Blind Equalization in Neural Networks

$\begin{aligned} w_{j p} (n + 1) = w_{j p} (n) - 2 μ_{3} \tilde{x} (n) [| \tilde{x} (n) |^{2} - R_{2}] {f^{'}}_{4} (\sum_{k = 1}^{K} w_{k} (n) v_{k}^{K} (n)) \\ \times (\sum_{p = 1}^{P} w_{p k} (n) v_{p}^{P} (n)) {f^{'}}_{2} (\sum_{j = 1}^{J} w_{j p} (n) v_{j}^{J} (n)) w_{k} (n) w_{p k} (n) v_{j}^{J} (n) (3.76) \end{aligned}$

where μ₃ is the weight iterative step size between the first hidden layer and the second hidden layer.

Weight iterative formula between the input layer and the first hidden layer

The connection weight between the input layer and the first hidden layer is w_ij(n), so

$w_{i j} (n + 1) = w_{i j} (n) - 2 μ_{4} \tilde{x} (n) [| \tilde{x} (n) |^{2} - R_{2}] \frac{\partial \tilde{x} (n)}{\partial w_{i j} (n)} (3.77)$

$\begin{aligned} \frac{\partial \tilde{x} (n)}{\partial w_{i j} (n)} = {f^{'}}_{4} (\sum_{k = 1}^{K} w_{k} (n) v_{k}^{K} (n)) \frac{\partial u (n)}{\partial w_{i j} (n)} \\ = {f^{'}}_{4} (\sum_{k = 1}^{K} w_{k} (n) v_{k}^{K} (n)) w_{k} (n) \frac{\partial v_{k}^{K} (n)}{\partial w_{i j} (n)} \\ = {f^{'}}_{4} (\sum_{k = 1}^{K} w_{k} (n) v_{k}^{K} (n)) w_{k} (n) {f^{'}}_{3} (\sum_{p = 1}^{P} w_{p k} (n) v_{p}^{P} (n)) \\ \times w_{p k} (n) {f^{'}}_{2} (\sum_{j = 1}^{J} w_{j p} (n) v_{j}^{J} (n)) \frac{\partial v_{j}^{J} (n)}{\partial w_{i j} (n)} \\ = {f^{'}}_{4} (\sum_{k = 1}^{K} w_{k} (n) v_{k}^{K} (n)) w_{k} (n) {f^{'}}_{3} (\sum_{p = 1}^{P} w_{p k} (n) v_{p}^{P} (n)) w_{p k} (n) {f^{'}}_{2} (\sum_{j = 1}^{J} w_{j p} (n) v_{j}^{J} (n)) \\ \times w_{j p} (n) {f^{'}}_{1} (\sum_{i = 1}^{I} \end{aligned}$

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

Weight iterative formula between the input layer and the first hidden layer

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