## Beamforming Overview

Beamforming is the spatial equivalent of frequency filtering and can be grouped into two classes: data independent (conventional) and data-dependent (adaptive). All beamformers are designed to emphasize signals coming from some directions and suppress signals and noise arriving from other directions.

Phased Array System Toolbox™ provides nine different beamformers. This table summarizes the main properties of the beamformers.

`phased.PhaseShiftBeamformer`ConventionalNarrowbandTime domain
`phased.TimeDelayBeamformer`ConventionalWidebandTime domain
`phased.SubbandPhaseShiftBeamformer`ConventionalWidebandFrequency domain
`phased.LCMVBeamformer`AdaptiveNarrowbandFrequency domain
`phased.MVDRBeamformer`AdaptiveNarrowbandFrequency domain
`phased.FrostBeamformer`AdaptiveWidebandTime domain
`phased.GSCBeamformer`AdaptiveWidebandTime domain
`phased.TimeDelayLCMVBeamformer`AdaptiveWidebandTime domain
`phased.SubbandMVDRBeamformer`AdaptiveWidebandFrequency domain

### Conventional Beamforming

Conventional beamforming, also called classical beamforming, is the easiest to understand. Conventional beamforming techniques include delay-and-sum beamforming, phase-shift beamforming, subband beamforming, and filter-and-sum beamforming. These beamformers are similar because the weights and parameters that define the beampattern are fixed and do not depend on the array input data. The weights are chosen to produce a specified array response to the signals and interference in the environment. A signal arriving at an array has different times of arrival at each sensor. For example, plane waves arriving at a linear array have a time delay that is a linear function of distance along the array. Delay-and-sum beamforming compensates for these delays by applying a reverse delay to each sensor. If the time delay is accurately computed, the signals from each sensor add constructively.

Finding the compensating delay at each sensor requires accurate knowledge of the sensor locations and signal direction. The delay-and-sum beamformer can be implemented in the frequency domain or in the time domain. When the signal is narrowband, time delay becomes a phase shift in the frequency domain and is implement by multiplying each sensor signal by a frequency-dependent compensatory phase shift. This algorithm is implemented in the `phased.PhaseShiftBeamformer`. For broadband signals, there are several approaches. One approach is to delay the signal in time by a discrete number of samples. A problem with this method is that the degree of resolution that you can distinguish is determined by the sampling rate of your data, because you cannot resolve delay differences less than the sampling interval. Because this technique only works if the sampling rate is high, you must increase the sampling frequency well beyond the Nyquist frequency so that the true delay is very close to a sample time. A second method interpolates the signal between samples. Time delay beamforming is implemented in `phased.TimeDelayBeamformer`. A third method Fourier transforms the signals to the frequency domain, applies a linear phase shift, and converts the signal back into the time domain. Phase-shift beamforming is performed at each frequency band (see `phased.SubbandPhaseShiftBeamformer`).

Beamforming is not limited to plane waves but can be applied even when there is wavefront curvature. In this case, the source lies in the near field. Perhaps the term beamforming is no longer appropriate. You can use the source-array geometry to compute the phase shift for each point in space and then apply this phase shift at each sensor element.

The advantage of a conventional beamformer is simplicity and ease of implementation. Another advantage is its robustness against pointing errors and signal direction errors. A disadvantage is its broad main lobe which decreases resolution of closely spaced sources or targets. A second disadvantage is that it has large sidelobes that allow interference sources to leak into the main beam.

The second class of beamformers consists of the data-dependent beamformers. The terms optimal or adaptive beamformers are sometimes used for this class interchangeably but they are not quite the same. Optimal beamformers apply weights that are determined by optimizing some quantity. The MVDR beamformer determines the beamforming weights, w, by maximizing the signal-to-noise+interference ratio of the array output

`$\frac{{|{w}^{\prime }s|}^{2}}{{w}^{\prime }{R}_{n}w}={A}^{2}\frac{{|{w}^{\prime }a|}^{2}}{{w}^{\prime }{R}_{n}w}$`

where s represents the signal values at the sensors, a represents the source steering vector, and A2 represents the source power at the array. Rn is the noise+interference covariance matrix. Because the SNR is invariant under any scale factor applied to the weights, an equivalent formulation of this criterion is to minimize the noise output w'R.nw subject to a constraint

The solution of this equation is

`${w}_{\text{opt}}=\frac{{R}_{\text{n}}{}^{-1}a}{a\text{'}{R}_{\text{n}}{}^{-1}a}$`

and yields the minimum variance distortionless response (MVDR) beamformer. Because of the constraint, beamformer preserves the desired signal while minimizing contributions to the array output due to noise and interference. The MVDR beamformer is implemented in `phased.MVDRBeamformer`. A broadband version is implemented in `phased.SubbandMVDRBeamformer`.

There are several advantages to the MVDR beamformer.

• The beamformer incorporates the noise and interference into an optimal solution.

• The beamformer has higher spatial resolution than a conventional beamformer.

• The beamformer puts nulls in the direction of any interference sources.

• Sidelobes are smaller and smoother.

There are two major disadvantages to the MVDR beamformer. The MVDR beamformer is sensitive to errors in either the array parameters or arrival direction. The MVDR beamformer is susceptible to self-nulling. In addition, trying to use MVDR as an adaptive beamformer requires a matrix inversion every time the noise and interference statistics change. When there are many array elements, the inversion can be computationally expensive.

In practical applications, an accurate steering vector and an accurate covariance matrix are not always available. Generally, all that is available is the sampled covariance matrix. This deficiency can lead to both inadequate interference suppression and distortion of the desired signal. In this case, the true signal direction is slightly off from the beam pointing direction. Then the actual signal is treated as interference.

However it often turns out that the noise is not separable from the signal and it is impossible to determine Rn. In that case, you can estimate a sample covariance matrix from the data.

`${\stackrel{^}{R}}_{x}=\frac{1}{K}\sum _{k=1}^{K}x\left(k\right){x}^{\prime }\left(k\right)$`

and minimizes w'Rxw instead. Minimizing this quantity leads to the minimum power distortionless response (MPDR) beamformer. If the data vector, x, contains the signal and the estimated data covariance matrix is perfect and the steering vector of the desired signal is known exactly, the MPDR beamformer is equivalent to the MVDR beamformer. However, MPDR degrades more severely when Rx is estimated from insufficient data or the signal arrival vector is not known precisely.

Rewrite the direction constraint in the form a’w = 1 by transposing both sides. This equivalent form suggests that it possible to include multiple constraints by using a matrix constraint Cw = d where C is now a constraint matrix and d represents the signal gains due to the constraints. This is the form used in the linear constraint minimum variance (LCMV) beamformer. The LCMV beamformer is a generalization of MVDR beamforming and is implemented in `phased.LCMVBeamformer` and `phased.TimeDelayLCMVBeamformer`. There are several different approaches to specifying constraints such as amplitude and derivative constraints. You can, for example, specify weights that suppress interfering signals arriving from a particular direction while passing signals from a different direction without distortion. The optimal LCMV weights are determined by the equation

`${w}_{\text{opt}}={R}_{\text{n}}{}^{-1}{C}^{\prime }\left(C{R}_{\text{n}}{}^{-1}C\text{'}{\right)}^{-1}d$`

The advantages and disadvantages of the MVDR beamformer also apply to the LCMV beamformer.

While MVDR and LCMV are adaptive in principle, re-computation of the weights requires the inversion of a potentially large covariance matrix when the array has many elements. The Frost and generalized sidelobe cancelers are reformulations of LCMV that convert the constrained optimization into minimizing an unconstrained form and then compute the weights recursively. This approach removes any need to invert a covariance matrix. See `phased.FrostBeamformer` and `phased.GSCBeamformer`.