Blind Equalization in Neural Networks

where μ₃ and μ₄ are the expansion factor and the translation factor, respectively.

The iteration step-size factors are

$\frac{\partial J\left(n\right)}{\partial a\left(n\right)}=2\left[{\left|\tilde{x}\left(n\right)\right|}^2-R_2\right]{\left|\tilde{x}\left(n\right)\right|}^2\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial a\left(n\right)}\left(7.50\right)$

$\frac{\partial J\left(n\right)}{\partial b\left(n\right)}=2\left[{\left|\tilde{x}\left(n\right)\right|}^2-R_2\right]{\left|\tilde{x}\left(n\right)\right|}^2\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial b\left(n\right)}\left(7.51\right)$

$a\left(n\right)=a_R\left(n\right)+j{a}_I\left(n\right)\left(7.52\right)$

$b\left(n\right)=b_R\left(n\right)+j{b}_1\left(n\right)\left(7.53\right)$

$\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial a\left(n\right)}=\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial a_R\left(n\right)}+j\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial a_I\left(n\right)}\left(7.54\right)$

$\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial b\left(n\right)}=\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial b_R\left(n\right)}+j\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial b_I\left(n\right)}\left(7.55\right)$

$\begin{array}{c}\begin{array}{l}\frac{\partial \left|\tilde{x}\left(n\right)\right|}{\partial a_R\left(n\right)}=\frac{\partial \sqrt{\tilde{x}\left(n\right){\tilde{x}}^{*}\left(n\right)}}{\partial a_R\left(n\right)}=\frac{1}{2\left|\tilde{x}\left(n\right)\right|}\frac{\partial \tilde{x}\left(n\right){\tilde{x}}^{*}\left(n\right)}{\partial a_R\left(n\right)}=\frac{1}{2\left|\tilde{x}\left(n\right)\right|}\frac{\partial \left(f^2\left(v_R\left(n\right)\right)+f^2\left(v_I\left(n\right)\right)\right)}{\partial a_R\left(n\right)}\\ =\frac{1}{\left|\tilde{x}\left(n\right)\right|}[f\left(v_R\left(n\right)\right)\end{array}\end{array}$