A Low-Complexity PARAFAC Decomposition for Underdetermined Blind System Identification with Complex Mixtures

Abstract

A new method to effectively reduce the complexity for underdetermined blind system identification with complex mixtures is proposed in this paper. Generally speaking, the identifiability of a MIMO system can be guaranteed by decomposing a tensor constructed by the cumulant of the observations in an appropriate order, but higher order statistics will bring heavier computation load as well as more estimation errors. In the proposed method, by stacking two order-K tensors with $K-1$ identical factor matrices, a new tensor can be constructed followed by a PARAFAC decomposition, in which way the required minimum order of statistics can be reduced for a given underdetermined system. Experiments conducted on both the order-3 and order-4 tensors-stacking demonstrate the merits of the proposed algorithm in reducing the computational complexity while not degrading the identification performance in comparison with the standard alternating least squares algorithm. Specifically, in the $4 \times 3$ and $6 \times 3$ underdetermined systems, the numerical complexity can be reduced by more than $50\%$ and $90\%$, respectively. Furthermore, the proposed algorithm outperforms the classical blind source separation algorithms in the determined and overdetermined cases especially in low SNRs.