19CM798
ROBUST SYNTHESIS OF BEAMFORMING WEIGHTS IN
LINEAR ANTENNA ARRAY
A PROJECT REPORT
Submitted by
ATHIRA P S
(CB.EN.P2CST20004)
Under the guidance of
Dr. Natarajamani S
and Ms.Prabha G
In partial fulfillment for the award of the degree of
MASTER OF TECHNOLOGY
IN
COMMUNICATION SYSTEMS ENGINEERING
DEPARTMENT OF ELECTRONICS AND COMMUNICATION
ENGINEERING
AMRITA SCHOOL OF ENGINEERING
AMRITA VISHWA VIDYAPEETHAM
AMRITANAGAR, COIMBATORE - 641 112
JANUARY 2022
AMRITA VISHWA VIDYAPEETHAM
AMRITA SCHOOL OF ENGINEERING
COIMBATORE - 641 112
BONAFIDE CERTIFICATE
This is to certify that the project report entitled ROBUST SYNTHESIS OF BEAMFORMING
WEIGHTS IN LINEAR ANTENNA ARRAY submitted by ATHI RA P S (CB.EN.P2CST20004)
in partial fulfillment of the requirements for the award of the Degree of Master of Technology in
COMMUNICATION SYSTEMS is a Bonafide record of the work carried out under my guidance
and supervision at Amrita School of Engineering, Coimbatore .
Project Advisor: Dr. Natarajamani S ,
Ms.Prabha G
Designation: Asst.Prof, ECE
Program Coordinator: Dr.Ramanadhan
Designation: Asst.Prof, ECE
Dr. M. Jayakumar
Chairperson
Department of Electronics and Communication Engineering
The project was evaluated by us on:
Internal Examiner
External Examiner
ACKNOWLEDGEMENT
I take this opportunity to humbly express my sincere gratitude and thanks to all those concerned
with my M.Tech Dissertation.
I express my sincere gratitude to Dr.Natarajamani, Ms.Prabha for providing valuable guidance
and appropriate suggestion during the course of the dissertation because of which I was able to
complete the first phase of the dissertation process satisfactorily.
I am obliged to Dr.Ramanadhan Program Coordinator, whose valuable inputs and guidance
helpedto improve the quality of the dissertation work. Also, I am thankful to the project review
panel consisting of Dr.Jayakumar.M (Chairperson, ECE),Dr.Ramanadhan (Asst.Prof ,ECE) and
Dr.GandhiRaj (Asst.Professor,ECE) for their ample inputs during each review process and
pointing out the areas to improve specifically, and thus guiding to improve the quality of
dissertation.
ABSTRACT
To meet future wireless communications networks increasing demand Smart Antenna technology will
have a widespread application. The ability of smart antennas to respond to interference in the real-time
environment is the major factor for the use of smart antennas. The ability to place nulls in an undesired
direction and major lobes in the desired direction is known as Adaptive beamforming.
Due to their ability to isolate noise signals from desired signals smart antennas have major applications in
wireless communications. Smart Antenna technology increases directivity towards the desired directions,
also noise resistive and better capacity for a wireless communication system.
Adaptive Beamforming algorithms are used by Smart Antennas to steer the main lobe towards the desired
direction, in turn, placing nulls in an undesired direction. The steering of the beam pattern is done by
calculating the weights required based on the angle of the desired direction. This is done by adaptive
algorithms that are programmed in the DSP.
This project presents an extensive study of the basics of Smart Antenna systems and the functioning of
Smart Antennas. Simulation of the Smart antenna algorithms for 5G frequency is carried out through
MATLAB software based on real-time antenna parameters like the distance between antenna elements,
several Antenna elements, etc. Comparison based on the simulation of different algorithms will be carried
and the conclusion is drawn for an algorithm that has better performance for a given environmental
condition.
i
TABLE OF CONTENTS
CONTENTS
PAGE NO.
1. INTRODUCTION .......................................................................................................................7
1.1 Problem definition …………………………………………………………………………………..7
1.2 Scope of the work …………………………………………………………………………………... 7
1.3 Overview of Antenna and Arrays .......................................................................................................8
1.3.1
Need of Array Antenna …………………………………………………………………….8
1.3.2
Block Diagram of Antenna Array and its working ..................................................................8
1.4 Smart Antenna System ........................................................................................................................9
1.4.1
Definition………………………………………………...…………………………………9
1.4.2
Analogy for Smart Antenna ………………………………………………………………..9
1.4.3
Smart Antenna………………………………………………………………………………10
1.4.4
Need for Smart Antenna…………………………………………………………………….10
1.4.5
Architecture of Smart Antenna System……………………………………………………10
1.5 Beamforming in Smart Antenna System………………………………………………………….11
1.5.1
Types of Beamforming……………………………………………………………………12
1.6 Features of Smart antenna Technology…………………………………………………………...14
1.7 Benefits of Smart antenna Technology …………………………………………………………..14
1.8 Summary ……………………………………………………………………………………………14
2.
LITERATURE SURVEY……………………………………………………………………16
3. PROPOSALS AND CONTRIBUTIONS……………………………………………………18
4. METHODOLOGY……………………………………………………………………………19
ii
4.1 Least Mean Square(LMS)……………………………………………………………………........19
4.2 Sample Matrix Inversion(SMI)........................................................................................................22
4.3 Recursive Least Square(RLS)...........................................................................................................22
4.4 Normalized Least Mean Square.......................................................................................................23
5. RESULTS AND DISCUSSIONS..................................................................................................25
5.1 Flow chart of an implementation of algorithms..............................................................................25
5.2 Comparative performance analysis of LMS, NLMS, SMI, RLS, LMS/SMI, and Combined
NLMS algorithms in the adaptive array.............................................................................................26
5.2.1 comparison of null depth,and maximum SLL for varying number of antenna elements............26
5.3 Results of implementation of LMS algorithm...............................................................................29
5.3.1 Effect of varying number of elements........................................................................................29
5.3.2 efEfect of varying inter elements spacing..................................................................................30
5.4 Summary...........................................................................................................................................31
5.5 Results and implementation of LMS-RLS algorithm...................................................................31
5.6 Comparison of null depth, and maximum SLL for LMS, LMS-RLS.........................................33
REFERENCES.......................................................................................................................................34
PUBLICATION STATUS..................................................................................................................... 36
iii
LIST OF ABBREVIATIONS
ABBREVIATIONS
EXPANSIONS
PAGE NO
LMS
Least Mean Square
14
SMI
Sample Matrix Inversion
14
RLS
Recursive Least Square
14
NLMS
Normalized Least Mean Square
14
SLL
Sidelobe level
14
SA
Smart Antenna
4
RF
Radio Frequency
5
DSP
Digital Signal Processing
12
MSE
Mean Square Error
19
iv
LIST OF FIGURES
FIGURE NO
TITLE
PAGE NO
1
Block diagram of the antenna array
8
2
Block diagram of the adaptive antenna array
11
3
Beamforming pattern for the desired signal
12
and interfering signals
4
Switched beamforming pattern
13
5
Adaptive beamforming pattern
14
6
LMS algorithm adaptive beamforming
22
network
7
RLS algorithm adaptive beamforming
25
network
8
Flow chart of algorithm implementation
27
9
Array elements N = 8
28
10
Array elements N = 16
29
11
Array elements N = 21
29
12
4 element linear array
31
13
8 element linear array
31
14
16 element linear array
31
15
21 element linear array
31
16
d = 0.3 lambda
32
17
d = 0.5 lambda
32
18
d = 2 lambda
33
19
Polar plot of LMS-RLS for d =0.5
34
20
Rectangular
34
v
LIST OF TABLES
TITLE
Array performance of different
TABLE NO
1
PAGE NO
30
algorithms with varying antenna
elements
Performance of combined adaptive
2
35
beamforming algorithms with LMS
vi
19CM798 Dissertation
ATHIRA P S and CB.EN.P2CST20004
JANUARY and 2022
CHAPTER 1
INTRODUCTION
1.1 Problem Definition
As the growing demand for wireless communication is constantly increasing, the need for better
coverage, improved capacity, and higher transmission quality rises. Thus, more efficient use of
the radio spectrum is required. One of the promising technologies is the use of a smart antenna
system. A smart antenna system is capable of efficiently utilizing the radio spectrum and is a
promise for an effective solution to the present wireless system problems while achieving
reliable and robust high-speed, high-data-rate transmission. Traditional antennas emit a
distributed beam whereas in smart antennas the beam of the antenna is highly directional and
pointed only towards the desired user and the interferences are rejected. Adaptive beamforming
is one of the main and most famous aspects of the development of smart antenna
technologies[1]. A smart antenna system consists of multiple numbers of elements. Using a
beamforming algorithm smart antenna is able to form the main beam towards the desired user
and null in the direction of interfering signals. This beam moves with the user, this improves
the power utilization of the antenna. The main focus of this project is a brief account of the
Smart Antenna (SA) system in the context of adaptive beamforming. The goal is put null in the
direction of the interferer and maximum in the desired direction.
1.2 Scope of the work
A smart antenna is an array of radiating antenna elements combined with digital signal
processing to transmit and receive in the adaptive manner. Adaptively in the sense, it
automatically adjusts the directionality of its radiation pattern in response to the signal
environment. Smart antennas can increase signal range, reduce signal fading, suppress
interfering signals, and increase the capacity of wireless systems.
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1.3 Overview of Antennas Arrays
1.3.1 Need of Array antenna.
Conventional single element antennas have various drawbacks compared to array antennas.
Single element antennas have no noise cancellation property. The following points represent
the need for array antennas over a conventional antenna system.[2]
i.
Signal strength increases.
ii.
The high directivity is obtained
iii.
Minor lobes are reduced
iv.
High signal to noise ratio achieved.
v.
High gain is obtained.
vi.
Power wastage is reduced.
vii.
Better performance is obtained
1.3.2 Block diagram of Antenna Array and its working
An antenna with more no of elements is called an Antenna array. The Antenna array consists
of individual radiators and elements.. The radiation pattern produced by them would be the
vector sum of individual ones. It can radiate individually but when it is an array, it sums up the
radiation of all the elements to form a radiation beam, which has high gain, high directivity,
and better performance with minimum losses.[4]. The below images indicate the Antenna array
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Figure 1: Block diagram of the Antenna Array.
The applications of array antennas are used in satellite communication, wireless
communication, radar communication.
1.4 Smart Antenna Systems.
1.4.1 Definition.
A smart antenna is a system with multiple antenna elements in an array with an ability to change
their beam pattern in response to interferer signals in the receiver and increase radiation towards
desired signals.
1.4.2 Analogy for Smart Antennas.
Adaptive smart antenna systems can be easily understood with a simple analogy. Consider when
we stand in a room with our eyes closed and conversing with a person. Consider the person is
moving into the room while speaking. We can receive the audio through our ears and be able
to detect the location of the person even with our eyes closed. As the person moves from side
to side, audio reaches to our both ears at different times. Our brain can process the audio and
be able to determine the side where the person is present. The working of Adaptive Antenna
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systems is also similar but with an antenna having multiple antenna elements arranged in an
array pattern.[5]
1.4.3 Smart Antenna.
The name smart antenna doesn’t mean that the antenna is smart. The antennas are usually
attached to the base station, but when the antenna is combined with a Digital Signal processor
with intelligent algorithms of beamforming to receive and transmit signals in the environment
make the antenna smart.[1]
This configuration increases the system capable of the wireless communication system. Various
parameters such as array gain, interference mitigation are considerably increasing with smart
antenna systems. Smart antenna systems can be more efficient when replaced with conventional
systems which have the effects of multipath fading and co-channel interference.
1.4.4 Need for Smart Antenna.
The main aim of a smart antenna system is to improve the signal quality of a receiving signal
by increasing beam pattern transmission towards the direction of the desired signal by taking
advantage of adaptive algorithms. Smart antenna makes a system more efficient and increases
the capacity of a conventional antenna system. Smart antennas are widely employed in modern
wireless systems to combat the increasing applications and increased capacity to which the
conventional antennas have to serve.[2]
1.4.5 Architecture of Smart Antenna System.
In smart antenna system, the radiation pattern of the antenna is controlled via certain algorithms
using digital signal processors. The inputs to the antenna arrays are assumed as the desired signal,
interfering signals. Using some adaptive algorithms, the array weights are controlled and
subsequently the output error is minimized.
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There are N antennas placed at the input side of the block diagram. All the antenna elements
together can be called as Antenna Arrays. The adaptive algorithm is used to minimize the error
e(n) between desired signal d(n) and array output y(n). The antennas combine with the weights
and form the output. The error e(n) gets the difference of the desired output d(n) and actual output
y(n).The error e(n) goes to the Adaptive Algorithm and updates the weights. This process goes
on until it gets the desired direction.
Figure 2: Block diagram of an Adaptive Antenna Array
The radiation pattern produced by them would be the vector sum of individual ones. The distance
between the elements and the length of the elements relative to the wavelength is important
during antenna design. It can radiate individually but when it is an array, it sums up the radiation
of all the elements to form a radiation beam, which has better gain, good directivity, and
increased performance with lower losses.[1]
1.5 Beamforming in Smart Antenna.
Beamforming is a type of technique used to send strong signals and focus the signal on a
targeted device. The process of cutting an interfering signal and constructing a pattern is called
Beamforming.[3]
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Figure 3: Beamforming pattern for desired and interfering signals
1.5.1 Types of Beamforming.
Beamforming is categorized into two types based on the method the antenna array uses to steer
the beam dynamically.
1. Switched Beamforming.
The switched beam system functions by having a set of predefined beam patterns of array
antennas combined and switching the beam in response to the signal environment. These
antennas will have high directivity, which helps to achieve some gain. In this, there will be
numerous amounts of fixed beams amongst which one beam will turn on or will be steered
towards the wanted signal. This can be done only with adjustment in the phase. The below
figure indicates the Switched beam
Figure 4: Switched beamforming pattern.
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Switched beam antenna systems form different fixed beams with heightened sensitivity in
particular directions. These antenna systems identify signal strength, choose from one of several
predetermined, fixed beams, and switch from one beam to another as the mobile moves
throughout the sector.[5] The main drawback of this beam is that the inter-cell interferences
among the beams are needed to be handled.
2. Adaptive Beamforming
In this type , there will be a change in the beam pattern according to the movement wanted by
the user and the movement of the interference. These are the antenna arrays that dynamically
change the antenna pattern to control the interference, multipath, and noise.[6].The significance
of using the Adaptive antenna is that the generated antenna beams can adaptively track the signal
direction. In this there is no inter-cell issue is found due to the continuous tracking of users. In
this, the capacity to a greater extent than switched beam. They can have extensive signal
processing through DSP. The main drawback of these antennas is that the cost of installation is
very high. The below figure shows the Adaptive array
Figure 5: Adaptive beamforming pattern
Using a variety of new signal-processing algorithms, the adaptive system takes advantage of its
capacity to effectively locate and track various types of signals to effectively minimize
interference and maximize intended signal reception.
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1.6 Features of Smart Antenna Technology.
1. Signal Gain
2. Interference Rejection
3. Spatial Diversity
4. Power Efficiency
5. Better range/ Coverage
6.
Increased Capacity
7. Multipath Rejection).
1.7 Benefits of Smart Antenna Technology
1. Reduction in co-channel interference.
2. Range improvement.
3. Increase in capacity.
4. Reduction in transmitted power
1.8 Summary
Work carried out investigates the performance of LMS (Least Mean Square), RLS (Recursive
Least Square), SMI(Sample Matrix Inversion), NLMS(Normalised Least Mean Square),
Combined NLMS, LMS/SMI ,LMS-RLS beamforming algorithms to steer the antenna beam in
the particular desired direction. Algorithms are simulated for a linear antenna array with different
sizes, and results are discussed in terms of their Max SLL, and Null depths . The simulation is
carried out using MATLAB.
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Chapter 1 explains the Adaptive arrays used in smart antennas and the problem overview.
There are various types of smart antennas present. But we used only the appropriate one i.e
Adaptive antenna array. The adaptive Antenna Array and its operation have been explained
with the help of a block diagram. Adaptive Beamforming algorithms are used by Smart
Antennas to steer the main lobe towards the desired direction, in turn, placing nulls in an
undesired direction. The steering of the beam pattern is done by calculating the weights
required based on the angle of the desired direction. This is done by adaptive algorithms that
are programmed in the DSP.
Chapter 2 explains what the literature survey has been carried through.
Chapter 3 explains the proposals and contributions of work.
Chapter 4 expalins different types of beamforming algorithms for smart antennas
Chapter 5 explains the implementation and the results of Adaptive beamforming Algorithms for
smart antennas. It explains and shows the factors that are taken for the implementation of the
results of the adaptive algorithms.
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CHAPTER 2
LITERATURE SURVEY.
[1] Presents the survey-styled introduction to the concept to the concept of adaptive beamforming
techniques. [2] gives the overall introduction to the various adaptive antenna beamforming
algorithms(LMS,RLS,NLMS,SMI)and
some
combined
algorithms(combined
NLMS,
LMS/SMI). Found that the RLS, SMI gives better performance, LMS, NLMS, Combined NLMS
introduce the lowest SLL, RLS, SMI, LMS/SMI show deep nulls and have the highest SLL.In
[3] LMS, RLS, SMI algorithms are analyzed by varying numbers of elements in the uniform
linear antenna array.It is observed that as the number of elements increases the improvement in
terms of narrow beam width and deepest null at interfere becomes superior.
[4] discusses two types of adaptive beamforming algorithms. The beamforming performance is
studied by varying the element spacing and the number of antenna array elements for each
algorithm. These algorithms are compared for their rate of convergence, beamforming, and null
steering performance (beamwidth, null depths, and maximum sidelobe level). From the results
obtained in the paper it is evident that d = 0.5 λ is used as optimal distance between antenna.This
paper,[5] , examined the performance analysis of LMS adaptive beamforming algorithm for
smart antenna system in the form of normalized array factor and mean square error. The authors
have presented performance of the antenna system for different spacing between array elements,
number of array elements, different geometry. The following are the conclusions made in this
paper

when increase spacing between elements beamwidth becomes narrower, but for higher
values sidelobe level
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 When increase number of elements ,beamwidth becomes narrower and also sidelobe
levels are decreases
 Beamwidth of planar array is narrower than other geometry and side lobe level decreases
than other geometry
 Planar array gives a good array factor compared to others
A full digital beamforming array with nonreciprocal TX/Rx beam patterns is proposed in [6].
The paper discusses how Nonreciprocal arrays can achieve a wider coverage in the uplink and
provide high gain beams for the downlink. Introduced a novel idea to incorporate the actual
nonisotropic radiation pattern produced by the array elements and the mutual coupling into two
popular deterministic BF methods in,[7]. Two popular beamforming methods minimum variance
distortionless response (MVDR) method and the null steering beamforming (NSB) method are
modified according to this new idea in order to be applied on a realistic microstrip linear antenna
array. In,[8], the linear arrays are evaluated on the basis of the number of elements and the
distance between the elements for beamforming. The analysis is done to find the relationship of
the gain and the beam width with the number of the elements and the spacing between the
elements.
In,[9], compared the two algorithm by varying different parameters like number of elements,
spacing between array elements and step-size.In,[10],the authors investigates the performance
of sample matrix inversion with least mean square (SMI-LMS), sample matrix inversion with
recursive least square (SMI-RLS) and least mean square with recursive least square (LMS-RLS)
algorithm. Side lobe level, null depth and error plot of adaptive beam formation for different
signal-to-noise (SNR) with different element spacing are simulated and compared. It is found
that performance of LMS-RLS algorithm is better than SMI-LMS and SMI-RLS algorithm.
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CHAPTER 3
PROPOSALS AND CONTRIBUTION
The project proposal is to design and implementation of antenna arrays , beamforming algorithms for
antenna arrays and evaluate the performance analysis of both.
Work carried out investigates the performance of LMS (Least Mean Square), RLS (Recursive
Least Square), SMI(Sample Matrix Inversion), NLMS(Normalised Least Mean Square),
Combined NLMS, LMS/SMI , beamforming algorithms to steer the antenna beam in the
particular desired direction. Algorithms are simulated for a linear antenna array with different
sizes, and results are discussed in terms of their Max SLL, and Null depths . The simulation is
carried out using MATLAB.
Comparing based on side lobe radiation and main lobe radiation of all adaptive algorithms, found
that some combined algorithms provide better results. Beams of a smart antenna are generated
by combined approaches using LMS-RLS. These combined algorithms are applied for
beamforming for various values of step size (for LMS), forgetting factor (for RLS) best results
are reported. In LMS-RLS, first weights are updated using LMS and then again updated using
RLS.
The contribution of this project includes the implementation of adaptive beamforming
algorithms, namely LMS, NLMS, SMI, and RLS, LMS/SMI, LMS-RLS, Combined NLMS.
Furthermore, a comparative performance analysis is carried out and evaluated in terms of null
depth, maximum sidelobe level for the number of antenna elements, and spacing.
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CHAPTER 4
METHODOLOGY
Beamforming algorithms can be classified into two classes, namely non-blind & blind
algorithms. In the case of nonblind adaptive algorithms, a reference signal is used in the process
of adjusting the array weights function. On the other hand, no reference signal is used in blind
adaptive algorithms. However, when compared with their non-blind counterparts, these
algorithms tend to be more computation-intensive, and often provide lower accuracy and slower
convergence rate.
In the non-blind algorithms, the adaptive weights of the array beamformer are usually adapted
according to a specified criterion, such as minimization of mean square error (MMSE). An error
signal, produced by comparing the output signal with a reference signal, is used to iteratively
adjust the weights of the beamformer to their optimal values, Wopt, to obtain the minimum
MSE. The trained algorithms could be classified according to their adaptive criterion: leastmean squares method (LMS), sample matrix inversion (SMI) , recursive least-squares method
(RLS),normalized least mean square method(NLMS).
4.1 Least Mean Square Algorithm (LMS).
The LMS algorithm is an adaptive algorithm that uses a gradient-based approach of steepest
descent. Minimum Square Error (MSE) criteria are employed by LMS Algorithm to the
derivation of the weight vector. In the training period, the reference signal is transmitted and
received at the other end which is used to calculate the error signal. Error is minimized by the
gradient-based method of steepest descent. LMS algorithm makes continuous correction of
weight vector by a small amount in an iterative manner based on the gradient vector, called a
step size, which outputs the minimum square error.
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LMS algorithm is simple compared to others since it does not require any complex calculations.
The LMS algorithm continuously calculates the error between the desired signal and the actual
signal until the error becomes minimum. When the error becomes minimum the weights
adaption reaches their optimal stage.
Array antenna receives the incoming signal x(n) and multiplied with the weight factors W,
which alters the phase and amplitude of the incoming signal. These altered weighted signals are
summed up and give the output signal y(n). Then the error is minimized by an adaptive
algorithm by calculating the difference between the output signal y(n) and the desired signal
d(n), by updating the weight vector.[1-5]
Figure 6 : LMS algorithm adaptive beamforming network
The algorithm is used to minimize the error e(n) between desired signal d(n) and array output
y(n), as
e(n) = d(n) – y(n)
(1)
The output of adaptive beamformer, at a time ‘n’, is given by a linear combination of the data at
the N antenna elements, can be expressed as
y(n) = wH x(n)
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w = [w1,w2,….wN]H
Where, H denotes Hermitian (complex conjugate) transpose. The weight vector w is a complex
vectors.
Signal received by multiple antenna elements is
x(n) = [x1(n),x2(n),…….xN(n)]
LMS algorithm updates the weight vectors according to the equation
w(n+1) = w(n) + µ x(n) e*(n)
(3)
Where μ is the step size varies from 0 to 1 which is relatively small.
The LMS calculation speed depends on the step size and it ranging between
0< µ< 1
𝜆𝑚𝑎x
𝜆𝑚𝑎𝑥 represents the maximum eigenvalue of the correlation matrix R which is the limit for a step
size value. The speed of the algorithm depends on the value of the step size. If the value of step
size is small then the speed of the algorithm is slow and if the value of step size is the large speed
of the algorithm is fast. The LMS algorithms adopt a gradient-based approach, where it also tries
to decrease the mean square error[MSE]. Based on the MSE the algorithm changes the values of
the weights to counter the MSE. If the MSE is more the values of weights are decreased and if
MSE is negative the weights will be more positive so that the error becomes optimal. If the step
size is smaller the convergence rate of the algorithm will be high and if the step size is larger the
convergence rate is faster making the system unstable at MSE value. Hence a proper step size
needs to be chosen.
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4.2 Sample Matrix Inversion (SMI).
In the SMI algorithm, the weight calculation is different from the LMS algorithm which makes
SMI faster than LMS. SMI algorithm uses its estimated value in place of the correlation matrix
to calculate its weight, wherein the LMS algorithm uses many iterations which ultimately
converge the error to the minimum and update its weights which is a slower computational
speed than the SMI.
Weights are calculated by equation
𝑤𝑆𝑀𝐼 = 𝑅𝑥𝑥(𝑛) −1 ∗ 𝑟(𝑛)
(4)
Where
𝑅𝑥𝑥 (𝑛) =
r(n) = ∑
∑
. x(n) ∗ x 𝐻(𝑛)
(5)
(6)
𝑥(𝑛) ∗ 𝑑(𝑛)
In the rapidly changing environmental conditions, this is the most suitable for beamforming
since the rate of convergence is faster than the LMS algorithm.[1-5]
4.3 Recursive Least Squares (RLS)
Even though the SMI algorithm is faster in convergence than the LMS algorithm its complexity
can cause potential singularities that can cause problems in calculations of weights which is a
drawback.[1-5]
The equation for the correlation matrix and the vector is given by equation
𝑅 xx(k) = 𝛼𝑅 xx(k − 1) + 𝑥̅(k) xH(k)
(7)
𝑟̌ 𝑥𝑥(k) = 𝛼𝑟̌ 𝑥𝑥(k − 1) + 𝑑* (k)𝑎̅(k)
(8)
where 𝛼 is the forgetting factor; that is a positive constant value ranging from 0 < 𝛼 < 1.
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The weights of the RLS algorithm is updated as given by
𝑤(k) = 𝑤(k − 1) + 𝑔(k)(𝑑* (k) − 𝑥 𝐻(k)𝑤(k − 1))
(9)
Where the gain vector 𝑔(𝑘) is defined as given by equation
𝑔(𝑘)
=
𝛼 −1𝑅 𝑥𝑥(k−1)𝑥(k)
(10)
1+𝛼−1𝑥̅𝐻 𝑅 𝑥𝑥(k−1)𝑥(k)
Figure 7: RLS adaptive beamforming network
4.4 NLMS
The LMS algorithm is the most basic method for calculating weights and the algorithm requires
large number of iterations before satisfactory convergence. Also,the stability and convergence
time of the LMS algorithm are dependent upon the step-size parameter. To overcome this
dependency, the NLMS algorithm is introduced. The NLMS algorithm is used to achieve good
stability and faster convergence.[1-5].The weight update equation for the NLMS algorithm is
given as
w(n+1) = w(n) +
μ NLMS
.x(n).
e*(n)
(11)
||x(n)||2 + α
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The equation represents the final weight update for the NLMS algorithm where the step-size is
divided by the normalized value of the input signal x(n). In Equation, to avoid the denominator
being zero for no signal condition, a small positive constant ‘α’ is added to the denominator.[910].
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CHAPTER 5
RESULTS AND DISCUSSIONS
Simulation of adaptive algorithms is carried out using MATLAB software. The array factor of
the array antenna is plotted against the angle of arrival of the signal. The Y-axis represents the
array factor of an array antenna and X-axis represents the angles of signals.
Various simulation input parameters are listed below:
1. The number of Antenna elements.
2. Distance between antenna elements.
3. The angle of the desired signal.
4. The angle of the interfering signal
5.1 Flow chart of Implementation of Algorithm
start
Initialize element spacing ,element number,angle of
desired signal ,interference signal
Initialize weights to
zero
Calculate weights
using algorithms
Updates new
weights
End
Figure 8: Flow chart for Algorithm Implementation
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5.2 Comparative performance analysis of LMS, NLMS, SMI, RLS,
LMS/SMI, and Combined NLMS algorithms in the adaptive array.
The performance studies of the above-mentioned algorithms are done by using MATLAB, in
varying parameters, for example, null depth, maximum SLL, rate of convergences. . Here,
initially smart antenna system is designed by taking a uniform linear array (ULA) with interelemental spacing (d) equal to half wavelength distance.
5.2.1Comparison of null depth, and maximum SLL for varying number of
antenna elements
For the variation of array elements of the LMS, NLMS, SMI, LMS/SMI, Combined NLMS and
RLS algorithms, have considered fixed element spacing at 0.5 of wavelengths (𝜆). Figures 9,
10, and 11 illustrate the array patterns of the desired signal and null steering toward interferers
with the variation of antenna elements for 8,16, and 21, respectively
Figure 9 : Array element N = 8
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Figure 10 : Array elements N = 16
Figure 11 : Array elements N = 21
Table 1 presents a summary of the results for varying antenna elements.
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Algorithms
LMS
RLS
SMI
NLMS
LMS/SMI
Combined NLMS
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Antenna
elements
Maximum
sidelobe
level(dB )
JANUARY and 2022
Maximum null
depth(dB)
8
6.2
-7.03
16
12.48
-14.14
21
6.05
-6.79
8
7.84
-9.85
16
15.68
-20.00
21
7.77
-9.85
8
7.84
16
15.68
21
7.77
8
7.84
-7.89
16
15.68
-15.78
21
7.77
-7.75
8
7.61
-8.79
16
15.25
-17.58
21
7.62
-8.79
8
7.4
-9.63
16
14.92
-19.26
21
7.44
-9.63
TABLE 1: Array performance of different algorithms with varying antenna elements
The plots indicate that if the number of antenna elements increases keeping element spacing at
0.5𝜆, the antenna beamwidth becomes narrower. The maximum SLL slightly varies and the
maximum null depths remain almost the same. It can be observed that the RLS, NLMS, and
LMS with SMI weights initialization show deep nulls, on the other hand, RLS, SMI, NLMS
have the highest SLL. Whereas LMS, and combined NLMS introduce the lowest SLL, where
both NLMS and combined NLMS have deeper null than LMS.
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5.3 Results of Implementation of LMS Algorithm
5.3.1 Effect of Varying Number of Elements
For this simulation, the inter-element spacing is set to be 𝜆 /2, where 𝜆 is the wavelength. Figure
12,13,14,15 illustrates the polar radiation pattern for 4, 8, 16, and 21 elements linear arrays.
Figure 12 : 4 element linear array
Figure 13 : 8 element linear array
Figure 14 : 16 element linear array
Figure 15 : 21element linear array
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Simulations results obtained have proved that an increase in the number of elements in a linear
array would result in higher directivity, as well as a sharper and narrower beamwidth, but this
brings about more side-lobes. It has been shown that the interference rejection capability of the
array improves as the number of antenna elements increases.
5.3.2 Effect of Varying Inter-elements Spacing
Figure 16,17,18 illustrates the polar radiation pattern for an 8-elements linear array for an interelement spacing (d) of 0.3 wavelengths, half wavelength, double wavelength . The angle of the
desired signal is 90° and the angle of the interference signal is take as 0° .
Figure 16 : d = 0.3 lambda
Figure 17:
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Figure 18 : d = 2 lambda
It is observed that an increase in inter-element spacing in a linear array will result in higher
directivity and smaller beamwidth, it is found that the number of undesirable side lobes also
increases with increasing inter-element spacing.
5.4 Summary
Comparing based on side lobe radiation and main lobe radiation LMS algorithm has shown the
lowest SLL, RLS showed deep nulls. As the distance between the antenna elements increases
more than 0.5 𝜆, the radiation pattern shows multiple major lobes that are not required. At an
optimal value of 0.5 𝜆 of element spacing, the algorithms have maximum radiation in the desired
direction.
5.5 Results of Implementation of LMS-RLS Algorithm
beams of smart antenna are generated by combined approaches using LMS-RLS. These
combined algorithms are applied for beamforming for various values of step size (for LMS),
forgetting factor (for RLS) best results are reported. In LMS-RLS, first weights are updated using
LMS and then again updated using RLS.
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Figure 19,20 illustrates the polar and linear radiation pattern for an 8-elements linear array for
an inter-element spacing (d) of half wavelength. The angle of the desired signal is 90° and the
angle of the interference signal is taken as 0°.
Figure 19 : Polar plot of LMS-RLS for d=0.5
Figure 20 : Rectangular plot of LMS-RLS
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5.6 Comparison of null depth, and maximum SLL for LMS, LMS-RLS
The parameters, like, maximum side lobe level (SLLmax), null depth, direction of user(90 0)
and direction of interferer(00 )of LMS and LMS-RLS algorithms in adaptive beamforming are
tabulated in Table 2.
Algorithms
Antenna
elements
Antenna spacing
Maximum
Maximum null
sidelobe level depth
(dB)
(dB)
LMS
8
0.5λ
6.2
-7.03
LMS-RLS
8
0.5λ
-31
-114.68
Table 2: Performances of combined adaptive beamforming algorithms with LMS
It is found that the lowest side lobe levels can be achieved by LMS-RLS algorithms for spacing
of 0.5λ. Also LMS-RLS algorithm produces deepest null as compared to LMS.
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REFERENCES
[1] Prathviraj Khande et al (2021). An overview of Algorithms for Adaptive Beam Forming
(ABF) for Smart Antenna System (SAS). South Asian Res J Eng Tech, 3(1): 23-30
[2] M. Abualhayja'a and M. Hussein, "Comparative Study of Adaptive Beamforming
Algorithms for Smart antenna Applications," 2020 International Conference on
Communications, Signal Processing, and their Applications (ICCSPA), 2021, pp. 1-5.
[3] D. N. Patel, B. J. Makwana and P. B. Parmar, "Comparative analysis of adaptive
beamforming algorithm LMS, SMI and RLS for ULA smart antenna," 2016
International Conference on Communication and Signal Processing (ICCSP), 2016, pp.
1029-1033.
[4] Ayodele S. Oluwole and Viranjay M. Srivastava, " Analysis and Synthetic Model of
Adaptive Beamforming for Smart Antenna Systems in Wireless Communication," Journal
of Communications, vol. 13, no. 8, pp. 436-442, 2018
[5] Ashwini D Pandhare and Khyati Zalawadia. Performance Analysis of LMS Adaptive
Beamforming Algorithm for Smart Antenna System. International Journal of Computer
Applications 179(28):34-37, March 2018.
[6] C. Guo et al., "Design and Implementation of a Full-Digital Beamforming Array With
Nonreciprocal Tx/Rx Beam Patterns," in IEEE Antennas and Wireless Propagation Letters,
vol. 19, no. 11, pp. 1978-1982, Nov. 2020
[7] Z. D. Zaharis, I. P. Gravas, P. I. Lazaridis, T. V. Yioultsis, C. S. Antonopoulos and T. D.
Xenos, "An Effective Modification of Conventional Beamforming Methods Suitable for
Realistic Linear Antenna Arrays," in IEEE Transactions on Antennas and Propagation, vol.
68, no. 7, pp. 5269-5279, July 2020,.
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[8] “Evaluation of Beam Forming Capability of Linear Antenna Array for Smart Antenna
System “ , Sarmistha Satrusallya and Mihir Narayan Mohanty , Research Article ,
Department of Electronics and Communication Engineering, Institute of Technical
Education and Research, Siksha O’ Anusandhan (Deemed to be University), Bhubaneswar,
Odisha, India.
[9] A. Udawat, P. C. Sharma and S. Katiyal, "Performance analysis and comparison of adaptive
beam forming algorithms for Smart Antenna Systems," International Conference on Next
Generation Networks, 2019, pp. 1-5,.
[10] Senapati, Anupama, and Jibendu Sekhar Roy. "Performances of some combined
algorithms for adaptive beamforming in smart antenna using linear array." Asian Journal of
AppliedSciences 4.3(2016).
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PUBLICATION STATUS
The literature that is to be submitted to a conference is to be written. The before hand works for
the same is in process and the writing of the literature will be started soon. Need to explore a
bit deeper to get some new results by manipulation of the considered parameters and to finalize
theconference in which the paper is to be published and the work for the same is in process.
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