Blind Equalization in Neural Networks

${\displaystyle \sum _{k=1}^K{\displaystyle \sum _{n=1}^N\Vert \Delta W\left(n\right)\Vert \le c_2\left(8.33\right)}}$

The mean value theorem is adopted for p(x) and p′(x), and there exists a constant c₃ satisfying the following equation:

$\left|\Vert \frac{\partial \mathrm{erf}\left(W\left(n+1\right)\right)}{\partial W}\Vert -\Vert \frac{\partial \mathrm{erf}\left(W\left(n\right)\right)}{\partial W}\Vert \right|\le c_3{\displaystyle \sum _{k=1}^K\Vert d_k\Vert }\left(8.34\right)$

where d_k = W_k(n + 1) – W_k(n).

From eqs. (8.33) and (8.34), it can be obtained:

$\underset{n\to \infty k}{\lim }\Vert \frac{\partial \mathrm{erf}\left(W_k\left(n\right)\right)}{\partial W_k\left(n\right)}\Vert =0\left(8.35\right)$

Resembling eq. (8.34), there is a constant c₄ satisfying the following equation:

$\Vert \frac{\partial \mathrm{erf}\left(W_{k+l}\left(n+1\right)\right)}{\partial W}-\frac{\partial \mathrm{erf}\left(W_k\left(n\right)\right)}{\partial W}\Vert \le \frac{c_4}{l}\left(8.36\right)$

According to eqs. (8.35) and (8.36), it can be obtained:

$\underset{n\to \infty }{\lim }\Vert \frac{\partial \mathrm{erf}\left(W_{k+1}\left(n\right)\right)}{\partial W}\Vert =0$