Sensor array
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.
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.
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.
(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.
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.
Simply summing the signals received by the sensors and calculating the mean value give the result
Heuristically, if we can find delays of each of the received signals and remove them prior to the summation, the mean value
In order to have a decent directional resolution the length of the array should be several times larger than the radio wavelength.
Summing these in-phase signals will result in constructive interference that will amplify the SNR by the number of antennas in the array.
at sufficiently high resolution, and calculate the resulting mean output signal of the array using Eq.
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.
Frequency domain beamforming algorithms use the spatial covariance matrix, represented by
Assuming zero-mean Gaussian white noise, the basic model of the spatial covariance matrix is given by
The Bartlett beamformer is a natural extension of conventional spectral analysis (spectrogram) to the sensor array.
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.
MUSIC uses the noise sub-space of the spatial covariance matrix in the denominator of the Capon algorithm
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.
In ML beamformers the quadratic penalty function is used to the spatial covariance matrix and the signal model.
(9) is minimized by approximating the signal model to the sample covariance matrix as accurate as possible.
In practice, the penalty function may look different, depending on the signal and noise model.
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.
Several well-known ML beamformers are described below without providing further details due to the complexity of the expressions.
