Editing Array processing (section)

=== Parametric–based solutions ===
While the spectral-based methods presented in the previous section are computationally attractive, they do not always yield sufficient accuracy. In particular, for the cases when we have highly correlated signals, the performance of spectral-based methods may be insufficient. An alternative is to more fully exploit the underlying data model, leading to so-called parametric array processing methods. The cost of using such methods to increase the efficiency is that the algorithms typically require a multidimensional search to find the estimates. The most common used model based approach in signal processing is the maximum likelihood (ML) technique. This method requires a statistical framework for the data generation process. When applying the ML technique to the array processing problem, two main methods have been considered depending on the signal data model assumption. According to the Stochastic ML, the signals are modeled as Gaussian random processes. On the other hand, in the Deterministic ML the signals are considered as unknown, deterministic quantities that need to be estimated in conjunction with the direction of arrival.<ref name="ref2"/><ref name="ref6"/><ref name="ref5"/>

==== 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>

==== Deterministic ML approach ====
While the background and receiver noise in the assumed data model can be thought of as emanating from a large number of independent noise sources, the same is usually not the case for the emitter signals. It therefore appears natural to model the noise as a stationary Gaussian white random process whereas the signal waveforms are deterministic (arbitrary) and unknown. According to the Deterministic ML the signals are considered as unknown, deterministic quantities that need to be estimated in conjunction with the direction of arrival. This is a natural model for digital communication applications where the signals are far from being normal random variables, and where estimation of the signal is of equal interest.<ref name="ref2"/><ref name="ref3"/>