Underdetermined Independent Component Analysis Based on First- and Second-Order Statistics

Abstract

This paper proposes a class of new algorithms based on first- and second-order statistics for independent source extraction of circular signals in underdetermined complex-valued mixture. The complex-valued mixing matrix is estimated by two extremely cost-effective novel methods based on the conditional mean of the mixtures which require some prior knowledge of the positive support of the real and/or imaginary parts of the sources. And the sources are recovered by combining the conventional minimum mean-squared error-based beamforming approach with the acquired prior knowledge. Based on how much prior knowledge is got, we propose several new algorithms. The complexity analysis about the proposed algorithms suggests that the algorithms which employ more prior knowledge have higher complexity, but their computational cost is significantly low. Two examples are provided for showing the possible applications of these proposed algorithms. Simulation results validate the effectiveness and reliability of all presented methods.

Appendix: Derivation of Formula (9)

We show the derivation from Eqs. (8) to (9). Note that Eq. (8) can be easily simplified in the real case, while it is more complicate for the complex case. We omit the t in following derivation for convenience and rewrite the object function as

$$\begin{aligned} - E\{ {| {{s_j} - {{\hat{s}}_j}} |^2}\}= & {} - \,E\{ {({\mathrm{Re}} \{ {s_j}\} - {\mathrm{Re}} \{ {{\hat{s}}_j}\} )^2} + {({\mathrm{Im}} \{ {s_j}\} - {\mathrm{Im}} \{ {{\hat{s}}_j}\} )^2}\} \nonumber \\= & {} - \,E\{ {{\mathrm{Re}} ^2}\{ {s_j}\} + {{\mathrm{Im}} ^2}\{ {s_j}\} \} - E\{ {{\mathrm{Re}} ^2}\{ {{\hat{s}}_j}\} + {{\mathrm{Im}} ^2}\{ {{\hat{s}}_j}\} \} \nonumber \\&+\,2E\{ {\mathrm{Re}} \{ {s_j}\} {\mathrm{Re}} \{ {{\hat{s}}_j}\} + {\mathrm{Im}} \{ {s_j}\} {\mathrm{Im}} \{ {{\hat{s}}_j}\} \}. \end{aligned}$$

(24)

The term $E\{ {{\mathrm{Re}} ^2}\{ {s_j}\} + {{\mathrm{Im}} ^2}\{ {s_j}\} \}$ in Eq. (24) is irrelevant to ${\mathbf{w}}_j$, so we cut out this term in the object function. By employing ${\widehat{s}_j}(t) = {\mathbf{w}}_j^\mathrm{{H}}(t){\mathbf{x}}(t)$, $E\{ {{\mathrm{Re}} ^2}\{ {{\hat{s}}_j}\} \}$ is computed by

$$\begin{aligned} E\{ {{\mathrm{Re}} ^2}\{ {{\hat{s}}_j}\} \}= & {} E\left\{ {({\mathrm{Re}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Re}} \{ {\mathbf{x}}\} - {\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Im}} \{ {\mathbf{x}}\} )^2} \right\} \nonumber \\= & {} {\mathrm{Re}} \left\{ {\mathbf{w}}_j^\mathrm{{H}}\} E\{ {\mathrm{Re}} \{ {\mathbf{x}}\} {\mathrm{Re}} \{ {{\mathbf{x}}^\mathrm{{T}}}\} \} {\mathrm{Re}} \{ {\mathbf{w}}_j^*\right\} \nonumber \\&+\,{\mathrm{Im}} \left\{ {\mathbf{w}}_j^\mathrm{{H}}\} E\{ {\mathrm{Im}} \{ {\mathbf{x}}\} {\mathrm{Im}} \{ {{\mathbf{x}}^\mathrm{{T}}}\} \} {\mathrm{Im}} \{ {\mathbf{w}}_j^*\right\} \nonumber \\&-\,2{\mathrm{Re}} \left\{ {\mathbf{w}}_j^\mathrm{{H}}\} E\{ {\mathrm{Re}} \{ {\mathbf{x}}\} {\mathrm{Im}} \{ {{\mathbf{x}}^\mathrm{{T}}}\} \} {\mathrm{Im}} \{ {\mathbf{w}}_j^*\right\} , \end{aligned}$$

(25)

where the superscript ${\left( \cdot \right) ^*}$ denotes the conjugate operator. Similarly, $E\{ {{\mathrm{Im}} ^2}\{ {{\hat{s}}_j}\} \}$ is expressed as

$$\begin{aligned} E\{ {{\mathrm{Im}} ^2}\{ {{\hat{s}}_j}\} \}= & {} {\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} E\{ {\mathrm{Re}} \{ {\mathbf{x}}\} {\mathrm{Re}} \{ {{\mathbf{x}}^\mathrm{{T}}}\} \} {\mathrm{Im}} \{ {\mathbf{w}}_j^*\} \nonumber \\&+\,{\mathrm{Re}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} E\{ {\mathrm{Im}} \{ {\mathbf{x}}\} {\mathrm{Im}} \{ {{\mathbf{x}}^\mathrm{{T}}}\} \} {\mathrm{Re}} \{ {\mathbf{w}}_j^*\} \nonumber \\&+\,2{\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} E\{ {\mathrm{Re}} \{ {\mathbf{x}}\} {\mathrm{Im}} \{ {{\mathbf{x}}^\mathrm{{T}}}\} \} {\mathrm{Re}} \{ {\mathbf{w}}_j^*\}. \end{aligned}$$

(26)

Set ${\mathbf{D}} = E\{ {\mathrm{Re}} \{ {\mathbf{x}}\} {{\mathrm{Re}} ^\mathrm{{T}}}\{ {\mathbf{x}}\} \} + E\{ {\mathrm{Im}} \{ {\mathbf{x}}\} {{\mathrm{Im}} ^\mathrm{{T}}}\{ {\mathbf{x}}\} \}$${\mathbf{G}} = E\{ {\mathrm{Re}} \{ {\mathbf{x}}\} {{\mathrm{Im}} ^\mathrm{{T}}}\{ {\mathbf{x}}\} \}$. Stack up ${\mathbf{D}}$ and ${\mathbf{G}}$ into ${\mathbf{Q}} = \left( {\begin{array}{*{20}{c}} {\mathbf{D}}&{}\quad {\mathbf{0}}\\ {\mathbf{0}}&{}\quad {\mathbf{D}} \end{array}} \right)$ and ${\mathbf{U}} = \left( {\begin{array}{*{20}{c}} {\mathbf{0}}&{}\quad { -\, {\mathbf{G}}}\\ {\mathbf{G}}&{}\quad {\mathbf{0}} \end{array}} \right)$, respectively. The term $E\{ {{\mathrm{Re}} ^2}\{ {{\hat{s}}_j}\} + {{\mathrm{Im}} ^2}\{ {{\hat{s}}_j}\} \}$ in Eq . (24) becomes

$$\begin{aligned} E\{ {{\mathrm{Re}} ^2}\{ {{\hat{s}}_j}\} + {{\mathrm{Im}} ^2}\{ {{\hat{s}}_j}\} \}= & {} E\left\{ {{\mathrm{Re}} ^2}\{ {{\hat{s}}_j}\} \} + E\{ {{\mathrm{Im}} ^2}\{ {{\hat{s}}_j}\} \right\} \nonumber \\= & {} {\mathrm{Re}} \left\{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathbf{D}}\;{\mathrm{Re}} \{ {\mathbf{w}}_j^*\} + {\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathbf{D}}{\mathrm{Im}} \{ {\mathbf{w}}_j^*\right\} \nonumber \\&+\,2\left\{ {\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathbf{G}}{\mathrm{Re}} \{ {\mathbf{w}}_j^*\} - {\mathrm{Re}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathbf{G}}{\mathrm{Im}} \{ {\mathbf{w}}_j^*\} \right\} \nonumber \\= & {} \widetilde{\mathbf{w}}_j^\mathrm{{T}}({\mathbf{Q}} + 2{\mathbf{U}}){\widetilde{\mathbf{w}}_j}, \end{aligned}$$

(27)

where ${\widetilde{\mathbf{w}}_j} = {[ {\begin{array}{*{20}{c}} {{\mathrm{Re}} \{ {\mathbf{w}}_j^\mathrm{{T}}\} }&\quad { -\, {\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{T}}\} } \end{array}} ]^\mathrm{{T}}}$. Since ${\mathbf{x}} = {\mathbf{As}} = \sum \nolimits _{i = 1}^N {{{\mathbf{a}}_i}{s_i}}$, it can be easily got that

$$\begin{aligned}&{\mathrm{Re}} \{ {\mathbf{x}}\} = \sum \limits _{i = 1}^N {({\mathrm{Re}} \{ {{\mathbf{a}}_i}\} {\mathrm{Re}} \{ {s_i}\} - {\mathrm{Im}} \{ {{\mathbf{a}}_i}\} {\mathrm{Im}} \{ {s_i}\} )}, \nonumber \\&{\mathrm{Im}} \{ {\mathbf{x}}\} = \sum \limits _{i = 1}^N {({\mathrm{Im}} \{ {{\mathbf{a}}_i}\} {\mathrm{Re}} \{ {s_i}\} + {\mathrm{Re}} \{ {{\mathbf{a}}_i}\} {\mathrm{Im}} \{ {s_i}\} )}. \end{aligned}$$

(28)

Then, $E\{ \mathrm{{Re}}\{ {s_j}\} \mathrm{{Re}}\{ {{\hat{s}}_j}\} \}$ is calculated to be

$$\begin{aligned} E\{ \mathrm{{Re}}\{ {s_j}\} \mathrm{{Re}}\{ {{\hat{s}}_j}\} \}= & {} E\{ \mathrm{{Re}}\{ {s_j}\} (\;{\mathrm{Re}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Re}} \{ {\mathbf{x}}\} - {\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Im}} \{ {\mathbf{x}}\} )\} \nonumber \\= & {} E\{ \mathrm{{R}}{\mathrm{{e}}^2}\{ {s_j}\} \} ({\mathrm{Re}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Re}} \{ {{\mathbf{a}}_i}\} - {\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Im}} \{ {{\mathbf{a}}_i}\} ). \end{aligned}$$

(29)

Similarly, $E\{ {\mathrm{Im}} \{ {s_j}\} {\mathrm{Im}} \{ {{\hat{s}}_j}\} \}$ is obtained by

$$\begin{aligned} E\{ {\mathrm{Im}} \{ {s_j}\} {\mathrm{Im}} \{ {{\hat{s}}_j}\} \} = E\{ \mathrm{{I}}{\mathrm{{m}}^2}\{ {s_j}\} \} ({\mathrm{Re}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Re}} \{ {{\mathbf{a}}_i}\} - {\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Im}} \{ {{\mathbf{a}}_i}\} ). \end{aligned}$$

(30)

Thus, by combining Eqs. (29) and (30), the term $E\{ {\mathrm{Re}} \{ {s_j}\} {\mathrm{Re}} \{ {{\hat{s}}_j}\} + {\mathrm{Im}} \{ {s_j}\} {\mathrm{Im}} \{ {{\hat{s}}_j}\} \}$ in Eq . (24) turns to be

$$\begin{aligned} E\{ \mathrm{{Re}}\{ {s_j}\} \mathrm{{Re}}\{ {{\hat{s}}_j}\} + \mathrm{{Im}}\{ {s_j}\} \mathrm{{Im}}\{ {{\hat{s}}_j}\} \}= & {} E\{ \mathrm{{Re}}\{ {s_j}\} \mathrm{{Re}}\{ {{\hat{s}}_j}\} \} + E\{ \mathrm{{Im}}\{ {s_j}\} \mathrm{{Im}}\{ {{\hat{s}}_j}\} \} \nonumber \\= & {} \widetilde{\mathbf{w}}_j^\mathrm{{T}}{\widetilde{\mathbf{a}}_j}, \end{aligned}$$

(31)

where ${\widetilde{\mathbf{a}}_j} = {[ {\begin{array}{*{20}{c}} {{\mathrm{Re}} \{ {\mathbf{a}}_j^\mathrm{{T}}\} }&\quad { -\, {\mathrm{Im}} \{ {\mathbf{a}}_j^\mathrm{{T}}\} } \end{array}} ]^\mathrm{{T}}}$. Then, replacing the corresponding terms in Eq. (24) by the results in Eqs. (27) and (31), we achieve the final object function shown by $2\widetilde{\mathbf{w}}_j^\mathrm{{T}}{\widetilde{\mathbf{a}}_j} - \widetilde{\mathbf{w}}_j^\mathrm{{T}}({\mathbf{Q}} + 2{\mathbf{U}}){\widetilde{\mathbf{w}}_j}$.

Additionally, $\mathrm{{Re}}\{ {{\hat{s}}_j}\}$ is given by

$$\begin{aligned} \mathrm{{Re}}\{ {{\hat{s}}_j}\}= & {} {\mathrm{Re}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Re}} \{ {\mathbf{x}}\} - {\mathrm{Im}} \{ {\mathbf{w}}_j^\mathrm{{H}}\} {\mathrm{Im}} \{ {\mathbf{x}}\} \nonumber \\= & {} \widetilde{\mathbf{w}}_j^\mathrm{{T}}\widetilde{\mathbf{x}}, \end{aligned}$$

(32)

where $\widetilde{\mathbf{x}} = {[ {\begin{array}{*{20}{c}} {{\mathrm{Re}} \{ {{\mathbf{x}}^\mathrm{{T}}}\} }&\quad { -\, {\mathrm{Im}} \{ {{\mathbf{x}}^\mathrm{{T}}}\} } \end{array}}]^\mathrm{{T}}}$. So, the constraint in Eq. (8) becomes $- \widetilde{\mathbf{w}}_j^\mathrm{{T}}\widetilde{\mathbf{x}} < 0$.

This completes the proof.