Blind Identification of Underdetermined Mixtures with Complex Sources Using the Generalized Generating Function

Abstract

The generalized generating function (GGF), which can exploit the statistical information carried on complex-valued signals efficiently by treating its real and imaginary parts as a whole and offer an elegant algebraic structure, has been proposed by authors for blind identification of mixtures. In this paper, we extend the GGF-based method to be able to deal with the challenging underdetermined mixtures with complex-valued sources. A new algorithm named ALSGGF, in which the mixing matrix is estimated by decomposing the tensor constructed from the higher conjugate derivative of the second GGF of the observations with alternating least squares algorithm, is proposed. Furthermore, we show that the conjugate derivatives of different orders of the second GGF can be used jointly in a simple way to improve the performance. Simulation experiments validate the superiority of the proposed ALSGGF algorithms.

Appendix

In this Appendix, we show the computational details of core equation in (8).

First, the differentiation of (8) with respect to $u_{q_1 }^*$ gives

$$\begin{aligned} \frac{{\partial \Phi _{x} ({\mathbf {u}})}}{{\partial u_{{q_{1} }}^{*} }}&= \sum \nolimits _{{p = 1}}^{P} {\frac{{\partial \left( {\varphi _{p} (\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } )} \right) }}{{\partial \left( {\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } } \right) ^{*} }}\frac{{\partial \left( {\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } } \right) ^{*} }}{{\partial u_{{q_{1} }}^{*} }}} \nonumber \\&= \sum \nolimits _{{p = 1}}^{P} {A_{{q_{1} p}} \frac{{\partial \left( {\varphi _{p} (\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } )} \right) }}{{\partial \left( {\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } } \right) ^{*} }}} \end{aligned}$$

(19)

Similarly, the differentiation of (8) with respect to ($u_{q_1 }^*,u_{q_2 }^*,\ldots ,u_{q_K }^*)$ gives

$$\begin{aligned} \frac{{\partial ^{K} \Phi _{x} ({\mathbf {u}})}}{{\partial u_{{q_{1} }}^{*} \partial u_{{q_{2} }}^{*} \cdots \partial u_{{q_{K} }}^{*} }} \!&= \!\sum \nolimits _{{p = 1}}^{P} {A_{{q_{1} p}} \cdots A_{{q_{{K - 1}} p}} \frac{{\partial ^{K} \left( {\varphi _{p} (\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } )} \right) }}{{\partial \left( {\left( {\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } } \right) ^{*} } \right) ^{K} }}\frac{{\partial \left( {\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } } \right) ^{*} }}{{\partial u_{{q_{K} }}^{*} }}} \nonumber \\&= \sum \nolimits _{{p = 1}}^{P} {A_{{q_{1} p}} \cdots A_{{q_{K} p}} \frac{{\partial ^{K} \left( {\varphi _{p} (\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } )} \right) }}{{\partial \left( {\left( {\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } } \right) ^{*} } \right) ^{K} }}} \end{aligned}$$

(20)

Defining $G_{p} = \partial ^{K} \left( {\varphi _{p} (\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } )} \right) /\partial \left( {\left( {\sum \nolimits _{q} {A_{{qp}}^{*} u_{q} } } \right) ^{*} } \right) ^{K}$, we can rewrite (20) in a more compact form as in (9).