Blind Equalization in Neural Networks

In eq. (2.78), R = E[Y(n)Y^T(n)] is the autocorrelation matrix of the equalizer input sequence and P = E[$\tilde{x}$(n)Y^T(n)] is the correlation matrix. The MSE is generally composed of two parts, such as theoretical error and excess MSE.

2.5.7.1The theoretical error

The theoretical error refers to the error produced by adopting finite length transversal filter to replace the infinite length transversal filter.

For the infinite filter, if the noise superimposed in channel is ignored, then

$\begin{array}{l}\tilde{x}\left(n\right)=W^T\left(n\right)Y\left(n\right)={\displaystyle \sum _{i=-\infty }^{+\infty }{w}_i\left(n\right)y\left(n-i\right)}={\displaystyle \sum _{i=-\infty }^{+\infty }{\displaystyle \sum _{j=-\infty }^{+\infty }{w}_i\left(n\right)h_j\left(n\right)x\left(n-i-j\right)}}\\ ={\displaystyle \sum _{l=-\infty }^{+\infty }x\left(n-1\right)}{\displaystyle \sum _{i=-\infty }^{+\infty }{w}_i\left(n\right)h_{l-i}\left(n\right)}={\displaystyle \sum _{l=-\infty }^{+\infty }{\delta }_l\left(n\right)x\left(n-l\right)}=x\left(n\right)\left(2.79\right)\end{array}$

and required

${\displaystyle \sum _{i=-\infty }^{+\infty }{w}_i\left(n\right)}{h}_{l-i}\left(n\right)={\delta }_l\left(n\right)$