Blind Equalization in Neural Networks

Substituting eq. (2.33) into eq. (2.32), eq. (2.34) can be obtained:

$\begin{array}{l}E\left[{\left|\tilde{x}\left(n\right)\right|}^4\right]=E\left[{\left|x\left(n\right)\right|}^4\right]{\displaystyle \sum _i{\left|s_i\left(n\right)\right|}^4+2E^2\left[{\left|x\left(n\right)\right|}^2\right]}\left\{{\left[{\displaystyle \sum _i{\left|s_i\left(n\right)\right|}^2}\right]}^2-{\displaystyle \sum _i{\left|s_i\left(n\right)\right|}^4}\right\}\left(2.34\right)\\ +{\left|E\left[x^2\left(n\right)\right]\right|}^2\left[{\left|{\displaystyle \sum _i{s}_i^2\left(n\right)}\right|}^2-{\displaystyle \sum _i{\left|s_i\left(n\right)\right|}^4}\right]\end{array}$

Substituting eqs. (2.29) and (2.31) into eq. (2.34), (2.35) can be obtained:

$\begin{array}{r}E\left[{\left|\tilde{x}\left(n\right)\right|}^4\right]=E\left[{\left|x\left(n\right)\right|}^4\right]\sum _i{\left|s_i\left(n\right)\right|}^4+2E\left[\left|x{\left(n\right)}^2\right|\right]{\left[\sum _i{\left|s_i\left(n\right)\right|}^2\right]}^2\\ -2E^2\left[{\left|x\left(n\right)\right|}^2\right]\sum _i{\left|s_i\left(n\right)\right|}^4+\left|E{\left[x^2\left(n\right)\right]}^2\right|{\left|\sum _i{s}_i^2\left(n\right)\right|}^2\\ -{\left|E\left[x^2\left(n\right)\right]\right|}^2\sum _i{\left|s_i\left(n\right)\right|}^4\left(2.35\right)\\ =E\left[{\left|x\left(n\right)\right|}^4\right]\sum _i{\left|s_i\left(n\right)\right|}^4+2E\left[\left|\tilde{x}{\left(n\right)}^2\right|\right]-2E^2\left[{\left|x\left(n\right)\right|}^2\right]\sum _i{\left|s_i\left(n\right)\right|}^4\\ +{\left|E\left[{\tilde{x}}^2\left(n\right)\right]\right|}^2-{\left|E\left[x^2\left(n\right)\right]\right|}^2\sum _i{\left|s_i\left(n\right)\right|}^4\end{array}$

By moving the mathematical expectations of transmitted sequence and recovery sequence to both sides of the equal sign, respectively, eq. (2.36) can be obtained:

$\begin{array}{r}E\left[{\left|\tilde{x}\left(n\right)\right|}^4\right]-2E^2\left[{\left|\tilde{x}\left(n\right)\right|}^2\right]-{|E[{\tilde{x}}^2(}^{}\end{array}$