Republic of Iraq
Ministry of Higher Education & Scientific Research
Mustansiriayah University
College of Engineering
Electrical Engineering Department
Different Algorithms for Implementation of
Adaptive Beam Forming in Wireless System
A Thesis
Submitted to the Electrical Engineering Department,
College of Engineering, AL- Mustansiriayah University, in a Partial
Fulfillment of the Requirements for the Degree of Master of Science in
Electrical Engineering / Electronic and Communications
By:
Aseel Abdul-Karim Qasim
(B.Sc. Electrical Engineering)
Supervised By:
Prof. Dr. Adheed Hassan Sallomi
Rabi' al-Awwal 1442
November 2020
‫سورة المجادلة‪ :‬االية "‪"11‬‬
Acknowledgement
All praises and thanks to Allah for giving me the strength and wellbeing to complete this project.
I would like to thank sincerely my supervisor Prof. Dr. Adheed Hassan
Sallomi for his guidance, advice and encouragement throughout the making
of this project.
A special thanks to my family. Words cannot express how grateful I
am to my mother and father for all of the sacrifices that they’ve made on my
behalf. Your prayer for me was what sustained me thus far.
I would also like to thank my teachers, especially Dr. Raad Hamdan
Thaher for their encouragement and advice, and I would like to thank Dr.
Bashar Mundher Mansoor for his support throughout this work.
Lastly. I would like to thank all my friends who supported me and
helped me through my studies, because they were a one hand and they really
care about my success. Thank you very much to everyone who helped me.
Dedication
I dedicate this work to my Mother for her unending love….
Aseel Abdul-Karim Qasim
Table of Contents
Table of Contents
I
List of Abbreviations
V
List of Symbols
VIII
Abstract
X
Chapter one
General Introduction
1.1 Introduction
1
1.1.1 Smart Antenna System Structure and Principles
1
1.2 Literature Survey
2
1.3Aim of Work
7
1.4 Contribution
7
1.5 Thesis Outline
8
Chapter Two
Theoretical Background for Adaptive
Antenna Array System
2.1 Introduction
9
2.2 Antenna Fundamental
9
2.2.1 Radiation Pattern
9
2.2.2 Beamwidth
10
2.2.3 Directivity
10
2.2.4 Gain
11
2.3 Isotropic Antenna
11
2.4 Arrays Antennas
11
2.4.1 Introduction
11
2.5 Linear Arrays
12
2.5.1 Array Element
12
2.5.2 Two Element Array
12
2.5.3 N-Element Uniform-Linear Array
13
2.5.4 N-Element Non-Uniform-Linear Array
14
2.5.5 Another Array Geometry
15
I
2.5.5.1 Circular array
15
2.5.5.2 planer array
16
2.6 Smart Antenna
16
2.6.1 Introduction
16
2.6.2 Type of Smart Antenna
17
2.7 Array Weighting
18
2.8 Angle-of-arrival (AOA) estimation
19
2.9 Beam-forming technique
19
2.9.1 Fixed weight beam forming
20
2.9.2 Adaptive Beam Forming
20
2.10 General control algorithm for adaptive antenna array
21
2.10.1 Least Mean Square (LMS) Algorithm
21
2.10.2 Sample Matrix Inversion (SMI) Algorithm
22
2.10.3 Recursive Least Square (RLS) Algorithm
24
2.10.4 Conjugate Gradient Method (CGM) Algorithm
25
2.10.5 Particles Swarm Optimization Algorithm
25
2.11 Benefits and Features of Smart Antennas
26
2.12 Drawbacks of Smart Antenna
27
2.13 Description and Modeling of Wireless Channel
28
2.13.1 Path loss
29
2.13.2 Shadowing
29
2.13.3 Multipath Fading
29
2.13.4 Rayleigh Model
29
2.13.5 Rice model
29
2.13.6 Power Spectral Density
30
2.13.7 Power Delay Profile (PDP)
30
2.13.8 Mobile Station Mobility Model
31
Chapter Three
Proposal algorithm analysis
3.1 Introduction
32
II
3.2 The implementation of normalizing LMS algorithm with
maximum SIR factor
33
3.3 Variable step size via error controlling algorithm
36
3.4 Improvements for CMA (ICMA&BCM) algorithm
Performance
39
3.4.1 Conventional constant modulus algorithm CMA
39
3.4.2 Improvement proposed to the constant modulus algorithm
ICMA via variable step size vector
40
3.4.3 Proposed blocking constant modulus algorithm BCMA 41
3.5 Particles Swarm Optimization Algorithm (PSO)
42
3.6 Adaptive antenna array performance via REDS algorithm44
3.7 BER
Chapter Four
47
Simulation Results
4.1 Introduction
50
4.2 Characteristic studies of uniform linear arrays
50
4.2.1 Array factor interpretations with variation of array element
displacement
50
4.2.2 Array factor interpretations with variation of number of
array elements
52
4.2.3 Performance of comparison for smart antenna system
using different antenna array algorithms
54
4.2.4 Comparison of Performance for smart antenna system
using different antenna array geometry
55
4.3 Performance of smart antenna system via variable Step size
aspect
57
4.4 Performance of smart antenna system via SIR-LMS
algorithm
59
4.5 Comparison of smart antenna system executions using
different improvements for CMA (ICMA&BCM) algorithm 60
III
4.6
Smart
antenna
performance
with
particle
swarm
optimization (PSO) algorithm
64
4.6.1 Weight Vector Optimization
64
4.6.2 Inter Element Spacing Optimization
66
4.6.3 Weight Vector and Inter Element Spacing Optimizatio 68
4.7 Nulling Effect on the Radiation Pattern
69
4.8 Smart antenna array system performance via REDS
algorithm
71
4.8.1 (AWGN) Channel Results
72
4.8.2 Rayleigh Fading Channel Results
75
4.9 BER
79
4.10 SNR
81
Chapter Five
Conclusions and Future Works
5.1 Conclusion
83
5.2 Future work
84
Refrences
86
IV
List of Abbreviation
ABF
Adaptive beam forming
AF
Array factor
AOA
Angle of arrival
AWGN
Additive white Gaussian noise
BCMA
Blocking constant modulus algorithm
BER
Bit error rate
bps
Bit per second
BPSK
Binary phase-shift keying
BS
base station
BW
Band width
CA
Circular array
CCM
Constrained constant modulus
CDMA
Code division multiple access
CGM
Conjugate gradient method algorithm
CMA
Constant modulus algorithm
CSLMS
Constrained-stability LMS
dB
Decibel
DBF
Digital beam-formed
DF
Direction finding techniques
DMI
Direct matrix inversion
DOA
Direction of arrival
DS-CDMA
Direct sequence code division multiple access
ECVSS
Error control variable step size LMS algorithm
EDS
Euclidean direction search
EF
Element factor
FBGA
Field programmable gate array
FEDS
Fast Euclidean direction search
V
FF
Fitness function
FM
Frequency modulation
FNBW
First null beam-width
FSK
Frequency-shift keying
GA
Genetic algorithm
HPBW
Half power beam width
ICMA
Improvement constant modulus algorithm
IPSO
Improvement Particle swarm optimization
ISI
Inter-symbol-interference
Km/h
Kilometer per hour
LA
Linear array
LMS
Least mean square algorithm
Mbits
Mega bits
MHz
Mega hertz
MIMO
Multiple-input and multiple-output
MRVSS
Modified robust variable step size
MS
Mobile Station
MSE
Mean square error
MUSIC
Multiple signal classification
MVDR
Minimum variance distortion-less response
NLMS
Normalized least mean square algorithm
NULA
Non uniform linear array
PA
Planner array
PDP
Power delay profile
PSO
Particle swarm optimization
REDS
Rabid Euclidean direction search
RLS
Recursive least square algorithm
SDMA
Space division multiple access
VI
SINR
Signal to interference plus noise ratio
SIR
Signal to interference ratio
SLL
Side lobe level
SMI
Sample matrix inversion
SMI-LMS
Simple matrix inversion - least mean square algorithm
SNOIs
Signals not of interest
SNR
Signal to noise ratio
SOI
Signal of interest
SSF
Step size factor
ULA
Uniform linear array
VSSLMS
Variable step size LMS
Rnn
Noise correlation matrix
Rii
Interference correlation matrix
Rss
Signal correlation matrix
VII
List of Symbols
θs
Angle of arrival of the desired signal
θi
Angle of arrival of the interfering signal
G
Array element gain
̅w
̅̅
Array weight vector
φ
Azimuth angle
L
Block length of data
fc
Carrier frequency
(*)
Conjugate operator
α
Constant value less than one for RLS
̅
d
Desired signal
fd
Doppler frequency
d
Elements spacing of the uniform linear array
e
Error signal
E|•|
Expectation operator
∇w
Gradient with respect to w.
(•)
̅
n
H
Hermitian of matrix
Input noise vector
x̅
Input signal vector, [x1 x2 … . xM ]
i, i1 , i2
Interfering signals
Fj0(k)
M x 1 vector, where M is element array
λmax
maximum eigenvalue of the autocorrelation matrix of the
input vector
̅
R
nn
N x K zero mean Gaussian noise
M
Number of array elements
m
count for number of antenna elements array from 1-M
K
Number of sample
Wopt
Optimum weights vector of an algorithm
VIII
y
Output signal of an algorithm
p
p parameter
C
Speed of light
‖ •‖ 2
Squared Euclidean norm operator
̅ xx
R
The array correlation matrix
r̅
The correlation vector
r̂
The estimate of correlation vector
̂ xx
R
The estimate of the array correlation matrix
S(k)
The function of desired signal
j0 (k)
The index of the weight
̅
A
The K x M matrix of array snapshots
nth
The 𝑘𝑡ℎ element of n
µ
Step size parameter
δ
The phase shift from element to element
a̅(θ)
The steering vector
σ2
The variance of the noise signal
t
Time
τ
Time delay
n
Time index
trace[R] Trace of autocorrelation matrix [R]
I̅
Unity matrix
µ(n)
Variance step size
𝜆
Wavelength of the carrier signal
vid
Velocity of swarm
xid
Position of swarm
(. )T
Transposition.
IX
Abstract
There is a certain variance when comparing different sorts of
algorithms like Least Mean Square (LMS) which is largely used because of
the low complexity of computational and ease of execution. The least-squares
algorithms or (algorithm that content step size factor), such as Recursive
Least Squares (RLS), Conjugate Gradient (CG), and Rapid Euclidean
Direction Search (REDS), would converge more rapidly and have smaller
mean square error (MSE), and another algorithm will be introduced also like
blind algorithm such as Constant Modulus Algorithm (CMA) and blocking
algorithm such as Sample Matrix Inversion (SMI), also presents an
optimization algorithm like the Particle Swarm Optimization algorithm
(PSO). This thesis also discussed different tests for different array geometry.
Initially, this thesis studies the effects of changing the number of
elements, and the spacing distance between elements, and study the effect of
changing the array geometry on smart system performance. The first
improvement in this thesis will apply on the least square algorithm, by
changing the step size factor (LMSSI) that enhance the LMS convergence
from more than 80 to about 20 iterations also provided reducing in the MSE.
The second development would be the SIR-LMS that gives zero iteration
number. The CMA algorithm reinforcement investigates into two-manner,
firstly by apply the variable step size algorithm (ICMA) and secondly by
blocking the CMA algorithm (BCMA) the result illustrates that the ICMA
improve the number of iteration from 50 for traditional CMA to 23 and 6
iterations for BCMA with keeping the directional beam pattern to desired
signals and perfectly nulling in the inference directions.
PSO is discussed in three-part, firstly to optimization, the weight
vector. Then the second part will be the inter-element spacing optimization,
most important achievements acquired an economic and effeteness system
using the same or less number of elements besides reducing the array size.
The improvement PSO (IPSO) that combine the weight and spacing
optimization, and third part studied the nulling effect on the radiation pattern
with different number of elements and spacing.
REDS algorithm utilizing on smart antenna with three array geometry
linear, circular and planner gave more stable and acceptable performance, it
can be seen that the number of iteration reduce to 5 for circular, 7 for planner
and 8 iteration for linear array as compared with other algorithm, also gave
improvement in MSE as appear in chapter four early, the effectiveness of
REDS algorithm also appear more robust and remain stable when apply
X
another type of noise like Rayleigh fading noise in single beam and multi
beam signals.
The BER and the SNR will be studying for linear and circular array
and shows that the BER is approximately close for both arrays at the same
number of elements, and it is clear to observe that the SNR curve of overall
system increase with the number of elements used in the array regardless the
geometry of antenna array. The subsequent simulations on arrays are
performed using the MATLAB program version R2017a.
XI
Chapter One
General
Introduction
Chapter One
General Introduction
1.1 Introduction
The expansion of adaptive arrays initiates in more than 60s years. In
the 1940s, the concept of an antenna array was first used in military
applications and significantly evolved in wireless communication. In 1959,
Van Atta first coined the term adaptive array to describe a self-phased array.
Production of arrays plays a significant role in many different fields, most
modern radar and sonar systems depend on arrays antenna as an essential part
of the system [1].
The rising demand for mobile communications is gradually growing,
reaching about one billion mobile phone users globally. Mobile phones have,
of course, been one of the most essential components of ordinary life, and a
business-critical device across all developed nations. Therefore, further
effective use of the radio spectrum is demanded and the need for wider
coverage, improved efficiency, and higher quality of transmission increases.
Smart antenna array contains several distributed elements of an
antenna (dipoles, monopoles, or elements of the directional antenna) arranged
in several geometries [3]. Selected control algorithms, with predefined
parameters, give adaptive arrays the unique ability to adjust the
characteristics of the radiation pattern (nulls, sidelobe level, main beam
direction, and beam-width) [4], that relies on the form of an element, the
specific positions and the excitation at each element (amplitude and phase).
The term smart antenna encompasses all circumstances in which a system
uses an antenna array and the antenna pattern is dynamically changed as
required by the system, The beam pattern is therefore modified as desired and
the interference signals pass [5], there are two basic forms of smart antennas,
the first kind is a phased array or multi-beam antenna composed of either a
collection of fixed beams with one beam combining the signal output per
each element of the array with the required a single beam (established vie
phase-only modification) guided to the desired signal by correctly altering
the phase between the elements. Another model is the adaptive antenna array
with weighted and combining received signals to enhance the desired to
interference signal power ratio (SINR) [6].
1.1.1 Smart Antenna System Structure and Principles
The structure diagram of the adaptive array system is shown in Figure
(1.1).
1
Chapter One
General Introduction
Figure (1.1): smart antenna structure.
It consists of the following principal elements:
A. Array sensors and RF unit: This unit consists of antenna arrays that
receive radio frequency (RF) signals from the space, down conversion
chains that detach the carrier of the RF signals obtained by the antenna
array, and analog to digital converters (A / D) that transform the carrier
signals for more processing.
B. Adaptive antenna processor: The signal processing unit, depending
on the signal received, which subdivided into the signal processing
unit and adaptive algorithm. The signal processing unit calculates
weight vectors by which the received signal is multiplied of each array
element, and this weight vectors calculation can be implemented using
various adaptive algorithms and techniques.
C. Beamforming unit: It is a process that blends the radiation patterns of
each antenna element on the antenna arrays to form a directional and
focused energy beam [4]. The beamforming unit is responsible for
shaping and directing the main beam in the proper direction whereas
generating nulls in the direction of interference signals leading to an
increase of SINR [7]. In just the same unit (Digital Signal Processor
(DSP)) the beamforming and signal processing units can commonly
be integrated.
In general, to achieve an adaptive beamforming a necessary sequence
of steps must be carried out which is each element of the adaptive array has
to be independently weighted the received signal, the weight vector for each
of these signals has to be continuously controlled, all signals are then
combined to form the output and the decision about the amount of adaptation
in the beam unit must be made, based on the output level and then a control
signal is to be sent back to the weight control unit to modify the weight vector
[8].
1.2 Literature Survey
The literature and the study of adaptive array geometry became the
peak topic in last years, which, based on the development of smart antenna
array. Here are some of the important researches reviewed as follows:
2
Chapter One
General Introduction
(Ioannides and Constantine A. Balanis, 2005)[14] examined adaptive
beamforming with three different rectangular, circular and concentrated
geometries consisting of seven arrays each with different radii and uniformly
distrusted elements, and compared the results utilizing LMS and RLS
algorithms, it was found that the circular array provides the strongest
beamforming capabilities with directivity equal to 14.82 dB with LMS and
14.76 dB with RLS while, the concentrated circular show the deepest nulls
against interference.
(Mohammad and Zainol, 2006)[15] this paper introduced the “MINLMS”,
the MI-NLMS combines the SMI and NLMS algorithms. Simulation results
showed that the MI-NLMS algorithm provides remarkable improvements in
terms of interference suppression, convergence rate and BER performance
over that of LMS algorithms. With respect to LMS, MI-NLMS provides 15
dB improvements in interference suppression, 5 dB gain enhancement, and
the reduction of the BER for MI-NLMS is 76% compared to LMS algorithm
in the case of 4-element antenna array.
( K. R. Mahmoud, M. El-Adawy, S. M. M. Ibrahem, R. Bansal, K. R.
Mahmoud, and S. H. Zainud-Deen, 2007)[16] search many configurations
using 18 half-wave dipole elements in free space. The PSO algorithm has
been used to optimize the complex excitations, amplitudes, and phases of the
beamforming adaptive array components. This took the effects of mutual
coupling between the elements totally into consideration. The distinction
between circular and hexagonal array showed that hexagonal array
geometries had the same beam width as circular array geometries, giving
slightly deeper nulls which about -100 dB, while about -60 dB for circular
array.
(Atrouz, Alimohad, and Aissa, 2009)[17] established an adaptive
beamforming algorithm applied a rectangular array to a two-dimensional
elevation and azimuth angles. The algorithm developed was based on an
algorithm from LMS, produce a new algorithm called the Sequential Block
LMS algorithm (SBLMS), the simulation showed that this algorithm's
efficiency to accomplish the Jammers suppress tasks was very good.
( C. S. Rani, P. V. Subbaiah, K. C. Reddy, and S. S. Rani, 2009)[18] various
adaptive beamforming algorithms such as LMS and RLS algorithms used in
smart antennas were addressed. And showing the LMS algorithm
convergence rate depends on the array correlation matrix's Eigenvalues.
(Zuniga, Erdogan, and Arslan, 2010)[19] assessed the efficacy of PSO in
solving adaptive antenna array with rectangular antenna configuration and
compared the results with conventional genetic algorithm (GA). The
observation that the PSO obtained an average lower side lobe level
(SLL) equal to -13 dB suitable to prevent interference, while -8 dB for Ga.
3
Chapter One
General Introduction
Additionally, the mean minimum power at all the nulls for the PSO algorithm
is −35.7 dB whereas the power for the GA is −23.4 dB. These figures suggest
that the PSO algorithm tends to perform better than the GA algorithm in terms
of radiation power directed towards undesired signals.
{Formatting Citation}[20] the performance of various circular array
configurations for adaptive antennas was compared. Each design is made up
of 19 elements, and the distance between the elements is 0.65. Such
configurations include uniform circular arrays, uniform circular centered
arrays, planar uniform circular arrays, and planar uniform hexagonal array.
The PSO algorithm has been used here to determine the complex weights of
the elements to modify the antenna according to the condition.
(Amritpal Singh, Bhinder and Arvind Kumar.,2012) [21] presented planar
arrays of rectangular micro-strip patches with a spacing of 0.5 between the
elements. The outcome of the simulation showed that the main beam of the
bigger array elements can get the signal of interest more reliably and rejects
signal noise of interest but has a drawback that it increases the cost and
complexity of implementation and therefore increases the convergence rate
for the adaptive algorithm herewith minimize the bandwidth.
(Maina, Langat and Kihato, 2013)[22] developed adaptive beamforming
using for the null steering approach in a uniform rectangular array system.
PSO algorithm and Simulated Annealing (SA) algorithm had to implement
the beamforming procedure, to deliver optimal radiation in desired directions
and minimum radiation in undesired directions, the weights of beamformer
were synthesized.
(Seyed Abolfazl Hosseini, 2013)[23] described a new FEDS algorithm in
noise cancellation for speech enhancement, interference nulling, and system
identification purposes. The authors showed that the FEDS converges to the
true parameters and that its convergence rate is comparable to that of the RLS
but, at a much lower computational cost.
(Rao and Sarma, 2014)[24] considered three different geometries for the
adaptive antenna. Evaluate also the convergence rate for the LMS algorithm,
which would rely on the correlation matrix's Eigenvalues. It converges slowly
in a dynamic channel setting with major Eigenvalues distributed. RLS
Algorithm solves the problem, also a constant mudulos algorithm (CMA)
algorithm is used when the reference signal is not available.
(Venkatesan, Arunnehru and Umamaheswari, 2014)[25] the author used
LMS which is easy to implement with low computation and RLS algorithm
which usually converges faster than the LMS algorithm with less than 45
iterations but the price paid is added complexity. And show the rate of CGM
was faster than that of the conventional algorithm with less than 20 iterations.
4
Chapter One
General Introduction
Reliance on these algorithms on the SNR algorithm and the iteration numbers
required to obtain the desired signal, which minimizes the error faster than
most adaptive algorithms were tested.
(Vesa, Alexa, and Baltə, 2015)[26] given a brief overview of antenna array
aspects. States that the structures are more clearly in the case of the 2D planar
array and 3D array. Even, if the phase shift is applied between the current
inserted in the antenna elements the linear array becomes unworkable in
terms of the pattern of directivity. The author determines throughout the tests
that increasing the number of elements over 7 may not contribute to a
significant increase in directivity.
(Patel, Makwana and Parmar, 2016)[27] presents adaptive algorithms LMS,
SMI, and RLS model and evaluate the radiation pattern. Which demonstrates
the LMS convergence rate function as one of the algorithm's weaknesses as
it is directly dependent on the step size amount, and also provided the SMI
algorithm that overcomes the LMS limits but raises the complex correlation
matrix calculation. Finally learning the RLS algorithm where the array
weight vectors are updated very fast since the convergence variance is
defined by the information of the Eigenvalue of the signal correlation matrix.
(Banerjee and Dwivedi, 2016)[28] PSO algorithm was implemented for a
uniform linear array of 16 elements, output analyzes are performed using the
specific variable tests. To confirm this method, the mean SLL, null depth, and
first null beamwidth (FNBW), it has been shown that the PSO based former
beam provides better SLL with direct main beam and null placement.
{Formatting Citation}[29] established a PSO algorithm for evaluating and
optimizing the element positioning and excitation amplitude, which included
HPBW, SLL, directivity, and null steering in some spatial locations for the
optimization goals output of this thesis. Those goals would be achieved with
two fitness functions applying in the PSO algorithm.
(S. C. Swati Patidar, Kishor Kumbhare,2017)[30] the genetic algorithm (GA)
has been used in smart antenna to locate the beam pattern with optimum
signal gain for the given direction. And by using this method the SLL was
restricted and the direction of the angle array switched.
(P. N. Chuku, Olwal, and Djouani, 2018)[31] examined smart antenna
system based on RLS. The RLS algorithm's gain factor has been improved to
boost its efficiency in terms of lower MSE resulting in a higher convergence
rate. The improved RLS algorithm also holds weights that are identical to the
calculated weights as compared to the standard RLS algorithm and the LMS
algorithm.
(Zhang et al., 2019)[32] this paper focuses on maximizing the performance
of an array in transmitting and receiving data on how to optimize the positions
5
Chapter One
General Introduction
of antenna array elements using a contraction adaptive particle swarm
optimization (CAPSO) algorithm. Two functions analyze the convergence
rate and compare it with three other methods. The CAPSO shows reasonable
performance for the various array element numbers.
(Dakulagi and Alagirisamy, 2019)[33] describes the use of an innovative
two-dimensional (2D) uniform linear array for adaptive algorithms. This new
antenna design has been used to research the common beamforming
algorithms, i.e. the least mean square algorithm (2D-LMS) and the least
normalized mean square algorithm (2D-NLMS) and also suggested a stepsize variable NLMS (2D-VSSNLMS) algorithm. The suggested 2DVSSNLMS algorithm involves fewer beamforming iterations and high
interference resistance. That allowed MIMO-WLAN, WI-MAX, 4 G LTE,
low-power, low-cost anti-interference systems, and other advanced
communication systems.
(Dakulagi and Bakhar, 2020)[34] presented and comprehension many kinds
of beamforming and DOA schemes for wireless communications. This work
mostly indicates that the RLS algorithm utilizes the array correlation matrix
Eigen spread to measure the weights, as the result of which accuracy and
convergence rate of this scheme is poor and influenced by the Eigen spread
value. The conjugate (orthogonal) for every new iteration for conjugate
gradient method (CGM) computes the antenna array, by comparing it with
LMS, SMI, and RLS schemes, the convergence rate of this approach is
enhanced. Smart antenna using CGM also given improved system capability
and can minimize the interference effect by creating very small beams in the
looking position, as well as produced the newest LMS algorithm that shows
a better convergence rate compared to any LMS algorithm version.
( M. U. Shahid, A. Rehman, M. Mukhtar, and M. Nauman, 2020)[35] a
methodical evaluation of the performance of fixed beamforming algorithms
for smart antennas such as Multiple Sidelobe Canceller (MSC), Maximum
Signal-to interference ratio (MSIR) and minimum variance (MVDR) has
been comprehensively presented in this paper. Simulation results show that
beamforming is helpful in providing optimized response towards desired
directions. And show that the MVDR beamformer provides the most optimal
solution and more general application beamformer.
6
Chapter One
General Introduction
1.3 Aims of Work
The aims of this thesis could be summarized as the following points:
o Test the performance of smart antenna system by applying nine
different algorithms (least mean square LMS, normalized least mean
square NLMS, recursive least mean square RLS, constant modules
algorithm CMA, simple matrix inversion SMI, conjugate gradient
method CGM, particle swarm optimization PSO, rabid direct
Euclidian search REDS), using MATLAB program then looking for
suitable one that leads to optimum weight to apply maximum power
in the desired user direction and eliminates interference users by
steering nulls in that direction.
o Analyzing and choosing the most suitable array antenna configuration
by made the comparison between three-antenna array configuration
(linear array (1D), circular and planner array (2D) and cubic array
(3D)) with most proper algorithm to achieve the goal of the work to
track the sources of signal in two planes (azimuth and elevation)
simultaneously.
o Comparing the efficiency of the smart antenna system with traditional
antennas by observing the effect on the beam pattern of varying
element number and displacement spacing, and showing the
relationship between changes on the beamwidth.
1.4 Contribution
a. Applied nine algorithms on the antenna array system and comparing the
performance result on antenna array between them, and calculate some of
the important antenna characteristic like the gain, sidelobe level,
beamwidth and finally bit error rate and signal to noise ratio.
b. Proposed a new adaptive system design by using a particle swarm
algorithm by producing a nonuniform array.
c. Produce an enhancement and improvement on the conventional algorithm
like least square algorithms or algorithms that has a step size factor by two
methods which are variable and hybrid aspects. Also, improve the
constant modules algorithm by two different aspects which are the
variable and blocking method.
d. Introduce a comparison between different array geometry performances
on a rapid euclidian direction search algorithm, and apply two types of
noise on it which is an AWGN and Rayleigh fading.
7
Chapter One
General Introduction
1.5 Thesis Outline
This thesis is organized into five chapters, established as follows:
Chapter One: Presented a general presentation to the growth of the adaptive
antenna array system and how the smart antenna could address some
limitations factor for mobile communication systems and afforded a brief
literature survey for relevant work in this field.
Chapter Two: Provides theoretical history introduction to the smart antenna,
beamforming adaptive array techniques, and structure for various geometries
that would be used with the proposed algorithm, also produce a system
measuring parameters.
Chapter Three: where describes the implementation for proposed
algorithms are derived and presented their mathematical modal.
Chapter Four: provide the simulation results of the system module
described in chapter three, which compares the quality of different adaptive
algorithms and array geometry concerning convergence errors and resultant
beam patterns using the MATLAB program.
Chapter Five: require conclusions and recommendations for further work.
8
Chapter Two
Theoretical
Background for
Adaptive
Antenna Array
System
Chapter Two
Theoretical Background for Adaptive Antenna System
2.1 Introduction
An antenna array is a system consisting of a series of several antennas
that are distributed by spacing distance to a specific given point, with phase
and amplitude adjustments of the exciting currents in each of the antenna
elements, the main beam and position nulls in either location could be
scanned electronically [39]. An adaptive array managed by sophisticated
signal processing[40], using a combination of signal processing algorithms,
allows the adaptive system to efficiently locate and monitor various types of
signals to dynamically reduce interference and optimize the intended
reception, the different algorithms vary in the determination of certain
weights. To get into an antenna and its performance and characteristics some
parameters need to be summarized. The section below provides a brief
introduction of an array feature.
2.2 Antenna Fundamental
To get into an antenna and its performance, characteristics, and some
parameters need to be defined. Parameter definitions will be given most of
them based on internationally well-known Standards like the IEEE Standard
Definitions of Terms for Antennas (IEEE Std 145-1993), and the IEEE Std
145-1983.
2.2.1 Radiation Pattern
The pattern might be dependent on the feature representing the
electrical or magnetic fields. Thus, when the amplitude or relative amplitude
of a specific portion of the electrical field vector is graphically plotted, an
amplitude pattern, field pattern, or voltage pattern is identified. When the
amplitude square or relative amplitudes are plotted a typical two-dimensional
power pattern is labeled, the field pattern plot for rectangular and polar
coordinates can be shown in Figures 2.1.
9
Chapter Two
Theoretical Background
(a)
(b)
Figure 2.1 (a) Rectangular (b) Polar, Coordinates for Field Patterns [37]
2.2.2 Beamwidth:
Which is determined from a radiation pattern's -3dB points that
represent the position of the highest radiation-intensity lobe, the normalized
pattern is -3dB points that equal to √1/2= 0.707 from the maximum point of
the pattern.
Figure 2.2 Main lobe of an antenna identifying the HPBW [37].
2.2.3 Directivity
The directivity represents the reflection of the capability of the antenna
to systemically locate the energy in some direction and the ratio of radiation
intensity in a given direction from the antenna to the average radiation
intensity in all directions. The average radiation intensity is equal to the total
power radiated by the antenna divided by 4π [38]
𝐷=
4𝜋𝑈𝑚𝑎𝑥
(2.1)
𝑃𝑟𝑎𝑑
2𝜋
𝑃𝑟𝑎𝑑 = ∫0
𝜋
∫0 𝑈(𝜃, 𝜑) sin(𝜃) 𝑑𝜃 𝑑𝜑
(2.2)
Where 𝑃𝑟𝑎𝑑 , is the total radiation power and 𝑈𝑚𝑎𝑥 represents the
ultimate radiation intensity.
10
Chapter Two
Theoretical Background
2.2.4 Gain
Antenna's gain is an adjustment of the directivity which is the potential
of the antenna to directing the energy in desired directions, in many other
phrases is the ratio of the radiation intensity in a given direction to a radiation
intensity that would have been obtained if the power received by the antenna
were isotropically radiated, the antenna gain can express as[38]
𝐺(𝜃, 𝜑) = 𝑒𝐷(𝜃, 𝜑)
(2.3)
Where e presents the complete efficiency of antenna plus loss and
mismatch impact. When an antenna has no dissipative loss then its gain is
equal to its directivity under any specific direction.
2.3 Isotropic Antenna
An isotropic antenna is a theoretical ideal antenna and a hypothetical
lossless antenna may be described as an isotropic radiator, which radiates in
all directions equally-horizontally and vertically by the same strength.
Figure2.3 Isotropic Radiation Pattern
The antenna has quite a gain of 0 dB in the sphere around it and a 100
% efficiency. The definition of an isotropic is ideal and yet physically
realizable, it is also used as a guideline for presenting the actual antennas'
directive properties [38].
2.4 Arrays Antennas
2.4.1 Introduction
The architecture of the array consists of two parts the first is the
geometry of the array. Antenna arrays can be one, two, and threedimensional, the array geometry defines the physical locations of the antenna
11
Chapter Two
Theoretical Background
elements. The second aspect is array pattern in general, all elements of the
array are considered to provide isotropic patterns, combined element patterns
form the total array pattern which depends on the antenna array geometry and
exciting current and radiation pattern thus acquired the array factor [39].
2.5 Linear Arrays:
The linear array is the most simple array geometry. In which the
elements are aligned on a straight line and have a uniform or non-uniform
inter-element distance space.
2.5.1 Array Element
A single radiating element in a linear array or a convenient grouping
of radiation elements that have fixed relative excitations, this is a single
radiating antenna.
2.5.2 Two Element Array
The array with two-element is the main basic and easiest array to
evaluate. Figure 2.4 illustrates the array consisting of 2 identical polarized
vertically aligned placed arrays in the direction of the y-axis with keeping a
specific displacement (d) between them.
Figure 2.4 Two Infinitesimal Elements [37]
The complete radiation field by with two elements , which is the sum
of the two, and would be offered within an x-y plane by[37]
𝐸𝑡 = 𝐸1 + 𝐸2
(2.4)
𝐸𝑡 =
𝑗𝑘𝜂𝐼0 𝑒
−𝑗
𝛿
2 𝐿 𝑠𝑖𝑛𝜃
4𝜋𝑟1
𝑒 −𝑗𝜅𝑟1 +
𝑗𝑘𝜂𝐼0 𝑒
+𝑗
𝛿
2 𝐿 𝑠𝑖𝑛𝜃
4𝜋𝑟2
𝑒 +𝑗𝜅𝑟2
(2.5)
The field point is placed in r distance away from the reference with r
>>d therefore assuming the vectors of the distance 𝑟1 , r, and 𝑟2 are parallel
with each other. So the approximations could write as follow:
𝑑
𝑟1 ≈ 𝑟 − 𝑠𝑖𝑛𝜃
(2.6)
2
𝑑
𝑟2 ≈ 𝑟 + 𝑠𝑖𝑛𝜃
(2.7)
2
12
Chapter Two
=
𝑗𝑘𝜂𝐼0 𝐿 𝑠𝑖𝑛𝜃
4𝜋𝑟
𝑒
−𝑗𝜅𝑟1
Theoretical Background
[𝑒
−𝑗
(𝜅𝑑 𝑠𝑖𝑛𝜃+𝛿)
2
+𝑒
+𝑗
(𝜅𝑑 𝑠𝑖𝑛𝜃+𝛿)
2
]
(2.8)
Where δ the electrical phase variation between the two adjacent
elements (in radians), L is the dipole length,𝜃 is the angle at the z-axis in
spherical coordinate, d is the spacing between elements.
By further simplify equation (2.8) would get [37]
𝐸𝑡 =
𝑗𝑘𝜂𝐼0 𝐿 𝑒 −𝑗𝜅𝑟
4𝜋𝑟
𝑠𝑖𝑛
𝜅𝑑 𝑠𝑖𝑛𝜃+𝛿
(2 𝑐𝑜𝑠(
2
))
(2.9)
From (2.9) it is obvious that the total array field is equal to the field
product of a single element placed at the origin element factor (EF) by the
array factor (AF) in general, the AF is a function of the number of elements,
their geometrical arrangement, their relative magnitudes, their relative phases
and their spacing. A quite accurate representation of the radiation in the array
must also include the coupling impact between neighboring elements,
whatever this subject is out of the scope of this thesis, more information of
element coupling has been produced by Balanis [38].
2.5.3 N-Element Uniform Linear Array
The two-element array has been demonstrated so that the array of Nelements has also been implemented, which is the more general array. Figure
2.5 Indicates a linear N-element array assuming all elements are perfectly
aligned and all elements had the same amplitudes but that each corresponding
element has quite a progressive phase lead current excitation relative to the
previous one (δ represents the phase by which the current in each element
leads the current of the preceding element).
Figure 2.5 N-Element Linear Array [40]
The array factor is given in [40]. If the elements in the array were not
isotropic sources, the complete field could be created by multiply the array
factor of isotropic sources by the single element field
AF = [1 + ej(kdsin θ+δ) + ej2(kdsin θ+δ) … . ej(N−1)(kdsin θ+δ) ]T
13
(2.10)
Chapter Two
Theoretical Background
Where k is the phase constant, θ is arrival angle emitters.
Equation (2.10) can be more concisely expressed by
j(n−1)(kdsinθ+δ)
j(n−1)
AF = ∑N
= ∑N
ψ
n=1 e
n=1 e
(2.11)
Where ψ = kd sinθ + δ
From equation (2. 10), the array vector can be derived as [40]
1
a̅(θ) = [
j(kdsin θ+δ)
e
⋮
] = [1 ej(kdsin θ+δ) … . ej(N−1)(kdsin θ+δ) ]T (2.12)
ej(N−1)(kdsin θ+δ)
The array vector was alternatively can be called: array steering vector
array propagation vector, array response vector, or array manifold vector.
2.5.4 N-Element Non-Uniform Linear Array
The symmetrical radiation pattern can be generated by the
unsymmetrical placement of the antenna array. The symmetry situation,
therefore, decreases the complexity of the computation. The linear array
would consist of N equal radiating elements. The element is positioned
asymmetrically around the origin on the x-axis. The schematic of the array
structure presented in Figure 2.6, the N elements is assumed to be even
number. The elements are subjected to two divides of M elements where N =
2M. The elements numbering are given as −𝑀, −𝑀 + 1 · · ·
− 1, 1, 2, . . . 𝑀 − 1, 𝑀 .The spaces of elements 1, 2, . . . . 𝑀 from the
reference point are named as 𝑑1, 𝑑2, . . . 𝑑𝑀.
Figure 2.6 N-Element Non-Uniform Linear Array [37]
The symmetrical elements, −1, −2, . . . . −𝑀 have the same values of
spacing distance. The array factor is specified by
j(n−1)(k𝑑𝑛 sinθ+δ)
AF = 2 ∑M
(2.13)
n=1 e
From the schematic diagram, it is clear that the AF's amplitude and
phase can be controlled in uniform arrays by carefully choosing the relative
phase between the elements; in non-uniform arrays, the distance, amplitude,
14
Chapter Two
Theoretical Background
and phase could be utilized to monitor the composition and distribution of
the total array element.
2.5.5 Another Array Geometry
Array antennas can be one, two, and three-dimensional shapes,
depending on the dimension of space one wants to access. Figure (2.7) shows
a different array of geometries that can be applied in adaptive antennas
applications.
a)Circular antenna
array in x-y
c)Cubic antenna array in
x-y-z plane
b)rectangular antenna
array in x-y plane
Figure 2.7 Different Array Antenna Geometries [24]
Figure (2.7a) shows two-dimensional circular array in the X-Y plane,
with uniform angular distribution between elements of a value ∅𝑛 =
2𝜋(𝑛−1)
𝑁
where N represents the number of elements and n= 1,2,…, N . Because of its
symmetry, this structure can produce beamforming in any direction and is
more suitable for beamforming in one and two-dimensions. A twodimensional rectangular array with horizontal element spacing of ∆X and
vertical element spacing of ∆Y is shown in Figure (2.7b). This type is used
to perform two-dimensional beamforming (i.e. in both azimuth and elevation
angles). Beamforming in an entire space, where there are all angles, requires
some sort of cubic, cylindrical, conical, or spherical structure (threedimensional configuration) as shown in Figure (2.7c). This figure shows a
3D cubic structure with elements distributed in X-Y-Z planes. It’s a type of
a conformal array antenna used for special applications [40].
2.5.5.1 Circular Array
The array factor or circular array in the x-y plane, with uniform angular
distribution between elements of value ∅𝑛 =
15
2𝜋(𝑛−1)
𝑁
. The nth array element
Chapter Two
Theoretical Background
is located at the radius (a) with the phase angle𝜑𝑛 . The AF can be found in
a similar procedure as was calculated with the LA as [9]
−𝑗(𝑘𝑎𝜌̂ .𝑟̂ +𝛿𝑛 )
−𝑗(𝑘𝑎𝑠𝑖𝑛𝜃cos (𝜑−𝜑𝑛 )+𝛿𝑛 )
𝐴𝐹 = ∑𝑁
= ∑𝑁
𝑛=1 𝑤𝑛 𝑒
𝑛=1 𝑤𝑛 𝑒
(2.14)
2.5.5.2 Planar Array
This type is used to perform 2D beamforming (in both azimuth and elevation
angles) with horizontal element spacing of ∆X and vertical element spacing
of ∆Y. The AF for PA can be expressed as combining the AF of two LAs
[10]. Pattern multiplication can be used to find the pattern of the entire M×N
element array. Using pattern multiplication would have
𝐴𝐹 = 𝐴𝐹𝑥 . 𝐴𝐹𝑦 =
𝑗(𝑛−1)(𝑘𝑑𝑦 𝑠𝑖𝑛𝜃sin𝜑+𝛽𝑦 )
𝑗(𝑚−1)(𝑘𝑑𝑥 𝑠𝑖𝑛𝜃cos𝜑+𝛽𝑥 ) ∑𝑁
∑𝑀
𝑚=1 𝑎𝑚 𝑒
𝑛=1 𝑏𝑛 𝑒
(2.15)
𝐴𝐹𝑥𝑦 =
𝑗[(𝑚−1)(𝑘𝑑𝑥 𝑠𝑖𝑛𝜃cos𝜑+𝛽𝑥 )+(𝑛−1)(𝑘𝑑𝑦 𝑠𝑖𝑛𝜃sin𝜑+𝛽𝑦 )]
𝑁
∑𝑀
𝑚=1 ∑𝑛=1 𝑤𝑚𝑛 𝑒
(2.16)
The AF for cube array can be expressed as combining the AF of three LAs,
Pattern multiplication can be used to find the pattern of the entire M×N×Z
elements array as follow
𝐴𝐹𝑥𝑦𝑧 =
𝑂
𝑗[(𝑚−1)(𝑘𝑑𝑥 𝑠𝑖𝑛𝜃cos𝜑+𝛽𝑥 )+(𝑛−1)(𝑘𝑑𝑦 𝑠𝑖𝑛𝜃sin𝜑+𝛽𝑦 )+(𝑜−1)(𝑘𝑑𝑧 𝑠𝑖𝑛𝜃+𝛽𝑧 )]
𝑁
∑𝑀
𝑚=1 ∑𝑛=1 ∑o=1 𝑤𝑚𝑛𝑜 𝑒
(2.16)
2.6 Smart Antenna
2.6.1 Introduction
A smart antenna pattern as mention earlier controlled by algorithms
according to certain requirements. These requirements may be maximizing
or enhance received the signal to interference ratio (SINR) and may also be
considered as forming beams for transmission, reducing the variance,
elimination for the mean square error (MSE), tracking the desired signals and
interfering signals extinctions. Smart antenna involves the digitizing of the
array outputs by using an A / D converter, such digitization would be
completed by either at IF frequencies or at baseband [40]
Smart antennas offer the potential for advanced radar systems,
enhanced mobile wireless network capabilities, and amended wireless
communication systems by mean of the enforcement of space division
multiple access (SDMA).
16
Chapter Two
Theoretical Background
2.6.2 Type of Smart Antenna
Smart antenna systems may be classified into the following three
types, as describes in Figure 2.8.
Figure 2.8 Different Smart Antenna Concepts
The following points present a distinction between the three sorts of
smart antenna systems [38].
1. Switched Beam Antennas: Switched beam also can be called a
switched lobe antennas which is the extension of the cellular
sectorization system that has a collection of several predefined
patterns with it the cell is divided into three sectors with macro-sectors
of 120 degrees each. The switched beam method more partition
macro-sectors into multiple micro-sectors thereby enhancing extent
field and efficiency. First, the signal intensity is measured, then one
of these established fixed beams is chosen, if the mobile phone travels
through the field, the device switches from one beam to another beam
[38]. A few more gains are obtained due to the higher directivity
comparison with traditional antennas. Such an antenna is simpler to
incorporate than the more complex adaptive arrays in existing cell
structures but it offers a small device improvement.
2. Dynamically-Phased Arrays: In the previous status of a switched
beam antenna the beams are determined and fixed. A consumer
perhaps within the zone of one beam at a given time, however as
travels the source of the beam away and passes the beam's
circumstance, the obtained signal would become weakened and an
intra-cell transfer exists. However, by the dynamically phased array,
17
Chapter Two
Theoretical Background
a direction of arrival (DOA) algorithm monitors the wanted signal as
the traverses inside the beam range that follows.
3. Adaptive Antenna Arrays: A going to be lots smartest is adaptive
antenna. Adaptive array systems can locate and monitor consumer
(SOI) and interferer (SNOI) signals and dynamically change antenna
pattern, this structure also can know as the adaptive beamforming or
the digital beamforming. Figure 2.10 Shows a functional device block
that transforms and digitizes down the received signals to the
baseband, then set the SOI used the direction-of-arrival (DOA)
algorithm and simultaneously detects the SOI and SNOI[38].
2.7 Array Weighting
The appearance of sidelobes means the array radiates energy in an
unwanted direction. Furthermore, the array absorbs energy from unexpected
directions due to mutuality. This is the basis on which communications
encountered fading. Weighting, shading, or windowing of the array elements
would eliminate the side lobes. There are various applications for array
weighting in areas like digital signal processing (DSP), radio astronomy,
radar, sonar, and communication. Figures 2.9(a) & (b) presents an even and
an odd a symmetric linear array elements M.
(a)
(b)
Figure 2.9 (a) Even Array and, (b) Odd Array with Weights [40]
The distant field from an array for equivalent elements may be
subdivided into the element factor (EF) by the array factor (AF) products
[40].
In general, the array factor is found by summing the weighted outputs
of every element. The array factor for the weighted even and odd element
arrays can be found as:
𝐴𝐹𝑒𝑣𝑒𝑛 = 𝑤𝑀 𝑒 −𝑗
⋯ + 𝑤𝑀 𝑒
(2𝑀−1)
𝑘𝑑
2
(2𝑀−1)
𝑗
𝑘𝑑
2
𝑠𝑖𝑛𝜃
1
+ ⋯ + 𝑤1 𝑒 −𝑗2𝑘𝑑 𝑠𝑖𝑛𝜃 + 𝑤𝑀 𝑒 𝑗
𝑠𝑖𝑛𝜃
(2𝑀−1)
𝑘𝑑
2
𝑠𝑖𝑛𝜃
+
(2.17)
18
Chapter Two
Theoretical Background
Here 2M = N, equal to the total element numbers.
The weights 𝑤𝑛 would choose to meet the requirements. Symmetrical
scalar weights, therefore, may be employed to form side lobes; to
reformulating the even array factor, Euler’s identity for the cosine would
utilize such as:
𝐴𝐹𝑒𝑣𝑒𝑛 = 2 ∑𝑀
𝑛=1 𝑤𝑛 cos (
(2𝑛−1)
2
𝑘𝑑 𝑠𝑖𝑛𝜃)
(2.18)
The 2 was being omitted from the statement in equation (2.18). with
no loss of commonality to produce a quasi-normalization
𝐴𝐹𝑒𝑣𝑒𝑛 = 2 ∑𝑀
(2.19)
𝑛=1 𝑤𝑛 cos ((2𝑛 − 1)𝑢)
Where 𝑢 =
𝜋𝑑
𝜆
𝑠𝑖𝑛𝜃
If the statement is zero, the array factor is maximum, inclusion θ = 0.
Figure 2.11(b) presents the odd array, to have a quasi-normalized odd array
factor, it will also summing up each exponential inputs in the array.
𝐴𝐹𝑜𝑑𝑑 = ∑𝑀+1
𝑛=1 𝑤𝑛 cos (2(𝑛 − 1)𝑢)
(2.20)
Likewise; the array factor could be written in terminology of vector
like
AF = w
̅ T . a̅(θ)
(2.21)
Where
w
̅ T = [wM , wM−1 … w1… wM−1 , wM ], and a̅(θ) Is the steering vector that
was described in equation (2.21)
2.8 Angle of Arrival (AOA) Estimation:
The estimation of the angle of arrival (AOA) was also called as spectral
estimation, the direction of arrival (DOA) or bearing of the impinging waves
was used for the synthesis of beams steered at the desired signal with nulls
guided to another signal interference [41]. Through more precisely detecting
the arrival angles (AOA) smart antennas have been used to improve direction
finding (DF) techniques [2].
2.9 Beamforming Techniques:
Beamforming is the technique used to describe the application of
weights to the inputs of an array of antennas to focus the reception of the
19
Chapter Two
Theoretical Background
antenna array in a certain direction. Beamforming presents several
advantages to antenna design:1) Space Division Multiple Access (SDMA) is the main aim in the
advancement of cellular radio systems as it ensures that more than one
user can be assigned to the same frequency in the same cell [41].
Because the former beam is attained, it can direct its focus toward a
certain signal the other signals from multiple directions can reuse the
same carrier frequency.
2) Since the beamformer is directed in a specific direction, the sensitivity
of the antenna can be increased, particularly when receiving weak
signals, for a better signal-to-noise ratio.
3) Signal interference is reduced due to the minimum pattern gain
oriented toward undesired signal angles [42].
2.9.1 Fixed Weight Beamforming:
The fixed weight of M element array and N input signals beamformer,
as shown in Figure 2.10, is a smart antenna in which fixed weight is used to
study the signal arriving from a specific direction.
Figure 2.10 Block Diagram of Fixed Weight Beamformer [2]
In the fixed weight beamforming approach, the arrival angles do not
change with time, so the optimum weight needn’t be adjusted [2].
2.9.2 Adaptive Beamforming
The key benefit of digital beamforming was that phase shifting and
array weighting can be achieved on the digitized data instead of the hardware
implementation [40]. Different methods of implementing an adaptive
algorithm could be used, the following are common adaptive algorithms that
are of special interest to this thesis.
20
Chapter Two
Theoretical Background
2.10 General control algorithm for adaptive antenna array
There are several adaptive algorithms used for the adaptive antenna
system, they are typically characterized in terms of their convergence
properties and computational complexity [43]. Adaptive algorithms can be
classified as non-blind adaptive, blind adaptive and optimized algorithms as
shown in Figure 2.11.
Figure 2.11 Classification of Adaptive Array Algorithms
1. Non-blind adaptive algorithms - Such algorithms employ a guide
signal to change the array weights iteratively, thus that weight output
is compared with the target signal at the end of each iteration and the
produced error signal are exercised in these algorithms to alter the
weights.
2. Blind adaptive algorithms - These algorithms don't use the guide
signal, and therefore no array weight adjustment is required.
3. Optimization algorithm - The purpose of optimization is to allows
comparison of the different choices to achieve the “best” design
relative to a set of prioritized criteria or constraints. These include
maximizing some function such as productivity, strength, reliability,
longevity, efficiency, and utilization .
2.10.1 Least Mean Square (LMS) Algorithm
The weight vector w
̅ = [w1 w2 … . wM ]T must be revised to minify
the error while iterating the adaptive weights.
The desired signal d(𝑗) and the interferers I1 (j), I2 (j), … IN (i) which is
received by each antenna array elements (M), times are noted by the jth time
samples [37].
21
Chapter Two
Theoretical Background
e(j) = d(j) − w
̅ H (j) x̅(j)
(2.22)
The squared error is [37]
|e(j)|2 = |d(j) − w
̅ H (j) x̅(j)|
2
(2.23)
The cost function is given as
J(w
̅) = D − 2 w
̅ H r̅ + w
̅ H ̅Rxx w
̅
(2.24)
Where 𝐷 = 𝐸[|d|2 ]
(2.35)
Taken the gradient of equation (2.35) and equate it to zero to reducing
the cost function.
̅ xx −1 r̅
w
̅ opt = R
(2.26)
The prompt estimates of these values are provided as
𝑅𝑥𝑥 ≈ 𝑥̅ (𝑗)𝑥̅ 𝐻 (𝑗)
(2.27)
𝑟(𝑗) ≈ 𝑑 ∗ (𝑗)𝑥̅ 𝐻 (𝑗)
(2.28)
After the cost function gradient, the LMS weight vector would be [37]
w
̅ (j + 1) = w
̅ (j) − μ [𝑅𝑥𝑥 𝑤
̅ − r]
=w
̅ (j) + μ e∗ (j) x̅(j)
(2.29)
As appear in equation (2.29), the convergence of the LMS algorithm
is the proportional indirect way with step size factor μ [38]. So ought to select
step size within a range which compensates for convergence rate, such as
0≤μ≤
1
(2.30)
2λmax
̅ xx. If there is only one
Where λmax is the largest eigenvalue of R
desired signal and the rest interfering signals are noise, equation (2.10) could
be
approximateas
0≤μ≤
1
(2.31)
2𝑡𝑟𝑎𝑐𝑒[𝑅̂𝑥𝑥 ]
Three main factors guided the response of the LMS algorithm which
is, μ, number of weights, and the Eigen-value for the input vector of the
correlation matrix. Many iterations ought to before satisfying to achieving
the convergence rate, that drawback of the conventional LMS adaptive
because of using fixed μ.
22
Chapter Two
Theoretical Background
2.10.2 Sample Matrix Inversion (SMI) Algorithm
One of the limitations of the LMS adaptive structure will be that the
algorithm should have to go through numerous iterations until effective
convergence is achieved the eigenvalue spread of the array correlation matrix .
The Sample Matrix Inversion (SMI (also known as direct matrix
inverse (DMI))) algorithm is the fastest and one of the simplest adaptive
algorithms compared to the LMS algorithm and had the minimum MSE
algorithms, where the optimal wiener solution offers the optimal array
weights. SMI is a time average estimate of an array correlation matrix using
K-time samples [49, 50], and the K–length block of data was employ, so this
method named a block-adaptive algorithm adapted the weight block by block.
The correlation matrix present by determined the time average as [13]
̂ xx(k) = 1 ∑𝐾
̅
̅H
R
(2.32)
1 X K (k) X K (k)
K
Where K is the observation interval. The correlation vector r̅ can be obtained
by
1
̅∗
̅
r̂(k) = ∑𝐾
(2.33)
1 d (k) XK (k)
K
Vector X is given by M×K matrix such as [7]
x̅1 (1 + kK) ⋯ x̅1 (K + kK)
̅(k) = [
⋮
⋱
⋮
X
]
x̅M (1 + kK) ⋯ x̅m (K + kK)
(2.34)
Where k is the block number and K is the block-length. Then the
estimate of the array correlation matrix is
̂ xx(k) =
R
1
K
̅
XK (k) ̅
XKH (k)
(2.35)
Also, the desired signal vector can be defined as
d̅(k) = [d(1 + kK) d(2 + kK) … . d(K + kK)]
(2.36)
The results of correlation vector is
̂r(k) =
1
K
d̅∗ (k) ̅
XK (k)
(2.37)
Measure the SMI weights for kth block of length K as
̅ xx −1 (k)r̅(k)
w
̅ SMI = R
̅ K (k) ̅
= [X
XKH (k)]−1 d̅∗ (k) ̅
XK (k)
23
(2.38)
Chapter Two
Theoretical Background
The benefit of the approach is that the convergence rate does not
depend on the signal level. However it has several drawbacks and the two
principle troubles that related with matrix inversion which is the high
computational complexity can't be overcome facilely through using of
integrated circuits so the correlation matrix may be ill-conditioned resulting
in errors or singularities when inverted, and the exercise of finite-precision
arithmetic and the necessity of inverting a large matrix where to invest.
2.10.3 Recursive Least Square (RLS) Algorithm:
As noted previously that the SMI procedure does have many
disadvantages, the computing time and theoretical singularities can trigger
problems, even if SMI faster than the LMS algorithm. The necessary
correlation matrix and the correct correlation vector could be recursively
calculated the weight vector w
̅ is chosen in the least-squares approach to
reduce the cost function consisting of several error squares over some time.
No matrix inversion computations are needed for the RLS algorithm, as the
inverse correlation matrix would be direct measuring. The following
equations will characterize RLS [49, 50]
̂ −1
Through product R
̂(k) canceling the division by K, so the
xx (k) r
matrix of correlation and the correlation vector removing K has been
rewritten as
̂ xx(k) = ∑𝐾
̅
̅H
R
(2.39)
1 X K (k) XK (k)
𝐾
∗
r̂(k) = ∑1 d̅ (k) ̅
XK (k)
(2.40)
̂ −1
The block length presents by k and last time sample k and R
̂(k)
xx (k), r
is the predictions of the correlation stopping at k sample of time [37]
̂ xx (k) = α R
̂ xx (k − 1) + x̅(k) x̅ H (k)
R
(2.41)
−1 ̂ −1 (k
̂ −1
̂−1
R
R xx − 1) − α−1 g̅(k) x̅ H (k) R
(2.42)
xx (k) = α
xx (k − 1)
̂ −1
g̅(k) = R
̅ (k)
xx (k) x
(2.43)
w
̅ (k) = w
̅ (k − 1) + g̅(k)[d̅H (k) − x̅ H (k)w
̅(k − 1)]
(2.44)
̂ xx (k) would be setting to [37]
The initial evaluate value of R
̂ xx (0) = x̅(1) x̅ H (1) + R
̅ nn
R
(2.45)
̂ nn = N x K zero-mean Gaussian noise
R
24
Chapter Two
Theoretical Background
This efficiency enhancement is done at the cost of greater
computational complexity [39].
2.10.4 Conjugate Gradient Method (CGM) Algorithm
The Conjugate Gradient Method is an effective method for symmetric
positive definite systems. For addressing a system of linear equations, the
CGM algorithm aims to search iteratively for the objective functions by
selecting conjugate (perpendicular) paths at each new iteration. Conjugacy
background is meant to mean orthogonal, in certain terms the CGM method
generates orthogonal search directions which lead in the quickest
convergence [45]. The CGM is aiming to minimize the quadratic cost
function
1 H
J(w
̅) = w
̅ ̅A w
̅ − d̅H w
̅
(2.46)
2
x̅1 (1) ⋯ x̅M 1)
̅= [ ⋮
⋱
⋮ ]
Where, A
x̅1 (K) ⋯ x̅M (K)
̅ is the K x M matrix of array snapshots, (M, K equals to numbers of the
A
element in an array and the number of snapshots, respectively)
d̅ = [d(1) d(2) … . d(K)]T represents the desired vector of K snapshots
Mathematically, simple CGM algorithm can be summarized as [40]
̅ (k) = The general weight update
w
̅ (k + 1) = w
̅ (k) + µCGM (k)D
µCGM (k) =
r̅H (k) ̅
A̅
AH r̅(k)
=
̅D
̅H ̅
̅ (k)
D
AH A
step-size of CGM
(2.47)
(2.48)
̅ (k) = the updates for the residuals
r̅(k + 1) = r̅(k) + µ(k)A̅ D
(2.49)
̅ (k + 1) = ̅AH r̅(k + 1) − α(k)D
̅ (k) = the direction vector update
D
(2.50)
The linear search is used to determine α (n) which reduces 𝑗(w
̅ (n)) [40]
α(k) =
r̅H (k+1) ̅
A̅
AH r̅(k+1)
r̅H (k) ̅
A̅
AH r̅(k)
(2.51)
Thus, locating the residual and the corresponding weights and updating
till achieved the convergence rate is the technique to use CGM.
2.10.5 Particles Swarm Optimization Algorithm
Particles swarm optimization PSO is often used to solve prosperous
complex multidimensional optimization problems in a different domain, such
25
Chapter Two
Theoretical Background
as antenna design and device modeling, 𝑒𝑡𝑐 [46].PSO algorithm can be
applied to enhance the adaptive antenna radiation pattern in every iteration.
The converged value can be achieved by a different parameter such as
controlling the amplitude, phase, position and complex weights (Includes
amplitude and phase control, this paper deals with complex weight
(amplitude and phase) and also position control. The representation of (PSO)
algorithm is based on the following steps below to find the optimal radiation
pattern of an adaptive antenna system [47]:
Stage 1: Parameters initializations that require to be optimized and give them
an appropriate range like initialize population, number of iterations,
regulating parameters (φ1andφ2) and weights (w)
Stage 2: Initialize random swarm position and velocities of the 𝑘𝑡ℎ variable
which becomes each particle's respective individual best and then selected,
the first global best among these initial positions.
Stage 3: Compute particle fitness 𝐹𝐹(𝑖, 𝑘)which is assigned to the present
locations.
Stage 4: Update the personage best and global best by comparing their fitness
value at the current position with the best fitness value which has ever gating
at each time 𝑝𝑖𝑑 (𝑖, 𝑘) = 𝐹𝐹(𝑖, 𝑘)and also, the global best can be defined
by𝑝𝑔𝑑 (𝑖)= max (𝑝𝑖𝑑 (𝑖, 𝑘)) which it's the best position through all of the
personage best positions.
Stage 5: Update velocity and position using
𝑣𝑖𝑑 (𝑖 + 1, 𝑘) = 𝑤 ∗ 𝑣𝑖𝑑 (𝑖, 𝑘) + 𝐶1 ∗ 𝑅𝑎𝑛𝑑( . ) ∗ (𝑝𝑖𝑑 (𝑖, 𝑘) − 𝑥𝑖𝑑 (𝑖, 𝑘)) +
𝐶2 ∗ 𝑅𝑎𝑛𝑑( . ) ∗ (𝑝𝑔𝑑 − 𝑥𝑖𝑑 (𝑖, 𝑘))
(2.52)
𝑥𝑖𝑑 (𝑖 + 1, 𝑘) = 𝑥𝑖𝑑 (𝑖, 1) + 𝑣𝑖𝑑 (𝑖 + 1, 𝑘)
(2.53)
Stage 6: Update fitness function to the𝐵𝐹(𝑖 + 1, 𝑘)
Stage 7: If fitness function for 𝐵𝐹(𝑖 + 1, 𝑘) > 𝑓𝑖𝑡𝑛𝑒𝑠𝑠 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
for 𝐵𝐹(𝑖, 𝑘)then 𝑝𝑖𝑑 (𝑖, 𝑘) = fitness function for 𝐵𝐹(𝑖 + 1, 𝑘)
Stage 8: Update 𝑝𝑔𝑑 (𝑖, 𝑘) = max (𝑝𝑖𝑑 (𝑖 + 1, 𝑘)).
Stage 9: Termination criteria. If 𝑖 < 𝑖𝑚𝑎𝑥 then boost𝑖 and go to step 5. The
number of iterations reaches the maximum permissible limit, else stop.
2.11 Benefits and Features of Smart Antennas
The operating benefits for the network operator can be
summarized as follows by accessing the spatial domain through smart
antenna systems [44]:
26
Chapter Two
Theoretical Background
• Capacity improvement: The central cause for developing an interest
in smart systems was the increased capability. Smart antennas allow
multiple cell users to use the same frequency without interfering with
one another because the smart antenna beams of base station are split
to hold different users in separate beams at the same frequency.
Coverage extension. Smart antennas should concentrate its energies
on expected users, in-state wasting energy in unnecessary directions
like a traditional antenna, which is as well as indicate that fewer base
stations may be required to serve a wide range of areas and longer
battery life in mobile stations and high data rates could be enabled.
• Locating users and due to the nature of spatial detection of the smart
system, the DOA algorithms are also used to locate human beings in
an emergency for any location-specific operation.
• Security is another advantage of smart-antenna systems. Security is a
significant problem in a society that is more focused on doing business
and forwarding on personal information and made it intractable to tap
the connection, as the attacker should be located in the same
orientation as the consumer from a base station to tap the connection
efficiently.
The features of a smart antenna are [2]
• Signal gain: Multiple antenna inputs are coupled to maximize the
available power necessary to create a given coverage rate.
• Interference rejection: Antenna pattern can be produced towards points
of interference, improving the signal-to-interference ratio of the received
signals. This eliminates interference said by the base station on the
inverting link or uplink. That also would limit the amount of downlink or
forward link interference distributed in the network. These carrier
enhancements to the interference ratio provide an increase in efficiency.
• Spatial diversity: complex array information was utilized to reduce
fading and other unwanted multipath propagation impacts.
• Power efficiency: join inputs with many elements to maximize the
processing gain available in the downlink (against customers).
2.12 Drawbacks of Smart Antenna
Even though the advantages of using smart antennas are considered
quite so many, some significant limitations still occur [39].
27
Chapter Two
Theoretical Background
• A smart transceiver antenna is more complicated than a conventional
transceiver to the base station. For each of the array antenna elements,
separate transceiver chains are required and correct real-time calibration
of each is necessary.
• Adaptive beamforming and DOA is a computationally demanding
process therefore, quite powerful numeric processors and control systems
should be included in the smart antenna base station.
• Smart antenna base stations would probably cost a lot more than
traditional base stations.
2.13 Description and Modeling of Wireless Channel
To analyze the efficiency of a smart antenna system, specific
knowledge of the channels and the parameters of channel parameters is
substantial, since the propagation channel is the main benefactor of
difficulties and drawbacks which are an effect on the mobile radio systems.
In many forms, the physical channel influences the transmission of radio
signals on both the straight-ahead (from the base station into mobile) and
inverted (from the mobile into the base station) connections. The direct path
with the multipath channel is indicated in Figure 2.12 [40].
Figure 2.12 Multipath with a Direct Path Channel.
Every propagation path has its own power, time delay, and angle of
arrival. The signal received is abundant poorer than the signal transmitted on
account of factors like the loss of mean propagation, gradual fading, and fast
fading. The mean loss of propagation derives from square-law spreading,
water and foliage absorption, and ground-reflections impact, mean loss of
propagation depends on the range and variations quietly slow also for quick
mobiles. Fading may also be described as flat or selective fading of
frequencies [40]. The following two central categories could be classified
28
Chapter Two
Theoretical Background
into parameterized of the physical model applied to describe the wireless
channel between the mobile station and base station [13]:Firstly: Temporal parameters that include the path loss, Shadowing,
Multipath fading, Power spectral density, and Power delay profile. Secondly:
Spatial parameters that include the angle of arrival and mobile station
mobility. Such parameters are modeled in detail in the coming sub-sections.
2.13.1 Path loss:
It is such an average reduction in signal strength of path length between
certain transmitters and the receiver. Studies illustrate that the average signal
power received reduces exponentially together with distance [39].
2.13.2 Shadowing:
Shadowing or slow fading due to changes in propagation factors
related to buildings, pathways, trees, and other obstacles in a specific region,
which reflects the difference in the average path loss in a local area [39].
2.13.3 Multipath Fading
This form of fading is comparatively quick and thus accountable for
shortage signal fluctuations [13]. That's induced due to the constructive and
destructive combination which arbitrarily slowed, reflected, scattered, and
the diffracted subpath signal sources through distances of the range for the
little wavelengths. Many signal replicas arrived at the receiver, having
traversed several propagation paths, combining constructively and
destructively.
2.14.4 Rayleigh Model
The Rayleigh distribution had to paradigm the multi-path fading in
non-line of sight (NLOS) fading conditions without direct line of sight (LOS)
direction between the mobile station and base station [13]
2.13.5 Rice model
The Rice distribution concepts the propagation paths composed of one
direct LOS consequence and several randomly weaker NLOS consequences
[13]. The Rayleigh (non-line-of-sight) fading models are considered in this
study, which is widely was using to define the statistical time-varying form
of the personage multipath envelope.
29
Chapter Two
Theoretical Background
2.13.6 Power Spectral Density
The shift in the frequency of the received signal can be apparent when
there is proportional movements occur between the transmitter and receiver,
because of the Doppler shift [13]. The Doppler shift is different for every
sub-path as it depends on the Angle of Departure of the subpath related to the
direction of movement of the mobile station. The Power Spectral Density
(PSD) produced by the sum of all the scattered and reflected sub-paths, which
cause a continuous spectrum of Doppler frequencies. The inverse Fourier
transform of the PSD is the autocorrelation function of the fading signal [2].
Depending on the particular propagation environment and the underlying
communication scenario different PSDs have been proposed, [48]. A popular
selection for land mobile communications that indicates in this research is the
Jake PSD [2]. For the Rayleigh fading case, the normalized Jakes PSD and
the coincide autocorrelation function would be defined by [13]
S(f) =
α2
f
π fD √1−(f )2
|f| ≤ fd
(2.54)
d
R(τ) = j0 (2π fD τ)
(2.55)
Where j0 (. ) is the zero-order Bessel function of the first kind and fD is the
maximum Doppler shift that the signal undergoes, given by[48]
v
v fc
λ
c
fD = =
(2.56)
Where v is the user speed (in m/s), λ is the wavelength of the
transmitted signal, c is the velocity of light and fc is the carrier frequency. In
this thesis, the maximum Doppler frequency used is 117 Hz. This
corresponds to a speed of v = 140 km/hr for fc = 900 MHZ.
2.13.7 Power Delay Profile (PDP)
There is more than one propagation path for each transmitter and
receiver in several mobile channels and a received signal composed of two
or more different elements and each moves a separate path from the
transmitter. Each multipath aspect arrives with a delay depending on the
length of the path [41]. Based on the sort of environment (indoor or outdoor)
as well as the common conditions for propagation, the PDP model can take
various forms. In this work, for simple, the time delays relay with the variable
30
Chapter Two
Theoretical Background
resolvable multipath would be independent of the Angles of Arrival (AOA’s)
[13].
2.13.8 Mobile Station Mobility Model
Modeling the mobility of mobile stations is a central topic of spacetime channel simulations. MS motion affects the properties of both spatial
and temporal channels. The smart antenna should monitor and steer its beam
towards the desired target in the situation of adaptive beamforming. Hence,
the efficiency of a smart antenna could not be realistically measured without
mobile station mobility simulations. The subsequent criteria have been used
to define the impact of MS mobility on execution efficiency [13]:• With mobile moving for a particular Monte Carlo simulation run,
numbers of sub-path and multipath components should behold
constant.
• As mobile motions, the sub-path parameters (departure angles, random
phases, and angular spread) are held constant.
31
Chapter Three
Proposal
Algorithm
Analysis
Chapter Three
Proposed Algorithm Analysis
3.1 Introduction
Adaptive algorithms have been used either in block mode or in iterative
mode to adjust the weight vector 𝑤(𝑛). Block processing techniques measure
a new solution using the statistical estimates obtained out of the latest data
block. In iterative algorithms, the current weight vector is adjusted by an
incremental amount to form a new weight vector that approximates the
optimal solution.
There is a certain variance when comparing as an example the
algorithm Least Mean Square(LMS) which is largely used by many
researcher because of the low complexity of computational and ease of
execution. The least-squares algorithms (an algorithm that content step size
factor), like Recursive Least Squares (RLS), Conjugate Gradient (CG), and
Rabid Euclidean Direction Search (REDS), would converge more rapidly and
also have smaller mean square error (MSE) versus LMS. Nevertheless, their
complexity enables them to intrinsically unfit to real-time applications.
In this chapter, different tests of common antenna array geometry like
linear, circular, planar, and cube antenna array comparing for various antenna
array performance also, overall system performance had been inspected
along with changing in some system parameters.
Figure 3.1 illustrates the flowchart of implementation for different
antenna array geometry and different algorithms, it should be noted that the
flowchart bellow gave a prime vision of smart antenna algorithm function.
Throughout this chapter, several developments on adaptive beamforming
algorithms will be presented to enhance the achievement of the smart antenna
in terms of rate convergence, interference suppression capabilities, and the
tracking capabilities of the desired signal. Such techniques use both the block
adaptive and sampled methods.
32
Chapter Three
Proposal Algorithm Analysis
Figure3.1. Flowchart for a conventional smart antenna.
3.2 The Implementation of Normalizing LMS Algorithm with Maximum
SIR Factor
The benefit of this process is converges better than the traditional
process of adaptation. The proposed adaptive beamforming algorithm is LMS
normalize by maximum SIR would be introduced here, the maximization of
the SIR is based on one principle that could apply to maximize the received
signal and eliminate the signals of interferences. It is obvious that by putting
nulls at their angles of arrival, the cancelation of all interference would
automatically optimize the SIR, to boost the performance of adaptive
33
Chapter Three
Proposal Algorithm Analysis
algorithms along with convergence speed, weight stability, and interference
suppression.
The SIR strategies were used to calculate the optimum vectors of
weights allocated for each element in an array by considering a fixed beam
pattern instead of random value before computing the ultimate weight vectors
by the least square algorithm. The weight coefficients extracted from
standardization were set as initial coefficients and then updated by the LMS
algorithm to improve device stability and convergence speed. The proposed
optimum updated weight vectors of combining the two aspects of
𝑆𝐼𝑅𝑚𝑎𝑥 and LMS algorithm introduces according to the following equations:
The desired signal weighted array output power is provided as [37]
𝜎𝑠 2 = 𝐸[|𝑤
̅ 𝐻 . 𝑥̅𝑠 |2 ]= 𝑤
̅ 𝐻 . 𝑅̅𝑠𝑠 . 𝑤
̅
(3.1)
𝑅̅𝑠𝑠 = 𝐸[𝑥̅𝑠 𝑥̅𝑠 𝐻 ] presents the signal correlation matrix
The power of undesired signal weighted array output is provided as [37]
𝜎𝑢 2 = 𝐸[|𝑤
̅ 𝐻 . 𝑥̅𝑢 |2 ]= 𝑤
̅ 𝐻 . 𝑅̅𝑢𝑢 . 𝑤
̅
(3.2)
𝑅̅𝑢𝑢 = 𝑅̅𝑖𝑖 + 𝑅̅𝑛𝑛
(3.3)
Where 𝑅̅𝑖𝑖 , 𝑅̅𝑛𝑛 presents the correlation matrix to the interferers and
noise. The SIR described the proportion of the desired signal power split by
undesired signal power.
𝑆𝐼𝑅 =
𝜎𝑠 2
𝜎𝑢 2
=
̅ 𝐻 .𝑅̅𝑠𝑠 .𝑤
̅
𝑤
𝐻
̅
̅ .𝑅𝑢𝑢 .𝑤
̅
𝑤
(3.4)
With applying the derivative relative to 𝑤
̅ and returns the impact to
zero. The procedure of optimization was given in Harrington. Rearranging
terms, the relationship derives as follow
−1
𝑅̅𝑢𝑢 𝑅̅𝑠𝑠 . 𝑤
̅ = 𝑆𝐼𝑅. 𝑤
̅
(3.5)
𝑆𝐼𝑅𝑚𝑎𝑥 is equal to the largest eigenvalue 𝜆𝑚𝑎𝑥 for the 𝐻𝑒𝑟𝑚𝑖𝑡𝑖𝑎𝑛
matrix 𝑅̅𝑢𝑢 , 𝑅̅𝑠𝑠 . The eigenvector associated with the largest eigenvalue is
the optimum weight vector̅̅̅
𝑤𝑜𝑝𝑡 .Thus
−1
𝑅̅𝑢𝑢 𝑅̅𝑠𝑠 . 𝑤
̅𝑆𝐼𝑅 = 𝜆𝑚𝑎𝑥 . 𝑤
̅ 𝑜𝑝𝑡 = 𝑆𝐼𝑅𝑚𝑎𝑥 . 𝑤
̅𝑆𝐼𝑅
(3.6)
The weight vector can pose in terms of the optimum Wiener solution [37]
−1
𝑤
̅𝑆𝐼𝑅 = 𝛽. 𝑅̅𝑢𝑢 . 𝑎̅0
(3.7)
The correlation matrix is defined as
34
Chapter Three
Proposal Algorithm Analysis
𝑅̅𝑠𝑠 = 𝐸[|𝑠|2 ]𝑎̅0 . 𝑎̅0 𝐻
Where, 𝛽 =
𝐸[|𝑠|2 ]
𝑆𝐼𝑅𝑚𝑎𝑥
(3.8)
. 𝑎̅0 𝐻 . 𝑤
̅𝑆𝐼𝑅
(3.9)
The following part means with LMS algorithm, the error square gave as [37]
|e(k)|2 = |d(k) − w
̅ H (k) x̅(k)|
2
(3.10)
The LMS weight vectors would give as
w
̅ (k + 1) = w
̅ (k) + μ e∗ (k) x̅(k)
(3.11)
The proposed algorithm can be drawn in the flowchart as indicated in Figure
3.2.
35
Chapter Three
Proposal Algorithm Analysis
Figure3.2. Normalized LMS algorithm
3.3 Variable Step Size via Error Controlling Algorithm
Many algorithms evolved from the regular LMS algorithm to enhance
the convergence rate, the LMS algorithm is implemented by the
normalization known as normalized LMS (NLMS). To improve the LMS
algorithm's convergence the Variable Step Size LMS (VSSLMS) algorithms
is produced while maintaining stable achievements. The NLMS algorithm is
36
Chapter Three
Proposal Algorithm Analysis
an application of the LMS step size improvement (LMSSI) algorithm that
selects for each iteration of the algorithm a different step size value 𝜇(𝑛) this
factor of step size is proportional to the opposite of the cumulative predicted
energy from the array instant value of the inputs signal. The NLMS
minimizes the step size μNLMS (k) to allow major improvements to the weight
vectors of the updates. This avoids the update weight vectors from diverging
and allows the convergence algorithm more stable and quicker than when
using a fixed step size [15]
µ
0
μNLMS (k) = ‖x̅(k)‖
2
(3.12) Where µ0 is a constant small positive value between (0-1).
The final weight vector could be updated by [15]
µ
0
w
̅ (k + 1) = w
̅ (k) + ‖x̅(k)‖
e∗ (k)x̅(k)
2
(3.13)
Where e(k) = d(k) − w
̅ H (k)x̅(k) present the error signal
w
̅ (k + 1) = w
̅ (k) +
μ0
q+‖x̅(k)‖2
e∗ (k)x̅(k)
(3.14)
A small constant q must be added to the denominator to avoid
denominator being zero when the data at any instant is zero. A new way of
choosing the step size for the LMSSI algorithm which is the error driving
algorithm to improve the performance of smart antenna by modifying the step
size of the NLMS algorithm. In the NLMS algorithm, the small constant q
has a fixed effect in the step size factor and can lead to a decrease in its value.
This decrease in step size has an impact on the NLMS algorithm's
convergence rate and weight stability. The error signal can be utilized to
prevent the denominator from going to zero for every iteration and to control
the step size. Under this approach, the q parameter could put as
σ
n
q = ‖e(k)‖
2
(3.15)
Where σn is the square root of the noise variance σ2n . Thus, the step size
would be
μSS (k) =
μ0
σn
+‖x̅(k)‖2
‖e(k)‖2
(3.16)
Therefore, the weight vector of LMSSI algorithm is
w
̅ (k + 1) = w
̅ (k) +
μ0
σn
+‖x̅(k)‖2
‖e(k)‖2
e∗ (k)x̅(k)
37
(3.17)
Chapter Three
Proposal Algorithm Analysis
Figure3.3. Flowchart of LMSSI
The step size μ(n), change its value with respect to the array signals of
error, and updating the weight vector to detect any updates in the smart
antenna environments. Also can say, at the beginning of the adaptive cycle,
when the error signal is high, q is low, and step size is high for rapidly down
the error. Nevertheless, in steady-state the signal of error is small, q is high
and the step size is low for a low degree of mismatch. This avoids divergent
update weights and made the step size more constant and converges quicker
than the least square algorithms.
38
Chapter Three
Proposal Algorithm Analysis
3.4 Improvements for CMA (ICMA&BCM) algorithm Performance
3.4.1 Conventional Constant Modulus Algorithm CMA
The CMA algorithm is useful because it requires no carrier
synchronization and can be successfully applied to a non-constant modulus
signal.
Many wireless communication and radar signals are frequency or
phase-modulated signals. Some examples of phase and frequency modulated
signals are FM, PSK, FSK, QAM, and polyphase. This being the case,
ideally, the signal amplitude should be a constant. Accordingly, the signal
has a fixed amplitude or modulus. However, the received signal in fading
channels, since there are multipath terms, is the combination of all multipath
aspects. So the channel incorporates a difference in amplitude on its
magnitude signal. Dominique Godard [39] primarily rely on the concept of a
constant modulus (CM) to establish a group of blind equalization algorithms
used in 2D wireless systems. Godard was using a cost function named order
p dispersion function and the optimum weights are allocate after reduction
[39]
𝐽(𝑘) = 𝐸[(|𝑦(𝑘)|𝑝 − 𝑅𝑝 ) 𝑞 ]
(3.18)
Where p and q is the positive integer equal to 1.
Godard suggested that the cost function gradient is zero when 𝑅𝑝 is described
by
𝑅𝑝 =
𝐸[|𝑠(𝑘)| 2𝑝 ]
(3.19)
𝐸[|𝑠(𝑘)| 𝑝 ]
Where 𝑠(𝑘) is calculated Zero memory of 𝑦(𝑘) and the error signal is
produced as
𝑒(𝑘) = 𝑦(𝑘)|𝑦(𝑘)𝑝−2 (𝑅𝑝 − |𝑦(𝑘)|𝑝 )|
(3.20)
When p = 1 case reduces
𝐽(𝑘) = 𝐸[((|𝑦(𝑘)|) − 𝑅1 ) 2 ]
the
cost
function
would be
(3.21)
Where
𝑅1 =
𝐸[|𝑠(𝑘)| 2 ]
(3.22)
𝐸[|𝑠(𝑘)|]
If the output calculated 𝑠(𝑘) set to unity, the signal of error can produce as
𝑦(𝑘)
𝑒(𝑘) = (𝑦(𝑘) − |𝑦(𝑘)|)
(3.23)
Here when the p = 1 case, the vector of weight, would be [40]
39
Chapter Three
Proposal Algorithm Analysis
1
𝑤(𝑘 + 1) = 𝑤(𝑘) + 𝜇(1 − |𝑦(𝑘)|)𝑦 ∗ (𝑘)𝑥(𝑘)
(3.24)
And when p = 2 case decreases the cost function to the formula
𝐽(𝑘) = 𝐸[(𝑦(𝑘)2 − 𝑅2 ) 2 ]
Where, 𝑅2 =
(3.25)
𝐸[|𝑠(𝑘)| 4 ]
(3.26)
𝐸[|𝑠(𝑘)| 2 ]
The weight vector with p = 2 cases would be as [40]
𝑤(𝑘 + 1) = 𝑤(𝑘) + 𝜇(1 − 𝑦(𝑘)2 )𝑦 ∗ (𝑘)𝑥(𝑘)
(3.27)
The case p= 1 was found to converge somewhat more fast than the case p =
2 [40]
3.4.2 Improvement Proposed to the Constant Modulus Algorithm ICMA
via Variable Step Size vector
Improvement that applies in this section replaced the step size factor
(𝜇) of CMA algorithm by variable step size value that produced in previous
section 3.3.1 above so, the new update weight vector equation would be as
follow:
1
𝑤(𝑘 + 1) = 𝑤(𝑘) + 𝜇(1 − |𝑦(𝑘)|)𝑦 ∗ (𝑘)𝑥(𝑘)
Where, 𝜇 = μSS (k) =
μ0
σn
+‖x̅(k)‖2
‖e(k)‖2
(3.28)
(3.29)
Where the error equation gave as
𝑦(𝑘)
𝑒(𝑘) = (𝑦(𝑘) − |𝑦(𝑘)|)
(3.30)
So, the weight vectors equation gave as
𝑤(𝑘 + 1) = 𝑤(𝑘) +
μ0
σn
+‖x̅(k)‖2
‖e(k)‖2
1
(1 − |𝑦(𝑘)|)𝑦 ∗ (𝑘)𝑥(𝑘)
(3.31)
The advantage of this step would enhance the coverage speed, the
number of iteration, MSE value, and more stability for output signal
amplitude. The result of this part will be shown in chapter 4.
40
Chapter Three
Proposal Algorithm Analysis
3.4.3 Proposed Blocking Constant Modulus Algorithm BCMA (Static &
Dynamic)
One important drawback of Godard CMA was its weak convergence
rate, which restricted the efficiency in dynamic environments of the
algorithm when the signal should captivate rapidly. Therefore this limits
CMA's quality if channel conditions happen fast. The BCMA is a
resemblance to the SMI that discussed previously in chapter Two. The static
BCMA was incorporated with only one block of data, the algorithm is iterated
through n values before it converges. The input array block of K length
vectors give as
𝑋̅ = [𝑥(1) 𝑥(2) 𝑥(3) . . . 𝑥(𝐾)]
(3.32)
The newly updated weight vector would produce as
1
𝑤(𝑛 + 1) = 𝑤(𝑛) + 𝜇(1 − |𝑦(𝑛)|)𝑦 ∗ (𝑛)𝑋
(3.33)
So, the initial weights w
̅ (1) were select the complex to restrict output data
vector r̅(1) would name as
y̅(n) = [𝑤 𝐻 (𝑛)𝑥(1) 𝑤 𝐻 (𝑛)𝑥(2) . .. 𝑤 𝐻 (𝑛)𝑥(𝐾)]
(3.34)
After that the following weight w
̅ (2)was found, then the iteration will
continue until sufficient convergence is achieved. This will be termed a static
BCMA algorithm as this iteration method involves just one static block, of
the length K.
The main achievements of the static BCMA is could converge faster
than the conventional CMA algorithm to converge after just a few iterations.
The static BCMA easily calculated the weights based upon a sampled data
fixed block. To ensure the adaptation in a dynamic environment for the data,
it should update the blocks of data for every iteration. Let us describe a
dynamic data block as the output of the array before weights apply to nth
iteration, the nth block of length K will be
𝑋(𝑛) = [𝑥(1 + 𝑛𝐾) 𝑥(2 + 𝑛𝐾) 𝑥(3 + 𝑛𝐾) . . . 𝑥(𝐾 + 𝑛𝐾) ]
(3.35)
The output of weighted array to the nth iteration describes as
y̅(n) = [𝑤 𝐻 (𝑛)𝑥(1 + 𝑛𝐾) 𝑤 𝐻 (𝑛)𝑥(2 + 𝑛𝐾) . ..
(3.36)
𝑤 𝐻 (𝑛)𝑥(𝐾 + 𝑛𝐾)]
Then the vector of the complex restricted output data is produced as
𝑦(𝑛) = [𝑦(1 + 𝑛𝐾) 𝑦(2 + 𝑛𝐾) 𝑦(3 + 𝑛𝐾) . . . 𝑦(𝐾 + 𝑛𝐾) ] 𝑇
1
𝑤(𝑛 + 1) = 𝑤(𝑛) + 𝜇(1 − |𝑦(𝑛)|)𝑦 ∗ (𝑛)𝑋(𝑛)
41
(3.37)
(3.38)
Chapter Three
Proposal Algorithm Analysis
So the dynamic BCMA algorithm is more suitable due to the constancy
and coverage rate and speeds iterative improvement.
3.5 Particles Swarm Optimization Algorithm (PSO)
Particles swarm optimization PSO is often used to solve prosperous
complex multidimensional optimization problems in a different domain, such
as antenna design and device modeling 𝑒𝑡𝑐. PSO algorithm which could be
used to obtain the enhancement of recent standards in various parameters to
improve the pattern of adaptive antenna radiation in each iteration, PSO
flowchart can be shown in Figure 3.4.
The converged value can be achieved by a different parameter such as:
Controlling the Amplitude, Phase, Position and Complex Weights (Includes
amplitude and phase control), this thesis deals with complex weight
(amplitude and phase) and also position control.
42
Chapter Three
Proposal Algorithm Analysis
Figure3.4. Particle Swarm Optimization Flowchart.
The PSO algorithm needs appropriate fitness function for optimizing
so, two fitness function that uses and should have the possibility to directivity
improvement, SLL reduction, and minify the HPBW, as follow
𝐹1 = 𝑀𝑎𝑥{𝐴𝐹(𝜃𝑑 )}
𝑤1𝑓1+𝑤2𝑓2
𝐹2 = 𝑀𝑖𝑛{
|𝐴𝐹𝑚𝑎𝑥 |
(3.39)
}
(3.40)
𝑓1 = |𝐴𝐹(𝜃1 )|2 + |𝐴𝐹(𝜃2 )|2
(3.41)
𝑓2 = 𝑀𝑎𝑥{|𝐴𝐹(𝜃𝑆1 )|2 + |𝐴𝐹(𝜃𝑆2 )|2 }
(3.42)
43
Chapter Three
Proposal Algorithm Analysis
Where AF is the array factor equation, which taking from equation
(2.11) in chapter two. 𝜃𝑑 is the arrival angle of desired, 𝜃1 &𝜃2 are the two
interference angle and 𝜃𝑆1 & 𝜃𝑆2 are the angles where the maximum sidelobe
level is acquired during the optimization process. Thus, 𝑓1 function is to
force the null at a specific angle and 𝑓2 used to keep the side lobe minimum
as possible. The specification of each function on antenna array would be
introduced later in chapter 4.
3.6 Adaptive Antenna Array Performance via REDS Algorithm
Figure 3.5 illustrated the block diagram of the smart antenna
beamforming system.
𝑆(𝑘)
𝑦(𝑘)
I(k)
𝑒(𝑘)
d(k)
Figure3.5. Block Diagram of Smart Antenna System.
The weight vector w
̅ = [w1 w2 … . wM ]T as shown in Figure 3.5
Antenna array adaptive weights should be adjusted in this manner to reduce
the error when iterating.
The desired signal S̅(k) and interferers I1 (k), I2 (k), … IN (k) are
received with M potential weights by M elements array. The noise also
includes in each received signal at m element. The k th represents the time
samples for the time.
The weighted array output is given by
y= w
̅ H (k). x̅(k)
(3.43)
Where
I1 (k)
I (k)
x̅(k) = a̅0 S(k) + [a̅1 a̅2 … . a̅N ] . [ 2 ] + n̅(k)
⋮
I𝑁 (k)
= x̅S (k) + x̅I (k) + n̅(k) = input signal
(3.44)
x̅S (k) is the desired signal vector,x̅I (k) is the interfering signals vector.
n̅(k) = zero-mean Gaussian noise for each channel.
44
Chapter Three
Proposal Algorithm Analysis
a̅i = M-element array steering vector for θ𝐼 the direction of arrival. In this
thesis three type of array factor would be apply with this algorithm.
An error signal is, e(k) = d(k) − w
̅ H (k) x̅(k)
Utilizing the cost function gradient, the weight vector becomes as follow
w
̅ (k + 1) = w
̅ (k) + μ e∗ (k) x̅(k)
(3.45)
The reformulation will be written with the use of the REDS
algorithm, the error signal and the weight vector can give as
e(k) = d(k) − w
̅ H (k)x̅(k)
w
̅ (k + 1) = w
̅ (k) + μ e∗ (k)x̅(k)
Where,
(3.46)
(3.47)
x̅(k) = [x1 x2 … . xM ]T = input signal
d̅ = [d(1) d(2) … . d(K)]T = desired signal
The error signal e(k) can be write as
e(k) = d(k) − ∑M
i=1 wi (k)xi (k)
(3.48)
Considering the samples k-L, k-L+1, k-L+2 …... k, where L>M, equation
(3.48) can be written as
e̅(k) = d̅(k) − U(k)w
̅(k)
Where
(3.49)
d̅(k) = [d(k), d(k − 1), d(k − 2) … d(k − L + 1)]T
(3.50)
̅ (k) = [u̅1 (k), u̅2 (k) … . u̅M (k)]
U
(3.51)
u̅j (k) = [xj (k), xj (k − 1) … xj (k − L + 1)]T
The new approaches error at k is provided as
(3.52)
̅ (k − 1)w
e̅0 (k) = d̅(k) − U
̅(k − 1)
(3.53)
By update just one weight in w
̅ (k − 1) the new error can be written as
update
e̅1 (k) = d̅(k) − [U(k)w
̅ (k − 1) + U(k) w
̅ j0(k) (k) Fj0(k) ]
(3.54)
j0 (k) Is the index of the weight to be updated in the zeroth P-iteration
iteration at time k and Fj0(k) is M x 1 vector with 1 in position j and 0 in all
other positions.By choosing index j0 (k), the updated weight of the element
is expressed as
update
w
̅ j0(k) (k) = w
̅ j0(k) (k − 1) + w
̅ j0(k) (k)
(3.55)
update
Where w
̅ j0(k) (k) a projection value of the error e̅0 (k) and can be givens as
45
Chapter Three
update
w
̅ j0(k) (k) =
Proposal Algorithm Analysis
̅j0 (k) (k)>
<e̅0 (k)u
̅j0 (k) (k)‖
‖u
(3.56)
2
When P-iteration is equal to zero with step-size the updates the array weight
would be introduced as follow
update
w
̅ o (k) = w
̅ (k − 1) + μREDS w
̅ j0(k) (k) Fj0(k)
(3.57)
The optimize value of μREDS would be established by applying the same step
size parameter (μ) that used in Equation (2.42). (LMS). Figure 3.6
demonstrates the flowchart for the proposed algorithms.
46
Chapter Three
Proposal Algorithm Analysis
Figure 3.6. REDS algorithm flowchart
3.7 BER
By modeling a communication system, it includes a smart antenna
system to display the enhancement in received power (SNR and BER), so
creates array factor and guides the pattern for both Linear, Circular and Planar
47
Chapter Three
Proposal Algorithm Analysis
Arrays, generates the signal, modulates it and transmits it with added Noise
through the channel.
Figure 3.7 presented the flowchart of simulate for a communication
system module that included transmitter of signal message, modulator for the
signal, beamformer, transmission channel, receiver, demodulator, and
sampler. It transmits and create a pattern in the direction of DOA and receive
message signal as user/target and measure received signal quality and BER.
Figure 3.7 Processing Flowchart
48
Chapter Three
Proposal Algorithm Analysis
Different tests of common antenna arrays like a linear array and
circular array antenna are comparing between performances also overall
system performance had been inspected along with changing some
parameters in the system.
Testing BER and impact of an increasing number of elements in three
types of the antenna array, linear array, circular and planar array besides the
effect of increasing the noise value added to the system. So the BER had been
measured by sending and generating a random message of 1000 Bits using
MATLAB software package and comparing the received single with the
transmitted one.
49
Chapter Four
Simulation
Results
Chapter Four
Simulation Results
4.1 Introduction
In this chapter, the proposed system simulation results had been
obtained and discussed, the simulation results were described to compare the
potential of different algorithms to form beams in the direction of the desired
signal and to locate position null in the direction of interference signal with
the expectation of the desired signal angle and interference angle, and their
production was evaluated by the beam distance, high sidelobe level, null
depth and convergence rate. The algorithm that studied is LMS, RLS, NLMS,
SMI, CMA, CGM, REDS, GA and PSO algorithms, and also improvement
in CMA algorithm and algorithm with step size factor and normalized SIR
enhancement algorithms.
Another aspect deliberated in this section would be exploring the
radiation patterns of different antenna array geometry which can be classified
like one dimension (1D) linear array, tow dimension (2D) planar and circular
array and three dimensions (3D) cubic array by varying various parameters,
likewise the displacement spacing and the number of elements in array
system. The subsequent simulations on arrays are performed using the
MATLAB software package version R2017a.
4.2 Characteristic Studies of Uniform Linear Arrays
This simulations are presented using a uniform linear array
configuration. All simulations are assumed to have the signal input (s(k) =
sin (2πft)) with f=900 MHz, the signal of desired is (d(k) = s(k)) and the
number of sample intervals K is set at 100. The interfering signals and noise
signals are negligible. In this chapter (M) would represented the total number
of ements.
4.2.1 Array Factor Interpretations with Difference Element
Displacement
The impact of varying element spacing on the beamwidth had been
simulated with a range from 0.1λ to1λ. Figure 4.1 represents the radiation
pattern for a linear array of 10 elements for different element spacing. It can
be observed that smaller element spacing produces a wider beamwidth in
comparison to a half-wavelength spaced linear array. Also, for an element
spacing of 1λ, a grating lobe (undesirable radiation pattern) that is equal in
the magnitude of the main radiation lobe is generated. This is the large
spacing between elements such that having full wavelength spacing will
generate secondary lobes with large magnitudes.
50
Chapter Four
Simulation Results
Radiation pattern 10-element array for DOA = 30, d=0.1λ
Radiation pattern 10-element array for DOA = 30, d=0.25λ
Radiation pattern 10-element array for DOA = 30, d=0.5λ
Radiation pattern 10-element array for DOA = 30, d=1λ
Figure 4.1 Radiation Pattern for Different Element Spacing 𝟎. 𝟏𝛌, 𝟎. 𝟐𝟓𝛌, 𝟎. 𝟓𝛌, 𝟏𝛌
51
Chapter Four
Simulation Results
As it is observed that element spacing below 0.5λ produces wide
beamwidth and would not be as useful in this instance. This is because the
increase in beamwidth increases both fading and interference levels. Also,
when element spacing is increased to 1λ, the grating lobe begins to appear.
The beamwidth of the radiation pattern is measured at the -3dB point where
the radiated signal magnitude is 0.707 of the maximum power radiated.
Typical values for -3dB beamwidth and SLL for varying number of elements
can be seen in Table 4.1. This table shows the results obtained from a
constrained iterative process of conventional code.
Table 4.1 Summary result for different inter-element spacing values
Displacement space
d
0.1
0.25
0.5
1
SLL(dB)
HPBW(degree)
-13
-13
-13
0
67.3
24
11.7
5.8
The standard spacing between elements is half-wavelength to
minimize mutual coupling and avert grating lobes.
4.2.2 Array Factor Interpretations with Variation of Number of Array
Elements
Array for 300 DOA desired user and elements spacing of 0.5λ. The
simulation was produced to examine the behavior of the beamwidth as the
number of elements is increased from 2 to 25. Figure 4.2 illustrated both
linear and polar plots of radiation pattern for 2, 6, 12, and 25 array elements,
where the relationship of beamwidth and the increasing number of radiating
elements can be shown for a various number of elements.
Radiation Pattern 2-Element Linear Array for DOA=30, d=0.5λ
52
Chapter Four
Simulation Results
Radiation Pattern 6-Element Linear Array for DOA=30, d=0.5λ
Radiation Pattern 12-Element Linear Array for DOA=30, d=0.5λ
Radiation Pattern 25-Element Linear Array for DOA=30, d=0.5.
Figure 4.2 Linear and polar plots of radiation pattern for 2, 6, 12 and 25 array elements
As seen in Figure 4.2, as the number of elements increases, the
beamwidth will decrease. This increase in accuracy and control is obtained
at an increase in cost and complexity. Typical values for -3dB beamwidth
and SLL for varying number of elements can be seen in Table 4.2. This table
shows the results obtained from a constrained iterative process of
conventional code. The table includes the beamwidth values for different
element number of radiation patterns.
53
Chapter Four
Simulation Results
Table 4.2 Summary result for a different number of element values
Number of elements
M
2
6
12
25
SLL(dB)
HPBW(degree)
-3
-12.45
-13.1
-13.25
86.014
19
8.9
4.5
4.2.3 Performance of Comparison for Smart Antenna System Using
Different Antenna Array Algorithms
This section would be concerned with studying or comparing several
types of algorithms and the comparison would be relying on the same
parameter that examined in recent subjects above. Figure 4.3 shows linear
and polar array pattern plot sequentially for LMS, RLS, NLMS, SMI, CMA,
CGM, REDS, GA, and PSO algorithms.
Figure 4.3 linear & polar plot, for radiation pattern for linear array using a different type
of algorithms with N=5, d=0.5λ and DOA user at 𝟎° .
54
Chapter Four
Simulation Results
All algorithm has a DOA 0° and also an interference DOA at 45° , for
linear array using a different type of algorithms with 5 elements and d=0.5λ,
except the PSO and GA algorithm which would be discussed there
interference effect on radiation pattern in the following section.
Table 4.3 Summary result for applying a different type of algorithms
Algorithm type
LMS
RLS
NLMS
SMI
CMA
CGM
PSO
GA
REDS
SLL(dB)
-11.1
-10.9
-9.8
-9.5
-8.2
-9
-9
-6.5
-10
HPBW(degree)
20.76
25.89
20.6
20.1
31.6
30.4
22.2
26.5
20.9
The results of simulated algorithms are compared to various
parameters like computation complexity, convergence rate, SLL, and HPBW.
Accordingly, the analysis showed that, given computational complexity, RLS
was the best algorithm followed by LMS, CMA, and SMI based on their
magnitude of error. The LMS algorithm is a common solution used for the
technique of beamforming. Generally, the RLS algorithm converges with
magnitude level faster than the LMS algorithm but complexity is applied to
the price paid. The Conjugate Gradient Method is expected to converge with
less iterations but still has a wide HPBW compared with the other type of
algorithm, the previous disadvantages of each algorithm could be solved with
REDS algorithm. As can appear in Table 4.3 the LMS, NLMS and the REDS
algorithms incorporate lower degree of sidelobe. Also, REDS, LMS, RLMS
and SMI algorithms give the deepest nulls at the directions of interfering
signals. PSO produce the most stable beam pattern and also gave initial
adaption aspect.
4.2.4 Comparison of the Performance of Smart Antenna System Using
Different Antenna Array Geometry
The comparison in this part was based on the changing in antenna array
geometry, where the shapes that would be used are linear, rectangular,
circular and cube arrays, this test done with standard LMS algorithm by
N=5.and d=0.5. Figure 4.4 illustrates the polar plot for radiation pattern using
different array shapes.
55
Chapter Four
Simulation Results
Figure 4.4 Polar plot for comparison between different antenna array geometry
Figure 4.5 shows the mean square error curve for each antenna
geometry, and also to compare and differentiate between each shape Table
4.4 below clarify the summary results for each antenna array radiation
pattern.
Figure 4.5 Mean square error curves for LA, CA, PA and CUA arrays
a circular array can be considered as the fastest convergence array,
which is about 10 iteration, while in linear array iteration reaches 60 as
shown in Table 4.4.
Table 4.4 Summary result for applying linear and circular array
geometry
Antenna type
SLL
HPBW(dB)
MSE
Linear (M=10)
Circular
(M=10)
-12.9
-7.8
10.11
21.23
0.0029
1.5679e-04
Iteration
numbers
60
11
From the results of the Table 4.4 and Table 4.5, it could be illustrated
that the circular array and cube array had the lowest percentage of error, and
under the same number of iterations.
56
Chapter Four
Simulation Results
Table 4.5 Summary result for planner and cube array geometry
Antenna type
SLL
HPBW(dB)
MSE
Planar (M=5*5)
Cube
(M=5*5*5)
-23.8
-54
14.65
9.72
0.0024
1.0036e-04
Iteration
numbers
55
45
The beamwidth (HPBW) and side lobe level (SLL) per every
configuration had shown in Table 4.4 and Table 4.5 and the result clarifies
that the linear array has a suitable result but it takes a lot of time to start with
adaption process comparing with the other geometry
4.3 Performance of Smart Antenna System via Variable Step Size Aspect
This slit gives the achievements of proposed μ(n) improvement
algorithm. In this simulation, the type of algorithm that chose to improve is
the LMS algorithm, and it is worth mentioning that this proposed
improvement could apply in any algorithm had step size factor (μ), the
number of sample intervals applied equally to (K=100). Figure 4.6 produced
the linear and polar plots of the array pattern in dB for the LMSSI algorithm,
the main beam pattern of the system here steers the direction at 00 .
Figure 4.6 Linear & polar radiation pattern plots of LMSSI algorithm at the number of
element M=10, DOA at 0, and DOA interference at 30,-10.
The proposed step size algorithm in Figure 4.7 converges fast and
without fluctuation compared with the LMS algorithms. On the other hand,
the variation of weight values using the SSF algorithm in the steady-state is
more stable than LMS in converging weights. From Figure 4.7, it can be
observed that, the number of samples needed for variable μ(n) algorithms to
converge are much lower than those for the LMS algorithm.
57
Chapter Four
Simulation Results
(a)
(b)
Figure 4.7 Tracking of the desired signal for (a) LMS (b) LMSSI algorithms
Figure 4.7 display the obtaining and tracking of the array output for the desire
signals after about 80 iterations for LMS and about 20 iterations for variable
step size algorithm.
Figure 4.8 Magnitude of linear array weights for LMSSI algorithm
The proposed algorithm LMSSI reduces and increases the step size
μSS (k) to track any changes in the smart antenna environment.
Figure 4.9 Mean square error curves for LMSSI
58
Chapter Four
Simulation Results
Moreover, from the MATLAB software, at the steady-state, the LMS
error is almost 4.5028e-04 whereas the LMSSI error is almost 2.7274e-04 at
around 100 iterations.
4.4 Performance of Smart Antenna System via SIR-LMS Algorithm
The SIR-LMS algorithm is developed for SA application which
combined the individual pretty side of maximum SIR and LMS algorithms.
This algorithm presented when the number of samples applied is (K=100).
The linear and polar radiation pattern plots for the SIRLMS algorithms are shown in Figure 4.10 This figure shows the null depth
provided by the SIR-LMS at −61.5dB. As shown in this Figure 4.10 the main
beam pattern of the system steers the direction at 00 .
Figure 4.10 Linear & polar Radiation pattern plots of SIR- LMS algorithm at the number
of element M=10, DOA=𝟎° , and DOA interference at−𝟒𝟓° .
Figure 4.11 illustrates the MSE curve for the SIR-LMS algorithm. It
can be shown that the SIR-LMS algorithm can achieve the fastest
convergence than any previous algorithms.
Figure 4.11 Mean square error curves for LMSIR algorithm
Moreover, comparison with some algorithms like conventional LMS
and NLMS algorithms start converging, simultaneously, from iteration
numbers 80 and 25, while the converge in the SIR-LMS algorithm starts to
59
Chapter Four
Simulation Results
from the primer or initial stat of iterations. In this case, the SIR-LMS error is
7.2038e-28 with zero iteration number.
Figure 4.12 Tracking of desired signal
Figure 4.13 Magnitude weights for LMSIR
Figures 4.12&4.13 display that, the SIR-LMS start adjusting from the initial
weight vector values to optimal weights. On the opposite, an algorithm such
as LMS, NLMS, ... etc, is beginning to converge from arbitrary weight
towards optimum weight values the simulation results illustrated that the
proposed SIR-LMS algorithm delivers noticeable improvements in the
elimination of interference, the convergence rate and weight stabilizing
which quickly reach to their optimum values and without fluctuation
compared with the other algorithms like LMS and NLMS algorithms.
4.5 Comparison of Smart Antenna System Executions Using Different
Improvements for CMA (ICMA&BCM) Algorithm
A fastness of converging CMA can be improved with many sides, one
of them that examined in this section was using variable step size factor
(μSS (k)), that was introduced in section 4.3 above, Figures 4.19&4.20 shows
the beam pattern and the effect of the ICMA algorithm on the received multipath combined signal to achieve the desired signal path1.
The second aspect to improve the converging speed of CMA is the
Least Square CMA (LS-CMA) which is a block update iterative algorithm
that is guaranteed to be stable and easily implemented. Typically LS-CMA
converges in 5 to 10 iteration regardless of the block size and as seen the
static LS-CMA can converge 100 times faster than the conventional CMA.
However, the computational load makes the LSCMA impractical for a realtime application, but this algorithm gave a disadvantage on the effect of the
algorithm on the received multipath combined signal for getting the desired
signal, so the proposal refinement is to blocking the conventional CMA
algorithm that leads to convergence rate at 6 iterations only and also obtained
obvious desire signal (path 1)
60
Chapter Four
Simulation Results
The simulation was used on a uniform linear array of 8 elements.
Described the direct path for which the signal arrives as a 32-bit binary
chipping series with a value of ±1, at 45 °, the first multi-path signal reaches
at 0 ° and the second multi-path signal reaches at -30 °. Figure 4.14 shows
the arriving signal paths of the direct path, the first multipath, the second
multipath, and the combined path as seen by the receiver. Figure 4.15
indicates the plot of the array factor as well as how the CMA algorithm
minimized the multipath signals whereas guiding the maximum signal to a
direct path. Figure 4.16 shows the effect of the algorithm on the received
combined signal.
a
b
c
d
Figure 4.14 Arriving signal amplitude ((a) Direct path (desire signal), (b) path 2, (c) path 3,
and (d) combined signals)
Figure 4.15 Linear & polar Radiation pattern plots for conventional CMA algorithm at 50
iterations number of element M=8, DOA=𝟒𝟓° , and DOAs interference at𝟎° &−𝟑𝟎° .
61
Chapter Four
Simulation Results
a
Figure 4.16 Arriving signal and Output signal for CMA
Figure 4.19Linear & polar Radiation pattern plots for ICMA algorithm at 23 iterations
number of element M=8, DOA=𝟒𝟓° , and DOAs interference at𝟎° &−𝟑𝟎° .
Figure 4.20 Arriving signal and Output signal for ICMA
Figure 4.21show the comparison between two error curve (a), for
conventional CMA and (b) for ICMA, and as could be obvious the number
of iteration for each algorithm.
62
Chapter Four
Simulation Results
Figure 4.21 Error signal amplitude for CMA and ICMA algorithms
Figure 4.20 shows that the performance of the CMA algorithm on
output signal improvement when the number of iteration n is decreasing from
50 to 23.
Figure 4.22 Linear & polar Radiation pattern plots for BCMA algorithm at 6 iterations
number of element M=8, DOA=𝟒𝟓° , and DOAs interference at𝟎° &−𝟑𝟎° .
The combination of three algorithms associated with the CMA
algorithm shown below.
Figure 4.23 Linear & polar radiation pattern plots for CMA, ICMA and BCMA algorithm
at number of element M=8, DOA=𝟒𝟓° , and DOAs interference at𝟎° &−𝟑𝟎° .
As can be seeing in Figure 4.23 the two proposal algorithm achieved
more accentuate result in terms of depth nulling beside the fastest
convergence.
63
Chapter Four
Simulation Results
4.6 Smart Antenna Performance with Particle Swarm Optimization
(PSO) Algorithm
For an adaptive array antenna simulation, the training sequence's 200
input signals having values marked as ±1to represent a transmitted
dispatching binary quantities as shown in Figure 4.24.
Figure 4.24 Binary Signal.
Initialize the parameters that will be used for the system development
in MATLAB shown in Table 4.5 below.
Table 4.5 Initialize simulation parameter
Parameter
Frequency of operation
spacing between element
Value
900 MHz
𝑁𝑜.Of element
Type of antenna
The phase between two element
Variable(8-12-18-25)
Isotropic
0 Radian
0.5𝜆
For PSO algorithm design, the necessary parameters were described as
supervening:
Size of population P equal to 10, Maximum number of generation
equals 50, Maximum and minimum value of inertia weight range are (0.80.3), Acceleration constantsC1 and C2 are 2. The previous parameters should
be given before algorithm applications. Their values affect the process of
optimal solution search and results.
4.6.1 Weight Vector Optimization
The radiation pattern results according to the equation for uniform
element spacing for the optimized weighting vector (phase and amplitude) of
ULA could be shown in Figure 4.25.
64
Chapter Four
Simulation Results
Figure 4.25 Linear and polar for PSO algorithm with uniform array (ULA) Radiation
Patterns at 8, 12, 18, and 25 elements and DOA user at𝟎°.
Table 4.6 indicates that increasing the element numbers lead to an
increase in the numbers of sidelobe also that companion with reducing in SLL
and HPBW leading to increasing in directivity calculation were performed
using fitness function equation.
Table 4.6 (ULA) Parameter result
No.elements
Directivity
Sidelobe level
reduction (𝑑𝐵)
Beamwidth (𝑑𝑒𝑔)
8
21.043
-10
12
22.983
-12.1
18
24.888
-15
25
26.27
-14
12.8
9-8.8
5.3
3.97
As observed in Table 4.6 that the directivity is increased to 24.888,
HPBW is decreased to 5.3 °and the minimal SLL up to -15 dB with N
increased to 18.
65
Chapter Four
Simulation Results
Figure 4.26 Linear and polar for PSO algorithm with Uniform Antenna Array (ULA)
Radiation Patterns at λ/4, λ /2, 1.5 λ and 3 λ displacement spacing element and DOA user
at𝟎°.
Figure 4.26 shows that with an increase in displacement distance
between element the sidelobe level (SLL) would decrease with the same
number of elements, as can visualizing at λ /2 the SSL equal to -13.2 dB,
while at 3 λ the SLL equal to -8.6 dB.
4.6.2 Inter Element Spacing Optimization
The displacement distance between neighboring two elements is the
one of influential LA performances. PSO algorithm used to find the optimal
elements distance value for the element with the equal excitation
amplitude 𝑎𝑛 = 1 and excitation phase 𝜃𝑖 = 0° employing the equation of
fitness function. The simulation result of the radiation pattern of the
optimized weighting vector is viewed in Figure 4.27. The array elements
number suggested by the author is equal to 4, 8, 12, 18 and 25.
66
Chapter Four
Simulation Results
Figure 4.27 Linear and polar for PSO algorithm with non-uniform array (NLA) Radiation
Patterns at 4, 8, 12, 18, and 25 elements and DOA user at𝟎°.
Table 4.7 (NLA) Parameter result
No.elements
Directivity
Side lobe level
reduction (dB)
Beam-width
(𝑑𝑒𝑔)
8
21.716
-10.3
12
23.542
-12.2
18
25.845
-10.52
25
26.02
-13.5
9.5
6.87
4.19
3.75
As shown above, there is a clear and good improvement in the
difference between patterns of the two types of optimization variables, so by
changing the dimensions of antenna array system the enhancement appears
in both directivity and beamwidth, and the number of sidelobe also. Table
4.8, shows the resulting displacement position of each element only by using
the (PSO) optimization algorithm process.
67
Chapter Four
Simulation Results
Table 4.8 (NLA) antenna array position on x-axis normalize to
𝒍𝒂𝒎𝒅𝒂(λ)
[𝑥0 , 𝑥1 , 𝑥2 , . . . . . . . . , 𝑥𝑛−1 ]
8element
12element
18element
[0.1983,0.3274,0.0958,0.1994,0.
1782 ,0.1571 ,0.0922 ,0.0374]
[0.2342,0.1929,0.1822,0.1925,0.
2207,0.1992,0.1980,0.0995
,0.1326,0.284,0.2366,0.1774]
[0.2715,0.3019,0.04232,0.3044,
0.2107,0.03251,0.0928,0.1822,0
.3191,0.3216,0.0525,0.3235,0.3
190,0.16179,0.2667,0.04729,0.1
4058,0.3052]
∑ [𝑥0 , 𝑥1 , 𝑥2 , . . . . . . .
. . , 𝑥𝑛−1 ]= (total
length of NLA)
(1.2858)
∑ [𝑥0 , 𝑥1 , 𝑥2 , . . . . . . .
. , 𝑥𝑛−1 ]= (total length
with 0.5 spacing )
(4)
(2.3498)
(6)
( 3.14)
(9)
4.6.3 Weight Vector and Inter Element Spacing Optimization
PSO algorithm workout even to study the possibility of developing
weights and spacing distance between elements together and that will be
called improved particle swarm optimization (IPSO). The algorithm will be
operated by two parts first one uses to calculate the distances and then
combined with the original program to update the weights. The fitness
function are given in chapter 3, design for this purpose, observations are
made for N = 8 and 20 in Table 4.9.
68
Chapter Four
Simulation Results
Figure 4.28 Linear and polar for IPSO Radiation Patterns at 4, 8, 20 elements and DOA
at𝟎°
Table 4.9 IPSO parameter results
No .element
Directivity
8
20
22.48
24.39
Sidelobe level
reduction (dB)
-15.7
-15
Beamwidth (𝑑𝑒𝑔)
10
4.77
As observed in Table 4.9 that at 8 elements the directivity is increased
to 22.48, HPBW is reduced to 10° and the SLL is minimizing up to -15.7 dB
So, from the comparison with the old test, it's clear that an enhance of -5.7
dB in SLL and also in the number of the side lobe, 2.8° in the HPBW by
using IPSO algorithm shown in Figure 4.28.
4.7 Nulling Effect on the Radiation Pattern
One of the most important applications of a smart antenna is the nulling
effect. The null control was designed to transmit minimal power in locations
where the eavesdropper is present. Controlling the parameters, the excitation
amplitude, the phase, the array distance spacing, and the number of the
elements will achieve the null control, but this thesis deal with a number of
elements for uniform LA and non-uniform LA by using the fitness function
equation. Two cases are considered using a different number of antenna
elements and nulls at specified directions.
Figures 4.29 & 4.30 showed an improvement in the main beam
direction and reduction in the other direction which contains the interference
according to Table 4.10 results.
69
Chapter Four
Simulation Results
Figure 4.29 Radiation Pattern for 8 elements at DOA at 𝟎° and 𝟓𝟎°&−𝟓𝟎° and 𝟑𝟎°&−𝟑𝟎°
Interferences Respectively
Figure 4.30 Radiation Pattern for 8, 12 and 20 elements at DOA at 𝟎° and 𝟑𝟎°&−𝟑𝟎°
Interference Respectively
Table 4.10 SLL, HPBW AND depth null results
𝑁𝑜.element
HPBW
SLL
Null depth at 30
Null depth at 30
8
12
12.4
4.15
-11.8
-12.8
-40
-46.1
-43.4
-43
Figure 4.31 shows that with an increase in displacement distance
between elements, the sidelobe level would decrease with the same number
of elements, as can visualizing at λ /2 the SLL equal to -13 dB, while at 1 λ
the SLL equal to 0 dB.
70
Chapter Four
Simulation Results
Figure 4.31 Radiation Pattern at λ /2, 0.75 λ and 1 λ displacement spacing element at 8
elements and DOA user at 𝟎° and 𝟓𝟎°&−𝟓𝟎° DOA Interference Respectively
4.8 Smart Antenna Array System Performance via REDS Algorithm
REDS algorithm is applied to different geometries of the smart antenna
array to investigate its potentials in beamforming convergence. The essential
benefit of this smart system is that it appears to have the quickest rate of
convergence. The results of the simulation are comparing with three classical
geometries of the adaptive array, which is linear, circular, and rectangular
antenna array. In each shape, the weight vector of the array, mean square
error, and radiation pattern performance are presented.
The number of elements in all discussion is presented in Table 4.11
for comparison purposes.
Table 4.11 Initialization parameters
𝑁𝑜. 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠
Displacement distance
Linear array
10
0.5𝜆(x-axis)
Circular array
10
2π/𝑁
Planar array
5*5
0.5𝜆 ∗ 0.5𝜆(x-y plane)
A linear array made up of isotropic elements that applied according to
the below requirements:
• Input signal S(k) = sin(2πft(k)) with f =
(1: K) ∗
T
K
1
T
= 900 MHZ and t =
Where K represents the sample interval numbers, and T is
the period and that will be the desired signal d(k).
(d(k) = S(k)).
• Desired DOA is θ0 = 00 and interfering signal, I1 each of the
rndn(1, K) with DOA, θ1 = 450 .
71
Chapter Four
Simulation Results
• Step size initial parameter µ0 = 1
• Every element in the array combined with zero mean Gaussian noise
with variance σ2n = 0. 001 added to the input signal in the array.
• • Signal to Noise Ratio (SNR) and Signal to Interference Ratio (SIR)
are placed at 30 dB and 10 dB
• The step size parameter is calculated according to the equation. (2.42)
and within the MATLAB software package as follows
μ = 1/(4 ∗ real(trace(R xx ))), where R xx = X ∗ X′
• In the beginning, every weight vector is set to zero.
Two types of noise are discussed here to clarify the performance of the
REDS algorithm under the effect of different noise channel, which is,
Additive White Gaussian Noise channel (AWGN) and Rayleigh fading
channel, respectively.
4.8.1 (AWGN) Channel Results
The basic noise model used in information theory to minimize the
effect of many random processes that occur in nature. To measure the
efficiency of the smart antenna system, for each received signal at element M
an (AWGN) channel model is authorized Figure 4.32 contains just a zero
mean AWGN.
The curves of mean square error are presents in Figures 4.33& 4.35&
4.37 (c) provide that the convergence speed utilize PA geometry is quicker
than LA and CA configuration. Moreover, the array output using PA was
good predicts of the input signal comparing with LA and CA but, at the same
time, it can see that the error magnitude of CA is less than both LA and PA.
Figure 4.32 Linear and polar radiation patterns for linear array (LA), all geometry worked
out with 50 sample/AWGN.
72
Chapter Four
(a)
Simulation Results
(b)
(c)
Figure 4.33, (a) Signal tracking of desired and output curves (b)Weight magnitudes
(c)MSE for linear array (LA) worked out with 50 sample/AWGN.
Figure 4.34 Linear and polar radiation patterns for circular array (CA), all geometry
worked out with 50 sample/AWGN.
(a)
(b)
(c)
Figure 4.35, (a) Signal tracking of desired and output curves (b) Weight magnitudes (c)
MSE for circular array (CA) worked out with 50 sample/AWGN.
73
Chapter Four
Simulation Results
Figure 4.36 Linear and polar radiation patterns for planar array (PA), all geometry
worked out with 50 sample/AWGN.
(a)
(b)
(c)
Figure 4.37, (a) Signal tracking of desired and output curves (b) Weight magnitudes (c)
MSE for planar array (PA) worked out with 50 sample/AWGN.
The studies indicate that the REDS algorithm has a more fast rate of
convergence by comparison with the previous algorithm like LMS which
needs about more than 80 iteration to start adaptation processes, NLMS
algorithms that need more than 25 iterations, CMA algorithm that need 50
iteration and so on. Moreover, Figures 4.33&4.35 &4.37 (a) can illustrate the
array output of PA and CA is a good estimate of the desired signal.
Also Figures 4.33&4.35&4.37 (b) shows the magnitudes of the
estimated complex weights of the, LA, CA and PA array configurations,
which is the magnitude of the weights for each element in array .the results
also show that the performance of the REDS have very good stability and it
can notice the adaptation processes starts approximately after 5 samples only.
As apparent in Figures 4.32&4.34&4.36 above that LA and PA
antenna configuration have more side lobe than CA array but at the same time
the null depth of LA and PA is larger than CA array, also the sidelobe
amplitude PA radiation pattern is less than LA and CA array configurations.
The numerical result can be shown in Table 4.12 below.
74
Chapter Four
Simulation Results
Table 4.12 AWGN results
LA
CA
PA
Depth null(dB)
SLL(dB)
Weight stability
-61
-60
-96
-13
-10
-23.6
8 iterations
5 iterations
7 iterations
Maximum mean
square error
5.9119e-04
0.0141
5.1338e-04
4.8.2 Rayleigh Fading Channel Results
Rayleigh
fading is
a statistical model for
the
effect
of
a propagation environment on a radio signal, is most applicable when many
objects in the environment scatter the radio signal and that lead to no
dominant propagation along a line of sight between the transmitter and
receiver, such as that used by wireless devices
The magnitude of a signal that has passed through such a transmission
medium (also called a communication channel) will vary randomly is
assumed. To evaluate the performance of an adaptive antenna with an
analytical model operating with AWGN and Rayleigh fading environment.
In the assessment of results the following criteria are considered:
• The signals arriving at every element undergo independent Rayleigh
fading, both for the desired and interfering signal.
• Both desired and interference signals have the same amplitude as is a
release from 0° and45°.
• Every simulation relates to a run (data rate) of 500 Kbits.
• Doppler frequency fd of 117 Hz, corresponding to the speed of 140
km/h at 900 MHz will be used in this simulation to present a worstcase scenario.
• All antenna geometry is implemented when K=60 is set as the sample
number.
Figure 4.38 Linear and polar radiation patterns for linear array (LA), all geometry worked
out with 60 sample/ Rayleigh channel.
75
Chapter Four
(a)
Simulation Results
(b)
(c)
Figure 4.39, (a)Signal tracking of desired and output curves (b)Weight magnitudes (c)MSE
for linear array (LA),all geometry worked out with L=6,K=60 sample/ Rayleigh channel.
Figure 4.40 Linear and polar radiation patterns for circular array (CA), all geometry
worked out with 60 sample/ Rayleigh channel.
(a)
(b)
(c)
Figure 4.41, (a)signal tracking of desired and output curves (b)Weight magnitudes (c)MSE
for circular array (CA),all geometry worked out with L=6,K=60 sample/ Rayleigh channel.
76
Chapter Four
Simulation Results
Figure 4.42 Linear and polar radiation patterns for planar array (PA), all geometry
worked out with 60 sample/ Rayleigh channel.
(a)
(b)
(c)
Figure 4.43, (a)Signal tracking of desired and output curves (b)Weight magnitudes (c)MSE
for planar array (PA), all geometry worked out with L=6, K=60 sample/ Rayleigh channel.
In Figures 4.38&4.40&4.42 the results for the REDS algorithm are
shown at block length set to 6. The results show that the performance of the
REDS is working better when compared with planar array than the LA&CA
array shapes. As shown in Figures 4.39&4.41&4.43 the convergence speed
using LA antennas is slower than the CA antenna configuration. Also, the
output of the array using CA is a good predictor of the input signal comparing
with the LA configuration. Figure 4.43 indicates the faster convergence speed
and the strong estimate of the output array in the PA configuration.
From Figures 4.39&4.41&4.43 (c), the LA antenna converges after
more than 30 iterations, CA antenna converges after 20 iterations, whereas
the PL antenna convergence can occur at about less than 10 iterations.
Figures 4.39&4.41&4.43 (b) shows the magnitudes of the complex
weights of LA, CA, and PA configurations. These figures show the
magnitude of the weights for each element in the array. And as illustrated in
the Figures 4.39&4.41&4.43 above, the data show that PA efficiency is better
and faster than other types of configuration. Numerical simulation results of
array factor with Rayleigh channel are shown in Table 4.13.
77
Chapter Four
Simulation Results
Table 4.13 Rayleigh Fading results
LA
CA
PA
Depth null(dB)
SLL(dB)
Weight stability
-62
-48
-90
-13
-6
-24
40
20
10
Maximum mean
square error
8.7810e-04
0.0033
0.0014
Figures 4.38&4.40&4.42 show that the PA array introduced a deep
null which about −90 dB in the direction of the interferer, However, the
simulations show, the LA and CA array configurations provide a deeper null
of about -62 dB and -48 dB at K=60.
It is apparent from this array pattern in Figures 4.38&4.40&4.42 the
CA array gives the minimal sidelobe number while PL array gives the lowest
array SLLs compared with the other array configurations. Figures
4.38&4.40&4.42 presents that the amplitude of the side lobes of the radiation
pattern from the PA array is greater than the amplitude of the side lobes of
the radiation pattern from LA and CA array.
Multipath and their effects on antenna array pattern with using REDS
algorithm also have been introduced, Figure 4.44 illustrates a linear and polar
plot for antenna array pattern curve under multipath signals.
78
Chapter Four
Simulation Results
Figure 4.44 Linear & polar radiation pattern plots with 8 element linear array (LA)where
a-desire(S)=0/Interference(I)=45,-20,b-S=0,35/I=60,-30,c-S=0,30/I=50,-30,-70,d-S=0,20,30/I=50,-45,-70, worked out with 50 sample/AWGN.
Table 4.14 Analyze the multipath effect
a
b
c
d
Desired DOA(S)
0°
0° &35°
0° &30°
0° &30° & − 20°
Interference DOA(I)
45° & − 20°
60° & − 30°
°
50 & − 30° & − 70°
50° & − 45° & − 70°
Depth null (dB)
-48.5&-59.2
-61&-41.7
-38.1&-42.3&-62.7
-36.7&-51.2&-55.1
As illustrated in Table 4.14 above, the effectiveness of the REDS
algorithm appears to guidance or steering the mean beam in the direction of
users and steering the depth extinct to interference directions.
4.9 BER
In this section, the estimated BER had been measured, by sending a
message of 1000 Bit in random way using MATLAB program and comparing
the received single with the transmitted one. With AWGN as a source of noise
in the channel.
This section studied the effect of the number of elements on the overall
system performance. Figure 4.45 shows at lower noise values there are no
errors when receiving a signal message. However, errors occur when the
noise starts to reach high levels and start to increase by increasing the values
of noise giving much higher error values. This figure shows the comparison
of the different number of elements (m=2,4,8,16) and shows also that with
79
Chapter Four
Simulation Results
two elements give a lot of errors than using 8 elements in the array as so on
when the number of elements increased, better results are given.
Figure 4.45. BER curve for different number of the linear array
The circular array represents 2D antenna arrays. The circular array is
also very resistive to noise and the BER is approximately close to that of the
linear array approximately with the same number of elements (M=4, 8, 16)
as shown in Fig.46
Figure 4.46. BER curve for different number of the circular array
Table 4.15 and Figure 4.47 shows the result value of the arrays with
8 elements for linear array and circular array.
Table 4.15. Number of errors concerning noise
Noise (10−16)
1
5
10
50
100
500
No. error of linear
array
0
0
0
0
0
0
80
No. error of circular
array
0
0
0
0
0
0
Chapter Four
Simulation Results
1000
5000
10000
30000
50000
100000
200000
300000
400000
500000
600000
800000
1000000
1500000
0
0
0
0
0
1
13
39
68
103
121
162
204
274
0
0
0
0
1
9
62
111
142
188
216
266
310
386
Figure 4.47. BER curve for linear and circular antenna array geometry with 8 elements
4.10 SNR
The output Results of the signal to noise ratio with applying Noise of
1*10
for 1000 Bits message signal to test the benefit of changing the
element numbers on SNR values, are given in Table 4.16
−16
Figure 4.48 Relation between number of elements and SNR (dB) for a linear and circular
array
81
Chapter Four
Simulation Results
Table 4.16 SNR to a different number of elements
Number of elements
SNR(dB)
Linear array
SNR(dB)
Circular array
2
4
8
12
16
20
-13.8067
-7.7861
-1.7655
1.7563
4.2551
6.1933
-22.7126
-16.6925
-4.3701
0.5706
3.5825
5.7610
Figure 4.48 and Table 4.16 shows that there is a small difference in
the SNR curve for linear and circular array and it is clear to observe that the
SNR curve of overall system increase with the number of the elements of the
array regardless of the geometry of antenna array.
82
Chapter Five
Conclusions and
Future Work
Chapter Five
Conclusion and Future Works
5.1 Conclusion
From the work presented in this thesis some of the points could be
concluded as follow:
1-It was noticed that by increasing the number of array elements the main
lobe will be (smaller beamwidth) and more director, but the sidelobe numbers
will increases. And shows that lead to an increase in the distance between
elements leads to produce a grating lobe which is typical to the main lobe but
in the reverse direction.
2-The analysis showed that RLS was the best algorithm despite the
computational difficulty followed by LMS, CMA, and SMI based on their
error magnitude. LMS algorithms are well known for their simplicity and
robustness. The Conjugate Gradient Method is expected to converge with
fewer iterations but has a high HPBW relative to the other algorithms, the
previous disadvantages of each algorithm could be solved with the REDS
algorithm. PSO produce the most stable beam pattern and also gave an initial
adaption aspect.
3- Through using different antenna arrays geometries which are linear,
rectangular, circular and cube with LMS algorithm, it could be found that the
circular array and cube array had the lowest percentage of error, and under
the same number of iterations circular array can be considered as the fastest
convergence array, which is about 10 iteration while in linear array reach to
60 iterations. Despite the result, it clarifies that the linear array has the desired
outcome in terms of SLL and HPBW, but it takes a lot of time to begin with
the phase of adaptation compared to the other geometries.
4-Step size one of most effective factor on some of the algorithm that
dependent on step size factor through adaption process such as LMS, NLMS,
RLS, CMA, as appear in get a more stable and fasten convergence rate and
display how the output of the antenna array acquires and chase the desired
signal with about 80 iterations for LMS and about 20 iterations for the
algorithm with a variable step size.
5-Hybridtation between the good aspect of fixed beam with maximum SIR
algorithm and LMS algorithm produce (SIR-LMS) algorithm with the initial
adaptation process and give an enhancement in MSE.
6-The improvement would also expiate on CMA algorithm with two aspects
firstly by applying variable step size (ICMA) which produce an enhancement
on the amplitude of array output and also on convergence rate so the number
of iteration change from 50 by basic CMA to 23 with (ICMA), secondly by
83
Chapter Five
Conclusions and Future Work
applying the blocking idea (BCMA) on basic CMA and that lead to enhance
the convergence rate to 6 iterations only with lower MSE value as appear
earlier.
7- Applying PSO on smart antenna came in two parts, at the beginning PSO
used to find the adaptive weight vector. The second part of applying PSO on
smart antenna which designs non-uniform linear array which presented an
effective array with appearance improvement in SLL -6 dB, BW 3˚and
directivity in 2 dB as compared with standard linear array and the most
important thing have created an array with less antenna number and antenna
with less length with the same number of antennas would economically
benefit us.
8-REDS algorithm utilizing on the smart antenna with three array geometry
linear, circular and planner gave a more stable and acceptable performance,
it can be seen that the number of iteration reduce to 5 for circular, 7 for a
planner and 8 iterations for the linear array as compare with other algorithms
that reach to 85 and 25 for LMS and NLMS resistivity, also gave an
improvement in MSE as appearing in chapter four early, the effectiveness of
REDS algorithm also appear more robust and remain stable when applying
another type of noise like Rayleigh fading noise as comparing with another
algorithm in the single beam and multi-beam signals.
9-Calculating BER by different antenna array geometry shows the
effectiveness of smart antenna to receive the data with great resistivity to the
noise and fading signal. It also demonstrates BER's progress about a growing
number of array components.
10- SNR estimated in terms of different number of elements with linear and
circular array configuration, which presents an increasing in SNR with
increase in number of elements and that gives a great performance to a smart
antenna to receive the transmitted signal with the lower number of error.
5.2 Future Work
1. Currently, proposed algorithms have been implemented using MATLAB
software alone. Executing an algorithm using either a Digital Signal
Processor (DSP) chip or Field Programmable Gate Arrays (FPGA) would
make it useful to check the actual complexity of the computation.
2. Using the PSO algorithm to design a more non-uniform antenna array with
an acceptable performance like a circular array.
3. Design different array geometry like hexagonal and planner circular,
cylindrical...etc., and examine their performance through various array
algorithms.
84
Chapter Five
Conclusions and Future Work
4. Analysis of the impact of mutual coupling between the array's
components.
5. Study the effect of each algorithm on power reduction property, by
improve the base station sensitivity or reduce their radiated power, this
would translate into cell phone battery life extension and reduced the base
stations density at the required coverage area to reduce RF pollution
levels.
85
References
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List of Publication
• A. A. K. Qasim and A. H. Sallomi, “Optimisation of Adaptive Antenna
Array Performance Using Particle Swarm Algorithm,” 1st Int. Sci. Conf.
Comput. Appl. Sci. CAS 2019, pp. 137–142, 2019.
• A. A. Qasim, “Performance improvement for smart antenna system least
square beamforming algorithm,” pp. 155–165, 2020.
• A. A.-K. Qasim, “Analysis of Adaptive Antenna System using Rabid
Euclidean Direction Search (REDS) Method for Different Antenna
Geometry,” Bull. Electr. Eng. Informatics, vol. 10, no. 2, 2020.
• Adheed Hassan Sallomi , Aseel Abdul-karim Qasim ‘’Constant modulus
algorithm (CMA) improvement by variable step size and blocking
technique”, International Journal of Interactive Mobile Technologies
(IJIM)(ISSN: 1865-7923, SCOPUS Indexed).(Acceptance)
• Aseel Abdul-Karim Qasim, Adheed Hassan Sallomi ‘’ Design and
Analysis of Phased Array System by MATLAB Toolbox”, published in
the Al Kitab Journal for pure sciences Vol.4, Issue 1, June 2020.
(Acceptance)
‫جمهورية العراق‬
‫وزارة التعليم العالي و البحث العلمي‬
‫الجامعة المستنصرية‬
‫كلية الهندسة‬
‫قسم الهندسة الكهربائية‬
‫تطبيق خوارزميات مختلفة في تشكيل الحزمة التكيفية في نظام الالسلكي‬
‫رسالة مقدمة‬
‫الى قسم الهندسة الكهربائية‪،‬كلية الهندسة‪،‬الجامعة المستنصرية‬
‫كجزء من متطلبات نيل درجة ماجستير علوم في الهندسة الكهربائية‪/‬‬
‫االلكترونيك واالتصاالت‬
‫من قبل‪:‬‬
‫اسيل عبد الكريم قاسم‬
‫(بكالوريوس هندسة كهربائية)‬
‫باشراف‬
‫أ‪.‬د‪ .‬عضيد حسن سلومي‬
‫ربيع أول ‪ 1442‬ه‬
‫تشرين ثاني ‪ 2020‬م‬
‫الخالصة‬
‫هناك اختالف معين عند مقارنة أنواع مختلفة من الخوارزميات مثل ‪Least Mean Square‬‬
‫)‪ (LMS‬والتي تستخدم إلى حد كبير بسبب التعقيد المنخفض للحسابات وسهولة التنفيذ‪ .‬وغيرها من‬
‫الخوارزميات التي تعتمد على مربع الخطأ (‪ ، )algorithm that content step size factor‬مثل‬
‫((‪ ، ) Recursive Least Squares)RLS‬و ( ‪ ، )Gradient)CG(Conjugate‬و()‪)REDS‬‬
‫‪ ، )Rapid Euclidean Direction Search‬وتكون أسرع ولها مربع خطأ صغير مقارنا"‬
‫ضا مثل الخوارزمية العمياء مثل‬
‫بخوارزمية ‪ .LMS‬سيتم تقديم بعض الخوارزميات األخرى أي ً‬
‫خوارزمية ()‪ ) Constant Modulus Algorithm)CMA‬وخوارزمية ((‪(Sample )SMI‬‬
‫‪ ، Matrix Inversion‬كما وتوجد خوارزميات التحسين مثل خوارزمية (( ‪Particle Swarm‬‬
‫ضا اختبارات مختلفة لعدد من‬
‫‪ .)Optimization algorithm (PSO‬سوف تناقش هذه الرسالة أي ً‬
‫مصفوفات الهوائيات ذات الترتيب الهندسي مثل )‪Planner ،Circular (CA) ،Linear (LA‬‬
‫)‪ ، (PA‬وصفيف هوائي مكعب )‪ )Cube‬ومقارنة األداء هذه االشكال عن طريق تغيير بعض‬
‫خصائص النظام التكيفي‪.‬‬
‫في البداية ‪ ،‬درست هذه الرسالة تأثيرات تغيير عدد العناصر ‪ ،‬كما تدرس مسافة التباعد بين العناصر‬
‫تأثير تغيير هندسة المصفوفة على أداء النظام الذكي‪ .‬سيتم تطبيق التحسين األول في هذه الرسالة على‬
‫الخوارزمية المربعة‪ ،‬حيث تُظهر خوارزمية ‪ LMSSI‬أن سرعة تقارب ‪ LMS‬تقل من ‪ 80‬إلى‬
‫تكرارا ‪،‬وتحقق ايضا" انخفاضا" ملحوضا" في قيمة ‪ .MSE‬سيكون التطوير الثاني هو‬
‫حوالي‪20‬‬
‫ً‬
‫‪ SIR-LMS‬والذي حقق التكيف للمصفوفة الهوائية عند نقطة البداية للنظام‪ .‬يدرس تعزيز خوارزمية‬
‫‪ CMA‬في اتجاهين ‪ ،‬أوالً عن طريق تطبيق خوارزمية حجم الخطوة المتغيرة ‪ ICMA‬وثانيًا‬
‫خوارزمية ‪ ، BCMA‬توضح النتيجة أن ‪ ICMA‬يحسن عدد التكرار من ‪ 50‬لـ ‪ CMA‬التقليدية إلى‬
‫‪ 23‬و ‪ 6‬تكرارات لـ ‪ BCMA‬مع الحفاظ على نمط الحزمة االتجاهية لإلشارات المطلوبة وإبطالها‬
‫تما ًما في اتجاهات االستدالل‪.‬‬
‫وتم مناقشة خوارزمية ‪ PSO‬في ثالثة أجزاء ‪ ،‬أوالً للتحسين متجه الوزن الذي تظهره النتيجة بالمقارنة‬
‫مع ‪ Ga‬عند ‪ 8‬عناصر يتحسن الوضع مع ‪ 2.3‬ديسيبل في ‪ SLL‬و ‪ ° 3.77‬في ‪ HPBW‬بأستخدم‬
‫‪ .PSO‬كان الجزء الثاني هو تحسين التباعد بين العناصر‪ ،‬وهذا يمثل أهم منجز لتوفير نظا ًما اقتصاديًا‬
‫بفاعلية استخدام نفس العدد أو أقل من العناصر إلى جانب تقليل حجم المصفوفة ‪ ،‬المرحلة التالية‬
‫ستكون في تحسين )‪ PSO (IPSO‬الذي يجمع بين تحسين الوزن‪ ,‬سيكون الجزء الثالث هو دراسة‬
‫تأثير االتداخل على نمط اإلشعاع لعدد مختلف من العناصر ولمسافات التباعد المختلفة‪.‬‬
‫وستتم تطبيق خوارزمية ‪ REDS‬مع هندسيات مصفوفة مختلفة وستتم المحاكاة باستخدام ضوضاء‬
‫(‪ )Additive White Gaussian Noise (AWGN‬و( ‪Rayleigh fading multipath‬‬
‫‪ .)channels‬يمتلك ‪ REDS‬معدل تقارب بين ‪ 8‬إلى ‪ 5‬تكرارات وهو أسرع مقارنة بخوارزمية‬
‫تكرارا‬
‫تكرارا لبدء عمليات التكيف وأكثر من ‪ 25‬و ‪50‬‬
‫‪ LMS‬التي تحتاج إلى أكثر من ‪50‬‬
‫ً‬
‫ً‬
‫لخوارزمية ‪ NLMS‬و ‪ CMA‬وما إلى ذلك‪ .‬إخراج الصفيف ‪ PA‬و ‪ CA‬يظهر جيدا" لإلشارة‬
‫المطلوبة ‪ ،‬وكذلك ‪ MSE‬من ‪ PA‬حول (‪ )1.2625e-04‬وهو األفضل على اإلطالق‪ ,‬تُظهر نتيجة‬
‫تكرارا ‪ ،‬يتقارب هوائي ‪ CA‬و ‪PA‬‬
‫محاكاة قناة ‪ Rayleigh‬أن هوائي ‪ LA‬يتقارب بعد أكثر من ‪30‬‬
‫ً‬
‫بعد ‪ 20‬وحوالي أقل من ‪ 10‬تكرارات ‪ ،‬كما يقدم صفيف ‪ PA‬صفرا ً أعمق والذي يبلغ حوالي ‪90‬‬
‫ديسيبل تجاه المسبب للتداخل مع توفير ‪ 62-‬ديسيبل و ‪ 48-‬ديسيبل قيمة خالية أعمق بأستخدام مصفوفة‬
‫‪ CA‬و ‪ LA‬على التوالي‪.‬‬
‫وتمت ايظا" دراسة معدل الخطأ (‪ )BER‬ونسبة اإلشارة إلى الضوضاء (‪ )SNR‬للمصفوفة الخطي‬
‫والدائري ويوضح أن معدل الخطأ في البتات (‪ )BER‬متقارب تقريبًا لكال المصفوفتين في نفس العدد‬
‫من العناصر ‪ ،‬في حين يُظهر المعدل النشط (‪ )SNR‬وجود اختالف بسيط في منحنى ‪ SNR‬في‬
‫المصفوفة الخطي والدائري و من الواضح أن منحنى ‪ SNR‬للنظام اإلجمالي يزيد مع عدد العناصر‬
‫المستخدمة في الصفيف بغض النظر عن هندسة مصفوفة الهوائي‪.‬‬
‫و قد تم استخدام الحقيبه البرمجية (‪ )MATLAB‬طراز(‪ )2017a‬في تنفيذ عمليات المحاكات في‬
‫هذه الرسالة‪.‬‬