deletions | additions       diff --git a/Waveform optimization.tex b/Waveform optimization.tex  index e7d6ec6..a405ffe 100644  --- a/Waveform optimization.tex  +++ b/Waveform optimization.tex 
...
Denoting the coefficient of the waveform $w(\phi[m]) = f[m]$ :  \begin{align*}   \partial/\partial{f[u]} &\sum_i \sum_t \sum_l &\sum_{i,t,l}  \sum_m (d_t^i - \exp(\lambda[l])f[m])^2 P_t^i[l,m] = 0\\ &\sum_i \sum_t \sum_l &\sum_{i,t,l}  \exp(\lambda[l])(d_t^i - \exp(\lambda[l])f[u]) P_t^i[l,u] = 0\\ &\sum_i \sum_t \sum_l &\sum_{i,t,l}  \exp(\lambda[l])d_t^i P_t^i[l,u] = \sum_i \sum_t \sum_l \exp(2\lambda[l])f[u] P_t^i[l,u]\\ \end{align*}   Finally:  \begin{equation}   f[u] = (\sum_i \sum_t \sum_l (\sum_{i,t,l}  \exp(\lambda[l])d_t^i P_t^i[l,u] ) / ( \sum_i \sum_t \sum_l \sum_{i,t,l}  \exp(2\lambda[l]) P_t^i[l,u] ) \end{equation}