Editing Array processing (section)

=== 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.