Adaptive beamforming using lms algorithm

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 05 | May-2014, Available @ http://www.ijret.org 589
ADAPTIVE BEAMFORMING USING LMS ALGORITHM
Revati Joshi1
, Ashwinikumar Dhande2
1
Student, E&Tc Department, Pune Institute of Computer Technology, Maharashtra, India
2
Professor, E&Tc Department, Pune Institute of Computer Technology, Maharashtra, India
Abstract
The smart antennas are widely used for wireless communication, because it has a ability to increase the coverage and capacity of
a communication system. Smart antenna performs two main functions such as direction of arrival estimation (DOA) and
beamforming. Using beamforming algorithm . smart antenna is able to form main beam towards desired user and null in the
direction of interfering signals. This paper evaluate the performance of LMS (Least Mean Square) beamforming algorithm in the
form of normalized array factor (NAF) and mean square error(MSE) by varying the number of elements in the array and the
placing between the sensor elements. The simulations are carried out using MATLAB.
Keywords: DOA (Direction of Arrival), LMS (Least Mean Square), AF (Array Factor).
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1. INTRODUCTION
Adaptive beamforming is a technique in which an array of
antennas are used to achieve maximum reception in the
direction of desired user while signals of same frequency
from other directions are rejected[1]. This is achieved by
varying the weights of the each of antennas used in the
array. A smart antenna system combines multiple antenna
elements with a signal- processing capability to optimize its
radiation and or reception pattern automatically in response
to the signal environment[2]. Multiple antennas have ability
to enhance the capacity and performance without the need of
of additional power or spectrum. In adaptive beamforming
the optimum weights are iteratively computed using
complex algorithms based upon different criteria. The
criteria for choosing the adaptive beamforming algorithm is
depends on it’s performance and convergence rate.
Adaptive beamforming algorithm can be classified in to two
main categories such as Non blind and blind adaptive
algorithms[3]. Non blind adaptive algorithms require the
statistical knowledge of the transmitted signal in order to
converge to a weight solution. This is achieved by using a
pilot training sequence sent over the channel to receiver to
help to identify the desired user. Whereas ,blind algorithms
do not need any training sequence, hence the term ’blind’.
They attempt to restore some characteristic of the
transmitted signal in order to separate it from the other users
in the surrounding environment.This paper focus on the
implementation of Least Mean Square(LMS) algorithm
which is a type of non blind algorithm.
Fig-1: Adaptive Beamforming
2. BEAMFORMING
In beamforming each user’s signal is multiplied by complex
weight that adjust the magnitude and phase of the signal to
and from each antenna[4]. The phases and amplitudes are
adjusted to optimize the received signal. This causes the
output of the arrays of antenna to form transmit or receive in
a particular direction and minimizes the output in other
direction
Fig-2: Beamforming block diagram
3. ARRAY SENSOR SYSTEM.
The system consists of ULA with M antenna elements that
are linearly spaced along with equal distance.
Fig 2: System model for uniform linear array of M elements

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The array factor for uniform linear array is given in eqn
1.
𝐴𝐹 ∅ = 𝐴 𝑚 𝑒 𝑗𝑚 (
2𝜋𝑑
𝜆𝑀−1
𝑚=0 (1)
The phase shift between inter element is given in eqn
2.
𝛼 = −
2𝜋𝑑
𝜆0
𝑐𝑜𝑠∅0 (2)
Where, ∅0 is the desired beam direction. At wave length
𝜆0,the phase shift corresponds to a time delay that will steer
the beam to∅0.
4. BASIC CONCEPT FOR LMS ALGORITHM
4.1 Weiner Optimum Solution
Fig-3: LMS Adaptive Array.
The LMS adaptive array is shown in figure 3.
By using a feedback loop the weigths, 𝑤1,….,, 𝑤 𝑁, are updated
by the time sampled error signal is given in eqn
3.
e(n)=d(n)-y(n) (3)
Where d(n) is training sequence or replica of the desired
signal. And y(n) is the output of the adaptive array described
in eqn
4
y(n)=𝑤 𝐻
x(n) (4)
Where x(m) is the input signal.
The adaptive algorithm adjusts the weight vector to
minimize the mean square error(MSE) of the error
signal,e(n) is given in eqn
5.
E e(n)= E d(n)-y(n) (5)
Where E is the expectation operator
Substituting eqn
3 &4 in to eqn
5 and expanding the augment
of the MSE term
E[ 𝑒(𝑘) 2
]=E[{d(n)-y(n)}{d(n)-𝑦(𝑛)}∗
] (6)
Solving eqn
6 obtain the value of MSE term
E[ 𝑒(𝑘) 2
]=E[ 𝑑(𝑘) 2
-𝑟𝑥𝑑
𝐻
w-𝑤 𝐻
𝑟𝑥𝑑 +𝑤 𝐻
𝑅 𝑥𝑥 w] (7)
E[x(n)𝑥 𝐻
(𝑛)]=𝑅 𝑥𝑥 is the MXM covariance Matrix of the
input data vector.x(n).
E[x(n)𝑑∗
(𝑛)]=𝑟𝑥𝑑 is the Mx1 cross correlation vector
between the input data vector ,x(n),and the training
sequence,d(n).
Taking the gradient operator of the mean square error with
respect to the array weights and setting the result equal to
zero.
∇E[ 𝑒(𝑘) 2
]=
𝜕
𝜕𝑤 ∗E[ 𝑒(𝑘) 2
]=0 (8)
Substituting the value of eqn
7 in to eqn
8 .
-2𝑟𝑥𝑑 +2𝑅 𝑥𝑥 𝑤𝑜𝑝𝑡 = 0 (9)
The optimum weight eqn
is obtain using eqn
10.
𝑤𝑜𝑝𝑡 =𝑅 𝑥𝑥
−1
𝑟𝑥𝑑 (10)
Disadvantages using weiner optimum solution
1. If the number of elements in ULA is large ,then it is
complex to invert the MXM covariance matrix ,𝑅 𝑥𝑥 [5].
2. It requires the use of expectation operator in both
𝑅 𝑥𝑥 and𝑟𝑥𝑑 .
5. LMS ALGORITHM
LMS algorithm is a type of Non-blind algorithm, because it
uses the training signal or reference signal. During training
period training signal is transmitted from transmitter to
receiver. It uses a gradient based method of steepest decent.
It follows 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[6]. LMS algorithm is relatively
simple; it does not require correlation function calculation
and matrix inversions [7].
Let nw represent the Mx1 weight vector at time sample
n.The weight vector can be updated at time sample n+1
which is given in eqn
11.
w(n+1)=w(n)+µ[-∇(J(n))] (11)
Where J(n)= E[ 𝑒(𝑛) 2
] defines the MSE cost function.
Using eqn
9,the value of the error signal at time sample n is
given as in eqn
12.
∇(J(n))= -2𝑟𝑥𝑑 +2𝑅 𝑋𝑋 𝑤(𝑛) (12)

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Substituting eqn
12 in to eqn
11 result in eqn
13
w(n+1)=w(n)+2µ[ 𝑟𝑥𝑑 − 𝑅 𝑥𝑥 𝑤(𝑛)]] (13)
Where µ is step size parameter. It is a real valued positive
constant generally less than one. The initial weight w(0) is
assumed to be zero. The successive corrections of the
weight vector eventually leads to the minimum value of the
mean squared error. The step size varies from 0 to λmax,
whereλ max is the largest Eigen value of the correlation
matrix R[8].
Substituting the value of 𝑟𝑥𝑑 and 𝑅 𝑥𝑥 in to eqn
13 result in
eqn
14.
w(n+1)=w(n)+2µE[x(n)𝑒∗
(n)] (14)
6. SIMULATION RESULTS
For simulation purposes the uniform linear array with M
number of element and input signal is modulated by using
BPSK modulation is considered. simulation of LMS
algorithm is carried out using MATLAB to illustrate how
various parameters such as number of antenna element, inter
element spacing ,number of interferes and variation in SNR
parameter affect the beam formation and convergence of the
algorithm.
6.1 General case
Consider that the desired user is arriving at an angle of 30
degree and an interfere user at an angle of -50 degree. The
spacing between the individual element is half wavelength
and the signal to noise ratio(SNR) is 30db.
The array factor for 8 element antenna antenna array is
computed.Figure.4.Shows rectangular array factor plot for
different angle of arrival. Figure.5.Shows polar array factor
plot for different angle of arrival
Fig-4: Array factor plot when desired user is 30degree and
the interfere is -50degree
Fig-5: Polar array factor plot when desired user is 30degree
and the interfere is -50degree
The optimum complex weights in the case for which the
algorithm converges is as follows.
w1 = 1.
w2 = 0.039512+0.97006i
w3 = -1.0194+0.03523i
w4 = -0.001461-0.98315i
w5 = 0.98244-0.051033i
w6 = 0.018318+1.0197i
w7 = -0.97044+0.012622i
w8 = -0.052976-0.99924i
Fig-5: Magnitude of array weights when no.of array element
is 8.
Fig-6: Acquisition and tracking of desired signal and actual
array output

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Fig-7: Mean square error Vs iteration no.
6.2 Special case
6.2.1 Effect of the Number of Array Elements on
the Array Factor and MSE.
Figure 8 shows the array factor plot of LMS algorithm when
number of antenna array element is 8,12,18.And desired
user is arriving at an angle 60 degree and interefer is at an
angle of -50 degree.
The simulation result shows that as the antenna array
element goes on increasing from 8,12 and 18 the beam
width becomes narrow and the number of side lobe goes on
increasing.But level of these side lobes is low compared to
those generated by small number of elements.
Fig 8: Effect of no.of array element on Array Factor.
Fig 9: Effect of no.of array element on Mean square error.
The increasing number of antenna array element produce an
increase in systen noise.And overall MSE tends to be almost
the same for the given values of antenna element.
6.2.2 Effect of the Separation Distance on the Array
Factor and MSE.
Figure 9 shows the performnce of LMS algorithm when the
distance between array element d=0.5 and d=1.And desired
user is arriving at an angle 30 degree and interefer is at an
angle of -50 degree.
Fig 10: Effect of interelement seperation distance on Array
Factor.
The simulation result shows that increasing interelement
distance between array element produce narrow beams,but
this also increases the number of side lobes.when the
seperation distance between array element is equal to wave
length granting lobes are created.
Fig 11: Effect of inter element seperation on Mean square
error.
It is also oberved that when distance between array element
is half wave length then granting lobes are avoided.From the
graph of MSE it is oberved that spacing between array
elements equal to half wavelength gives an optimum error in
a particular iteration.
7. CONCLUSIONS
This paper evaluate the performance of LMS algorithm for
two different cases such as general case and special case.In
general case the LMS algorithm is compared on the basis of
normalized array factor and Mean square error(MSE).It is
observed that LMS algorithm is converging after 50

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iteration.And it has negligible error.In special case the the
performance of LMS algorithm is compared on the basis of
observing the effect of varying the number of antenna
elements and distance between array element on the array
factor and Mean square error(MSE).As number of antenna
element increases the beam width of array factor becomes
narrow and number of side lobe increases ,but level of side
lobe is less than those generated by an array of small
number of elements and overall MSE tends to be almost the
same for the given values of antenna element. And spacing
between array element equal to half wavelength ,then MSE
gives an optimum error in a particular direction.
REFERENCES
[1]. D.M.M.Rahaman., Md.M.Hossair, “Least Mean Square
(LMS) For Smart Antenna”, Universal Journal of
Communications and Network”,pp-16-21,2013.
[2]. M.Jain., V.Gupta., “Performance Analysis of MUSIC
LMS Algorithm for Smart Antenna”, International Journal
of Scientific Engineering and Technology,Vol.2,Issue
10,pp1004-1007,2013.
[3]. U.Chalva., Dr.P.V.Humgund., “Performance Study of a
Non-blind Algorithm for Smart Antenna System”,
International Journal of Electronics and Communication
Engineering”, Vol.5,Issue 4,pp.447-455,2012.
[4]. V.Kumar., Dr.Rajouria., “Performance Analysis of LMS
Adaptive Beamforming Algorithm”, International Journal
of Engineering and Communication Technology,Vol.4,Issue
5,July-Sept-2013.
[5]. Douglas,S.C, “Introduction to Adaptive Filters”, CRC
Press LLC,PP-18,1999.
[6]. K.S.Kumar and T.Gunasekaran., “Performance Analysis
of Adaptive Beamforming Algorithms for Microstrip Smart
Antennas”, International Journal of Computing Science and
Communication Technologies, Vol.2,Issue 1,July2009.
[7]. Simon Haykins, “Adaptive Filter Theory”, 3rd
edition,
2001.
[8]. K.K.Shetty, “A Novel Algorithm for Uplink
Interference Suppression using Smart Antennas in Mobile
Communications “ Master’s Thesis, Famu-Fsu College of
Engineering, THE Florida state university, February 2004.

Adaptive beamforming using lms algorithm

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