Editing Array processing (section)

== Estimation techniques==
In general, parameters estimation techniques can be classified into: '''spectral based and parametric based methods'''. In the former, one forms some spectrum-like function of the parameter(s) of interest. The locations of the highest (separated) peaks of the function in question are recorded as the DOA estimates. Parametric techniques, on the other hand, require a simultaneous search for all parameters of interest. The basic advantage of using the parametric approach comparing to the spectral based approach is the accuracy, albeit at the expense of an increased computational complexity.<ref name="utexas1"/><ref name="ref2"/><ref name="ref6"/>

=== Spectral–based solutions ===
Spectral based algorithmic solutions can be further classified into beamforming techniques and subspace-based techniques.

==== Beamforming technique ====
The first method used to specify and automatically localize the signal sources using antenna arrays was the beamforming technique. The idea behind beamforming is very simple: steer the array in one direction at a time and measure the output power. The steering locations where we have the maximum power yield the DOA estimates. The array response is steered by forming a linear combination of the sensor outputs.<ref name="ref2"/><ref name="ref6"/><ref name="ref5"/><br>
''' ''Approach overview'' '''<br>
<math>\textstyle 1.\ R_{x}= \frac{1}{M}\sum_{t=1}^M \mathbf{x}(t) \mathbf{x}^{*}(t)</math><br>
<math>\textstyle 2.\ Calculate\ B(W_{i})=F^{*}R_{x}F(W_{i})</math><br>
<math>\textstyle 3.\ Find\ Peaks\ of\ B(W_{i})\ for\ all\ possible\ w_{i}'s.</math><br>
<math>\textstyle 4.\ Calculate\ \theta_{k},\ i=1, .... q.</math><br>
<br>
Where '''Rx''' is the sample [[covariance matrix]]. Different beamforming approaches correspond to different choices of the weighting vector '''F'''. The advantages of using beamforming technique are the simplicity, easy to use and understand. While the disadvantage of using this technique is the low resolution.

==== Subspace-based technique ====
Many spectral methods in the past have called upon the spectral decomposition of a covariance matrix to carry out the analysis.  A breakthrough came about when the eigen-structure of the covariance matrix was explicitly invoked, and its intrinsic properties were directly used to provide a solution to an underlying estimation problem for a given observed process. A class of spatial spectral estimation techniques is based on the eigen-value decomposition of the spatial covariance matrix. The rationale behind this approach is that one wants to emphasize the choices for the steering vector a(θ) which correspond to signal directions. The method exploits the property that the directions of arrival determine the eigen structure of the matrix.<br>
The tremendous interest in the subspace based methods is mainly due to the introduction of the [[MUSIC (algorithm)|MUSIC (Multiple Signal Classification)]] algorithm. MUSIC was originally presented as a DOA estimator, then it has been successfully brought back to the spectral analysis/system identification problem with its later development.<ref name="ref2"/><ref name="ref6"/><ref name="ref5"/>

''' ''Approach overview'' '''<br>
<math>\textstyle 1.\ Subspace\ decomposition\ by\ performing\ eigenvalue\ decomposition:</math><br>
<math>\textstyle R_{x}=\mathbf A \mathbf R_{s} \mathbf A^{*} + \sigma^{2}I=\sum_{k=1}^M \lambda_{k}e_{k}r_{k}^{*}</math><br>
<math>\textstyle 2.\ span\{\mathbf A\}=spane\{e1,....,e_{d}\}=span\{\mathbf E_{s}\}.</math><br>
<math>\textstyle 3.\ Check\ which\ a(\theta)\ \epsilon span\{\mathbf E_{s}\}\ or\ \mathbf P_{A}a(\theta)\ or\ P_{\mathbf A}^{\perp}a(\theta),\ where\ \mathbf P_{A}\ is\ a\ projection\ matrix.</math><br>
<math>\textstyle 4.\ Search\ for\ all\ possible\ \theta\ such\ that: \left | P_{\mathbf A}^{\perp}a(\theta) \right |^{2} = 0\ or\ M(\theta)=\frac{1}{P_{A}a(\theta)} =\infty</math><br>
<math>\textstyle 5.\ After\ EVD\ of\ R_{x}:</math><br>
<math>\textstyle P_{A}^{\perp}=I-E_{s}E_{s}^{*}=E_{n}E_{n}^{*}</math><br>
where the noise [[eigenvector matrix]] <math>E_{n}=[e_{d}+1, .... , e_{M}]</math>

MUSIC spectrum approaches use a single realization of the stochastic process that is represent by the snapshots x (t), t=1, 2 ...M. MUSIC estimates are consistent and they converge to true source bearings as the number of snapshots grows to infinity. A basic drawback of MUSIC approach is its sensitivity to model errors. A costly procedure of calibration is required in MUSIC and it is very sensitive to errors in the calibration procedure. The cost of calibration increases as the number of parameters that define the array manifold increases.

=== 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"/>