Sensor array

A sensor array is a group of sensors, usually deployed in a certain geometry pattern, used for collecting and processing electromagnetic or acoustic signals. The advantage of using a sensor array over using a single sensor lies in the fact that an array adds new dimensions to the observation, helping to estimate more parameters and improve the estimation performance. For example an array of radio antenna elements used for beamforming can increase antenna gain in the direction of the signal while decreasing the gain in other directions, i.e., increasing signal-to-noise ratio (SNR) by amplifying the signal coherently. Another example of sensor array application is to estimate the direction of arrival of impinging electromagnetic waves. The related processing method is called array signal processing. Application examples of array signal processing include radar/sonar, wireless communications, seismology, machine condition monitoring, astronomical observations fault diagnosis, etc.

Using array signal processing, the temporal and spatial properties (or parameters) of the impinging signals interfered by noise and hidden in the data collected by the sensor array can be estimated and revealed. This is known as parameter estimation.

Principles

Figure 1 illustrates a six-element uniform linear array (ULA). In this example, the sensor array is assumed to be in the far-field of a signal source so that it can be treated as planar wave.

Parameter estimation takes advantage of the fact that the distance from the source to each antenna in the array is different, which means that the input data at each antenna will be phase-shifted replicas of each other. Eq. (1) shows the calculation for the extra time it takes to reach each antenna in the array relative to the first one, where c is the velocity of light.

$\Delta t_{i}={\frac {(i-1)d\cos \theta }{c}},i=1,2,...,M\ \ (1)$

Each sensor is associated with a different delay. The delays are small but not trivial. In frequency domain, they are displayed as phase shift among the signals received by the sensors. The delays are closely related to the incident angle and the geometry of the sensor array. Given the geometry of the array, the delays or phase differences can be used to estimate the incident angle. Eq. (1) is the mathematical basis behind array signal processing. Simply summing the signals received by the sensors and calculating the mean value give the result

$y={\frac {1}{M}}\sum _{{i=1}}^{{M}}{\boldsymbol x}_{i}(t-\Delta t_{i})\ \ (2)$ .

Because the received signals are out of phase, this mean value does not give an enhanced signal compared with the original source. Heuristically, if we can find weights multiplying to the received signals to set them in phase prior to the summation, the mean value

$y={\frac {1}{M}}\sum _{{i=1}}^{{M}}[{\boldsymbol w}_{i}{\boldsymbol x}_{i}(t-\Delta t_{i})]\ \ (3)$

will result in an enhanced signal. The process of multiplying a well selected set of weights to the signals received by the sensor array so that the signal is added constructively while suppressing the noise is called beamforming. There are a variety of beamforming algorithms for sensor arrays, such as the delay-and-sum approach, spectral based (non-parametric) approaches and parametric approaches. These beamforming algorithms are briefly described as follows.

Array design

Sensor arrays have different geometrical designs, including linear, circular, planar, cylindrical and spherical arrays. There are sensor arrays with arbitrary array configuration, which require more complex signal processing techniques for parameter estimation. In uniform linear array (ULA) the phase of the incoming signal $\omega \tau$ should be limited to $\pm \pi$ to avoid grating waves. It means that for angle of arrival $\theta$ in the interval $[-{\frac {\pi }{2}},{\frac {\pi }{2}}]$ sensor spacing should be smaller than half the wavelength $d\leq \lambda /2$ . However, the width of the main beam, i.e., the resolution or directivity of the array, is determined by the length of the array compared to the wavelength. In order to have a decent directional resolution the length of the array should be several times larger than the radio wavelength.

Types of sensor arrays

Antenna array

Antenna array (electromagnetic), a geometrical arrangement of antenna elements with a deliberate relationship between their currents, forming a single antenna usually to achieve a desired radiation pattern
Directional array, an antenna array optimized for directionality
Phased array, An antenna array where the phase shifts (and amplitudes) applied to the elements are modified electronically, typically in order to steer the antenna system's directional pattern, without the use of moving parts
Smart antenna, a phased array in which a signal processor computes phase shifts to optimize reception and/or transmission to a receiver on the fly, such as is performed by cellular telephone towers
Interferometric array of radio telescopes or optical telescopes, used to achieve high resolution through interferometric correlation
Watson-Watt / Adcock antenna array, using the Watson-Watt technique whereby two Adcock antenna pairs are used to perform an amplitude comparison on the incoming signal

Acoustic arrays

Microphone array is used in acoustic measurement and beamforming
Loudspeaker array is used in acoustic measurement and beamforming

Other arrays

Geophone array used in Reflection seismology
Sonar array is an array of hydrophones used in underwater imaging

Delay-and-sum beamforming

If a time delay is added to the recorded signal from each microphone that is equal and opposite of the delay caused by the additional travel time, it will result in signals that are perfectly in-phase with each other. Summing these in-phase signals will result in constructive interference that will amplify the SNR by the number of antennas in the array. This is known as delay-and-sum beamforming. For direction of arrival (DOA) estimation, one can iteratively test time delays for all possible directions. If the guess is wrong, the signal will be interfered destructively, resulting in a diminished output signal, but the correct guess will result in the signal amplification described above.

The problem is, before the incident angle is estimated, how could it be possible to know the time delay that is 'equal' and opposite of the delay caused by the extra travel time? It is impossible. The solution is to try a series of angles ${\hat {\theta }}\in [0,\pi ]$ at sufficiently high resolution, and calculate the resulting mean output signal of the array using Eq. (3). The trial angle that maximizes the mean output is an estimation of DOA given by the delay-and-sum beamformer. Adding an opposite delay to the input signals is equivalent to rotating the sensor array physically. Therefore, it is also known as beam steering.

Spectrum-based beamforming

Delay and sum beamforming is a time domain approach. It is simple to implement, but it may poorly estimate direction of arrival (DOA): if the signal is contaminated with strong noise, it might be difficult to implement the algorithm. The solution to this is a frequency domain approach. The Fourier transform transforms the signal from the time domain to the frequency domain. This converts the time delay between adjacent sensors into a phase shift. Thus, the array output vector at any time t can be denoted as ${\boldsymbol x}(t)=x_{1}(t){\begin{bmatrix}1&e^{{-j\omega \Delta t}}&\cdots &e^{{-j\omega (M-1)\Delta t}}\end{bmatrix}}^{T}$ , where $x_{1}(t)$ stands for the signal received by the first sensor. Frequency domain beamforming algorithms use the spatial covariance matrix, represented by ${\boldsymbol R}=E\{{\boldsymbol x}(t){\boldsymbol x}^{T}(t)\}$ . This M by M matrix carries the spatial and spectral information of the incoming signals. Assuming zero-mean Gaussian white noise, the basic model of the spatial covariance matrix is given by

${\boldsymbol R}={\boldsymbol V}{\boldsymbol S}{\boldsymbol V}^{H}+\sigma ^{2}{\boldsymbol I}\ \ (4)$

where $\sigma ^{2}$ is the variance of the white noise, ${\boldsymbol I}$ is the identity matrix and ${\boldsymbol V}$ is the array manifold vector ${\boldsymbol V}={\begin{bmatrix}{\boldsymbol v}_{1}&\cdots &{\boldsymbol v}_{k}\end{bmatrix}}^{T}$ with ${\boldsymbol v}_{i}={\begin{bmatrix}1&e^{{-j\omega \Delta t_{i}}}&\cdots &e^{{-j\omega (M-1)\Delta t_{i}}}\end{bmatrix}}^{T}$ . This model is of central importance in frequency domain beamforming algorithms.

Some spectrum-based beamforming approaches are listed below.

Conventional (Bartlett) beamformer

The Bartlett beamformer is a natural extension of conventional spectral analysis (spectrogram) to the sensor array. Its spectral power is represented by

${\hat {P}}_{{Bartlett}}(\theta )={\boldsymbol v}^{H}{\boldsymbol R}{\boldsymbol v}\ \ (5)$ .

The angle that maximizes this power is an estimation of the angle of arrival.

MVDR (Capon) beamformer

The Minimum Variance Distortionless Response beamformer, also known as the Capon beamforming algorithm, has a power given by

${\hat {P}}_{{Capon}}(\theta )={\frac {1}{{\boldsymbol v}^{H}{\boldsymbol R}^{{-1}}{\boldsymbol v}}}\ \ (6)$ .

Though the MVDR/Capon beamformer can achieve better resolution than the conventional (Bartlett) approach, but this algorithm has higher complexity due to the full-rank matrix inversion. Technical advances in GPU computing have begun to narrow this gap and make real-time Capon beamforming possible.^[1]

MUSIC beamformer

MUSIC (MUltiple SIgnal Classification) beamforming algorithm starts with decomposing the covariance matrix as given by Eq. (4) for both the signal part and the noise part. The eigen-decomposition of is represented by

${\boldsymbol R}={\boldsymbol U}_{s}{\boldsymbol \Lambda }_{s}{\boldsymbol U}_{s}^{H}+{\boldsymbol U}_{n}{\boldsymbol \Lambda }_{n}{\boldsymbol U}_{n}^{H}\ \ (7)$ .

MUSIC uses the noise sub-space of the spatial covariance matrix in the denominator of the Capon algorithm

${\hat {P}}_{{MUSIC}}(\theta )={\frac {1}{{\boldsymbol v}^{H}{\boldsymbol U}_{n}{\boldsymbol U}_{n}^{H}{\boldsymbol v}}}\ \ (8)$ .

Therefore MUSIC beamformer is also known as subspace beamformer. Compared to the Capon beamformer, it gives much better DOA estimation.

Parametric beamformers

One of the major advantages of the spectrum based beamformers is a lower computational complexity, but they may not give accurate DOA estimation if the signals are correlated or coherent. An alternative approach are parametric beamformers, also known as maximum likelihood (ML) beamformers. One example of a maximum likelihood method commonly used in engineering is the least squares method. In the least square approach, a quadratic penalty function is used. To get the minimum value (or least squared error) of the quadratic penalty function (or objective function), take its derivative (which is linear), let it equal zero and solve a system of linear equations.

In ML beamformers the quadratic penalty function is used to the spatial covariance matrix and the signal model. One example of ML beamformer penalty function is

$L_{{ML}}(\theta )=\|{\hat {{\boldsymbol R}}}-{\boldsymbol R}\|_{F}^{2}=\|{\hat {{\boldsymbol R}}}-({\boldsymbol V}{\boldsymbol S}{\boldsymbol V}^{H}+\sigma ^{2}{\boldsymbol I})\|_{F}^{2}\ \ (9)$ ,

where $\|\cdot \|_{F}$ is the Frobenius norm. It can be seen in Eq. (4) that the penalty function of Eq. (9) is minimized by approximating the signal model to the sample covariance matrix as accurate as possible. In other words, the maximum likelihood beamformer is to find the DOA $\theta$ , the independent variable of matrix ${\boldsymbol V}$ , so that the penalty function in Eq. (9) is minimized. In practice, the penalty function may look different, depending on the signal and noise model. For this reason, there are two major categories of maximum likelihood beamformers: Deterministic ML beamformers and stochastic ML beamformers, corresponding to a deterministic and a stochastic model, respectively.

Another idea to change the former penalty equation is the consideration of simplifying the minimization by differentiation of the penalty function. In order to simplify the optimization algorithm, logarithmic operations and the probability density function (PDF) of the observations may be used in some ML beamformers.

The optimizing problem is solved by finding the roots of the derivative of the penalty function after equating it with zero. Because the equation is non-linear a numerical searching approach such as Newton–Raphson method is usually employed. The Newton–Raphson method is an iterative root search method with the iteration

$x_{{n+1}}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}\ \ (10)$ .

The search starts from an initial guess $x_{0}$ . If the Newton-Raphson search method is employed to minimize the beamforming penalty function, the resulting beamformer is called Newton ML beamformer. Several well-known ML beamformers are described below without providing further details due to the complexity of the expressions.

Deterministic maximum likelihood meamformer: In deterministic maximum likelihood meamformer (DML), the noise is modeled as a stationary Gaussian white random processes while the signal waveform as deterministic (but arbitrary) and unknown.

Stochastic maximum likelihood meamformer: In stochastic maximum likelihood meamformer (SML), the noise is modeled as stationary Gaussian white random processes (the same as in DML) whereas the signal waveform as Gaussian random processes.

Method of direction estimation: Method of direction estimation (MODE) is subspace maximum likelihood beamformer, just as MUSIC, is the subspace spectral based beamformer. Subspace ML beamforming is obtained by eigen-decomposition of the sample covariance matrix.

References

↑ Asen, Jon Petter; Buskenes, Jo Inge; Nilsen, Carl-Inge Colombo; Austeng, Andreas; Holm, Sverre (2014). "Implementing capon beamforming on a GPU for real-time cardiac ultrasound imaging". IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. 61: 76. doi:10.1109/TUFFC.2014.6689777.

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Sensors

Acoustic, sound, vibration	Geophone Hydrophone Microphone Seismometer

Automotive, transportation	Air–fuel ratio meter Blind spot monitor Crankshaft position sensor Curb feeler Defect detector Engine coolant temperature sensor Hall effect sensor MAP sensor Mass flow sensor Omniview technology Oxygen sensor Parking sensors Radar gun Speed sensor Speedometer Throttle position sensor Tire-pressure monitoring system Torque sensor Transmission fluid temperature sensor Turbine speed sensor Variable reluctance sensor Vehicle speed sensor Water sensor Wheel speed sensor

Chemical	Breathalyzer Carbon dioxide sensor Carbon monoxide detector Catalytic bead sensor Chemical field-effect transistor Electrochemical gas sensor Electrolyte–insulator–semiconductor sensor Electronic nose Fluorescent chloride sensors Holographic sensor Hydrocarbon dew point analyzer Hydrogen sensor Hydrogen sulfide sensor Infrared point sensor Ion selective electrode Microwave chemistry sensor Nitrogen oxide sensor Nondispersive infrared sensor Olfactometer Optode Oxygen sensor Pellistor pH glass electrode Potentiometric sensor Redox electrode Smoke detector Zinc oxide nanorod sensor

Electric, magnetic, radio	Current sensor Electroscope Galvanometer Hall effect sensor Hall probe Magnetic anomaly detector Magnetometer MEMS magnetic field sensor Metal detector Planar Hall sensor Radio direction finder Test light

Environment, weather, moisture	Actinometer Bedwetting alarm Ceilometer Dew warning Electrochemical gas sensor Fish counter Frequency domain sensor Gas detector Hook gauge evaporimeter Humistor Hygrometer Leaf sensor Psychrometer Pyranometer Pyrgeometer Rain gauge Rain sensor SNOTEL Snow gauge Soil moisture sensor Stream gauge Tide gauge Weather radar

Flow, fluid velocity	Air flow meter Anemometer Flow sensor Gas meter Mass flow sensor Water metering

Ionising radiation, subatomic particles	Bubble chamber Cloud chamber Geiger counter Neutron detection Particle detector Scintillation counter Scintillator Wire chamber

Navigation instruments	Airspeed indicator Machmeter Altimeter Attitude indicator Depth gauge Fluxgate compass Gyroscope Inertial navigation system Inertial reference unit Magnetic compass MHD sensor Ring laser gyroscope Turn coordinator Variometer Vibrating structure gyroscope Yaw-rate sensor

Position, angle, displacement	Accelerometer Auxanometer Capacitive displacement sensor Capacitive sensing Gravimeter Inclinometer Integrated circuit piezoelectric sensor Laser rangefinder Laser surface velocimeter Lidar Linear encoder Linear variable differential transformer Liquid capacitive inclinometers Odometer Photoelectric sensor Piezoelectric accelerometer Position sensor Rate sensor Rotary encoder Rotary variable differential transformer Selsyn Sudden Motion Sensor Tachometer Tilt sensor Ultrasonic thickness gauge Variable reluctance sensor Velocity receiver

Optical, light, imaging	Charge-coupled device Contact image sensor Electro-optical sensor Flame detector Infrared Kinetic inductance detector LED as light sensor Light-addressable potentiometric sensor Nichols radiometer Optical fiber Photodetector Photodiode Photoelectric sensor Photoionization detector Photomultiplier Photoresistor Photoswitch Phototransistor Phototube Position sensitive device Scintillometer Shack–Hartmann wavefront sensor Single-photon avalanche diode Superconducting nanowire single-photon detector Transition edge sensor Tristimulus colorimeter Visible-light photon counter Wavefront sensor

Pressure	Barograph Barometer Boost gauge Bourdon gauge Hot-filament ionization gauge Ionization gauge McLeod gauge Oscillating U-tube Permanent Downhole Gauge Piezometer Pirani gauge Pressure gauge Pressure sensor Tactile sensor Time pressure gauge

Force, density, level	Bhangmeter Force gauge Hydrometer Level sensor Load cell Magnetic level gauge Nuclear density gauge Piezoelectric sensor Strain gauge Torque sensor Viscometer

Thermal, heat, temperature	Bimetallic strip Bolometer Calorimeter Exhaust gas temperature gauge Flame detection Gardon gauge Golay cell Heat flux sensor Infrared thermometer Microbolometer Microwave radiometer Net radiometer Quartz thermometer Resistance thermometer Silicon bandgap temperature sensor Special sensor microwave/imager Thermistor Thermocouple Thermometer

Proximity, presence	Alarm sensor Doppler radar Motion detector Occupancy sensor Passive infrared sensor Proximity sensor Reed switch Stud finder Touch switch Triangulation sensor Wired glove

Sensor technology	Active pixel sensor Back-illuminated sensor Biochip Biosensor Capacitance probe Carbon paste electrode Catadioptric sensor Digital sensors Displacement receiver Electromechanical film Electro-optical sensor Fabry–Pérot interferometer Fisheries acoustics Image sensor Image sensor format Inductive sensor Intelligent sensor Lab-on-a-chip Leaf sensor Machine vision Microelectromechanical systems Photoelasticity Quantum sensor Radar Ground-penetrating radar Synthetic aperture radar Radar tracker Sensor array Sensor fusion Sensor grid Sensor node Soft sensor Sonar Staring array Transducer Ultrasonic sensor Video sensor technology Visual sensor network Wheatstone bridge Wireless sensor network

Related	List of sensors

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