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.
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Acknowledgments
The authors would like to thank the anonymous reviewers for their careful reading and helpful remarks, who have contributed to improve the clarity of the paper. This work is supported in part by the Natural Science Foundation of China under Grant 61001106, the National Program on Key Basic Research Project of China under Grant 2009CB320400, and the Major Projects of the National Natural Science Foundation of China under Grant 91338105.
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Appendix
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
Similarly, the differentiation of (8) with respect to (\(u_{q_1 }^*,u_{q_2 }^*,\ldots ,u_{q_K }^*)\) gives
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).
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Gu, F., Zhang, H., Wang, S. et al. Blind Identification of Underdetermined Mixtures with Complex Sources Using the Generalized Generating Function. Circuits Syst Signal Process 34, 681–693 (2015). https://doi.org/10.1007/s00034-014-9858-6
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DOI: https://doi.org/10.1007/s00034-014-9858-6