DIGITAL BEAMFORMING (DBF)
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demand for increased capacity is a major driving force for incorporating DBF
marriage between antenna technology and digital technology
3 major components: antenna array, digital transceiver, digital signal processor
based on capturing the RF signals at each of the antenna elements and converting them into two
streams of binary baseband signals (I & Q). Included in the digital baseband signal are the
amplitude and phase of signals received at each of the elements of array. Beamforming is carried
out by weighting these digital signals, thereby adjusting this amplitude and phases such that when
added together they form the desired beam. This process can be carried out by special purpose
digital signal processor
Attractive features:
1. A large number of independently steered high-gain beams can be formed without any
resulting degradation in signal-to-noise ratio.
2. All of the information arriving at the antenna array is accessible to the signal processors so
that system performance can be optimized.
3. Beams can be assigned to individual users, thereby assuring that all links operate with
maximum gain
4. Adaptive beamforming can be easily implemented to improve the system capacity by
suppressing cochannel interference. Any algorithm that can be expressed in mathematical
form can be implemeneted. As a byproduct, adaptive beamforming can be used to enhance the
system immunity to multipath fading.
5. DBF systems are capable of carrying out antenna system real-time calibration in the digital
domain. Therefore, one can relax the requirements for a closematch of amplitude and phase
between transceivers, because variation in these parameters can be corrected in real time.
6. DBF has potential for providing a major advantage when used in satellite communications. If,
after the launch of the satellite, it is found that the performance of the beamformer needs to be
upgraded, a new suite of software can be telemetered up to the satellite. This means that the
life of the satellite can be expanded by retrofits at various intervals, during which the
satellite’s capabilities are upgraded.
Adaptive beamforming
- adaptive beamformer: device that is able to separate signals collocated in the frequency band but
separated in the spatial domain, separating a desired signal from interfering signals
- algorithm based on maximization of SNR at the array output & least mean squares (LMS) errors
- Minimum-variance distortionless response (MVDR)
- Sample matrix inversion – fast adaptivity
Benefits of using adaptive antennas
1. Coverage
- increase the cell coverage range substantially through antenna gain and interference rejection.
- Fewer sites required with adaptive antennas employed in base stations
- Larger coverage if antenna at greater height above average terrain. Can be eased by using the
number of antenna elements
2.
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Capacity
It is possible to have multiple mobiles on the same RF channel but different spatial channels at a
particular cell site
allows a reuse factor of unity, that is a single frequency can be used in all cells.
Can increase the number of available voice channels through directional communication links,
depends on the propagation environment, the number of antenna elements and the amount of
dynamic channel assignment allowed.
Transmission bit rate can be increased due to the improved SIR at the output of the adaptive
beamformer
Allow RF channels to be adjusted through link power control to meet the requirements of userselectable data transfer rates.
3.
4.
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Signal quality
in noise-limited environment, minimum receiver thresholds are reduced by 10 logM dB on
average.
In interference-limited environments, the additional improvement in tolerable SIR at a single
element results from interference rejection afforded against directional interferers.
Can be considered as spatial equalizers
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Access technology
In uplink, paths from different angles of arrival are separated by using a particular adaptive
beamforming technique
Downlink: energy can be focused at the mobile so that long delay multipath components can be
reduced substantially
>>combat ISI through spatial discrimination of “interfering” signals on both links
5.
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Power control
Eased thru the inclusion of adaptive antenna technology
6.
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Handover
antenna tech provides mobile unit location information that can be used by the system to
substantially improve handoffs in both the low and high tiers. Accurate position estimates,
prediction of velocities is possible.
7.
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Base station transmit power
the maximum peak EIRP required per user on a particular channel is decreased compared to
without adaptive beamforming
8.
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Portable terminal transmit power
with adaptive antennas at cells, the transmit power levels from and to the mobile can be kept
minimum to provide the requested service.
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DIFFERENT TYPES OF ADAPTIVE BEAMFORMING
1. Adaptive beamforming for uplink
Reasons studies for uplink:
 traditionally used for radar, remote sensing and sonar reception system
 spatial channel information available on the uplink
Adaptive criteria
 optimum weights using different criteria are all given by the Wiener solution
because it provides the upper limit on the theoretical adaptive beamforming
steady-state performance
Adaptive Algorithms
 LMS algorithm: simple to implement , but limited in dynamic range  over
which it operates. Required power control or alternatively use normalized LMS
algorithm
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SMI technique:  fast convergence rate but  increase computational
complexity & numerical instability
RLS algorithm:  reduce computational complexity while maintaining similar
performance, convergence rate faster than LMS provided that SNR is high, but
 has forgotten factor that is very dependent on fading rate of the channel
Conjugate gradient method
Eigenanalysis algorithm
Rotational invariance based method
Linear least squares error algorithm (LSSE)
Hopfield neural network
Reference Signals
 If explicit reference signal available in communication it should be used as much
as possible for less complexity, high accuracy, fast convergence
a) spatial reference:
 referred to as angle of arrival(AOA) information of desired signal and its
multipath components
 AOA estimation techniques:
 wavenumber estimation: based on decomposition of a
covariance matrix whose terms consist of estimates of the
correlation between the signals at the elements of an array
antenna. Example: Multiple signal classification (MUSIC),
modified forward-backward linear prediction (FBLP),
Principal Eigenvector Gram-Schmidt (PEGS), Estimation of
Signal Parameters by Rotational Invariance Techniques
(ESPRIT)
 parametric estimation: variety of maximum likelihood
estimation (MLE) : particular likelihood function is formulated
for the given radio signals, high computational complexity 
  requirement for array calibration, extra processing load required for
estimating AOA
b) temporal reference: may be a pilot signal that is correlated with the wanted signal,
or known PN code in CDMA
Blind Adaptive Beamforming
 when explicit reference signal is not available
a) Constant Modulus Algorithm (CMA)
 For both compensating fading and canceling cochannel interference
 Applied to advanced mobile phone services ( AMPS), IS-54 signals,
GMSK signals, 16-QAM signals
b) Decision-Directed Algorithm
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 Low cost since not computationally intensive, and no array calibration
required
 fast convergence, typically within 50 symbols
locks on desired signal with probability of 99.9% at SIR levels as low as 1dB
cochannel rejection is typically more than 20 dB
implementation based on incoherent differential binary phase-shift keying
(DBPSK) demodulation and LMS algorithm
converge faster than CMA and SCORE
c) Cyclostationary Algorithm
 Developed and applied to AMPS
 AMPS exhibit cyclostationary properties due to presence of supervisory audio
tone
 Show considerable improvement in MSE compared to the case of
omnidirectional antennas
 Cylic beamforming can be applied to GMSK signals, only require that
cochannel users have slightly different frequencies. Used in GSM and DECT.
Shown capacity improvement
2. Adaptive Beamforming for Downlink
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Objective: to maximize the received signal strength at the desired mobile and to
minimize the interference to other mobiles and adjacent base stations, thereby
maximizing the downlink SINR
If transfer function of the channel at the downlink is known, the downlink SINR
can be maximized by multiplying the desired signal with a set of downlink
weights.
The weights are a scale version of the uplink weights, provided that the frequency
of both links is same and the channel is relatively static during reception and
transmission. Weight reuse can be applied to TDD systems ( CT2/CT2+, DECT,
PHP, DCS1800).
In FDD system (IS-54, IS-95, GSM), weight reuse cannot be used because far
frequency separation
Essence of the problem: to estimate the downlink transfer function
Feedback technique was proposed. Using probing signal transmitted by BS.
The mobile measure it own response to the probe signal and report them back to
BS. The transfer function is estimated using the report. Simple but require
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complete redesign of protocols and signaling & applicable only for slowly change
environments
Other way: mobile directly transmit a narrowband testing signal at downlink
frequency so that the BS can directly estimate the downlink channel transfer
function from that. Not interrupt the normal uplink transmission but still require
complete protocol redesign & require additional hardware in MS.
Approached using AOA info. Downlink weights are derived by maximizing
SIRN based on the same AOA.
Use fixed multiple beams for both reception and transmission at the BS. On
uplink, BS determine the direction of the path on which the strongest component
of the desired signal arrived. On downlink, the BS points the beam in the
corresponding direction. Not optimal, but SINR improved since narrowband
signal is pointed, and can use high power beam to boost the SINR.
Ref:
John Litva, Titus Kwok-Yeung Lo, “ Digital Beamforming in Wireless
Communications”, pg 157-184, Artech House Publisher, 1996
Ref:
John Litva, Titus Kwok-Yeung Lo, “ Digital beamforming In Wireless Communications”, Artech
House, London, 1996
Receive beamforming concept
- radiation pattern should match the energy profile in order to merge all the radiated power
- In wide angular spread case, pointing to a specific direction with narrow beam pattern is not
optimal because some part of power spills over
Transmit beamforming
- It is suitable to transmit pointing towards the most significant reflector in order to minimize the
interference between different users located at different angles.
- Suitable for narrowband transmission but not for MC-CDMA schemes where many carriers need
to be considered.
Ref:
Santiago Zazo, Ivana Raos, “Transmit Beamforming Design in Wide Angle Spread Scenarios for
B3G MC-CDMA Systems”, IEEE Workshop on Signal Processing Advances in Wireless
Communications, 2004
BLAST
- each antenna transmit an independently modulated signal simultaneously and on the same carrier
frequency
- minimize redundancy between the various antenna signals in order to favor maximum data rate
Space-Time Coding
- introduce a lot of redundancy in an effort to maximize the diversity gain and achieve a minimum
bit error rate
Space selectivity
- occurs when the received signal amplitude depends on the spatial location of the antenna, and is a
function of the spread of angles of departure of the multipaths from the transmitter, and the spread
of angles of arrival of multipaths at the receiver
General principle of LA is to:
- define a channel quality indicator, or so-called channel state information (CSI), that provides some
knowledge on the channel. Metrics used as CSI : SNR & SINR(available from physical layer),
PER & BER (Link layer)
- adjust a number of signal transmission parameters to the variations of that quality indicator over
the signaling dimension explored (time, freq, space or combination thereof)
Adaptation based on Mean SNR
1. Measure SNR at receiver (assessment of CSI)
2. Convert the SNR info into BER info for each mode candidate (computation of adaptation
thresholds, the minimum required SNR for a given mode to operate at a given target BER)
3. Based on target BER, select for each SNR measurement the mode that yields the largest
throughput while remaining within the BER target bounds (selection of the optimal mode)
4. Feedback the selected mode to the transmitter
- this assuming ideal conditions (SNR can be measured instantaneously, ideal coherent detection,
fading over time only, SNR measured in very short window so it is effectively nonfading)
- in practice, feedback delays and other implementation limitations will not allow instantaneous
mode adaptation. Conversion of SNR to BER is not simple because the channel may exhibit some
fading within the SNR window. > use of second and higher order statistics of SNR instead of
Mean.
Adaptation based on Multiple Statistics of Received SNR
- If multicarrier modulation is used, a two dimensional time- frequency window may be used.
- The mapping between SNR and average BER is determined using pdf of the SNR over that
window
- In physical channel this pdf cannot be obtained via simple analysis because it is a function of
many parameters
- It can be simplify by estimating limited statistical info such as the k-order moment over the
adaptation window, instead trying to estimate the full pdf
- Moment based CSI > simplicity and flexibility to LA algorithm, do not depends on any
assumption made on the number or transmit and receive antenna
“How effective these methods can be in realistic traffic and bandwidth constraints is an open research
problem. In particular, it is critical to measure the ability of the scheme to lend itself to a very fast
adaptation scenario without significant bandwidth loss”
Pros and cons of CSI
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SNR based : offer flexibility to adapt the modes on a very fast basis; however, it relies on the
computation and adaptation/switching thresholds that maybe inaccurate
Error-based : captures accurate information of the modes, however this accuracy is reach only
after a substantial amount of traffic observed.
“An important topic of current research is to combine all types of CSI together to yield both accuracy and
robustness over a wide range of channels, adaptation rates, and traffic conditions”
In multiple antenna system the SNR varies not only over time and frequencies but also depends on:
- the way the transmitting signals are mapped and weighed onto the transmit antennas
- the processing techniques used at the receiver
- antenna polarization and propagation
Space-time-frequency adaptation> the adaptation algorithm desired to be able to select the best way of
combining antennas at all time (choose between space-time coding approach, BLAST or beamforming
approach)
LA algorithm design challenges:
- determination of adaptation thresholds: picking the least amount of statistical information to be
computed while still describing the essence of channel behavior
- adaptation rate: fast adaptation consumes higher bandwidth, trade-off btw performance gain and
amount of resource allocated to control messages
Ref:
Adaptive algorithms for weight calculations in adaptive antenna arrays > determine the convergence rate
and hardware complexity
1. Time domain processing
 Applebaum algorithm: applicable only when DOA of the desired signal is known beforehead
 Least Mean Square (LMS): has been widely used for tap coefficient adaptations of an adaptive
processor in antenna array, but it causes signal acquisition and tracking problems due to its slow
convergence in multipath fading channel
 Constant Modulus Algorithm (CMA): useful when the constant envelope of modulated signal is
maintained.
 Direct Matrix Inversion (DMI): fast convergence, but computationally too complex and may cause
numerical instability
 Recursive Least Square (RLS): achieve faster convergence than LMS, less computational than
DMI
2. Spatial domain processing : focused on DOA estimation by spectral analysis in the space domain
 Discrete Fourier Transform (DFT)
 Maximum Entropy Method (MEM)
 Multiple Signal Classification (MUSIC)
 Estimation of Signal Parameters via Rotation Invariance Technique (ESPRIT)
Ref:
Jin Young Kim, Jae Hong Lee, “Performance of a Multicarrier DS/CDMA System with Adaptive
Antenna Array in Nakagami Fading Channel”, IEEE Conference paper, 1998
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Steering vector
contains the responses of all elements of the array to a narrow-band source of unit power
associated with each directional source.
For array of identical elements, each component of this vector has unit magnitude
The phase of its ith component is equal to the phase difference between signals induced on the ith
element and the reference element due to the source associated with the steering vector
Also known as space vector and array response vector
W= array weight vector = weights of the beamformer
X= array signal vector= signals induced on all elements
R= array correlation matrix = its element denote the correlation between various elements of the array
Si= steering vector associated with ith source with direction (Øi, θi)
It is useful to express R in terms of its eigenvalue and their associated eigenvectors
- eigenvalues can be divided by two sets when the environment consists of uncorrelated direction
sources and uncorrelated white noise
- noise eigenvalues: eigenvalues in one set are of equal values and it does not depend upon
directional sources and is equal to the variance of the white noise
- 2nd set: signal eigenvalues: a function of the parameters of directional sources, their number is
equal to the number of these sources. Each eigenvalue of this set is associated with a directional
source, its value changes with the change in the source power, and bigger than those associated
with white noise
- R of an array of L elements immersed in M directional sources and the white noise has M
signal eigenvalues and L-M noise eigenvalues
- R can be represented in the form of spectral decomposition of R, matrix with eigenvalues as
diagonal matrix and multiplied by their corresponding unit-norm eigenvectors
BEAMFORMING METHODS
1. Conventional beamformer
- simple, sometimes known as delay-and-sum beam former, with all its weights of equal
magnitudes
- the phases are selected to steer the array in particular direction, known as the look direction
- has unity response in the look direction, that is, the mean output power of the processor due to a
source in the look direction is the same as the source power
- in environment consisting only uncorrelated noise and no directional interferences, this beam
former provides maximum SNR
- not effective in the presence of directional jammers, intentional or unintentional
2. Null-steering beam former
- used to cancel a plane wave arriving from a known direction and thus produces a null in the
response pattern in the DOA of the plane wave
- DICCANE : estimate the signal arriving from a known direction by steering a conventional beam
in the direction of the source and then substracting the output of this from each element. Very
effective for canceling strong interference and could be repeated for multiple interference
cancellation. Cumbersome when the number of interferences grow
- Requires knowledge of the directions of interference sources, the beam former using the
weights estimated does not maximize the outpur SNR
3. Optimal beamforming
- use ML filter
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the processor weights are selected by minimizing the mean output power of the processor while
maintaining unity response in the look direction
Minimum Variance Distortionless Response (MVDR) beamformer does not require the
knowledge of the directions and power levels of the interferences as well as the level of the
background noise power to maximize the output SNR. It requires only the direction of the
desired signal
4. Optimization using reference signal
- may be employed to acquire a weak signal in the presence of a strong jammer
5. Beam-Space processing
- two-stage scheme where the first stage takes the array signals as input and produces a se of
multiple outputs, which are then weighted and combined to produce the array output
- first stage: fixed weighting of the array signal and amounts to produce multiple beams steered in
different directions. The weights are not adaptive
- weights applied to different beam outputs to produce the array outputs are optimized to meet a
specific optimization criteria and are adjusted during the adaptation cycle
- for an L element array, the processor consist of a main beam steered in the signal direction and a
set of not more than L-1 secondary beams
- weighted output of secondary beams is subtracted from the main beam
- weights are adjust to produce an estimate of interference present in the main beam, subtracted
to remove it
- Secondary beams (auxiliary beams) designed such that they do not contain the desired signal
form the look direction to avoid the signal cancellation in the subtraction process
- Other names: Howell-Applebaum, partitioned processor, partially adaptive arrays, adaptiveadaptive arrays, multiple-beam antennas
- Quiescent pattern (main beam pattern) chosen such that it has desired shape.
- Schemes to generate the output of auxiliary beams such that no signal from the look direction is
contained in them: subtracting the array signals from presteered adjacent pairs
- If number of beams = number of elements of array, it is fully adaptive and has same capabilities
as those array using element-space processing
- Number of beams < number of elements: partially adaptive. Null-steering capabilities reduced to
that equal to the number of auxiliary beams. Estimate weights using adaptive schemes leads to
faster convergence. MSE higher than fully adaptive
- Useful when number of interferers much less than number of elements. Computational
advantage over element-space processing since only need to adjust M-1 weights compared to L
weights.
- Less computation because it solves an unconstrained optimization problem
 element-space: the constraints on weights to prevent the signal arriving from the look direction
from being distorted and to make array more robust against errors
 beam-space: these are transferred to the main beam, leaving the adjustable weights free from
constraints
- beam-space shown superior performance in case of look-direction errors (shown in case of one
interferer)
- element-space depends on knowledge of the look direction, when actual signal direction is
different from the one that is used to constrain weights it cancels this signal as if it was an
interference close to the look direction
- beam-space: designed to have the main beam steered in the known look direction and the auxiliary
beams are formed to have null in this direction. Very small signal cancellation in main beam
compared to element-space.
- Auxiliary beam-forming techniques: formation of M-1 orthogonal beams , formation of beams in
the direction of interferences if known
- Auxiliary beam outputs are weighted and summed and the result is subtracted from the main beam
output to cancel the unwanted interference present in the main beam.
6. Broad-band Beam Forming
- TDL structure normally used
- lattice structure consisting of a cascade of J simple lattice filters sometimes used
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7. Partitioned Realization
8. Frequency Domain Beamforming
9. Digital beamforming
10. Eigenstructure Method
- eigenvalues R can be divided into two sets when environment consists of uncorrelated directional
sources & uncorrelated white noise
- largest M eigenvalues correspond to M directional sources, the eigenvector associated : signal
eigenvectors
- L-M smallest eigenvalues : background noise power, eigenvector associated: noise eigenvectors
- eigenvectors of R are orthogonal to each other : as spanning an L-dimensional space, divided into two
orthogonal subspaces
-signal subspace: subspace spanned by signal eigenvectors
- noise subspace: subspace spanned by noise eigenvectors
- signal subspace spanned by M steering vectors associated with M directional sources. Exploited by
eigenstructure method in number of ways
- to cancel interference: array using a weight vector contained in the signal space such that it is orthogonal
to the interference-direction steering vector
- in case the direction of interference is not known the weight is estimated by minimizing a suitably
selected cost function
- application: estimating weights of beam-space processors using eigenvectors of the Rn (matrix with signal
component removed) as is done for secondary beams: effective to cancel interference in beam space and
for achieving the desired performance in short observation time
application: for correcting errors in steering vectors, forming beams associated with the largest eigenvalues
of R
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ADAPTIVE BEAM FORMING
1.
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SMI algorithm
Estimates the array weights by replacing R with its estimate
unbiased estimate of R using N samples of array signals
expression of optimal weights require inverse of R, using Matrix inversion Lemma
as the number of samples grows the matrix update approaches its true value, so the estimated
weights approaches it optimal value
2.
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LMS
Constrained LMS : when the weight are subjected to constraints at each iteration
3.
4.
5.
6.
RLS
CMA
Conjugate Gradient Method
Neural Network Approach
DOA ESTIMATION METHODS
1. Spectral Estimation Methods
2. MVDR Estimator
3. Linear Prediction Method
4. MEM
5. MLM
6. Eigenstructure Methods
7. MUSIC Algorithm
8. Min-Norm Method
9. CLOSEST Method
10. ESPRIT
11. WSF Method
12. Other Methods
13. Preprocessing Techniques
14. Estimating the number of Sources
15. Performance Comparison
EFFECT OF ERRORS
1. Correlated Arrivals
- Correlation between the desired signal and an unwanted interference exists in situations of
multipath arrivals and deliberate jamming
2. Look Direction and Steering Vector Error
- occur when the look direction is not the same as the desired signal direction.
- The processor treats the desired signal source as an interference and attenuates it. Amount of
attenuation depends upon the power of the signal and the amount of error. A stronger signal is
canceled more and larger error causes more cancellation of the signal
- Possible solutions: make the beam broader using multiple linear constraints and norm
constraints, not require broadening of the main beam: use direction finding technique combined
with a reduced dimensional ML formulation to estimate the direction of the desired signal
accurately
- Beam-space are more robust than element-spaced
- Pointing error causes error occur in the steering vector, which is used in weight calculation
- Factors of steering vector error: imperfection in the knowledge of the position of array elements,
errors caused by finite word length arithmetic,
3. Element Failure & Element Position Error
4. Weight Errors
Reference:
LAL C. Godara, “Application of Antenna Arrays to Mobile Communications, Part II: BeamForming and Direction-of-Arrival Considerations” Proceeding of IEEE, August 1997
----------------------------------------------------------------------------------------------------------------------------- ---1965:
First fully adaptive array was conceived by Applebaum, designed to maximize the SNR at the
array output
Widrow introduce Least Mean Squares (LMS) error algorithm: Cancelling unwanted interference
Frost and Griffiths further works on LMS algorithm to ensure that desired signals were not
filtered out along with the unwanted signals. OPtimisation takes place as before, but antenna
gain is maintained constant in the desired direction
Ref:
--------------------------------------------------------------------------------------------------------------------------------Summary of “Beamforming: A Versatile Approach to Spatial Filtering” by Barry D.Van Veen & Kevin
M. Buckley, IEEE ASSP Magazine April 1988
Purpose of this paper:
 provide the overview of beamforming from signal processing perspective. Data independent,
statistically optimum, adaptive and partially adaptive are discussed
 provide overview of beamformer design
 briefly discuss performance & implementation issues
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Beamformer:
 a processor used in conjunction with an array of sensors to provide a versatile form of spatial
filtering.The sensor arrays collects spatial samples of propagating wave fields, which are
processed by the beamformer
 objective: to estimate the signal arriving from desired direction in the presence of noise and
interfering signals.
 Performs spatial filtering to separate signals that have overlapping frequency content but originate
from differential spatial locations.
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Desired and interfering signals usually originated from different spatial locations. This separation
can be exploited to separate signal from interference using spatial filter at the receiver
Implementing a spatial filter requires processing of data collected over spatial aperture
Typically, beamformer linearly combines the spatially sampled time series from each sensor to
obtain a scalar output time series in the manner that an FIR filter linearly combines temporally
sampled data
2 advantages of spatial sampling with an array of sensors:
 Spatial discrimination depends on size of aperture. Aperture increases > discrimination improves
 Spatial filtering versatility offered by discrete sampling: many application need to change filtering
function in real time. It is easily implemented in discretely system by changing the way in which the
beamformer linearly combines the sensors data
Basic Terminology & Concept
A. Beamforming & Spatial Filtering
Weight vector, w
Data vector, x(k)
H = Hermitian (complex conjugate transpose)
Beamformer response, r(θ,ω):
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define as amplitude and phase presented to a complex plane wave as a function of location
and frequency.
Location is generally 3dimension quantity, but often only consider 1 or 2D direction of arrival
(DOA)
Array response vector (steering vector or direction vector), d(θ,ω)
Θ: direction, ω: temporal frequency
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the angles between w and d(θ,ω) determine the response r(θ,ω).
If the angle=90˚, the response is 0.
If angle close to 0˚, the response magnitude will be relatively large
The ability to discriminate between sources at different locations and/or frequencies, say (Θ1,ω1)
and (Θ2,ω2), is determined by the angle btw their array response vectors, d(θ1,ω1) and d(θ2,ω2)
Problems in spatial sampling:
- spatial aliasing: ambiguity in source locations. The source at different locations have same array
response vector. Happens if the sensors are spaced too far apart.
- If the sensors are too closed, spatial discrimination suffers as a result of smaller than necessary
aperture, array vectors are not well dispersed in the N dimensional vector space.
- Linear equispaced array when ω1 sinΘ1= ω2 sinΘ2 > in broadband signal when a source at one
location and frequency cannot be distinguished from a source at one location and frequency
FOCUS of this paper: designing response via weight selection.
B. Second Order Statistics
C. Beamformer Classification
 Data independent : the weights do not depend on the array data and are chosen to present a
specified response for all signal/interference scenarios
 Statistically optimum: the weights are chosen based on the statistics of the array data to optimize
the array response. In general, it place nulls in the direction of interfering sources in an attempt to
maxi
Second Order Statistics
- evaluation on beamformer usually involves power of variance, so the second
order statistics of the data play an important role
Beamformer Classification
 Classified as data independent or statistically optimum, depending
on how the weights are chosen
Data Independent: the weights do not depends on the array data and are
chosen to present a specified response for all signal/ interference scenarios
Statistically optimum: weights chosen based on the statistics of the array
data to “optimize” the array response. Places nulls in the directions of
interfering sources in an attempt to maximize the signal to noise ratio at
the beamformer output.
The statistics of array data are not usually known and may change over time
so adaptive algorithms are typically employed to determine the weights.
Adaptive algorithm is designed so that the response converges to statistically
optimum solution.
1. Data Independent Beamforming
the weights are designed so the beamformer response approximates a desired
response independent of the array data or data statistics
design objective: approximating a desired response
A.Classical beamforming
B. General Data Independent Response Design
apply to the design of beamformers that approximate an arbitrary desired
response
2. Statistically Optimum Beamforming
 weights are chosen based on the statistics of the data received at the array
 the goal is to “optimize” the beamformer response so the output contains minimal
contributions due to noise and signals arriving from directions other than the
desired signal direction
 Assumptions: data are wide-sense stationary & its second order statistics are
known
A.
B.
C.
D.
E.
Multiple Sidelobe Canceller
Use of Reference Signal
Maximization of Signal to Noise Ratio
Linearly Constrained Minimum Variance Beamforming
Signal Cancellation in Statistically Optimum Beamforming
Adaptive Algorithms for Beamforming
 Block adaptation: statistics estimated from a temporal block of array data and
used in an optimum weight equation
 Continuous adaptation: weights are adjusted as the data is sampled such that the
resulting weight vector sequence converges to the optimum solution
Interference Cancellation & Partially Adaptive Beamforming
A. Interference Cancellation Vs Degrees of Freedom
B. Partially Adaptive Beamformer Design
Beamformer Implementations