Sensor array

Last updated January 10, 2024

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. A third examples includes chemical sensor arrays, which utilize multiple chemical sensors for fingerprint detection in complex mixtures or sensing environments. Application examples of array signal processing include radar/sonar, wireless communications, seismology, machine condition monitoring, astronomical observations fault diagnosis, etc.

Plane wave, time domain beamforming

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 the wave.

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

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 delays of each of the received signals and remove them prior to the summation, the mean value

$y={\frac {1}{M}}\sum _{i=1}^{M}{\boldsymbol {x}}_{i}(t)\ \ (3)$

will result in an enhanced signal. The process of time-shifting signals using a well selected set of delays for each channel of the sensor array so that the signal is added constructively is called beamforming . In addition to the delay-and-sum approach described above, a number of spectral based (non-parametric) approaches and parametric approaches exist which improve various performance metrics. 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
Digital antenna array, this is smart antenna with multi channels digital beamforming, usually by using FFT.
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
Image sensors for digital cameras

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). 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,^[1] 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, 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.^[2]

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 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.

SAMV beamformer

SAMV beamforming algorithm is a sparse signal reconstruction based algorithm which explicitly exploits the time invariant statistical characteristic of the covariance matrix. It achieves superresolution and robust to highly correlated signals.

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 beamformer: In deterministic maximum likelihood beamformer (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 beamformer: In stochastic maximum likelihood beamformer (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.

Related Research Articles

In antenna theory, a phased array usually means an electronically scanned array, a computer-controlled array of antennas which creates a beam of radio waves that can be electronically steered to point in different directions without moving the antennas. The general theory of an electromagnetic phased array also finds applications in ultrasonic and medical imaging application and in optics optical phased array.

In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the E step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. It can be used, for example, to estimate a mixture of gaussians, or to solve the multiple linear regression problem.

In signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents. This is done by assuming that at most one subcomponent is Gaussian and that the subcomponents are statistically independent from each other. ICA is a special case of blind source separation. A common example application is the "cocktail party problem" of listening in on one person's speech in a noisy room.

In mathematical statistics, the Fisher information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information.

<span class="mw-page-title-main">Synthetic-aperture radar</span> Form of radar used to create images of landscapes

Synthetic-aperture radar (SAR) is a form of radar that is used to create two-dimensional images or three-dimensional reconstructions of objects, such as landscapes. SAR uses the motion of the radar antenna over a target region to provide finer spatial resolution than conventional stationary beam-scanning radars. SAR is typically mounted on a moving platform, such as an aircraft or spacecraft, and has its origins in an advanced form of side looking airborne radar (SLAR). The distance the SAR device travels over a target during the period when the target scene is illuminated creates the large synthetic antenna aperture. Typically, the larger the aperture, the higher the image resolution will be, regardless of whether the aperture is physical or synthetic – this allows SAR to create high-resolution images with comparatively small physical antennas. For a fixed antenna size and orientation, objects which are further away remain illuminated longer – therefore SAR has the property of creating larger synthetic apertures for more distant objects, which results in a consistent spatial resolution over a range of viewing distances.

In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make statistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information.

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.

Beamforming or spatial filtering is a signal processing technique used in sensor arrays for directional signal transmission or reception. This is achieved by combining elements in an antenna array in such a way that signals at particular angles experience constructive interference while others experience destructive interference. Beamforming can be used at both the transmitting and receiving ends in order to achieve spatial selectivity. The improvement compared with omnidirectional reception/transmission is known as the directivity of the array.

In statistics, the theory of minimum norm quadratic unbiased estimation (MINQUE) was developed by C. R. Rao. MINQUE is a theory alongside other estimation methods in estimation theory, such as the method of moments or maximum likelihood estimation. Similar to the theory of best linear unbiased estimation, MINQUE is specifically concerned with linear regression models. The method was originally conceived to estimate heteroscedastic error variance in multiple linear regression. MINQUE estimators also provide an alternative to maximum likelihood estimators or restricted maximum likelihood estimators for variance components in mixed effects models. MINQUE estimators are quadratic forms of the response variable and are used to estimate a linear function of the variances.

In signal processing, direction of arrival (DOA) denotes the direction from which usually a propagating wave arrives at a point, where usually a set of sensors are located. These set of sensors forms what is called a sensor array. Often there is the associated technique of beamforming which is estimating the signal from a given direction. Various engineering problems addressed in the associated literature are:

An adaptive beamformer is a system that performs adaptive spatial signal processing with an array of transmitters or receivers. The signals are combined in a manner which increases the signal strength to/from a chosen direction. Signals to/from other directions are combined in a benign or destructive manner, resulting in degradation of the signal to/from the undesired direction. This technique is used in both radio frequency and acoustic arrays, and provides for directional sensitivity without physically moving an array of receivers or transmitters.

Space-time adaptive processing (STAP) is a signal processing technique most commonly used in radar systems. It involves adaptive array processing algorithms to aid in target detection. Radar signal processing benefits from STAP in areas where interference is a problem. Through careful application of STAP, it is possible to achieve order-of-magnitude sensitivity improvements in target detection.

MUSIC is an algorithm used for frequency estimation and radio direction finding.

In statistics, the generalized linear array model (GLAM) is used for analyzing data sets with array structures. It based on the generalized linear model with the design matrix written as a Kronecker product.

3D sound localization refers to an acoustic technology that is used to locate the source of a sound in a three-dimensional space. The source location is usually determined by the direction of the incoming sound waves and the distance between the source and sensors. It involves the structure arrangement design of the sensors and signal processing techniques.

Sonar systems are generally used underwater for range finding and detection. Active sonar emits an acoustic signal, or pulse of sound, into the water. The sound bounces off the target object and returns an “echo” to the sonar transducer. Unlike active sonar, passive sonar does not emit its own signal, which is an advantage for military vessels. But passive sonar cannot measure the range of an object unless it is used in conjunction with other passive listening devices. Multiple passive sonar devices must be used for triangulation of a sound source. No matter whether active sonar or passive sonar, the information included in the reflected signal can not be used without technical signal processing. To extract the useful information from the mixed signal, some steps are taken to transfer the raw acoustic data.

A seismic array is a system of linked seismometers arranged in a regular geometric pattern to increase sensitivity to earthquake and explosion detection. A seismic array differs from a local network of seismic stations mainly by the techniques used for data analysis. The data from a seismic array is obtained using special digital signal processing techniques such as beamforming, which suppress noises and thus enhance the signal-to-noise ratio (SNR).

Beamforming is a signal processing technique used to spatially select propagating waves. In order to implement beamforming on digital hardware the received signals need to be discretized. This introduces quantization error, perturbing the array pattern. For this reason, the sample rate must be generally much greater than the Nyquist rate.

SAMV is a parameter-free superresolution algorithm for the linear inverse problem in spectral estimation, direction-of-arrival (DOA) estimation and tomographic reconstruction with applications in signal processing, medical imaging and remote sensing. The name was coined in 2013 to emphasize its basis on the asymptotically minimum variance (AMV) criterion. It is a powerful tool for the recovery of both the amplitude and frequency characteristics of multiple highly correlated sources in challenging environments. Applications include synthetic-aperture radar, computed tomography scan, and magnetic resonance imaging (MRI).

Digital antenna array(DAA) is a smart antenna with multi channels digital beamforming, usually by using fast Fourier transform (FFT). The development and practical realization of digital antenna arrays theory started in 1962 under the guidance of Vladimir Varyukhin (USSR).

References

↑ Capon, J. (1969). "High-resolution frequency-wavenumber spectrum analysis". Proceedings of the IEEE. 57 (8): 1408–1418. doi:10.1109/PROC.1969.7278.
↑ 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 (1): 76–85. doi:10.1109/TUFFC.2014.6689777. PMID 24402897. S2CID 251750.