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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.   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
    -    Allow RF channels to be adjusted through link power control to meet the requirements of user-
         selectable data transfer rates.
   3.   Signal quality
   -    in noise-limited environment, minimum receiver thresholds are reduced by 10 logM dB on
   -    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

   4.   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.   Power control
   -    Eased thru the inclusion of adaptive antenna technology

   6.   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.   Base station transmit power
   -    the maximum peak EIRP required per user on a particular channel is decreased compared to
        without adaptive beamforming

   8.   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.


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
      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
                     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
           Low cost since not computationally intensive, and no array calibration
           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
      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

      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
      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
           complete redesign of protocols and signaling & applicable only for slowly change
          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.

           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
- each antenna transmit an independently modulated signal simultaneously and on the same carrier
     - 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

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
   - 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
       -   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

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


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
      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

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)

           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
         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

   1. Conventional beamformer
   - simple, sometimes known as delay-and-sum beam former, with all its weights of equal
   - 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
    -   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, adaptive-
       adaptive 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
    - 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
    - beam-space shown superior performance in case of look-direction errors (shown in case of one
    - 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

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
- 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



    1.   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.   LMS
        Constrained LMS : when the weight are subjected to constraints at each iteration

    3.   RLS
    4.   CMA
    5.   Conjugate Gradient Method
    6.   Neural Network Approach
  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


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
    - 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

        LAL C. Godara, “Application of Antenna Arrays to Mobile Communications, Part II: Beam-
        Forming 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



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

    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.

       -   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):
        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

Array response vector (steering vector or direction vector), d(θ,ω)
Θ: direction, ω: temporal frequency

    -    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

    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

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

   A.   Multiple Sidelobe Canceller
   B.   Use of Reference Signal
   C.   Maximization of Signal to Noise Ratio
   D.   Linearly Constrained Minimum Variance Beamforming
   E.   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

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