Blind Equalization in Neural Networks

where μ_k is the step between output layer and hidden layer.

8.3.2.2Weights iterative formula of the hidden layer

The weight between the kth node in the hidden layer and the lth node in the input layer is w_lk(n) in the complex feed-forward neural network. The iteration formula is

$w_{l k} (n + 1) = w_{l k} (n) - μ_{l k} \frac{\partial J (n)}{\partial w_{l k} (n)} (8.58)$

$\frac{\partial J (n)}{\partial w_{l k} (n)} = 2 [{| \tilde{f} (n) |}^{2} - R_{2}] | \tilde{f} (n) | [\frac{\partial | \tilde{f} (n) |}{\partial w_{l k, R} (n)} + \frac{\partial | \tilde{f} (n) |}{\partial w_{l k, I} (n)}] (8.59)$

According to eqs. (8.47) and (8.53),

$\begin{aligned} \frac{\partial | \tilde{f} (n) |}{\partial w_{l k, R} (n)} = \frac{\partial \sqrt{\tilde{f} (n) {\tilde{f}}^{*} (n)}}{\partial w l l k, R} = \frac{1}{2 | \tilde{f} (n) |} \frac{\partial | \tilde{f} (n) {\tilde{f}}^{*} (n) |}{\partial w_{l k, R} (n)} \\ = \frac{1}{2 | \tilde{f} (n) |} \frac{d^{2} {q^{2} [v_{R} (n)] + q^{2} [v_{I} (n)]}}{\partial w_{l k, R} (n)} \\ = \frac{1}{| \tilde{f} (n) |} q [v_{R} (n)] q' [v_{R} (n)] \frac{\partial v_{R} (n)}{\partial w_{l k, R}} + \frac{1}{| \tilde{f} (n) |} q [v_{I} (n)] q' [v_{I} (n)] \frac{\partial v_{I} (n)}{\partial w_{l k, R} (n)} \\ (8.60) \end{aligned}$

$\begin{array}{l} \frac{\partial v_{R} (n)}{\partial w_{l k, R} (n)} = w_{k, R} (n) q' [u \end{array}$

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

8.3.2.2Weights iterative formula of the hidden layer

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