Array processing

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 sensors that are spatially separated, e.g. radio antenna and seismic arrays. The sensors used for a specific problem may vary widely, for example microphones, accelerometers and telescopes. 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 (SNR) or "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 arrivals, and their signal waveforms.

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.

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. 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.

Communications (wireless)

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).

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

Array processing techniques got on much attention from medical and industrial applications. In medical applications, the medical image processing field was one of the basic fields that use array processing. Other medical applications that use array processing: diseases treatment, tracking waveforms that have information about the condition of internal organs e.g. the heart, localizing and analyzing brain activity by using bio-magnetic sensor arrays.

Array Processing for Speech Enhancement

Speech enhancement and processing represents another field that has been affected by the new era of array processing. Most of the acoustic front end systems became fully automatic systems (e.g. telephones). However, the operational environment of these systems contains a mix of other acoustic sources; external noises as well as acoustic couplings of loudspeaker signals overwhelm and attenuate the desired speech signal. In addition to these external sources, the strength of the desired signal is reduced due to the relatively distance between speaker and microphones. Array processing techniques have opened new opportunities in speech processing to attenuate noise and echo without degrading the quality of and affecting adversely the speech signal. In general array processing techniques can be used in speech processing to reduce the computing power (number of computations) and enhance the quality of the system (the performance). Representing the signal as a sum of sub-bands and adapting cancellation filters for the sub-band signals can reduce the demanded computation power and lead to a higher performance system. Relying on multiple input channels allows designing systems of higher quality comparing to systems that use single channel and solving problems such as source localization, tracking and separation, which cannot be achieved in case of using single channel.

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.

thumb|One antenna of the Allen 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.

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 its autocorrelation function by a Fourier transform:

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

where the autocorrelation function <math>R_{\text{XX(\tau)</math> for signal X as a function of time delay <math>\tau</math> is

(\tau) = \left( V_X(t) V_X(t + \tau)\right)</math>|

Cross-correlation spectroscopy with spatial interferometry, is possible by simply substituting a signal with voltage <math>V_Y(t)</math> in equation to produce the cross-correlation <math>R_{\text{XY(\tau)</math> and the cross-spectrum <math>S_{\text{XY(f)</math>.

Example: spatial filtering

In radio astronomy, RF interference must be mitigated to detect and observe any meaningful objects and events in the night sky. thumb|An array of radio telescopes with an incoming radio wave and RF interference

Projecting out the interferer

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:

<math>\mathbf{R} = \mathbf{R}_v + \sigma_s^2 \mathbf{a} \mathbf{a}^{\dagger} + \sigma_n^2 \mathbf{I}</math>

where <math>\mathbf{R}_v</math> is the visibilities covariance matrix (sources), <math>\sigma_s^2</math> is the power of the interferer, and <math>\sigma_n^2</math> is the noise power, and <math>\dagger</math> denotes the Hermitian transpose. One can construct a projection matrix <math>\mathbf{P}_a^{\perp}</math>, which, when left and right multiplied by the signal covariance matrix, will reduce the interference term to zero.

<math>\mathbf{P}_a^{\perp} = \mathbf{I} - \mathbf{a}(\mathbf{a}^{\dagger} \mathbf{a})^{-1} \mathbf{a}^{\dagger}</math>

So the modified signal covariance matrix becomes:

<math>\tilde{\mathbf{R = \mathbf{P}_a^{\perp} \mathbf{R} \mathbf{P}_a^{\perp} = \mathbf{P}_a^{\perp} \mathbf{R}_v \mathbf{P}_a^{\perp} + \sigma_n^2 \mathbf{P}_a^{\perp}</math>

Since <math>\mathbf{a}</math> is generally not known, <math>\mathbf{P}_a^{\perp}</math> can be constructed using the eigen-decomposition of <math>\mathbf{R}</math>, in particular the matrix containing an orthonormal basis of the noise subspace, which is the orthogonal complement of <math>\mathbf{a}</math>. The disadvantages to this approach include altering the visibilities covariance matrix and coloring the white noise term.

Spatial whitening

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.

<math>\tilde{\mathbf{R = (\sigma_s^2 \mathbf{a} \mathbf{a}^{\dagger} + \sigma_n^2 \mathbf{I})^{-{\frac{1}{2} \mathbf{R}(\sigma_s^2 \mathbf{a} \mathbf{a}^{\dagger} + \sigma_n^2 \mathbf{I})^{-{\frac{1}{2}</math>

The calculation requires rigorous matrix manipulations, but results in an expression of the form:

<math>\tilde{\mathbf{R = (\cdot)^{-{\frac{1}{2} \mathbf{R}_v(\cdot)^{-{\frac{1}{2} + \mathbf{I}</math>

This approach requires much more computationally intensive matrix manipulations, and again the visibilities covariance matrix is altered.

Subtraction of interference estimate

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:

<math>\tilde{\mathbf{R = \mathbf{R} - \sigma_s^2 \mathbf{a} \mathbf{a}^{\dagger}</math>

By right and left multiplying <math>\mathbf{R}</math>:

<math>\tilde{\mathbf{R \approx (\mathbf{I} - \alpha \mathbf{u}_1 \mathbf{u}_1^{\dagger})\mathbf{R}(\mathbf{I} - \alpha \mathbf{u}_1 \mathbf{u}_1^{\dagger}) = \mathbf{R} - \mathbf{u}_1 \mathbf{u}_1^{\dagger} \lambda_1(2 \alpha - \alpha^2)</math>

where <math>\lambda_1(2 \alpha - \alpha^2) \approx \sigma_s^2</math> by selecting the appropriate <math>\alpha</math>. This scheme requires an accurate estimation of the interference term, but does not alter the noise or sources term.

Summary

Array processing technique represents a breakthrough in signal processing. Many applications and problems which are solvable using array processing techniques are introduced. In addition to these applications within the next few years the number of applications that include a form of array signal processing will increase. It is highly expected that the importance of array processing will grow as the automation becomes more common in industrial environment and applications, further advances in digital signal processing and digital signal processing systems will also support the high computation requirements demanded by some of the estimation techniques.

In this article we emphasized the importance of array processing by listing the most important applications that include a form of array processing techniques. We briefly describe the different classifications of array processing, spectral and parametric based approaches. Some of the most important algorithms are covered, the advantage(s) and the disadvantage(s) of these algorithms also explained and discussed.

References

Sources

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.
A.B. Gershman, array processing courseware
Prof. J.W.R. Griffiths, Adaptive array processing, IEEPROC, Vol. 130,1983.
N. Petrochilos, G. Galati, E. Piracci, Array processing of SSR signals in the multilateration context, a decade survey.