Array processing: Difference between revisions

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{{Short description|Area of research in signal processing}}

{{distinguish|Array processor|Array data structure}}

{{more footnotes|date=November 2012}}

'''Array processing''' is a wide area of research in the field of [[signal processing]] that extends from the simplest form of 1 dimensional line arrays to 2 and 3 dimensional array geometries. Array structure can be defined as a set of [[Sensor|sensors]] that are spatially separated, e.g. [[antenna (radio)|radio antenna]] and [[Seismic array|seimicseismic arrays]]. The sensors used for a specific problem may vary widely, for example [[microphone|microphones]]s, [[Accelerometer|accelerometersaccelerometer]]s and [[telescope|telescopes]]s. However, many similarities exist, the most fundamental of which may be an assumption of [[wave propagation]]. Wave propagation means there is a systemic relationship between the signal received on spatially separated sensors. By creating a physical model of the wave propagation, or in [[machine learning]] applications a [[training data]] set, the relationships between the signals received on spatially separated sensors can be leveraged for many applications.

Some common problem that are solved with array processing techniques are:

* determine number and locations of energy-radiating sources

* enhance the signal to noise ratio ([[Signal-to-noise_ratio|SNR]]) or "[[SINR|signal-to-interference-plus-noise ratio (SINR)]]"

* track moving sources

Array processing metrics are often assessed in noisy environments. The model for noise may be either one of spatially incoherent noise, or one with interfering signals following the same propagation physics. [[Estimation theory]] is an important and basic part of signal processing field, which used to deal with estimation problem in which the values of several parameters of the system should be estimated based on measured/empirical data that has a random component. As the number of applications increases, estimating temporal and spatial parameters become more important. Array processing emerged in the last few decades as an active area and was centered on the ability of using and combining data from different sensors (antennas) in order to deal with specific estimation task (spatial and temporal processing). In addition to the information that can be extracted from the collected data the framework uses the advantage prior knowledge about the geometry of the [[sensor array]] to perform the estimation task.

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=[[Peter Stoica]] (Eds)|title=MIMO Radar Signal Processing|year=2009|publisher=J Wiley&Sons|___location=USA}}</ref><ref name="ref2">{{cite book|last=[[Peter 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=[[Peter Stoica]] (Eds)|title=Robust Adaptive Beamforming|year=2006|publisher=J Wiley&Sons|___location=USA}}</ref>

[[File:Aray Prcessing Model.png|thumb|Sensors array]]

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"/>

==Applications==

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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 systems]]''' used basically to detect objects by using radio waves. The range, altitude, speed and direction of objects can be specified. Radar systems started as military equipments then entered the civilian world. In radar applications, different modes can be used, one of these modes is the active mode. In this mode the antenna array based system radiates pulses and listens for the returns. By using the returns, the estimation of parameters such as velocity, range and DOAs (direction of arrival) of target of interest become possible. Using the passive far-field listening arrays, only the DOAs can be estimated. '''[[Sonar|Sonar systems]]''' (Sound Navigation and Ranging) use the sound waves that propagate under the water to detect objects on or under the water surface. Two types of sonar systems can be defined the active one and the passive one. In active sonar, the system emits pulses of sound and listens to the returns that will be used to estimate parameters. In the passive sonar, the system is essentially listening for the sounds made by the target objects. ItThere is very important to note thea difference between the radar system that uses radio waves and the sonar system that uses sound waves, the reason why the sonar uses the sound wave is because sound waves travel farther in the water than do radar and light waves. In passive sonar, the receiving array has the capability of detecting distant objects and their locations. Deformable array are usually used in sonar systems where the antenna is typically drawn under the water. In active sonar, the sonar system emits sound waves (acoustic energy) then listening and monitoring any existing echo (the reflected waves). The reflected sound waves can be used to estimate parameters, such as velocity, position and direction etc. Difficulties and limitations in sonar systems comparing to radar systems emerged from the fact that the propagation speed of sound waves under the water is slower than the radio waves. Another source of limitation is the high propagation losses and scattering. Despite all these limitations and difficulties, sonar system remains a reliable technique for range, distance, position and other parameters estimation for underwater applications.<ref name="ref2"/><ref name="ref6">{{citation |first1=Hema | last1= Singh |first2=RakeshMohan |last2=Jha| title=Trends in Adaptive Array Processing |year=2012}}</ref>

[[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|archive-url=https://web.archive.org/web/20130620112533/http://www.norsar.no/norsar/about-us/|archive-date=20 June 2013|url-status=dead}}</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 |accessdate=2012-08-06 |archive-url=https://web.archive.org/web/20120821220803/http://www.norsar.no/pc-31-83-Improving-IMS-array-processing.aspx |archive-date=2012-08-21 |url-status=dead }}</ref>

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[[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-path communication problem in wireless communication systems]]

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, so called [[beamforming]]. Receiving array can be directed in the direction of one user at a time, while avoiding the interference from other users.

* Medical applications

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* Array Processing in Astronomy Applications

Astronomical environment contains a mix of external signals and noises that affect the quality of the desired signals. Most of the arrays processing applications in astronomy are related to image processing. The array used to achieve a higher quality that is not achievable by using a single channel. The high image quality facilitates quantitative analysis and comparison with images at other wavelengths. In general, astronomy arrays can be divided into two classes: the beamforming class and the correlation class. Beamforming is a signal processing techniques that produce summed array beams from a direction of interest – used basically in directional signal transmission or reception- the basic idea is to combine elements in a phased array such that some signals experience destructive inference and other experience constructive inference. Correlation arrays provide images over the entire single-element primary beam pattern, computed off-line from records of all the possible correlations between the antennas, pairwise.

[[File:C G-K - DSC 0421.jpg|thumb|One antenna of the AllanAllen Telescope Array]]

* Other applications

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 model and problem formulation==

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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<math>\textstyle \mathbf x(t) = A(\theta)s(t) + n(t)</math>

If we assume now that M snapshots are taken at time instants t1, t2 …... tM, the data can be expressed as:<br>

<math>\mathbf X = \mathbf A(\theta)\mathbf S + \mathbf N</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 ...etc. are there conditions that should be met first? Toward this goal we want to make the following assumptions:<ref name="utexas1"/><ref name="ref2"/><ref name="ref6"/><br>

# The number of signals is known and is smaller than the number of sensors, {{math|q < r}}.<br>

# The set of any q steering vectors is linearly independent.<br>

# Isotropic and non-dispersive medium – Uniform propagation in all directions.<br>

# Zero mean white noise and signal, uncorrelated.<br>

# Far-Field.<br>

::a. Radius of propagation >> size of array.<br>

::b. Plane wave propagation.

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<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 very important 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"/>

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<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>

<math>\textstyle 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...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 ===

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== Correlation spectrometer ==

The problem of computing pairwise correlation as a function of frequency can be solved by two mathematically equivalent but distinct ways. By using [[Discrete Fourier Transform]] (DFT) it is possible to analyze signals in the time ___domain as well as in the spectral ___domain. The first The first approach is "XF" correlation because it first cross-correlates antennas (the "X" operation) using a time-___domain "lag" convolution, and then computes the spectrum (the "F" operation) for each resulting baseline. The second approach "FX" takes advantage of the fact that convolution is equivalent to multiplication in Fourier ___domain. It first computes the spectrum for each individual antenna (the F operation), and then multiplies pairwise all antennas for each spectral channel (the X operation). A FX correlator has an advantage over a XF correlators in that the computational complexity is [[Big O notation|O]](N²). Therefore, FX correlators are more efficient for larger arrays.<ref>{{cite journal

| last1=Parsons |first1=Aaron |last2=Backer |first2=Donald |last3=Siemion |first3=Andrew |authorlink3=Andrew Siemion

| date = September 12, 2008

| title = A Scalable Correlator Architecture Based on Modular FPGA Hardware, Reuseable Gateware, and Data Packetization

|journal=Publications of the Astronomical Society of the Pacific |volume=120 |issue=873 |pages=1207–1221 | doi = 10.1086/593053

|arxiv = 0809.2266 |bibcode=2008PASP..120.1207P|s2cid=14152210 }}</ref>

Correlation spectrometers like the [[Michelson interferometer]] vary the time lag between signals obtain the power spectrum of input signals. The power spectrum <math>S_{\text{XX}}(f)</math> of a signal is related to the its autocorrelation function by a Fourier transform:<ref name="Harris">[http://www.sofia.usra.edu/det_workshop/papers/session4/4-04harris_edjw021022.pdf ''Spectrometers for Heterodyne Detection''] {{webarchive |url=https://web.archive.org/web/20160307051932/http://www.sofia.usra.edu/det_workshop/papers/session4/4-04harris_edjw021022.pdf |date=March 7, 2016 }} Andrew Harris</ref>

{{NumBlk|:|<math>S_{\text{XX}}(f) = \int_{-\infty}^{\infty} R_{\text{XX}}(\tau) \cos(2 \pi f \tau),\mathrm{d}\tau</math>|{{EquationRef|I}}}}

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| issue = 12

| pages = 64–67

| doi = 10.1109/97.991140|bibcode=2002ISPL....9...64R|url=http://resolver.tudelft.nl/uuid:26525a9b-0815-49e2-82c8-b3c69ed4867f }}</ref>

=== Spatial whitening ===

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| pages = 1730–1747

| doi = 10.1109/18.857787

|arxiv = astro-ph/0008239|s2cid=4671806 }}</ref>

=== Subtraction of interference estimate ===

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| pages = 355–373

| doi = 10.1086/317360

|arxiv = astro-ph/0005359|bibcode=2000ApJS..131..355L|s2cid=50311217 }}</ref>

== Summary ==

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* {{cite book |title=Array Signal Processing |last1=Johnson |first1=D. H. |last2=Dudgeon |first2=D. E. |year= 1993|publisher=Prentice Hall}}

* {{cite book|first1=H. L.|last1=Van Trees|title=Optimum Array Processing|publisher=Wiley|___location=New York|year=2002}}

* {{cite journal|first1=H.|last1=Krim|first2=M.|last2=Viberg|url=http://www.vissta.ncsu.edu/publications/ahk/spm1996.pdf|title=Two Decades of Array Signal Processing Research|journal=IEEE Signal Processing Magazine|pages=67–94|date=July 1996|doi=10.1109/79.526899 |accessdate=8 December 2010|archive-url=https://web.archive.org/web/20130909011148/http://www.vissta.ncsu.edu/publications/ahk/spm1996.pdf|archive-date=9 September 2013|url-status=dead}}

* S. Haykin and K.J.R. Liu (Editors), "Handbook on Array Processing and Sensor Networks", Adaptive and Learning Systems for Signal Processing, Communications, and Control Series, 2010.

* E. Tuncer and B. Friedlander (Editors), "Classical and Modern Direction-of-Arrival Estimation", Academic Press, 2010.