20120130406017

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN IN –
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH 0976
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 6, September – October (2013), © IAEME

ENGINEERING AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 4, Issue 6, September – October 2013, pp. 166-174
© IAEME: www.iaeme.com/ijaret.asp
Journal Impact Factor (2013): 5.8376 (Calculated by GISI)
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IJARET
©IAEME

AN LMS ADAPTIVE ANTENNA ARRAY
SMITA BANERJEE1 and VED VYAS DWIVEDI2
1

Research Scholar, School of Engineering, R K University, Rajkot, Gujarat, India
Assistant Professor, Department of Electronics and Communication Engineering,
V.V.P. Engineering College, Rajkot, Gujarat, India (Parent Institute)
2
Director / Pro Vice Chancellor, C U Shah University, Wadhwan City, Surendranagar,
Gujarat, India

ABSTRACT
The importance of adaptive antenna array to track the satellite automatically is described in
this paper. The adaptive antenna array adjusts their pattern automatically to signal environment. The
desired signal reception is maintained by steering the main beam. The self-steering capability of an
adaptive antenna array is achieved by implementing LMS (Least mean square) algorithm. It is done
by changing the phase of the element in antenna array and angle of beam steering. MATLAB
software has been used to obtain array patterns for different values of phases in antenna array for
different values of beam steering angle.
Keywords: Adaptive
Communication.

antenna

array,

Adaptive

beamforming,

LMS

algorithm,

Satellite

I. INTRODUCTION
An adaptive antenna array has been widely used in different areas [1-3]. Satellite
Communication Systems are advantageous in providing communication services to huge regions
especially where the sufficient infrastructure for communication may not be constructed. In most
satellite communication systems, interference remains a problem for reliable reception of signals.
Adaptive antenna array is better for satellite communication systems as it has the ability to track the
satellite automatically [4-5]. Hence for this reason we use adaptive antenna array that automatically
steer the beam in the direction of desired signal i.e. signal of interest (SOI).
An adaptive antenna array combines the outputs of antenna elements but controls the
directional gain of the antenna by adjusting both phase and amplitude of the signal at each individual
element [6-7]. The combined relative amplitude and phase shift for each antenna is called a complex
weight. These weights are calculated using different algorithms [8-13]. The weighted signals are
166

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –

summed and the output is fed to a controller that adjusts the weights to satisfy an optimization
criterion.
Adaptive antennas have the ability of separating automatically the desired signal from the
noise and the interference signals and continuously update the element weights to ensure that the best
possible signal is delivered in the look direction. It not only directs maximum radiation in the
direction of the desired mobile user but also introduces nulls at interfering directions while tracking
the desired mobile user at the same time [14-15]. The adaptation achieved by multiplying the
incoming signal with complex weights and then summing them together to obtain the desired
radiation pattern as shown in figure 1. These weights are computed adaptively to adapt to the
changes in the signal environment. An adaptive algorithm is then employed to minimize the error
between the desired signal and the array output [16].

Figure.1. Structure of Adaptive Array
The optimum weights can be estimated with LMS algorithm. It is the most common and
popular adaptive signal processing techniques. Array processing involves manipulation of signals
induced on various antenna elements. It is the simplest and robust algorithm used for continuous
adaptation [14, 17]. In this paper, analysis of adaptive techniques LMS, is done through MATLAB
simulation by varying different parameters like desired direction and interference direction. Different
complex weights are obtained using this LMS beamforming algorithm.
This paper presents 8-elements array with λ/2 inter-element spacing and the LMS (least mean
square) algorithms for the interference rejection of adaptive array antennas. An Adaptive
beamforming is achieved by implementing LMS algorithm for directing the main beam towards the
desired source signals and generating complex weights which can be used for interference rejection.
By combining the signals incident on the linear antenna array and by knowing their directions of
arrival, a set of weights can be adjusted to optimize the radiation pattern. In this paper MATLAB
software has been used to obtain array patterns for different values of phases of elements in antenna
array and angle of beam steering by using LMS algorithm. The performance of beamforming
algorithms has been studied by means of MATLAB simulation.
II.

ADAPTIVE BEAMFORMING USING LEAST MEAN SQUARE ALGORITHM

Beamforming is the term used to describe the application of weights to the inputs of an array
of antennas to focus the reception of the antenna array in a certain direction, called the look direction

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or the main lobe. These effects are all achieved electronically and no physical movement of the
receiving antennas is necessary.
In adaptive beamforming, the radiation pattern of adaptive antenna is controlled through
various adaptive algorithms. Adaptive algorithm dynamically optimizes the radiation pattern
according to the changing electromagnetic environment. The antenna array pattern is optimized to
have maximum possible gain in the direction of the desired signal and nulls in the direction of the
interferers. The Least Mean Square (LMS) algorithm, introduced by Widrow and Hoff in 1959 is an
adaptive algorithm, which uses a steepest decent technique of gradient-based method. LMS
algorithm operates with a prior knowledge of the direction of arrival and the spectrum of the signal
but with no knowledge of the noise and interference in the channel. It uses the estimates of the
gradient vector from the available data. LMS incorporates an iterative procedure that makes
successive corrections to the weight vector in the direction of the negative of the gradient vector
which eventually leads to the minimum mean square error [17-18].
An Adaptive Beamforming using least mean square algorithm consists of multiple antennas,
complex weights, the function of which is to amplify (or attenuate) and delay the signals from each
antenna element and a summer to add all of the processed signals, in order to tune out the signals of
interest as shown in figure 2. Hence it is sometimes referred to as spatial filtering, since some
incoming signals from certain spatial directions are filtered out, while others are amplified [19].

Figure.2. LMS Adaptive beam forming network
Consider a Uniform Linear Array (ULA) with N isotropic elements, which forms the integral
part of the adaptive beamforming system as shown in the figure 2 [20-21]. The narrow band incident
waves are defined as s(t).
s(t ) = A exp(2π f c t + φ )

(1)

Where A: Amplitude of signal, fc: carrier frequency, ɸ: phase difference between incident waves at
successive elements=(2Π/λ) dsin(θ), d: distance between successive antenna phase centers in the
array, θ: angle of arrival w.r.t normal
As they reach the antenna elements, the waves are converted to electrical signals x(t) . We define the
input signals as x0(t),x1(t),.........xN-1(t).

168


1


 exp − jkd sin(θ )
(
) 




:
xi (t ) = 
 s (t )
:




:


 exp(− jk ( N − 1)d sin(θ ) 
xi (t ) = a(θ ) s(t )

(2)

Where N is the number of antenna elements and a(θ) is the steering vector which controls the
direction of antenna beam.
For adaptive beamforming, each element output xi(t) is multiplied by the weights
w0,w1,w2, ........wN-1 that modify phase and amplitude of the incoming signal accordingly. This
weighted signal is summed to give output. Then output signal y(t) of the antenna array is given by,
N −1

y (t ) = ∑ xi (t ) wi

(3)

n=0

y (t ) = [ w0

w1

1


 exp − jkd sin(θ )
(
) 




:
.. .. wN −1 ] 
 s (t )
:




:


 exp( − jk ( N − 1) d sin(θ ) 

y (t ) = wi xT (t )

(4)

An adaptive algorithm is then employed to minimize the error e(t) between a desired signal
d(t) and the array output y(t). The overall antenna pattern is continuously modified by adjusting
weight vector. This is a classical Weiner filtering problem for which the solution can be iteratively
found using the LMS algorithm.

III. LMS ALGORITHM FORMULATION
The output at time n, y (n) is given by a linear combination of the data at the N sensors can be
expressed as [22-23]:
w = [ w1………wN]H

(6 )

y(n) = wHx(n)

(7)

x(n) = [ x1(n)…………………xN(n)]

(8)

where H denotes complex conjugate. The weighted signals are summed and the output is fed
to adaptive beamformer that adjusts the weights to satisfy an optimization criterion.
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The optimum weights can be estimated with LMS algorithm. The LMS algorithm is an
MMSE weight adaptation algorithm that uses the steepest descent algorithm and gradient based
approach. It is referred to as sample by sample techniques as the weight vectors are updated for each
new sample. The algorithm recursively computes and updates the weight vector. Successive
corrections to the weight vector in the direction of the negative of the gradient vector eventually lead
to the MMSE between the beamformer output and the reference signal. At this point the weight
vector assumes to be its optimum value. . The LMS algorithm avoids matrix inverse operation by
using the instantaneous gradient vector ɸJ (n) to update the weight vector. Let w (n) denotes the
value of the weight vector at time n. The update value of the weight vector at time n+1 is w (n+1)
can be written as,
1
(9)
w( n + 1) = w( n) + µ [ −∇J ( n) ]
2
where µ is the step size which controls the speed of convergence. Its value is usually between 0 and
1. An exact measurement of the instantaneous gradient vector is not possible since this would require
a prior knowledge of both the covariance matrix R and the cross-correlation vector r. Instead, an
instantaneous estimate of the gradient vector ɸJ (n )is used which is given by,
∇ J ( n ) = −2 r ( n ) + 2 R ( n ) w( n )

(10)

Correlation matrix R(n) = x(n) xH(n)

(11)

Signal correlation vector r (n) = d*(n)x(n)

(12)

where

and

are the instantaneous estimates of R and r defined in Equation respectively. Substituting Equations
(10), (11) and (12) into Equation (9), the weight vector can be found that
w( n + 1) = w( n) + µ [ r (n) − R ( n) w( n) ]
= w(n) + µ x( n)  d ∗ (n) − x H (n) w( n) 



(13)

= w(n) + µ x( n)e∗ (n)
∗

where e∗ ( n ) = d ( n ) − x H ( n ) w ( n ) = error signal
The LMS algorithm is initiated with an arbitrary value w(0) for the weight vector at n=0. The
successive corrections of the weight vector eventually leads to the minimum value of the mean
squared error.
Therefore the LMS algorithm can be summarized in following equations;
Output, y ( n ) = wH ( n ) x ( n )

(14)

Error, e ( n ) = d

(15)

( n) − y ( n)

Weight, w ( n + 1) = w ( n ) + µ e* ( n ) x ( n )

170

(16)


IV. SIMULATION RESULTS & STUDY
For simulation purposes a 4-element linear array at 6 GHz is used with its individual
elements spaced at half-wavelength distance i.e; 5 and wave number of 0.2. The amplitude and phase
of the initial pattern is determined by arbitrary choice of weights, w whereas the final pattern is
determined by the strength of desired signal, the direction and strength of the interfering signal in the
system. In this simulation three different angle of arrival of single user and double interferers is
considered in LMS algorithm. Figure shows that one can track the user and the interfering signals at
the same time. Maximizing output SINR and minimizing MSE are two commonly used design
criteria for beamforming. mu is considered as 1/(real(trace(Rx))).
It can be shown in the simulations how interfering signals can be suppressed by putting nulls
as well as how the beam steering can be done towards the desired signal with the LMS algorithm.
Figure 3 is the normalized array factor plot before applying LMS. The angle of arrival of
desired and interfering signals are known from the array factor plot. They are 0°, 60° and 90°.

Figure.3. Normalized Array Factor plot before applying LMS
Three cases studied keeping different angle of arrival of user. The direction of the user are at
0°, 30° and -45° while those of interferers are at (60°, 90°), (0°, 90°)and (0°, 90°).
Table shows the directivity and half power beamwidth (HPBW) calculated for uniform linear
antenna array before and after LMS at the specific steering lobe.

TABLE-1
Uniform Linear Antenna Array
Main lobe
Before LMS

Directivity

HPBW

3.9920

34.3775

Steering lobe at 0° (after LMS)

4.2180

26.3561

Sterring lobe at -45° (after LMS)

4.2451

40.1070

Steering lobe at 30° (after LMS)

3.9999

34.3775

171


The array factor plot in figure 4-6 shows that the LMS algorithm is able to iteratively update
the weights to force deep nulls at the direction of the interferers and achieve maximum in the
direction of the desired signal.

Figure.4. Normalized Array Factor plot for LMS adaptive antenna array having θ=0°

Figure.5. Normalized Array Factor plot for LMS adaptive antenna array having θ=30°

Figure.6. Normalized Array Factor plot for LMS adaptive antenna array having θ=-45°
172


V. CONCLUSION
In this paper non-blind adaptive beamforming algorithm LMS have been analyzed on a
adaptive antenna system for three different cases. The performance of the algorithm is evaluated
through radiation pattern. The results illustrate the fact the LMS algorithm is able to iteratively
update the weights to force deep nulls at the direction of the interferers and achieve maximum in the
direction of the desired signal. It can be summarized that in satellite communication system, beam of
adaptive antenna array can be steered in the direction of user and can suppress the interference by
LMS beamforming algorithm.

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