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{{distinguish|Array processor|Array data structure}}
{{more footnotes|date=November 2012}}
'''Array
* Determine number and locations of energy-radiating sources (emitters).
* Enhance the signal to noise ratio SNR "[[SINR|signal-to-interference-plus-noise ratio (SINR)]]".
* Track multiple moving sources.
Precisely, we are interested in solving these problems in noisy environments (in the presence of noise and interfering signals). [[Estimation theory
Array processing is used in [[radar]], [[sonar]], seismic exploration, anti-jamming and [[wireless]] communications. One of the main advantages of using array processing along with an array of sensors is a smaller foot-print. The problems associated with array processing include the number of sources used, their [[direction of arrival]]s, and their signal [[waveforms]].<ref name="utexas1">Torlak, M. [http://users.ece.utexas.edu/~bevans/courses/ee381k/lectures/13_Array_Processing/lecture13/lecture13.pdf Spatial Array Processing]. Signal and Image Processing Seminar. University of Texas at Austin.</ref><ref name="ref1">{{cite book|last=J Li|first=P Stoica (Eds)|title=MIMO Radar Signal Processing|year=2009|publisher=J Wiley&Sons|___location=USA}}</ref><ref name="ref2">{{cite book|last=P Stoica|first=R Moses|title=Spectral Analysis of Signals|year=2005|publisher=Prentice Hall|___location=NJ|url=http://user.it.uu.se/%7Eps/SAS-new.pdf}}</ref><ref name="ref3">{{cite book|last=J Li|first=P Stoica (Eds)|title=Robust Adaptive Beamforming|year=2006|publisher=J Wiley&Sons|___location=USA}}</ref>
[[File:Aray Prcessing Model.png|thumb|Sensors
There are four assumptions in array processing. The first assumption is that there is uniform propagation in all directions of isotropic and non-dispersive medium. The second assumption is that for far field array processing, the radius of propagation is much greater than size of the array and that there is plane wave propagation. The third assumption is that there is a zero mean white noise and signal, which shows uncorrelation. Finally, the last assumption is that there is no coupling and the calibration is perfect.<ref name="utexas1"/>
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* Radar and Sonar Systems:
array processing concept was closely linked to radar and sonar systems which represent the classical applications of array processing. The antenna array is used in these systems to determine ___location(s) of source(s), cancel interference, suppress ground clutter. '''[[Radar|Radar
[[File:Radar System.png|thumb|Radar System]]
NORSAR is an independent geo-scientific research facility that was founded in Norway in 1968. NORSAR has been working with array processing ever since to measure seismic activity around the globe.<ref name=NORSAR>{{cite web|title= About Us|url=http://www.norsar.no/norsar/about-us/|publisher=NORSAR|accessdate=6 June 2013}}</ref> They are currently working on an International Monitoring System which will comprise 50 primary and 120 auxiliary seismic stations around the world. NORSAR has ongoing work to improve array processing to improve monitoring of seismic activity not only in Norway but around the globe.<ref>{{cite web|url=http://www.norsar.no/pc-31-83-Improving-IMS-array-processing.aspx |title=Improving IMS array processing |publisher=Norsar.no |date= |accessdate=2012-08-06}}</ref>
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* Communications (wireless)
[[Communication theory|Communication]] can be defined as the process of exchanging of information between two or more parties. The last two decades witnessed a rapid growth of wireless communication systems. This success is a result of advances in communication theory and low power dissipation design process. In general, communication (telecommunication) can be done by technological means through either electrical signals (wired communication) or electromagnetic waves (wireless communication). Antenna arrays have emerged as a support technology to increase the usage efficiency of spectral and enhance the accuracy of wireless communication systems by utilizing spatial dimension in addition to the classical time and frequency dimensions. Array processing and estimation techniques have been used in wireless communication. During the last decade these techniques were re-explored as ideal candidates to be the solution for numerous problems in wireless communication. In wireless communication, problems that affect quality and performance of the system may come from different sources. The multiuser –medium multiple access- and multipath -signal propagation over multiple scattering paths in wireless channels- communication model is one of the most widespread communication models in wireless communication (mobile communication).
[[File:Multi-Path Communication.png|thumb|Multi-
In the case of multiuser communication environment, the existence of multiuser increases the inter-user interference possibility that can affect quality and performance of the system adversely. In mobile communication systems the multipath problem is one of the basic problems that base stations have to deal with. Base stations have been using spatial diversity for combating fading due to the severe multipath. Base stations use an antenna array of several elements to achieve higher selectivity. Receiving array can be directed in the direction of one user at a time, while avoiding the interference from other users.
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In addition to these applications, many applications have been developed based on array processing techniques: Acoustic Beamforming for Hearing Aid Applications, Under-determined Blind Source Separation Using Acoustic Arrays, Digital 3D/4D Ultrasound Imaging Array, Smart Antennas, Synthetic aperture radar, underwater acoustic imaging, and Chemical sensor arrays...etc.<ref name="ref2"/><ref name="ref3"/><ref name="ref6"/>
== General
Consider a system that consists of array of '''r''' arbitrary sensors that have arbitrary locations and arbitrary directions (directional characteristics) which receive signals that generated by '''q''' narrow band sources of known center frequency ω and locations θ1, θ2, θ3, θ4 … θq. since the signals are narrow band the propagation delay across the array is much smaller than the reciprocal of the signal bandwidth and it follows that by using a complex envelop representation the array output can be expressed (by the sense of superposition) as :<ref name="ref2"/><ref name="ref6"/><ref name="ref5"/><br>
<math>\textstyle x(t)=\sum_{K=1}^q a(\theta_k)s_k(t)+n(t)</math>
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'''“The target is to estimate the DOA’s θ1, θ2, θ3, θ4 …θq of the sources from the M snapshot of the array x(t1)… x(tM). In other words what we are interested in is estimating the DOA’s of emitter signals impinging on receiving array, when given a finite data set {x(t)} observed over t=1, 2 … M. This will be done basically by using the second-order statistics of data”'''<ref name="ref6"/><ref name="ref5"/>
In order to solve this problem (to guarantee that there is a valid solution) do we have to add conditions or assumptions on the operational environment and\or the used model? Since there are many parameters used to specify the system like the number of sources, the number of array elements
::a. Radius of propagation >> size of array.<br>
::b. Plane wave propagation.
Throughout this survey, it will be assumed that the number of underlying signals, q, in the observed process is considered known. There are, however, good and consistent techniques for estimating this value even if it is not known.
== Estimation
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 algorithmic solutions can be further classified into beamforming techniques and subspace-based techniques.
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MUSIC spectrum approach 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.
===
While the spectral-based methods presented in 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"/>
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Cross-correlation spectroscopy with spatial interferometry, is possible by simply substituting a signal with voltage <math>V_Y(t)</math> in equation {{EquationNote|Eq.II}} to produce the cross-correlation <math>R_{\text{XY}}(\tau)</math> and the cross-spectrum <math>S_{\text{XY}}(f)</math>.
== Example:
In radio astronomy, RF interference must be mitigated to detect and observe any meaningful objects and events in the night sky. [[File:Telescope array.png|thumb|An array of radio telescopes with an incoming radio wave and RF interference]]
=== Projecting
For an array of Radio Telescopes with a spatial signature of the interfering source <math>\mathbf{a}</math> that is not a known function of the direction of interference and its time variance, the signal covariance matrix takes the form:
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| doi = 10.1109/97.991140}}</ref>
=== Spatial
This scheme attempts to make the interference-plus-noise term spectrally white. To do this, left and right multiply <math>\mathbf{R}</math> with inverse square root factors of the interference-plus-noise terms.
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|arxiv = astro-ph/0008239}}</ref>
=== Subtraction of
Since <math>\mathbf{a}</math> is unknown, the best estimate is the dominant eigenvector <math>\mathbf{u}_1</math> of the eigen-decomposition of <math>\mathbf{R} = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^{\dagger}</math>, and likewise the best estimate of the interference power is <math>\sigma_s^2 \approx \lambda_1 - \sigma_n^2</math>, where <math>\lambda_1</math> is the dominant eigenvalue of <math>\mathbf{R}</math>. One can subtract the interference term from the signal covariance matrix:
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