Blind Equalization in Neural Networks

The weights iterative formula of the feed-forward units is

$c_{j} (n + 1) = c_{j} (n) - 2 μ_{c} [{| \tilde{x} (n) |}^{2} - R_{2}] K q^{*} (n) y^{*} (n - j) (7.101)$

where μ_c is the iteration step size factor of the feed-forward unit:

$K = f (v_{R} (n)) f' (v_{R} (n)) + j f (v_{I} (n)) f' (v_{I} (n)) (7.102)$

The weight iteration formula of recurrent unit

The connection weight between recurrent and output unit is a_i(n + 1) = a_i, _{_{_R}}(n)+ jd_i,I(n).

$\begin{aligned} \frac{\partial | \tilde{x} (n) |}{\partial a_{i, R} (n)} = \frac{\partial \sqrt{\tilde{x} (n) {\tilde{x}}^{*} (n)}}{\partial a_{i, R} (n)} = \frac{1}{2 | \tilde{x} (n) |} \frac{\partial \tilde{x} (n) {\tilde{x}}^{*} (n)}{\partial a_{i, R} (n)} = \frac{1}{2 | \tilde{x} (n) |} \frac{\partial [f^{2} (v_{R}) + f^{2} (v_{I} (n))]}{\partial a_{i, R} (n)} \\ = \frac{1}{| \tilde{x} (n) |} [f (v_{R} (n)) f^{'} (v_{R} (n)) \frac{\partial v_{R} (n)}{\partial a_{i, R} (n)} + f (v_{1} (n)) f^{'} (v_{1} (n)) \frac{\partial v_{I} (n)}{\partial a_{i, R} (n)}] \\ = \frac{1}{| \tilde{x} (n) |} [f (v_{R} (n)) f^{'} (v_{R} (n)) {\tilde{x}}_{R} (n - i) + f (v_{1} (n)) f^{'} (v_{1} (n)) {\tilde{x}}_{I} (n - i)] \end{aligned}$

$\frac{\partial | \tilde{x} (n) |}{\partial a_{j, I} (n)} = \frac{1}{| \tilde{x} (n) |} [- f (v_{R} (n)) f' (v_{R} (n)) {\tilde{x}}_{I} (n - i) + f (v_{I} (n)) f' (v_{I} (n)) {\tilde{x}}_{R}$

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Blind Equalization in Neural Networks by Liyi Zhang, Tsinghua University Tsinghua University Press

The weight iteration formula of recurrent unit

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