deletions | additions       diff --git a/Waveform optimization.tex b/Waveform optimization.tex  index 1e7ed29..032ccb8 100644  --- a/Waveform optimization.tex  +++ b/Waveform optimization.tex 
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substituting into Eq.\ref{eqn:Qem} we find:  $\int_X \begin{equation}   \int_X  \sum_i^N [ \sum_t^{T_i} (d_t^i - m(x_t^i))^2 ]P(X|D,\Theta_{k-1}) + \int_X \sum_i^N [ \log(P(X|\Theta_{k}))]P(X|D,\Theta_{k-1}) $ \end{equation}  Next we want to take the derivative of this expression with respect to the components of $\Theta_{k}$ that defines the waveform and equate it to zero to find its maximum. As the second integral does not depend on the waveform, it can be neglected, as well as any multiplicative or additive factors that are independent of the waveform. 
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$ \sum_i^N \sum_t^{T_i} \sum_l \exp(\lambda[l])d_t^i P_t^i[l,u] = \sum_i^N \sum_t^{T_i} \sum_l \exp(2\lambda[l])f[u] P_t^i[l,u]$  Finally:  $ \begin{equation}  f[u] = (\sum_i^N \sum_t^{T_i} \sum_l \exp(\lambda[l])d_t^i P_t^i[l,u] ) / ( \sum_i^N \sum_t^{T_i} \sum_l \exp(2\lambda[l]) P_t^i[l,u] )$ )  \end{equation}