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.
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.
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.
\Deltati=
| (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=
| 1 | |
| M |
| M | |
| \sum | |
| i=1 |
\boldsymbolxi(t-\Deltati) (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=
| 1 | |
| M |
| M | |
| \sum | |
| i=1 |
\boldsymbolxi(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.
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
\pm\pi
\theta
| [- | \pi | , |
| 2 |
| \pi | |
| 2 |
]
d\leqλ/2
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]
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
\boldsymbolx(t)=x1(t)\begin{bmatrix}1&e-j\omega\Delta& … &e-j\omega(M-1)\Delta\end{bmatrix}T
x1(t)
\boldsymbolR=E\{\boldsymbolx(t)\boldsymbolxT(t)\}
\boldsymbolR=\boldsymbolV\boldsymbolS\boldsymbolVH+\sigma2\boldsymbolI (4)
where
\sigma2
\boldsymbolI
\boldsymbolV
\boldsymbolV=\begin{bmatrix}\boldsymbolv1& … &\boldsymbolvk\end{bmatrix}T
\boldsymbolvi=\begin{bmatrix}1&
| -j\omega\Deltati | |
| e |
& … &
| -j\omega(M-1)\Deltati | |
| e |
\end{bmatrix}T
Some spectrum-based beamforming approaches are listed below.
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)=\boldsymbolvH\boldsymbolR\boldsymbolv (5)
The angle that maximizes this power is an estimation of the angle of arrival.
The Minimum Variance Distortionless Response beamformer, also known as the Capon beamforming algorithm,[1] has a power given by
\hat{P}Capon(\theta)=
| 1 | |
| \boldsymbolvH\boldsymbolR-1\boldsymbolv |
(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 (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
\boldsymbolR=\boldsymbolUs\boldsymbolΛs\boldsymbol
| H | |
| U | |
| s |
+\boldsymbolUn\boldsymbolΛn\boldsymbol
| H | |
| U | |
| n |
(7)
MUSIC uses the noise sub-space of the spatial covariance matrix in the denominator of the Capon algorithm
\hat{P}MUSIC(\theta)=
| 1 | ||||||||
|
(8)
Therefore MUSIC beamformer is also known as subspace beamformer. Compared to the Capon beamformer, it gives much better DOA estimation.
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.
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
LML(\theta)=\|\hat{\boldsymbolR}-\boldsymbol
| 2 | |
| R\| | |
| F |
=\|\hat{\boldsymbolR}-(\boldsymbolV\boldsymbolS\boldsymbolVH+\sigma2\boldsymbolI
| 2 | |
| )\| | |
| F |
(9)
where
\| ⋅ \|F
\theta
\boldsymbolV
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
xn+1=xn-
| f(xn) | |
| f'(xn) |
(10)
The search starts from an initial guess
x0