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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

56

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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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

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-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

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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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-10

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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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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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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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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-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.