deletions | additions       diff --git a/Waveform optimization.tex b/Waveform optimization.tex  index f2a7b45..ee62e87 100644  --- a/Waveform optimization.tex  +++ b/Waveform optimization.tex 
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In the context of our HMM the joint distribution $P(D,X|\Theta)=P(D|X,\Theta)P(X|\Theta)$ is given by $\prod_i^N P(d_1^i,..,d_{T_i}^i ,x_1^i,..,x_{T_i}^i|\Theta) = \prod_i^N P(d_1^i,..,d_{T_i}^i|x_1^i,..,x_{T_i}^i,\Theta)P(x_1^i,..,x_{T_i}^i|\Theta)$.   Here $P(d_1^i,..,d_{T_i}^i|x_1^i,..,x_{T_i}^i,\Theta)$ is the emission probability for a given trace and is proportional to $\prod_t^{T_i} \exp[ -(d_t^i - m(x_t^i))^2]$. m(x_t^i))^2]$, where $m$ is the model that relates the data to the phase trough the waveform and amplitude.  The joint distribution of hidden states $P(X|D,\Theta)$ is given by $\prod_i^N P(x_1^i,..,x_{T_i}^i|d_1^i,..,d_{T_i}^i,\Theta)$. 
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\int_X \sum_{i=1}^N \left[ \sum_{t=1}^{T_i} (d_t^i - m(x_t^i))^2 \right] P(X|D,\Theta_{k-1}) + \int_X \sum_{i=1}^N \left[ \log(P(X|\Theta_{k})) \right] 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. waveform parameters at iteration $k$.  We can now expand the integral over the hidden states: