Blind Equalization in Neural Networks

Appendix B:Iterative Formulas Derivation of Complex Blind Equalization Algorithm Based on BRNN

(1) Weight iterative formula of the feed-forward unit

The connection weight between the feed-forward unit and the output unit is c_j(n), so

$c_j\left(n+1\right)=c_j\left(n\right)-{\mu }_c\frac{\partial J\left(n\right)}{\partial c_j\left(n\right)}\left(\text{B}\text{.1}\right)$

$\frac{\partial J\left(n\right)}{\partial c_j\left(n\right)}=2\left[{\left|\tilde{x}\left(n\right)\right|}^2-R_2\right]\left|\tilde{x}\left(n\right)\right|\left[\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial c_{j,R}\left(n\right)}+j\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial c_{j,I}\left(n\right)}\right]\left(\text{B}\text{.2}\right)$

According to eqs. (4.11)–(4.13), we obtain

$\begin{array}{l}\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial c_{j,R}\left(n\right)}=\frac{\partial \sqrt{\tilde{x}\left(n\right){\tilde{x}}^{*}\left(n\right)}}{\partial c_{j,R}\left(n\right)}=\frac{1}{2\left|\tilde{x}\left(n\right)\right|}\frac{\partial \left[\tilde{x}\left(n\right){\tilde{x}}^{*}\left(n\right)\right]}{\partial c_{j,R}\left(n\right)}\\ =\frac{1}{2\left|\tilde{x}\left(n\right)\right|}\frac{\partial \left\{f^2\left[v_R\left(n\right)\right]+f^2\left[v_I\left(n\right)\right]\right\}}{\partial c_{j,R}\left(n\right)}\\ =\frac{1}{\left|\tilde{x}\left(n\right)\right|}\left\{f\left[v_R\left(n\right)\right]f\text{'}\left[v_R\left(n\right)\right]\frac{\partial v_R\left(n\right)}{\partial c_{j,R}\left(n\right)}+f\left[v_I\left(n\right)\right]f\text{'}\left[v_I\left(n\right)\right]\frac{\partial v_I\left(n\right)}{\partial c_{j,R}\left(n\right)}\right\}\left(\text{B}\text{.3}\right)\\ \end{array}$