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CHAPTER 3
ROBUST ADAPTIVE BEAMFORMING
3.1 INTRODUCTION
Adaptive beamforming is used for enhancing a desired signal while
suppressing noise and interference at the output of an array of sensors. It is
well known that adaptive beamformers can suffer significant performance
degradation, when the array response vector for the desired signal is not
known exactly. This degradation is especially noticeable at high signal-to-
noise ratio (SNR). Imperfect knowledge of the array response vector may be
due to
(i) uncertainty in the source direction-of-arrival (DOA) or
(ii) sensor characteristics or
(iii) improper modeling and variations in the propagation medium
between the source and array.
The array signal processing has been studied for some decades as
an attractive method for signal detection and estimation in harsh environment.
An array of sensors can be flexibly configured to exploit spatial and temporal
characteristics of signal, noise and has many advantages over single sensor.
There are two kinds of array beamformers: fixed beamformer and
adaptive beamformer. The weight of fixed beamformer is pre-designed and it
does not change in applications. The adaptive beamformer automatically
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adjusts its weight according to some criteria. It significantly outperforms the
fixed beamformer in noise and interference suppression. A typical
representative is the linearly constrained minimum variance (LCMV)
beamformer.
A famous representative of LCMV is Capon beamformer (Capon,
1969). In ideal cases, the Capon beamformer has high performance, in
interference and noise suppression, provided that the array steering vector
(ASV) is known. However, the ideal assumptions of adaptive beamformer
may be violated in practical applications.
The performance of the adaptive beamformers highly degrades
when there are array imperfections such as steering direction error, time delay
error, phase errors of the array sensors, multipath propagation effects and
wavefront distortions. This is known as target signal cancellation problem. To
overcome the problem caused by steering direction error, multiple-point
constraints (Hudson 1981) were introduced in adaptive array. The idea of this
approach is intuitive. With multiple gain constraints at different directions in
the vicinity of the assumed one, the array processor becomes robust in the
region where constraints are imposed. However, the available number of
constraints is limited because the constraints consume the degrees of freedom
(DOFs) of array processor for interference suppression.
Introducing the derivative constraints into the array processor leads
to another class of solution. With the derivative constraints, the array response
is almost flat in the vicinity of target direction. The beamformer has wide
beam width in the target direction. With a small steering direction error, the
beamformer does not cancel the target signal. However, the wide beam width
is achieved at the cost of reduced capability in interference suppression
because the additional derivative constraints consume the DOFs of
beamformer. Derivative constraints can be used to obtain not only a flat
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response of array processor, but also a flat null in the assumed signal direction
in blocking matrix design. Quadratic constraints can be used to minimize the
weighted mean square deviation between the desired array response and the
response of the processor over the variations in parameters, such as the
steering error, the phase errors and the array geometry error, etc.,
When array processor cancels target signal, the norm of the filter
coefficients grows to a large value beyond the normal value, for noise and
interference suppression. An inequality constraint is imposed on the
coefficients norm of adaptive beamformer to limit the growth of tap
coefficients. The excess coefficients growth problem can also be solved by
using noise injection method. Artificially generated noise is added to
reference signals of adaptive filters. Although the artificial noise causes
estimation errors in the beamformer coefficients, it prevents tap coefficients
from growing excessively, resulting in robustness against array imperfections.
A similar approach called, the leaky least mean square (LMS) algorithm, can
also be used for this purpose.
The calibration based approaches can eliminate the inherent error of
the array processor, such as geometry error, sensor response error, etc.
However, it cannot eliminate dynamic errors, such as steering error when the
source is moving in a vicinity of the assumed direction. In target tracking
methods, the look direction is steered to the continuously estimated direction-
of-arrival (DOA). One problem is that, this method may mis-track to the
interference, in the absence of target signal, unless some other methods are
used to limit the tracking region. Robust beamformer, for real time
applications, iteratively searches for the optimal direction. It maximizes the
mean output power of Capon beamformer, using first-order Taylor series
approximation, in terms of steering direction error. This method does not
suffer from performance loss in interference/noise suppression. However, its
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performance degrades when there exist multiple errors, such as the steering
direction error, the array geometry error and the array sensor phase error. The
array steering vector is assumed to be a vector function of steering direction
only. When multiple imperfections exist, the assumed model of the ASV is
violated. Recently, robust methods use uncertainty set of the ASV. The true
ASV is assumed to be an ellipsoid centered at the nominal ASV. The
designed beamformers are robust against arbitrary variation of true ASV
within an assumed uncertainty set. These beamformers are equivalent and
belong to diagonal loading approach. The diagonal loading factor can be
calculated from the constraint equation.
Traditional approaches for increasing robustness to DOA
uncertainty include linearly constrained minimum variance (LCMV)
beamforming, diagonal loading, quadratically constrained beamforming, and
combinations of these. These techniques allow desired signal to arrive from a
region in the DOA space rather than just from a single direction only. It relies
implicitly on assumptions about the strength of the desired signal and the
interval over which the DOA can vary. In these techniques, robustness to
DOA uncertainty is increased at the expense of a reduction in noise and
interference suppression.
A different approach is to use samples of the sensor data to estimate
the signal DOA or the signal subspace. Direction-finding (DF)-based
techniques, estimate the DOA of the desired and interference signals and
proceed as if they are known. Subspace techniques estimate the signal plus
interference subspace to reduce mismatch. These data-driven techniques are
more complex to implement but can have nearly optimal performance when
the data is sufficient to yield good estimates of the DOA or subspace.
However, they suffer significant performance degradation when these
estimates are not reliable. Techniques that improve the robustness of
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data-driven beamformers, in the presence of moving and spatially spread
sources, by incorporating additional linear constraints have also been already
proposed.
An adaptive beamformer using a Bayesian approach balances the
use of observed data and a priori knowledge about source DOA. In this
approach, DOA is assumed to be a discrete random variable with a known
a priori probability density function (pdf) that characterizes the level of
uncertainty about source DOA. The resulting beamformer is a weighted sum
of minimum variance distortionless response (MVDR) beamformers. It is
pointed at a set of candidate DOA’s, where the relative contribution of each
MVDR beamformer is determined from a posteriori pdf of the DOA
conditioned on observed data. A simple approximation to a posteriori pdf
allows for a straightforward implementation that is somewhat more complex
than LCMV beamformer but considerably less complex than data-driven
beamformers. Performance of Bayesian beamformer is better when compared
to both LCMV and data-driven beamformers, in a variety of scenarios.
The worst-case (WC) performance optimization has been shown as
a powerful technique which yields a beamformer with robustness against an
arbitrary signal steering vector mismatch, data non-stationarity problems and
small sample support. The WC approach explicitly models an arbitrary
mismatch in the desired signal array response and uses WC performance
optimization to improve the robustness of the minimum variance
distortionless response (MVDR) beamformer. In addition, the closed-form
expressions for the SINR are derived therein. Unfortunately, the natural
formulation of the WC performance optimization involves the minimization
of a quadratic function subject to infinity non-convex quadratic constraints.
WC optimization is also modeled as a convex second-order cone program
(SOCP) and solved efficiently via the well-established interior point method
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(IPM). Regrettably, the SOCP method does not provide a closed-form
solution for the beamformer weights and even it cannot be implemented
online, whereas the weight vector needs to be recomputed completely with the
arrival of a new array observation. Approaches based on eigen decomposition
of sample covariance matrix developed a closed-form solution, for a WC
robust detector using Lagrange method which incorporates the estimation of
the norm of weight vector and/or the Lagrange multiplier.
A binary search algorithm followed by a Newton-like algorithm can
be used to estimate the norm of the weight vector after dropping the Lagrange
multiplier. Although these approaches have provided closed-form solutions
for the WC beamformer, they unfortunately, incorporate several difficulties.
First, eigen decomposition for the sample covariance matrix is required with
the arrival of a new array observation. Second, the inverse of diagonally
loaded sample covariance matrix is required to estimate the weight vector.
Third, some difficulties are encountered during algorithm initialization and a
stopping criterion is necessary to prevent negative solution of the Newton-like
algorithm.
Two efficient ad hoc implementations of the WC performance
optimization problem are, first, the robust MVDR beamformer with a single
WC constraint implemented using an iterative gradient minimization
algorithm with an ad hoc technique. It estimates the Lagrange multiplier
instead of the Newton like algorithm. This algorithm exhibits several merits
including simplicity, low computational load and no need for either sample-
matrix inversion or eigen decomposition. A geometric interpretation of the
implementation has been introduced to supplement the theoretical analysis.
Second, a robust linearly constrained minimum variance (LCMV)
beamformer with multiple beam WC (MBWC) constraints is developed using
a novel multiple WC constraints formulation. The Lagrange method is
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exploited to solve this optimization problem, which reveals that the solution
of the robust LCMV beamformer with MBWC constraints entails solving a
set of nonlinear equations. As a consequence, a Newton-like method is
mandatory to solve the ensuing system of nonlinear equations which yields a
vector of Lagrange multipliers. It is worthwhile to note that these approaches
adopt ad hoc techniques to optimize the beamformer output power with
spherical constraint on the steering vector. Unfortunately, the adaptive
beamformer is sensitive to noise enhancement at low SNR and additional
constraint is required to bear the ellipsoidal constraint.
3.2 ROBUST ADAPTIVE BEAMFORMING
Adaptive beamforming is a complementary means for signal-to-
interference-plus-noise-ratio (SINR) optimization (Van Trees 2002, Dimitris
and Ingle 2005, Godara 1997). Our investigation starts with the formation of a
lobe structure those results from the dynamic variation of an element-space
processing. A weight vector is controlled by an adaptive algorithm, which is
the MVDR-Sample Matrix Inversion algorithm (Jiang and Zhu 2004, Dimitris
and Ingle 2005, Godara 1997). It minimizes cost function of a link’s SINR by
ideally directing beams toward the signal-of-interest (SOI) and nulls in the
directions of interference. In optimum beamformers, optimality can be
achieved in theory if perfect knowledge of the second order statistics of the
interference is available. It involves calculation of interference plus noise
correlation matrix i nR . In adaptive beamformer, the correlation matrix is
estimated from collected data. In sample matrix Inversion technique, a block
of data is used to estimate adaptive beamforming weight vector. The estimate
niR̂ is not really a substitute for true correlation matrix i nR . Hence there is
degradation in performance. The SINR which is a measure of performance of
the beamformer degrades as sample support (the number of data) is low.
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3.3 SYSTEM MODEL
An uniform linear array (ULA) of M elements or sensors is
considered in this investigation. A desired signal 0S from a point source from
a known direction 0 with steering vector ‘ 0a ’ and L number of J (jammer or)
interference signals from unknown directions 1 2 3, , ....... ,L specified by
the steering vectors 1 2 3, , ,..... ,La a a a respectively impinges on the array. The
white or sensor or thermal noise is considered as ‘n’.
A single carrier modulated signal 0 ( )S t is given by
0 0( ) ( )cos(2 )S t S t Fc t (3.1)
It is arriving from an angle 0 and is received by the ith sensor. The
signal 0( )S t is a baseband signal having deterministic amplitude and random
uniformly distributed phase with Fc as the carrier frequency. The symbol is
used to indicate that the signal is a pass band signal. Let X1(k) be the single
observation or measurement of this signal made at time instant k, at sensor 1,
which is given as
T1 0 0 1 2 L 1 2 LX (k)= a S (k)+ [a ,a ......a ][J (k),J (k)…J (k)] + n(k) (3.2)
0 0 1( ) *( ) ( )L
j jja S k a J k n k (3.3)
Hence the single observation or measurement made at the array of
elements at the time instant k, called array snapshot is given as a vector with
‘T’ as the transpose,
1 2 3( ) [ ( ) ( ) ( )....... ( )]TMX k X k X k X k X k (3.4)
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The general model of the steering vector is given as
2 2 2cos( ) 2 cos( ) ( 1) cos( )1 ......
d dj d j j Me e e
aM
(3.5)
It is assumed that the desired signal, interference signals and noise
are mutually uncorrelated.
3.4 ADAPTIVE BEAMFORMING TECHNIQUES
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 is a technique
in which an array of antennas is exploited to achieve maximum reception in a
specified direction by estimating the signal arriving from a desired direction
while signals of the same frequency from other directions are rejected. This is
achieved by varying the weights of each of the sensors used in the array.
Though the signals emanating from different transmitters occupy the same
frequency channel, they still arrive from different directions. This spatial
separation is exploited to separate the desired signal from the interfering
signals. In adaptive beamforming, the optimum weights are iteratively
computed using complex algorithms based upon different criteria. For an
adaptive beamformer, covariance or correlation matrix must be estimated
from unknown statistics of array snapshots to get the optimum array weights.
The optimality criterion is to maximize the signal-to-interference-plus-noise
ratio to increase the visibility of the desired signal at the array output. In this
investigation, it is assumed that the angle of arrival of the desired signal is
known.
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3.4.1 Estimation of Correlation Matrix
For p-dimensional data, the sample covariance matrix estimate
becomes singular, and therefore unusable, if fewer than p+1 sample will be
available, and it is a poor estimate of the true covariance matrix unless many
more than p+1 sample are available. The correlation matrix can be estimated
using different methods which would result in different performance and
behavior of the algorithm. In block adaptive Sample Matrix Inversion
technique, a block of snapshots are used to estimate the ensemble average ofRx , a MM matrix and is written as
1
1{ ( ) ( )} ( ) ( )KH Hx K
R E x k x k x k x kM
(3.6)
20 0
HS j nM a a R R (3.7)
where M is the number of snapshots used and k is the time index, 2s is the
power of the desired signal and jR and nR are the jammer and noise
correlation matrices, respectively and H is the complex conjugate transpose.
The interference-plus-noise correlation matrix is the sum of these two
matrices
j n j nR R R (3.8)
where 2n n nR R I , and 2
n is the thermal noise power, I is the identity
matrix. It is assumed that thermal noise is spatially uncorrelated.
3.4.2 Conventional Beamformer
The expectation value at the antenna elements is written as
1 2 1 2[ ( )] [ ( ) ( )..... ( )][ ( ) ( )..... ( )]TM ME x t X t X t X t X t X t X t (3.9)
where { ( ) ( ) }HR E x t x t .
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The output signal is
( ) ( )Hy t W x t (3.10)
This is the conventional beamformer’s output signal with
beamformer weight w which is shown in Figure 3.1.
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-80
-70
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0
angle in
Figure 3.1 Conventional beamforming showing the beam pattern
Maximizing the beamformer output problem will result in
2{ } ( )Max Max Hww
P y w Rw (3.11)
Solving this equation gives
H
awa a
(3.12)
where a is the steering vector.
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3.4.3 MVDR Beamforming
If ‘M’ number of sensors are used in a beamformer with spacing
between them as d= /2, at any instant
0
1*
00
( ) ( ).M
jkk
k
y n s n W e (3.13)
where 0 is the phase difference from the reference input and ‘ ’ may be
written as =(2 d/ ) sin = sin where is the angle of incidence. To
protect all signals which are received from the wanted direction, a linear
constraint may be defined as
0
1*
00
. ( ) ( )M
jk Hk
k
W e w n a g (3.14)
The constraint ‘g’ may be interpreted as gain at the look direction
which is to be maintained as constant. A spatial filter that performs this
function is called a linearly constraint minimum variance beamformer
(LCMV). If the constraint is g =1 then the signal will be received at look
direction with unity gain and the response at the look direction is
distortionless. This special case of LCMV beamformer is known as minimum
variance distortionless response (MVDR) beamformer which is shown in
Figure 3.2.
Mathematically, a weight vector ‘w’ is to be calculated for this
constrained optimization problem.
min *w w Rw Subject to 0* 1w a (3.15)
Now the optimal weight vector may be written as
1 1( ) / ( ) ( )Hx xw R a a R a (3.16)
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-80
-70
-60
-50
-40
-30
-20
-10
0
angle in
Figure 3.2 MVDR-the optimum beamformer – beam pattern
This beamforming method experiences the following drawbacks
1) Computational complexity in the order of 2 3( ) ( )O N toO N .
2) In the case of large array, low sample support i.e (M>>k), xR
may result in singular matrix or ill-conditioned.
3.4.4 Sample Matrix Inversion (SMI)
Sample matrix Inversion techniques solve the equation 0x dxR W r
directly by substituting the maximum likelihood estimates for the statistical
quantities xR and dxr to obtain
1ˆ ˆxW R rdx (3.17)
The maximum likelihood estimates of the signal correlation and
cross correlation are
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1
0
MH
x k Kk
R x x (3.18)
and1
0
ˆM
dx kk
r dx (3.19)
When the input signal is stationary, the estimates only need to be
computed once. However in cases where the signal statistics are time varying
the estimates must be continuously updated. In SMI, the convergence
performance is quantified in terms of number of statistically independent
sample outer products that must be computed for the weight vector to be
within 3dB of the optimum.
3.4.5 MVDR-SMI Beamformer
MVDR is an optimal minimum variance distortionless response
beamformer. It is also referred as the full rank solution as it uses all ‘M’
adaptive degrees of freedom. It resembles the Wiener filter of the form
1W R r (3.20)
MVDR weight vector can be derived as
12,
12,
i nMVDR H
i n
aw
a a= s (3.21)
w a (3.22)
where s is unit norm i.e 1Ha a and is the Hadamard product.
A standard method of estimating the covariance matrix is byconstructing the sample covariance matrix
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, , ,1
1ˆ ( ) ( )K
Hi n i n i n
k
R x k x kk
(3.23)
, ( )Hi nx k is the kth training sample and k is the total number of
training samples that are available. The sample covariance matrix ,ˆ
i nR is the
maximum likelihood estimate of the true covariance matrix ,i nR . Now the
approach is called sample matrix inversion with MVDR beamforming and the
weights are calculated as
1,
( ) 1,
ˆˆi n
MVDR SMI Hi n
R aW
a R a (3.24)
The beam response for MVDR_SMI beamformer is shown inFigure 3.3.
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0
angle in
Mvdr-smi beamforming
Figure 3.3 MVDR-SMI beamformer with beam response
MVDR method may suffer from significant performance
degradation when there are even small array steering vector errors. Several
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approaches for increasing robustness to array steering vector errors have beenproposed during the past few decades. Diagonal loading, linearly constrained
minimum variance (LCMV) beamforming, quadratically constrained
beamforming and second order cone programming (SOCP) are some of them.In this work, adaptive colored diagonal loading is proposed to improve theSINR and to eliminate the steering vector errors.
3.4.6 Diagonal Loading (DL)
To overcome the above mentioned drawback no. 2 in Section 3.4.3,
a small diagonal matrix is added to the covariance matrix. This process is
called diagonal loading (Li 2003) or white noise stabilization which is useful
to provide robustness to adaptive array beamformers against a variety of
conditions such as direction-of-arrival mismatch; element position, gain,
and/or phase mismatch; and statistical mismatch due to finite sample support
(Hiemstra 2003, Li and Stoica 2006). Because of the robustness that diagonal
loading provides it is always desirable to find ways to add diagonal loading to
beamforming algorithms. But little analytical information is available in the
technical literature regarding diagonal loading (Fertig 2000).
To achieve a desired sidelobe level in MVDR-SMI beamformer,
sufficient sample support ‘k’ must be available. However due to non-
stationarity of the interference, only low sample support is available to train
the adaptive beamformer. The beam response of an optimal beamformer can
be written in terms of its eigen values and eigen vectors. The eigen values are
random variables that vary according to the sample support ‘k’. Hence the
beam response suffers as the eigen values vary. This results in higher sidelobe
level in adaptive beam pattern. A means of reducing the variation of the eigen
values is to add a weighted identity matrix to the sample correlation matrix.
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The result of diagonal loading of the correlation matrix is to add the
loading level to all the eigen values. This in turn produces the bias in these
eigen values in order to reduce their variation which in turn produces side bias
in the adaptive weights that reduces the output SINR. Recommended loading
levels of 2 2n L < 210 n where 2
n is the noise power and 2L is the diagonal
loading level.
The minimum loading level must be equal to noise power.
Diagonal loading increases the variance of the artificial white noise by an
amount 2L . This modification forces the beamformer to put more effort in
suppressing white noise rather than interference. When the SOI steering
vector is mismatched, the SOI is attenuated as one type of interference as the
beamformer puts less effort in suppressing the interferences and noise.
However when 2L is too large, the beamformer fails to suppress strong
interference because it puts more effort to suppress the white noise. Hence,
there is a tradeoff between reducing signal cancellation and effectively
suppressing interference. For that reason, it is not clear how to choose a good
diagonal loading factor 2L in the traditional MVDR beamformer.
The conventional diagonal loading beamformer is shown in
Figure 3.4. This beamformer can be thought of as a gradual morphing
between two different behavior, a fully adaptive MVDR solution (L = 0, no
loading) and a conventional uniformly weighted beam pattern (L = , infinite
loading) (Hiemstra 2002).
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0
angle in
Figure 3.4 MVDR-Diagonal Loading
The conventional DL weight vector can be calculated as
2 1ˆ[ ] ( )MVDR DL MVDR DL LW R I a (3.25)
where MVDR DL is the normalization constant given by
2 1ˆ( ) [ ] ( )HMVDR DL La R I a (3.26)
and 2L reduces the sensitivity of the beam pattern to unknown uncertainties
and interference sources at the expenses of slight beam broadening. The
choice of loading can be determined from L-Curve approach (Hiemstra 2003)
or adaptive diagonal loading.
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3.4.7 Colored Diagonal Loading (CDL)
In the presence of colored noise, DL can be applied which is termed
as colored diagonal loading (CDL) and the morphing process may result in a
beam pattern of our choosing. The colored diagonal loading is similar to
MVDR DLW but the diagonal loading level of 2L = , end point, can be altered
by the term (Hiemstra 2002)
2 1ˆ[ ] ( )MVDR CDL MVDR DL L dqW R R a (3.27)
where Rdq is the covariance matrix that captures the desired quiescent
structure. It may be determined directly based on 1) a priori information –
where Rdq, need not be a diagonal or 2) desired weight vector – where Rdq
must be diagonal. It is given as
1([ ( )] ( ))dq dqR diag diag w a (3.28)
where wdq is the desired quiescent weight vector. The colored diagonal
loading shows no improvement in pattern shape as shown in Figure 3.5.
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0
angle in
Figure 3.5 MVDR-Colored Diagonal Loading
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3.4.8 Adaptive Diagonal Loading (ADL)
In this method the loading level is calculated assuming the a priori
information about the SNR is available. The SNR can be estimated from link
budget or using some SNR estimation algorithm. A variable loading
MVDR.(VL-MVDR) is proposed in (Gu and Wolfe 2006) in which the
loading level is chosen as ( 2 R) and the beam pattern is shown in Figure 3.6.
2 1MVDR-ADLW [ ] ( )MVDR ADL ADLR I a (3.29)
where .ADL M SNR
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0
angle in
Figure 3.6 MVDR- Adaptive Diagonal Loading beam pattern
3.4.9 MVDR-SMI Beamformer with Adaptive Colored Diagonal
Loading
In Adaptive Colored Diagonal Loading, which is our proposed
method, the loading level is calculated assuming the a priori information
about the Signal to Noise Ratio (SNR) is available.
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The SNR can be estimated from link budget or using some SNR
estimation algorithm. A variable loading MVDR (VL-MVDR) is proposed in
(Gu and Wolfe 2006) which the loading level is chosen as 2 ˆ( )R
2 1ˆ[ ] ( )MVDR ADL MVDR DL ADLW R I a (3.30)
where .ADL M SNR
White noise stabilization is nothing but diagonal loading in which
the adaptive colored loading technique is embedded to get a novel hybrid
method which is proposed as
1ˆ[ ] ( )MVDR ACDL MVDR DL dqW R R a (3.31)
The beam pattern for MVDR-ADCL is shown in Figure 3.7.
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0
angle in
Figure 3.7 MVDR- Adaptive Colored Diagonal Loading beam pattern
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3.5 COMPUTATION
For the proposed hybrid algorithm, a 10 element Uniform Linear
Array is considered with SNR of 20 dB for the desired signal coming from
s = 0° and INR of 70 dB for two jammer signals coming from the directions
i = -70°, and 30°. The element spacing is d = 0.5 . The beam patterns for
various methods of beamforming are obtained and compared with the
performance of MVDR-Adaptive Colored Diagonal Loading. It is observed
that the conventional beamformer performs well to get the maximum gain in
the desired look direction of 0°. But its performance is worst regarding the
cancellation of interferences.
Figure 3.5 shows the MVDR Colored Diagonal Loading beam
pattern which performs much better than the conventional beamformer. This
shows a greater improvement in SINR than the conventional. The null is
placed properly with out any angle deviation. Figure 3.6 shows MVDR-ADL
beam pattern. Figure 3.7 shows MVDR-ACDL beam pattern. This beam
pattern gives improvement in SINR when compared to other diagonal loading
methods. Figure 3.8 shows the beam patterns of the above mentioned
techniques. The interferers’ angle and their corresponding beam responses are
given below.
Interferer 1 at angle -70° : -60dB
Interferer 2 at angle 30° : -60dB
3.6 RESULTS
3.6.1 Number of Elements
For the ULA which is considered for experimental work, the beam
patterns are analyzed by changing the number of elements as 4, 8, 12, 16, 24,
50 and 100.
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-20
-10
0
MVDR-diagonal lodingMvdr-adaptive DLMvdr-adapt-col-DL
angle in
Figure 3.8 Beam pattern of various diagonal loading methods
As the number of elements increases, the beam pattern shows
higher resolution i.e the 3 dB beam width becomes much narrower from to
26° to 1° for conventional beamformer and 17° to 1° for adaptive diagonal
loading beamformer. Finer or sharper beams are obtained when more number
of elements is used. Sharper the beam, the beamformer is not susceptible to
jammers. But the numbers of sidelobes are also increased. The 3-dB beam
width of different beamformers is tabulated in Table 3.1. A trade off can be
obtained to reduce the cost and to have a compact size. Hence a maximum of
16 elements are chosen for further analysis.
3.6.2 Noise Effect
An ULA with 16 elements is considered for analyzing the effect of
noise on the peaks of the signal power. Signal to noise ratio (SNR) is varied in
steps of 10 dB starting from 10 dB till 60 dB. As SNR increases the peak
becomes sharper. It shows that the interference sources are suppressed to a
maximum extent, so that it will not be a disturbance while extracting the
signal even in the presence of strong interferers.
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Table 3.1 Effect of changing number of antenna elements
3-dB beam width
4 26.2 19.5 17.1 17 17 17 168 12.8 15.47 13.3 14.8 14.8 14.8 25.5
12 8.4 8.7 6.9 8.4 8.5 8.7 8.516 6.25 6.4 6.4 6.4 6.4 6.6 13.220 5.1 5.2 6 5.3 5.3 5.2 6.824 4.4 4.5 4.5 4.3 4.3 4.3 4.350 2 2 2 2 2 2 2100 1 1 1 1 1 1 1
3.6.3 Training Issues with the Number of Array Snapshots
Increasing the number of array snapshots lead to complexity and
computational cost but the performance of the beamformer increases. It is a
trade off between the cost and the performance. This is shown in Figure 3.9.
3.6.4 Element Spacing
The spacing between the elements for an 16 element ULA was
varied as /4, /2, 3 /4 and which in turn vary the effective aperture length
of the array. Among the four choices /2 showed the best performance for the
particular frequency used for expermiments. When the distance between the
elements is increased beyond /2, it resulted in spatial aliasing i.e a lot of
spurious peaks were obtained which correspond to different frequencies.
Below /2 the resolution of the beams was not satisfactory.
74
0 50 100 150 200 250 300-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
SINR-CDLSINR-ADLSINR-ACDL
number of snapshots k
Figure 3.9 Training issues with the number of snapshots
The analytical results of the response of different beamforming
methods are tabulated in Table 3.2.
Table 3.2 Beam response of the signals - desired and jammers – using
various methods
Beamformingmethod
Desired signal=0°
Beam responsePower (in dB)
Jammer1 = 20°
Beam responsePower (in dB)
Jammer2=-20°
Beam responsePower (in dB)
Jammer3 =-70° Beamresponse
Power(in dB)conventional 0 -20 -20 -26.5
MVDR 0 -91 -66 -91
MVDR-SMI 0 -58 -61 -72
DL 0 -72.5 -72.5 -85
CDL -6 -50 -57 -66.5
ADL 0 -72.5 -72.5 -85
ACDL 0 -52 -56.5 -62
75
3.7 CONCLUSION
Our investigation deals with adaptive array beamforming in the
presence of errors due to steering vector mismatch and finite sample effect.
Diagonal loading (DL) is one of the widely used techniques for dealing with
these errors. The diagonal loading techniques has the drawback that it is not
clear how to get the optimal value of diagonal loading level based on the
recognized level of uncertainty constraint. Recently, several DL approaches
proposed, the so-called automatic scheme, on computing the required loading
factor. In our investigation, this drawback is tackled while the diagonal
loading technique is integrated into the adaptive update scheme by means
of variable loading technique rather than fixed diagonal loading or ad hoc
techniques. The novelty is that the proposed method does not require any
additional sophisticated scheme to choose the required loading. We propose a
fully data-dependent loading to overcome the difficulties. The loading factor
can be completely obtained from the received array data. Analytical formulas
for evaluating the performance of the proposed method under random steering
vector error are further derived. Experimental results are proved the validity
of the proposed method and make comparison with the existing DL methods.