Analysis on Robust Adaptive Beamformers

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Analysis on Robust Adaptive Beamformers Powered By Docstoc
					                                              (IJCSIS) International Journal of Computer Science and Information Security,
                                              Vol. 9, No. 3, March 2011




                      Analysis on Robust Adaptive Beamformers
                         T.S.JEYALI LASEETHA 1, DR.(MRS) R.SUKANESH2
                                                   1. Professor
                       Department of Electronics and Communication Engineering,
                                    Anna University of Technology
                                        Tirunelveli, Tamil Nadu
                                                 INDIA
                                         email id: laseetha@gmail.com
                                                  2. Professor
                       Department of Electronics and Communication Engineering,
                                         Madurai,Tamil Nadu
                                                 INDIA


 Abstract: MVDR (minimum variance distortionless response) beamformer is the optimal beamformer which
utilizes the second order statistics of the actual data for obtaining the Covariance matrix from which the weight
vector of the antenna array is determined. In adaptive beamfomer which utilizes MVDR beamformer along
with SMI(sample matrix inversion), actual data is not available to calculate the covariance matrix. Instead,
covariance matrix is estimated from the available data. It includes finding the Matrix inversion. It may result in
bad conditioning. To avoid this, some amount of loading is introduced to the diagonal elements, which is called
diagonal loading. Diagonal loading can be inserted by adding a scaled version of identity matrix. Diagonal
loading imparts Robustness to the adaptive beamformer against signal mismatch due to low sample support and
helps to achieve desired sidelobe level and SINR improvement. Various methods in diagonal loading are
analyzed in this paper with different loading levels and a novel hybrid algorithm for MVDR-SMI beamformer
with colored adaptive diagonal loading is also proposed. The performance of the proposed methods is
compared with other methods such as Conventional, MVDR, MVDR-SMI, MVDR-SMI-Diagonal Loading,
MVDR-SMI-Colored –DL, MVDR-SMI-Adaptive DL by conducting simulations experiment. The proposed
method shows the improvement in directivity and SINR compared to other methods.



Key-words: Smart antennas, Adaptive beamforming, Uniform Linear Array, Minimum Variance Distortionless
Response Beamformer (MVDR), Sample-Matrix Inversion(SMI), Adaptive colored diagonal loading




1.Introduction
In Wireless Communications, smart technologies                 that results from the dynamic variation of an
are not only being applied at the antenna level, but           element-space processing weight vector as opposed
also at the receiver for direction of arrival                  to a switched-beam or beam-space antennas, is
estimation, detection, diversity combining and                 controlled by an adaptive algorithm, which is the
equalization and at baseband processing software               MVDR-Sample Matrix Inversion algorithm[2,7,10].
levels. The ultimate benefit of these techniques is to         It minimizes cost function reduction of a link’s
increase cellular capacity and range. Adaptive                 SINR by ideally directing beams toward the signal-
beamforming reveals to be a complementary means                of-interest (SOI) and nulls in the directions of
for signal-to-interference-plus-noise-ratio (SINR)             interference. Many algorithms differ largely in
optimization [6,7,10]. In this paper, at antenna array         complexity, correlation matrix, eigen value spread
elements level, the formation of a lobe structure,             dependence, inherent gradient noise, limited



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                                                                                         ISSN 1947-5500
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dynamic range and limited number of samples used                 "{ {      " { {IJJ{     I{ . It is arriving from an
[1], [2].                                                       angle θ0 and is received by the ith sensor. The signal
                                                                S0(t) is a baseband signal having a deterministic
In optimum beamformers optimality can be                        amplitude and random uniformly distributed phase
achieved in theory if perfect knowledge of the                  and Fc is the carrier frequency. The symbol           is
second order statistics of the interference is                  used to indicate that the signal is a pass band signal.
available. It involves calculation of interference plus
noise correlation matrix           . For real world             X1(k) is the single observation or measurement of
scenarios, the adaptive methods are followed to                 this signal made at time instant k, at sensor 1, which
obtain optimality. In adaptive beamformer, the                  is given as
correlation matrix is estimated from the collected
                                                                 I# { {    I" " { { - {I# I$          I {{H# { { H$ { {    H { {{ - J{ {   (1)
data. In sample matrix Inversion technique a block
                                                                    I"     "{   {-          (# I   H { { - J{ {
of data is used to estimate the adaptive beamforming
                                                                                                                                           (2)
weight vector. The estimate            is not really a
substitute for true correlation matrix         . Hence           Hence the single observation or measurement
there is loss in performance. The SINR which is a               made at the array of elements at the time instant k,
measure of performance of the beamformer                        called array snapshot is given as a vector
degrades as the sample support (the number of data)
is low. The lower band on sidelobe levels of the                I{ {        {I# { { I$ { { I% { {                         I { {{           (3)
beamformer when no interference sources were
found at an angle is also to be calculated. Training            The general model of the steering vector[13] is
issues like the presence of desired signal in the               given as
correlation matrix           is also dealt with. The
                                                                                  {{   –   {   È {     {{   %   {{ % {   È {   {{
desired signal in the training set results in the                      #

cancellation and subsequent lose in performance.                                                                                           (4)

The paper is organized as follows. In Section 2,                Also it is assumed that the desired signal,
Problem formulation and general model is                        interference signals and noise are mutually
presented. In Section 3 Adaptive beamforming with               uncorrelated.
various beamforming methods are presented along
with the Novel Hybrid algorithm -Adaptive colored
diagonal loading.       In Section 4 simulation
experiments are presented. Section 5 contains
Results and discussions. Section 6 presents the                 3. Adaptive Beamforming
conclusions.
                                                                In optimum beamformer, a priori knowledge of true
                                                                statistics of the array data is used to determine the
                                                                correlation matrix which in turn is used to derive the
                                                                beamformer weight vector. Adaptive Beamforming
2. Problem Formulation And General                              is a technique in which an array of antennas is
Model                                                           exploited to achieve maximum reception in a
                                                                specified direction by estimating the signal arrival
An uniform linear array (ULA) of M elements or                  from a desired direction while signals of the same
sensors is considered. Let a desired signal S0 from a           frequency from other directions are rejected. This is
point source from a known direction θ0 with                     achieved by varying the weights of each of the
steering vector ‘a0’ and L number of J(jammer or)               sensors used in the array. Though the signals
interference       signals       from        unknown            emanating from different transmitters occupy the
directions{ # $ %             {, specified by the               same frequency channel, they still arrive from
steering vectors {I# I$ I%          I { respectively            different directions. This spatial separation is
impinge on the array. The white or sensor or thermal            exploited to separate the desired signal from the
noise is considered as ‘n’.                                     interfering signals. In adaptive beamforming the
                                                                optimum weights are iteratively computed using
A single carrier modulated signal   "{   { is given by          complex algorithms based upon different criteria.
                                                                For an adaptive beamformer, covariance or
                                                                correlation matrix must be estimated from unknown



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                                                                                                ISSN 1947-5500
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statistics of the array snapshots to get the optimum                     This beamforming                                 method          experiences            the
array weights. The optimality criterion is to                           following drawbacks
maximize the signal-to-interference-plus-noise ratio
to increase the visibility of the desired signal at the                                      1) the computational complexity is more in the
array output. The determination of the presence of                                              order of { $ { J { % { .
signals of interest is known as detection while the                                          2) In the case of large array, low sample
inference of their parameters likes, the angle of                                               support i.e(M>>k),            may result in
arrival θ0, is referred to as estimation. In this paper it                                      singular matrix or ill-conditioned.
is assumed that the angle of arrival of the desired
signal is known

                                                                                                                          conventional beamforming
                                                                                                   0
3.1 Estimation Of Correlation Matrix
                                                                                                  -10

The correlation matrix can be estimated[6,7,8,9]                                                  -20
using different methods which would result in
different performance and behavior of the algorithm.                                              -30




                                                                           beam response in dB
In block adaptive Sample Matrix Inversion                                                         -40
technique, a block of snapshots are used to estimate                                              -50
the ensemble average of    and is written as[8]
                                                                                                  -60
                               #
           { { {       { {{          (#    { {   { {        (5)                                   -70


       $
           I" I" -      -
                                                                                                  -80
 =H                                                         (6)
                                                                                                  -90

where N is the number of snapshots used and k is
the time index, $ is the power of the desired signal
                                                                                                 -100
                                                                                                        -80   -60   -40     -20       0       20     40     60         80
                                                                                                                                  angle inθ
and and         are the jammer and noise correlation
matrices, respectively. The interference-plus-noise                        Fig 1 conventional beamforming showing the
correlation matrix is the sum of these two matrices                                        beampattern
                   -                                      (7)

 Where          $
                  H, and $ is the thermal noise                                                                             Mvdr beamforming
power, I is the identity matrix. It is assumed that                                                0
thermal noise is spatially uncorrelated.                                                          -10

                                                                                                  -20

                                                                                                  -30
3.2 MVDR Beamformer
                                                                           beam response in dB




                                                                                                  -40
The MVDR beamformer whose pattern is shown in
Fig(2) is an adaptive high resolution beamformer                                                  -50
that minimizes the output power while maintaining                                                 -60
unity response in the desired direction.
Mathematically a weight vector ‘w’ is to be                                                       -70

calculated with the constrained optimization of                                                   -80
problem
                                                                                                  -90
      c               J I I       J c I"                 (8)
                                                                                                 -100
                                                                                                        -80   -60   -40     -20       0       20     40     60         80
Now the optimal weight vector may be written as                                                                                   angle inθ

               #
                     { {È     { {    #
                                          { {            (9)                                     Fig 2 MVDR-the optimum beamformer-
                                                                                                            beampattern



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                          0
                                                 Mvdr-smi beamforming                              minimum loading level must be equal to noise
                                                                                                   power. Diagonal loading increases the variance of
                                                                                                   the artificial white noise by the amount $ . This
                         -10

                         -20
                                                                                                   modification forces the beamformer to put more
                         -30
                                                                                                   effort in suppressing white noise rather than
  beam response in dB




                         -40                                                                       interference. When the SOI steering vector is
                         -50                                                                       mismatched, the SOI is attenuated as one type of
                         -60                                                                       interference as the beamformer puts less effort in
                                                                                                   suppressing the interferences and noise[17].
                                                                                                   However when $ is too large, the beamformer
                         -70

                         -80

                         -90
                                                                                                   fails to suppress strong interference because it puts
                                                                                                   most effort to suppress the white noise. Hence, there
                        -100
                               -80   -60   -40   -20       0       20   40   60   80               is a tradeoff between reducing signal cancellation
                                                       angle inθ
                                                                                                   and effectively suppressing interference. For that
                                                                                                   reason, it is not clear how to choose a good diagonal
                                                                                                   loading factor $ in the traditional MVDR
                        Fig 3 MVDR-SMI beamformer with beam
                                     response
                                                                                                   beamformer.
3.3 Diagonal Loading (Dl)                                                                              This conventional diagonal loading can be
                                                                                                   thought of as a gradual morphing between two
To overcome the above mentioned drawback no. 2,                                                    different behavior, a fully adaptive MVDR solution
a small diagonal matrix is added to the covariance                                                 (H      , no loading) and a conventional uniformly
matrix. This process is called diagonal loading[15]                                                weighted beampattern (H ∞, infinite loading)[5].
or white noise stabilization which is useful to                                                    The conventional DL weight vector can be
provide robustness to adaptive array beamformers                                                   calculated as
against a variety of conditions such as direction-of-
arrival mismatch; element position, gain, and/or                                                                                                                    { -           $
                                                                                                                                                                                      H{   #
                                                                                                                                                                                                { { (10)
phase mismatch; and statistical mismatch due to
finite sample support[12,14].         Because of the                                               where                                             is the normalization constant
robustness that diagonal loading provides it is                                                    given by

                                                                                                                                               { { {        -      $
                                                                                                                                                                       H{     #
                                                                                                                                                                                  { {
always desirable to find ways to add diagonal
                                                                                                                                                                                                      (11)
loading to beamforming algorithms and appropriate
amount of loading. But little analytical information                                               and $ reduces the sensitivity of the beampattern to
is available in the technical literature regarding                                                 unknown uncertainities and interference sources at
diagonal loading [11]. To achieve a desired sidelobe                                               the expenses of slight beam broadening[3]. The
level in MVDR-SMI beamformer sufficient sample                                                     choice of loading can be determined from L-Curve
support ‘k’ must be available. However due to non-                                                 approach[12] or adaptive diagonal loading. Beam
stationarity of the interference only low sample                                                   shape for Diogonal loading as shown in Fig(2) is
support is available to train the adaptive                                                         better when compared to previously stated methods
beamformer. We know that the beam response of an
optimal beamformer can be written in terms of its                                                                                                    MVDR-diagonal loading
                                                                                                                              0
eigen values and eigen vectors. The eigen values are
random variables that vary according to the sample                                                                           -10

support ‘k’. Hence the beam response suffers as the                                                                          -20

eigen values vary. This results in higher sidelobe                                                                           -30
                                                                                                      beam response in dB




level in adaptive beam pattern. A means of reducing                                                                          -40

the variation of the eigen values is to add a weighted                                                                       -50
identity matrix to the sample correlation matrix.
                                                                                                                             -60
The result of diagonal loading of the correlation
matrix is to add the loading level to all the eigen                                                                          -70


values. This in turn produces the bias in these eigen                                                                        -80

values in order to reduce their variation which in                                                                           -90

term produces side bias in the adaptive weights that                                                                        -100
                                                                                                                                   -80   -60   -40    -20       0        20       40       60    80
reduces the output SINR. Recommended loading                                                                                                                angle in θ

levels of $ ≤ $             $
                              where $ is the noise                                                                                   Fig 4 MVDR-Diagonal Loading
               $
power and         is the diagonal loading level. The


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                                                                                                                                                                               { -            $
                                                                                                                                                                                                      H{   #
                                                                                                                                                                                                                { { (14)

3.4 Colored Diagonal Loading(Cdl)                                                                               where                                                  [4]

In the presence of colored noise, DL can be applied                                                                                       0
                                                                                                                                                                MVDR-adaptive diagonal loading

which is termed as colored diagonal loading (CDL)
and the morphing process may result in a                                                                                                 -10


beampattern of our choosing. The colored diagonal                                                                                        -20

loading is similar to              but the diagonal                                                                                      -30

loading level of $ = ∞ , end point, can be altered




                                                                                                                  beam response in dB
                                                                                                                                         -40

by the term[5]                                                                                                                           -50



                                                          { -             $
                                                                                   {   #
                                                                                            { { (12)
                                                                                                                                         -60

                                                                                                                                         -70


where        is the covariance matrix that captures                                                                                      -80


the desired quiescent structure. It may be                                                                                               -90

determined directly 1)based on apriori information –                                                                                    -100
                                                                                                                                               -80    -60     -40     -20       0        20       40       60     80
where       , need not be diagonal or 2) desired                                                                                                                            angle in θ

weight vector –where        must be diagonal. It is
given as                                                                                                                                Fig 6 MVDR- Adaptive Diagonal Loading
                                                                                                                                                    beampattern
                                                             #
                                I {? I                   C           { {{                          (13)
                                                                                                                3.6 Adaptive coloured                                                         White               Noise
where     is the desired quiescent weight vector.                                                               Stabilization (ACDL)
The colored diagoinal loading shows no                                                                          As already discussed, white noise stabilization is
improvement in pattern shape as shown in Fig.5                                                                  nothing but diagonal loading in which the adaptive
                                             MVDR -colored-Diagonal loading
                                                                                                                colored loading technique is imbedded to get a
                          0                                                                                     novel hybrid method proposed as
                                                                                                                                                                                                       #
                         -10

                         -20                                                                                                                                                   ? -                C        { { (15)
                         -30
  beam response in dB




                         -40

                         -50                                                                                                                     MVDR-adaptive colored diagonal loading- the proposed algorithm
                                                                                                                                          0
                         -60
                                                                                                                                         -10
                         -70
                                                                                                                                         -20
                         -80
                                                                                                                                         -30
                                                                                                                  beam response in dB




                         -90
                                                                                                                                         -40
                        -100
                               -80   -60   -40    -20       0        20       40       60     80
                                                                                                                                         -50
                                                        angle in θ
                                                                                                                                         -60

                         Fig 5 MVDR-Colored Diagonal Loading                                                                             -70

                                                                                                                                         -80

                                                                                                                                         -90

                                                                                                                                        -100
3.5 Adaptive Diagonal Loading (ADL)                                                                                                            -80    -60     -40     -20       0        20       40       60     80
                                                                                                                                                                            angle in θ

In this method the loading level is calculated
assuming the apriori information about the SNR is                                                                                       Fig 7 MVDR- Adaptive Colored Diagonal
available. The SNR can be estimated from link                                                                                                   Loading beampattern
budget or using some SNR estimated algorithm.Gu
and Wolf proposed a variable loading MVDR.(VL-
MVDR) in which the loading level is chosen
as( $ {[16]




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                                                                                                                            have a compact size. Hence a maximum of 16
                                                                                                                            elements are chosen for further analysis
4. Simulations and Experiments
                                                                                                                                       Table1. Effect of Changing number of antenna
For the proposed hybrid algorithm ,a 16 element                                                                                                           elements
Uniform Linear Array is considered with SNR of 20                                                                                                  Desired




                                                                                                                 Beam forming method
dB for the desired signal coming from θs = 0° and                                                                                                 signal         Jammer1          Jammer2         Jammer3
INR of 70 dB for three jammer signals coming from                                                                                                 θ=0°            θ = 20°          θ=-20°         θ =-70°
the directions θi = -20°, 20° and 70°. The element                                                                                                                                                Beam
spacing is d = 0.5 λ.                                                                                                                               Beam           Beam             Beam          response
The various methods of beamforming are obtained                                                                                                   response       response         response        Power(in
to compare them with the performance of the Mvdr-                                                                                                  Power          Power            Power          dB)
Adaptive colored Diagonal Loading.                                                                                                                 (in dB)        (in dB)          (in dB)

                                                                                                                 conventional                     0              -20              -20             -26.5
                                          comparison of various diagonal loading methods
                          0
                                                                                                                 MVDR                             0              -91              -66             -91
                         -10

                         -20                                                                                     MVDR-SMI                         0              -58              -61             -72
                         -30
                                                                                                                 DL                               0              -72.5            -72.5           -85
  beam response in dB




                         -40

                         -50                                                                                     CDL                              -6             -50              -57             -66.5
                         -60

                         -70
                                                                                                                 ADL                              0              -72.5            -72.5           -85
                                     MVDR-diagonal loding
                         -80
                                     MVDR-coloured DL
                                     Mvdr-adaptive DL
                                                                                                                 ACDL                             0              -52              -56.5           -62
                         -90
                                     Mvdr-adapt-col-DL
                        -100
                               -80     -60     -40     -20       0        20    40         60   80
                                                             angle in θ



                                                                                                                                       Table2: Beam response of the signals - desired
 Fig 8 Beampattern of various diagonal loading                                                                                            and jammers – using various methods
                  methods
                                                                                                                                                                 3dB beamwidth
                                                                                                                              elements




5. Results and Discussion
                                                                                                                                                      MVDR


                                                                                                                                                             MVDR


                                                                                                                                                                         Diagon
                                                                                                                                           conven




                                                                                                                                                                                            ve DL
                                                                                                                                                                                            Adapti


                                                                                                                                                                                                        Adapti
                                                                                                                                                                                   Colore
                                                                                                                                                             - SMI
                                                                                                                              No.of



                                                                                                                                           tional




                                                                                                                                                                                   d DL




                                                                                                                                                                                                        ve
                                                                                                                                                                         al




5.1 Number of elements
 For the ULA which is considered for simulation                                                                             4              26.2       19.5   17.1        17       17        17          16
work , the beampatterns were analyzed by changing
the number of elements as 4, 8, 12,16, 24, 50 and                                                                           8              12.8       15.4   13.3        14.8     14.8      14.8        25.5
100. As the number of elements increases, the                                                                                                         7
beampattern shows higher resolution i.e the 3 dB
beamwidth becomes much narrower from to 26° to                                                                              12             8.4        8.7    6.9         8.4      8.5       8.7         8.5
1° for conventional beamformer and 17° to 1° for
adaptive diagonal loading beamformer. The detailed                                                                          16             6.25       6.4    6.4         6.4      6.4       6.6         13.2
results are tabulated in Table1. Finer or sharper
beams are obtained when more number of elements                                                                             20             5.1        5.2    6           5.3      5.3       5.2         6.8
are used. Sharper the beam, the beamformer is not
                                                                                                                            24             4.4        4.5    4.5         4.3      4.3       4.3         4.3
susceptible to jammers. But the number of side
lobes also increased. The 3-dB beamwidth of the                                                                             50             2          2      2           2        2         2           2
different beamformers are tabulated in Table 2. A
trade off can be obtained to reduce the cost and to                                                                         100            1          1      1           1        1         1           1




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5.2 Noise effect                                                                                   6. Conclusion
 An ULA with 16 elements is considered for                                                         A new Hybrid Robust adaptive beamforming
analyzing the effect of noise on the peaks of the                                                  algorithm is proposed with Adaptive Colored
signal power. Signal to noise ratio (SNR) was varied                                               diagonal loading based on data dependent approach.
in steps of 10 dB starting from 10 dB till 60 dB. As                                               This method is computationally efficient.
SNR increases the peaks becomes sharper. It also                                                   Simulation results show that the proposed method
showed that the interference sources were                                                          provide robustness against steering vector errors and
suppressed to a maximum extent, so that it will not                                                random array position perturbations by comparing it
be a disturbance while extracting the signal even in                                               with various diagonal loading methods
the presence of strong interferers

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               -10
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  SINR in dB




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               -35
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