Abstract
This chapter deals with blind source separation and blind mixture identification methods intended for linear-quadratic mixtures (including their bilinear and purely quadratic restricted versions), which were defined in the previous chapter. Various structures may be used to this end, including nonlinear recurrent artificial neural networks. Part of these structures were derived by extending previously reported structures for linear mixtures. Both classes of structures are described in this chapter. Most of the structures for linear-quadratic mixtures are based on the inverse of the mixing transform and aim at restoring the source signals by adequately recombining the observations. The invertibility of mixing transforms and the required number of observations, depending on the considered approach, are therefore also discussed in this chapter. Other structures are based on the mixing transform itself and aim at jointly fitting this direct model and its input, that is, the source values. The adaptation of the parameters of both types of linear-quadratic “separating structures” is described in the subsequent chapters.
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Notes
- 1.
Unless additional constraints on the source signals and/or mixing matrix are added, to make it possible to derive BSS methods suited to underdetermined mixtures, as was done in the literature for originally linear mixtures.
- 2.
This version uses the normalized sources defined in (3.7).
- 3.
A dual normalization also exists: see [34].
- 4.
These successive output values therefore also depend on n. This index n is omitted in the notations y i(m), in order to improve readability and to focus on the recurrence on outputs for given input values x 1(n) and x 2(n).
- 5.
Again, these successive output values therefore also depend on n, but this index n is omitted in the notations.
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Deville, Y., Tomazeli Duarte, L., Hosseini, S. (2021). Invertibility of Mixing Model, Separating Structures. In: Nonlinear Blind Source Separation and Blind Mixture Identification. SpringerBriefs in Electrical and Computer Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-64977-7_3
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