Beamforming
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 e ...
is a
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
technique used to spatially select propagating waves (most notably
acoustic and
electromagnetic
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
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
In signal processing, the Nyquist rate, named after Harry Nyquist, is a value (in units of samples per second or hertz, Hz) equal to twice the highest frequency (bandwidth) of a given function or signal. When the function is digitized at a hig ...
.
Introduction
Beamforming aims to solve the problem of filtering signals coming from a certain direction as opposed to an omni-directional approach. Discrete-time beamforming is primarily of interest in the fields of
seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
,
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
,
sonar
Sonar (sound navigation and ranging or sonic navigation and ranging) is a technique that uses sound propagation (usually underwater, as in submarine navigation) to navigation, navigate, measure distances (ranging), communicate with or detect o ...
and low frequency
wireless communications
Wireless communication (or just wireless, when the context allows) is the transfer of information between two or more points without the use of an electrical conductor, optical fiber or other continuous guided medium for the transfer. The most ...
.
Antennas
In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies a ...
regularly make use of
beamforming
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 e ...
but it is mostly contained within the analog domain.
Beamforming begins with an array of sensors to detect a 4-D signal (3 physical dimensions and time). A 4-D signal
exists in the spatial domain at position
and at time
. The 4-D
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the signal yields
which exists in the wavenumber-frequency spectrum. The wavenumber vector
represents the 3-D spatial frequency and
represents the temporal frequency. The 4-D sinusoid
, where
denotes the transpose of the vector
, can be rewritten as
where
, also known as the slowness vector.
Steering the beam in a particular direction requires that all the sensors add in phase to the particular direction of interest. In order for each sensor to add in phase, each sensor will have a respective delay
such that
is the delay of the ith sensor at position
and where the direction of the slowness vector
is the direction of interest.
Discrete-time weighted delay-and-sum beamforming
Source:
The discrete-time beamformer output
is formed by sampling the receiver signal
and averaging its weighted and delayed versions.
where:
*
is the number of sensors
*
are the weights
*
is the sampling period
*
is the steering delay for the ''i''th sensor
Setting
equal to
would achieve the proper direction but
must be an integer. In most cases
will need to be quantized and errors will be introduced. The quantization errors can be described as
. The array pattern for a desired direction given by the slowness vector
and for a quantization error
becomes:
Interpolation
Source:
The fundamental problem of discrete weighted delay-and-sum beamforming is quantization of the steering delay. The interpolation method aims to solve this problem by
upsampling
In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'', or it can describe an ...
the receiving signal.
must still be an integer but it now has a finer control. Interpolation comes at the cost of more computation. The new sample rate is denoted as
. The beamformer output
is now
The sampling period ratio
is set to an integer to minimize the increase in computations. The samples
are interpolated from
such that
After
is upsampled and filtered, the beamformer output
becomes:
At this point the beamformer's sample rate is greater than the highest frequency it contains.
Frequency-domain beamforming
Source:
As seen in the discrete-time domain beamforming section, the weighted delay-and-sum method is effective and compact. Unfortunately quantization errors can perturb the array pattern enough to cause complications. The interpolation technique reduces the array pattern perturbations at the cost of a higher sampling rate and more computations on digital hardware. Frequency-domain beamforming does not require a higher sampling rate which makes the method more computationally efficient.
The discrete-time frequency-domain beamformer is given by
For linearly spaced sensor arrays
. The discrete
short-time Fourier transform
The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice, the procedure for computing STFTs is to divid ...
of
is denoted by
. In order to be computationally efficient it is desirable to evaluate the sum in as few calculations as possible. For simplicity
moving forward. An effective method exists by considering a 1-D FFT for many values of
. If
for
then
becomes:
where
. Substituting the 1-D FFT into the frequency-domain beamformer:
The term in brackets is the 2-D
DFT with the opposite sign in the exponential
if the 2-D sequence
and
is the (M X N)-point DFT of
then
For a 1-D linear array along the horizontal direction and a desired direction:
where:
*
and
are dimensions of the DFT
*
is the sensor separation
*
is the frequency index between
and
*
is the steering index between
and
and
can be selected to "steer the beam" towards a certain temporal frequency and spatial position
References
{{Reflist
Digital signal processing