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Using Derivatives of Second Generating Function for Underdetermined Blind Identification

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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.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61671472, 61172061, and 61201242 and the Natural Science Foundation of Jiang Su Province in China under Grant Nos. BK20160079, BK20181335, and BK2012057.

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Authors and Affiliations

  1. College of Communications Engineering, Army Engineering University of PLA, Nanjing, 210007, China

    Qiao Su, Yimin Wei, Kui Xu, Hui Li & Yuehong Shen

  2. Graduate School, Army Engineering University of PLA, Nanjing, 210007, China

    Changliang Deng

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Appendix: Proof of that the Limitation of N in (15) is More Relaxed Compared to the ALESCAF Algorithm

Appendix: Proof of that the Limitation of N in (15) is More Relaxed Compared to the ALESCAF Algorithm

The number of sources of the ALESCAF algorithm has the following limitation:

$$\begin{aligned} \left\{ \begin{array}{ll} N \le {((M - 1)q + S)} \big /2 &{}\quad \mathrm{if}\;S < N\\ N \le (M - 1)q&{}\quad \mathrm{if}\;S \ge N. \end{array} \right. \end{aligned}$$
(31)

If \(S < N\), we first compare the right side of the first row in (31) with that in (15) and have

$$\begin{aligned} \begin{array}{l} {{({{(M' - 1)q} \big /{2 + S}})} \big /2} - {{((M - 1)q + S)} \big /2}\\ \quad = {{(M - 2)(M - 1)q} \big /8}. \end{array} \end{aligned}$$
(32)

(32) suggests that when \(M\ge 2\), the upper bound of N in the first row of (15) is more relaxed than that of (31).

As for the case with \(S\ge N\), we compare the right side of the second row in (31) with that in (15) and get

$$\begin{aligned} \begin{array}{l} {{(M' - 1)q} \big /{2 - }}(M - 1)q\\ = {{(M - 2)(M - 1)q} \big /4} \end{array} \end{aligned}$$
(33)

It can be inferred from (33) that the second row in (15) has more relaxed upper bound of N than that in (31). In conclusion, it is true that compared to the ALESCAF algorithm, the limitation on N in (15) is more relaxed.

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Su, Q., Wei, Y., Xu, K. et al. Using Derivatives of Second Generating Function for Underdetermined Blind Identification. Circuits Syst Signal Process 39, 4578–4595 (2020). https://doi.org/10.1007/s00034-020-01385-y

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