Blind Equalization in Neural Networks

The time-varying optimal weight vector from the hidden layer to the output layer is

${\widehat{w}}_j\left(n\right)={\left[\begin{array}{ccc}{\widehat{W}}_1\left(n\right)& \cdots & {\widehat{W}}_J\left(n\right)\end{array}\right]}^T\left(3.96\right)$

So,

$\widehat{\chi }\left(n\right)=Y\left(n\right){\widehat{W}}_{ij}\left(n\right)\widehat{W}{j}_{}\left(n\right)+\xi \left(n\right)\left(3.97\right)$

where ξ(n) is the interference signal with zero mean and independent identically distributed.

Plugging eq. (3.97) into the expression of residual error (2.76), we obtain

$\begin{array}{l}e\left(n\right)=\widehat{x}\left(n\right)-\widehat{x}\left(n\right)-Y\left(n\right){\widehat{W}}_{ij}\left(n\right){\widehat{W}}_j\left(n\right)-Y\left(n\right)W_{ij}\left(n\right)+\xi \left(n\right)\\ =Y\left(n\right)\left[{\widehat{W}}_{ij}\left(n\right){\widehat{W}}_j\left(n\right)-W_{ij}\left(n\right)W_j\left(n\right)\right]+\xi \left(n\right)=Y\left(n\right)V\left(n\right)+\xi \left(n\right)\left(3.98\right)\end{array}$

where V(n) is the weight error vector. In the algorithm convergence process, the neural network weight vector gradually closes to the optimal weight vector, that is, V(n) gradually reduces until it tends to zero. ξ(n) is the interference signal. ...