Editing Array processing (section)

==== Stochastic ML approach ====
The stochastic maximum likelihood method is obtained by modeling the signal waveforms as a Gaussian random process under the assumption that the process x(t) is a stationary, zero-mean, Gaussian process that is completely described by its second-order covariance matrix. This model is a reasonable one if the measurements are obtained by filtering wide-band signals using a narrow band-pass filter.<br>
''' ''Approach overview'' '''<br>
<math>\textstyle 1.\ Find\ W_{K}\ to\ minimize:</math><br>
<math>\textstyle min_{a^{*}(\theta_{k}w_{k}=1)}\ E\{\left |W_{k}X(t) \right |^{2}\}</math><br>
<math>\textstyle=min_{a^{*}(\theta_{k}w_{k}=1)}\ W_{k}^{*}R_{k}W_{k}</math><br>
<math>\textstyle 2.\ Use\ the\ langrange\ method:</math><br>
<math>\textstyle min_{a^{*}(\theta_{k}w_{k}=1)}\ E\{\left |W_{k}X(t) \right |^{2}\}</math><br>
<math>\textstyle=min_{a^{*}(\theta_{k}w_{k}=1)}\ W_{k}^{*}R_{k}W_{k}+ 2\mu(a^{*}(\theta_{k})w_{k}\Leftrightarrow 1) </math><br>
<math>\textstyle 3.\ Differentiating\ it,\ we\ obtain</math><br>
<math>\textstyle R_{x}w_{k}=\mu a(\theta_{k}),\ or\ W_{k} = \mu R_{x}^{-1}a(\theta_{k})</math><br>
<math>\textstyle 4.\ since</math><br>
<math>\textstyle a^{*}(\theta_{k})W_{k}=\mu a(\theta_{k})^{*}R_{x}^{-1}a(\theta_{k})=1</math><br>
<math>\textstyle Then</math><br>
<math>\textstyle \mu=a(\theta_{k})^{*}R_{x}^{-1}a(\theta_{k})</math><br>
<math>\textstyle 5.\ Capon's\ Beamformer</math><br>
<math>\textstyle W_{k}=R_{x}^{-1}a(\theta_{k})/(a^{*}(\theta_{k})R_{x}^{-1}a(\theta_{k}))</math>