_\Gamma =\mathbf{I}+\mathbf{B}_\Gamma , \end{aligned}$$

(17)

where $\mathbf{I}$ is a unit matrix and

$$\begin{aligned} \parallel \mathbf{B}_\Gamma \parallel _p \le \delta ,\;p=1\;or\;p=\infty . \end{aligned}$$

(18)

As $\mathbf{B}_\Gamma$ is a symmetric matrix, by interpolation of operators, we obtain that

$$\begin{aligned} \parallel \mathbf{B}_\Gamma \parallel _2 =\rho (\mathbf{B}_\Gamma )\le \parallel \mathbf{B}_\Gamma \parallel _p \le \delta ,\quad p=1\;\hbox {or}\;p=\infty , \end{aligned}$$

(19)

where $\rho (\mathbf{B}_\Gamma )$ denote the spectral norm of $\mathbf{B}_\Gamma$. Since $\parallel \mathbf{A}+\mathbf{B}\parallel _2 \le \parallel \mathbf{A}\parallel _2 +\parallel \mathbf{B}\parallel _2$ for any matrices $\mathbf{A}$ and $\mathbf{B}$ [13], it follows that

$$\begin{aligned} \parallel \mathbf{D}_\Gamma \parallel _2 \le 1+\delta ,\;\parallel \mathbf{D}_\Gamma ^{-1} \parallel _2 \le (1-\delta )^{-1}. \end{aligned}$$

(20)

Therefore, $\mathbf{D}_\Gamma$ has eigenvalues in $[1-\delta ,\;1+\delta ]$. Therefore, we have proved that matrix $\varvec{\Pi }$ satisfies the RIP with $\delta =\mathop {\max }\limits _{i=1, 2, \ldots , L} \delta _i$.