Abstract
In this paper, we are interested in the problem of Blind Source Separation using a Second-order Statistics (SOS) method in order to separate autocorrelated and mutually independent sources mixed according to a bilinear (BL) model. In this context, we propose a new approach called Bilinear Second-order Blind Source Separation, which is an extension of linear SOS methods, devoted to separate sources present in BL mixtures. These sources, called extended sources, include the actual sources and their products. We first study the statistical properties of the different extended sources, in order to verify the assumption of identifiability when the actual sources are zero-mean and when they are not. Then, we present the different steps performed in order to estimate these actual centred sources and to extract the actual mixing parameters. The obtained results using artificial mixtures of synthetic and real sources confirm the effectiveness of the new proposed approach.
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Notes
Note that in the linear SOBI, at most one source may be temporally uncorrelated.
In the special case of \(L=2\) actual centred sources, it is also possible to identify the estimated actual centred sources among 3 estimated extended centred sources using a criterion measuring statistical independence, like mutual information. Actually, we know that \(\widetilde{s}_1\) and \(\widetilde{s}_2\) are mutually independent while \(\widetilde{s}_1\times \widetilde{s}_2\) is not independent from \(\widetilde{s}_1\) and \(\widetilde{s}_2\).
In fact, the columns of the matrix estimated by Algorithm 1 are also permuted. However, using the method explained in Sect. 3.5, we can identify the columns containing linear coefficients and the columns containing quadratic ones. It is then possible to arrange the columns of the estimated matrix so that the first L ones correspond to the linear part (their order is not really important) and the other ones are matched correctly to these first L columns.
Computation has been performed with Matlab, on a computer with an intel core i7 processor, a frequency of 2.7 GHz, and a RAM size of 16 GB.
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Jarboui, L., Deville, Y., Hosseini, S. et al. A second-order blind source separation method for bilinear mixtures. Multidim Syst Sign Process 29, 1153–1172 (2018). https://doi.org/10.1007/s11045-017-0493-9
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DOI: https://doi.org/10.1007/s11045-017-0493-9