Using Derivatives of Second Generating Function for Underdetermined Blind Identification

Abstract

In this paper, we propose a new family of algorithms for underdetermined blind identification based on the second generating function (SGF). For the real case, the proposed algorithm first takes advantage of the partial derivatives of the SGF of the observation signals to construct a tensor and then reconstructs this tensor by grouping the parallel factors. The reconstructed tensor, which expands the length of several dimensions of the original tensor, transforms the underdetermined case into an overdertermined or determined case when $$({M^2} + M)/2 \ge N$$ , where M is the number of sensors and N is the number of sources. Thus, obtaining the estimation of the mixing matrix by decomposing the reconstructed tensor results in the performance improvement. The analysis of the uniqueness of tensor decomposition and the complex analysis show that the proposed algorithm relaxes the limitation of the maximal number of sources and owns lower computational complexity order compared with the conventional ALESCAF algorithm. We also extend the proposed algorithm to the complex case by complex-to-real transformation. Simulation results verify the effectiveness of the proposed algorithms and show that the proposed algorithms are preferred when the signal-to-noise ratio is higher than 5 dB.