People s Democratic Republic of Algeria Ministry of Higher Education and Scientific Research University M Hamed BOUGARA Boumerdes

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1 People s Democratic Republic of Algeria Ministry of Higher Education and Scientific Research University M Hamed BOUGARA Boumerdes Institute of Electrical and Electronic Engineering Department of Electronics Final Year Project Report Presented in Partial Fulfilment of the Requirements for the Degree of MASTER In Telecommunication Option: Telecommunication Title: 2D Antenna Array Pattern Synthesis Using Biogeography Based Optimzation Presented by: - Nabil ARHAB - Imad Eddine ZEGHAD Supervisor: Mr.A.RECIOUI Registration Number:..../2017

2 Abstract Abstract Antenna arrays are widely used today for long distance communication with very high gains. The design of such antennas depends on certain parameters including the array excitation (amplitude and phase) and element positions. In this work, non-uniformly excited planar array optimization is done. Many shapes are considered: circular, Concentric Circular and hexagonal Arrays. The objective is to minimize the Sidelobe Level (SLL) for the arrays by finding the right element amplitude excitations. The study is performed on different number of elements for each shape. During this work the biogeography based optimization (BBO) has been used to minimize the sidelobe levels to converge to a designed mask. The optimal values obtained result in a good reduction of side lobe level for the different array shapes, although; the directivity had a small decrease depending on the shape, number of elements and other parameters. I

3 Dedication Dedication I would like to dedicate this work: To My Mother and Father, To my brothers and sister And All dear friends Whoever helped to achieve this work. Nabil II

4 Dedication Dedication I dedicate this thesis to my parents And sisters, for their support, encouragement and Understanding throughout the period of the work And to all my friends. Imad II

5 Acknowledgment First and foremost, we thank ALLAH, for helping us to finish this modest work. It is our belief in him that helped us persevere at times when it seemed impossible to go on. We would like to thank Mr A. Recioui for all the help he offered to us though the project until it was finished. Imad, Nabil V

6 Table of content Abstract I Dedication...II Acknowledgment III Table of content..iv List of figure..v INTRODUCTION... 1 Chapter One : Generalities about antennas Introduction Definition of an antenna Types of antennas Antenna Fundamentals The radiation pattern Isotropic, Directional, and omnidirectional patterns Antenna Pattern Parameters Beamwidth Radiation Intensity Directivity Antenna arrays Type of antenna arrays Rectangular arrays Uniform Circular arrays Uniform Concentric Circular Array The hexagonal array antenna Conclusion Chapter two : Biogeography Based Optimization Introduction Optimization Optimization Problem Optimization Techniques Classical Optimization Techniques Advanced Optimization Techniques Biogeography based optimization IV

7 Table of content BBO Main Operators Migration Operator Mutation Operator Algorithm Testing with Benchmark Functions: Conclusion Chapter Three : Discussion and Results Problem Formulation BBO Algorithm Parameters Rectangular Planar Array Circular array Concentric Circular array The hexagonal array CONCLUSION Refrences Appendix.40 IV

8 List of figures Figure 1.1: An illustrative diagram of wireless transmission... 2 Figure 1.2: Coordinate system for antenna pattern analysis... 3 Figure 1.3: Normalized Power Pattern... 4 Figure 1.4 : Rectangular array... 7 Figure 1.5: Geometry of a uniform circular array... 8 Figure 1.6 : uniform concentric circular array... 9 Figure 1.7 : Hexagonal Array (HA) Structure Figure 2.1 illustrates a model of species abundance in a single habitat where (a) illustrate the simplest migration model where (E=I) Figure 2.4: The flowchart of the main BBO Algorithm Figure 2.5: convergence to the global minima after 12 iterations Figure 2.6: convergence to the global minima 15 iterations Figure 2.6: convergence to the global minima 152 iterations Figure 2.8: convergence to the global minima after 225 iterations Figure 3.1: the shape of the mask used in the optimization Figure 3.2: pattern of 7 x 7 Planar Rectangular Array optimization Figure 3.3: pattern of 10 x 10 Planar Rectangular Array optimization Figure 3.4: pattern of a 50 element circular array optimization Figure 3.5: pattern of 100 Circular array with optimization Figure 3.6: pattern of 92 element Concentric Circular array with optimization Figure 3.7: pattern of 168 element Concentric Circular array with optimization Figure 3.8: pattern of 20 element hexagonal array with optimization Figure 3.9: pattern of 50 element hexagonal array with optimization V

9 General Introduction INTRODUCTION Since the revolution of the communication an enormous increase in the traffic has been experienced for mobile and personal communication systems, this is due to both the increased number of users as well as the bit data rate services being introduced. This increase in traffic will put demand on both manufactures and operators to provide enough capacity in the networks. For wireless communication systems, the antenna is one of the most critical components since it is responsible for the proper transmission and reception of electromagnetic waves. A good design of the antenna can relax system requirements and improve overall system performance. The choice of an antenna for specific application (cellular, satellitebased, ground-based, etc.), depends on the platform to be used (car, ship, building, spacecraft, etc.), the environments (sea, space, land), the frequency of operation, and the nature of application (video, audio data, etc.). Usually the radiation pattern of a single element is relatively wide, and each element provides low values of directivity (gain). In many applications it is necessary to design antennas with very directive characteristics (very high gains) to meet the demands of long distance communication and avoid the interference. This can only be accomplished by increasing the electrical size of the antenna. Enlarging the dimensions of single elements often leads to more directive characteristics. Another way to enlarge the dimensions of the antenna, without necessarily increasing the size of the individual elements is to use many number of elements. An antenna array consists of more than one elements. A single-element antenna is usually not enough to achieve technical needs. That happens because its performance is limited. A set of discrete elements, which constitute an antenna array, offers the solution to the transmission and/or reception of electromagnetic energy. The geometry and the type of elements characterize an antenna array. The designed array should allow signals from a desired direction to add constructively while simultaneously adding destructively in the undesired directions, hence an array may be regarded as a spatial filter with high gain in the desired signal direction and low gain elsewhere. Theoretically, the array should be designed with 1

10 General Introduction a maximum directivity and minimum side lobe level so as to achieve maximum signal to noise plus interference ratio at the output of the array antenna; However, this is only true if the interferences are evenly distributed or assume certain distribution patterns over the whole spatial domain. Therefore, if the designer of the array does not know the distribution of the directions of the interferences, design such as side-lobe level reduction may be preferred. In this project we concentrated on designing a maximum side-lobe levels reduction of uniformly and non-uniformly distributed antenna elements along a planar, circular, and concentric circular and hexagonal by varying the amplitude excitation using a specific optimization techniques.various optimization techniques have been developed in recent past by harnessing nature s activities. Since the early 1970s, various nature-inspired optimization algorithms have emerged starting with the Genetic Algorithm (GA), some of which have proven to be very efficient global optimization methods. Along with the GA, Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO), Clonal Selection Algorithm (CLONALG), Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES), Differential Evolution (DE). In our project, the Biogeography Based Optimization is used as a technique, an evolutionary algorithm (EA), that optimizes a function by improving candidate solutions with regard to a given measure of quality or fitness function, our problem is finding the optimum set of amplitude excitation, to get best reduction of SLL.The report is organized as follows: In chapter 1, we summarize generalities of antenna arrays, in chapter 2, we talk about optimization in general and then we give an overview about, the most known classical and modern optimization techniques, then we explain what the BBO (Biogeography Based optimization) are and we look at the rules that control their dynamics. The third chapter deals with design of non-uniform planar, circular, concentric circular and hexagonal array antenna using BBO. Finally, the conclusions of the work are drawn. 2

11 Chapter One Generalities About Antennas

12 Chapter One: Generalities about antennas Generalities About Antennas 1.1 Introduction Antennas go back to the mid-1800s and much evolution has occurred since. The first experiments with wireless communication are on record in 1867 but little details are available. A major breakthrough is noted for Guglielmo Marconi with a wireless call that traveled to boats across the Atlantic Ocean. Previous to this feat radio transitions had a very limited range such that houses or boats very close could talk to each other, but communication over great distances was unlikely. Radio communication was only a series of tones, known as mores code, but it now provided the ship industry with an element of safety that it didn t have before Marconi The next breakthrough was in 1916 when operators at Radio Arlington were able to transmit the sound of a human voice up and down the Atlantic coast. This was a major breakthrough and the beginning of AM (Amplitude Modulation) radio. This sparked an interest in radio and during the mid-1920s many citizens were putting wire array antennas on their roofs to talk to other people nearby. The frequencies used by citizens were in the high frequency range lower than 200 meters wavelength. Because of this people who use this band are known and shortwave or ham operators. People quickly realized that the shorter the wave the further it propagates through space. Therefore people operating in the range below 50 meters were able to make contact around the world in the early 1920s. [1] 1.2. Definition of an antenna An antenna is defined by Webster s Dictionary as a usually metallic device (as a rod or wire) for radiating or receiving radio waves. The IEEE Standard Definitions of Terms for Antennas (IEEE Std ), defines the antenna or aerial as a means for radiating or receiving radio waves. In other words the antenna is the transitional structure between free-space and a guiding device, the guiding device or transmission 3

13 Chapter One: Generalities about antennas line may take the form of a coaxial line or a hollow pipe (waveguide), and it is used to transport electromagnetic energy from the transmitting source to the antenna, or from the antenna to the receiver. In the former case, there is a transmitting antenna and in the latter a receiving antenna. [3]. [2]. 1.3 Types of antennas Figure 1.1: An illustrative Diagram of wireless transmission [6] In this section, we'll introduce the fundamental antenna types. We'll start with the simplest of all antennas, the short dipole antenna (which is basically a short wire), and work our way through to the more complicated antennas. [4] Wire Antennas: Short Dipole Antenna, Dipole Antenna, Wave Dipole Broadband Dipoles Monopole Antenna Folded Dipole Antenna Loop Antenna, Cloverleaf Antenna Travelling Wave Antennas: Helical Antennas, Yagi-Uda Antennas, Spiral Antennas Reflector Antennas : Corner Reflector, Parabolic Reflector (Dish Antenna) Microstrip Antennas : Rectangular Microstrip (Patch) Antennas, Planar Inverted- F Antennas Log-Periodic Antennas: Bow Tie Antennas, Log-Periodic Antennas, Log- Periodic Dipole Aperture Antennas : Slot Antenna, Cavity-Backed Slot Antenna, Inverted-F Antenna, Slotted Waveguide Antenna, Horn Antenna, Vivaldi Antenna, Telescopes 4

14 Chapter One: Generalities about antennas Other Antennas : NFC Antennas, Fractal Antennas, Wearable Antennas 1.4 Antenna Fundamentals The radiation pattern The radiation pattern of an antenna is a graphic representation of the radiation properties of an antenna, and could include information on the energy distribution, phase, and polarization of the radiated field. [5] often this radiation pattern is determined in the far field region and represented as a function of the directional coordinates, there can be field pattern (magnitude of the electric or magnetic field) or a power pattern (square or magnitude of the electric or magnetic field), this radiation pattern is often normalized with respect to their maximum value, a radiation patterns are conveniently represented in spherical coordinates. [5] Figure 1.2: Coordinate system for antenna pattern analysis Isotropic, Directional, and omnidirectional patterns An isotropic radiator: defined as a hypothetical lossless antenna having equal radiation in all directions. Although it is ideal and not physically realizable, it is often taken as a reference for expressing the directive properties of actual antenna. A directional antenna: defined as one having the property of radiating or receiving electromagnetic waves more effectively in some directions than in others An omnidirectional: defined as one having an essentially non-directional pattern in any orthogonal plane (an omnidirectional pattern is then a special type of a directional pattern). [6]. 5

15 Chapter One: Generalities about antennas Antenna Pattern Parameters Radiation Lobe- a clear peak in the radiation intensity surrounded by regions of weaker radiation intensity. Main Lobe- (major lobe, main beam) - radiation lobe in the direction of maximum radiation. Minor Lobe- any radiation lobe other than the main lobe. Side Lobe- a radiation lobe in any direction other than the direction(s) of intended radiation. Back Lobe- the radiation lobe opposite to the main lobe Beamwidth A beamwidth is a parameter which is related with the pattern of an antenna, the beamwidth of a pattern is defined as the angular separation between two identical points on opposite side of the pattern maximum Power Beamwidth (HPBW) the angular width of the main beam in which the radiation intensity is one-half value of the beam. First Null Beamwidth (FNBW) angular width between the first nulls on either side of the main beam. Figure 1.3: Normalized Power Pattern 6

16 Chapter One: Generalities about antennas Radiation Intensity Radiation intensity U in a given direction is the power radiated by the antenna per unit solid angle. It is given by the product of the radiation density and the square of the distance r.[8] Directivity A very important antenna characteristic that describes how much it concentrates energy in a specified direction in preference to radiation in other directions. It is equal to its power gain if the antenna is100% efficient. The directivity of the antenna is defined as the ratio of radiation intensity in a given direction from the antenna to the radiation intensity averaged over all direction. The average radiation intensity is equal to the total power radiated by the antenna divided by 4π. If the direction is not specified, the direction of maximum radiation is implied [2]. D = U U0 = 4πU prad (1.1) Where: D U = directivity (dimensionless) = radiation intensity (W/unit solid angle) U0 = radiation intensity of isotropic source (W/unit solid angle) Prad = total radiated power (W) 1.5 Antenna arrays Usually the radiation pattern of a single element is relatively wide, and each element provides low values of directivity (gain). In many applications it is necessary to design antennas with very directive characteristics (very high gains), to meet the demands of long distance communication. This can only be accomplished by increasing the electrical size of the antenna Enlarging the dimensions of single elements often leads to more directive characteristics. Another way to enlarge the dimensions of the antenna, without necessarily increasing the size of the individual elements, is to form an assembly of radiating elements in an 7

17 Chapter One: Generalities about antennas electrical and geometrical configuration. This new antenna, formed by multi elements, is referred to as an array. 3] The total field of the array is determined by the vector addition of the fields radiated by the individual elements. This assumes that the current in each element is the same as that of the isolated element. This is usually not the case and depends on the separation between the elements. To provide very directive patterns, it is necessary that the fields from the elements of the array interfere constructively in the desired directions and interfere destructively in the remaining space. Ideally this can be accomplished, but practically it is only approached. In an array of identical elements, there are at least five controls that can be used to shape the overall pattern of the antenna these are: [3] a. The geometrical configuration of the overall array (linear, circular, rectangular, spherical etc.) b. The relative displacement between the elements. c. the excitation amplitude of the individual elements d. 4. the excitation phase of the individual elements e. the relative pattern of the individual elements Type of antenna arrays Antenna arrays a configuration of multiple antennas (elements) arranged to achieve a given radiation pattern it can be: a- Linear array: antenna elements arranged along a straight line. b- Circular array: antenna elements arranged around a circular ring. c- Planar array: antenna elements arranged over some planar surface (example - rectangular array). d- Conformal array: antenna elements arranged to conform to some non-planar surface (such as an aircraft skin) Rectangular arrays N such arrays (Figure 1.4) are placed at even intervals along the y direction, a rectangular array is formed. We assume again that they are equispaced at a distance and there is a progressive phase shift βy along each row. We also assume that the normalized current 8

18 Chapter One: Generalities about antennas distribution along each of the x-directed arrays is the same but the absolute values correspond to a factor of I1n (n=1,, N). Then, the Array Factor (AF) of the entire MN array is N M AF = I 1n I m1. exp (j(m 1)(kd x sinθcosφ + β x )) exp (j(n 1)(kd y sinθsinφ + β y )) n=1 m=1 (1.2) In the case of a uniform planar rectangular array, Im1 and In1= I N M AF = I exp(j(m 1)(kdxsinθcosφ + βx)) exp(j(n 1)(kdysinθsinφ + βy)) (1.3) n=1 m=1 Figure 1.4 : Rectangular array Uniform Circular arrays A Uniform Circular Array (UCA), is an antenna array having its elements arranged in a circle. The antenna has N discrete elements placed at distance a from the center of the circle, therefore a is the radius of the array. The circular array is uniform if its elements are uniformly distributed along the circle and uniformly excited. The UCA geometry, as shown in Figure (1.5) very interesting due to its symmetry on the azimuth plane that allows electronic azimuthal plane scanning, while keeping a stable beam 9

19 Chapter One: Generalities about antennas shape which is an advantage over other geometries with edges such as the rectangular array. [3] The array factor of the UCA lying in the x-y plane is given by [1]: AF(θ, φ) = N n=1 Figure 1.5: Geometry of a uniform circular array I n exp (j(k a sin(θ)cos(φ φ n ) + a n )) (1.4) In And an are the amplitude and phase of the excitation for the n th element respectively. φn= 2pi n N is the angular position of the nth element in the x-y plane, φ is the azimuth angle measured from the x-axis Uniform Concentric Circular Array A concentric circular array antenna is an array that consists of many concentric rings of different radii and a number of elements on its circumference The Figure. (1.6) below shows the general configuration of CCAA with M concentric circular rings, where the m th (m = 1, 2,, M) ring has a radius rm and the corresponding number of elements is Nm. If all the elements (in all the rings) are assumed to be isotopic sources, the radiation pattern of this array can be written in terms of its array factor only. 10

20 Chapter One: Generalities about antennas Figure 1.6 uniform concentric circular array The array factor, AF (ϕ, I, d mi ) for the CCAA in x-y plane may be written as AF(ϕ, I, d mi ) = M m=1 Nm i=1 I mi exp (j(kr m sin(θ ) cos(ϕ ϕ mi ) + a mi ) (1.5) K rm = Nm i=1 d mi (1.6) dmi the inter elements spacing, Imi is the current excitation of the i th element of the m th ring, K=2*pi/ λ, λ being the signal wave-length, And θ and symbolize the zenith angle from the positive z axis and the azimuth angle from the positive x axis to the orthogonal projection of the observation point respectively. If we assume the elevation angle to be 90 degree i.e. θ = 90 0 then the array factor may be written as a periodic function of φ mi with a period of 2π radians.the radiation pattern will be a broadside array pattern. The azimuth angle to the i th element of the m th ring is mi, The elements in each ring are assumed to be uniformly distributed.[7] φ mi = 2π ( i 1 N m ) (1.7) 11

21 Chapter One: Generalities about antennas The residual phase term αmi is a function of angular separation φmi and ring radius m, α mi = Kr m. sin θ 0. cos(φ 0 φ mi ) (1.8) m = 1 M; i = 1 Nm and φ 0 is the value of φmi where peak of the main lobe is obtained The hexagonal array antenna The hexagonal array (HA) can be treated as consisting of two concentric N -element circular arrays of different radii r1, r2 as the peripheral curve of its vertices is a circle, figure (1.7) shows the geometry of a regular hexagonal array with 2N elements (N = 6), of which N number elements are located at the vertices of the hexagon and the other N elements are located at the midpoints of the sides of the hexagon. [9]. the array factor of the hexagonal array is expressed by: N AF(θ, ϕ) = A n exp(jkr 1 sin(θ) (cos(ϕ 1n ) cos(ϕ) + sin(ϕ 1n ) sin(ϕ))) n=1 + B n exp(jkr 2 sin(θ) (cos(ϕ 2n ) cos(ϕ) + sin(ϕ 2n ) sin(ϕ))) (1.9) Figure 1.7 : Hexagonal Array (HA) Structure 12

22 Chapter One: Generalities about antennas Where: r 2 = r 1 cos ( π ), r n 1 = de/sin ( π ) n Where de is the inter-element spacing along any side of the hexagonal array φ1n=2π(n-1)/n is the angle in the x - y plan e between the x axis and the n th element at the vertices of the hexagon, φ2n= φ1n +π/n is the angle in the x - y plan e between the x axis and the n th element at the middle of each line of the hexagon, An and Bn are the relative amplitudes of n th element placed at the vertices and the middle of the hexagon, respectively 1.6 Conclusion This chapter dealt with generalities of antenna and antenna array and basic properties of antenna arrays, in addition to detailed description of the planar, circular, concentric circular and hexagonal arrays which are the antennas of interest in this work. 13

23 Chapter Two Biogeography Based Optimization

24 Chapter Two: Biogeography Based Optimization Biogeography Based Optimization In this chapter, both classical and modern optimization methods are discussed with a focus on the Biogeography Based Optimization and then a small simulation of the BBO will be performed with a certain benchmark functions to ensure its functionality 2.1 Introduction The human desire and the need to optimize everything is the main factor that made the field of optimization grow and get to the advanced methods that is being used today, the starting can be traced back to 300 BC when a Greek mathematician named Euclid who considered The minimal distance between two points a line, and proved that a square has the greatest area among the rectangles with a given total length of edges, and it followed on until the breakthrough made by Isaac Newton and Gottfried Leibniz in the 17 th century where they Were responsible for the development of the differential calculus methods of optimization [11]. By the twentieth century, this same method could be implemented in an easier and a faster way due to the emergence of high speed computers and this again, made the implementation of other more complex methods possible, this was followed by producing a massive literature on optimization techniques which made the emergence of several well defined new areas in optimization theory possible today. 2.2 Optimization Optimization is the act of finding the best possible result under certain circumstances and in its simplest case, it consists of finding the minimum cost possible for the solution or the maximum efficiency possible to our solution. the effort or the benefit can be usually expressed as a function of certain design variables. Hence, we can mathematically define optimization as the process of finding the conditions that give the maximum or the minimum value of a function. Most of optimization algorithms are designed to only find the minimum but if a point x corresponds to the minimum value of a function f(x), 15

25 Chapter Two: Biogeography Based Optimization the same point corresponds to the maximum value of the function f(x) [2], Thus, optimization always can be taken to be minimization Optimization Problem The main form of an optimization problem is finding the maximum or the minimum of an objective function. The Objective Function : Takes the form : y = f(x 1, x 2,., x n 1, x n ) (2.1) and it expresses one or several quantities which are to be minimized or maximized, mainly the optimization problems can have one to several objective functions and it is possible to reformulate a problem with multi-objectives as a problem with a single objective by either forming a weighted combination of the different objectives or by putting some of the objectives as constraints. The Design Variables: Our set of unknowns that we should find to solve the optimization problem, this variable should satisfy our requirements and the constraints on them. Given the design variables: x 1, x 2,., x n 1, x n The constraints on the variables should be given:? < x 1 <?,? < x 2 <?,..,? < x n 1 <?,? < x n <? So, once the design variables, constraints, objectives and the relationship between them have been chosen, the optimization problem can be defined Optimization Techniques Classical Optimization Techniques Classical optimization techniques are useful in finding the optimum solution of continuous and differentiable functions. These methods are analytical and make use differential calculus in locating the optimum points. Since most of the practical problems involve objective functions that are not continuous or not differentiable, the classical optimization techniques have limited scope in practical applications. However, a study of the calculus methods of optimization forms a basis for developing most of the numerical techniques of optimization. 16

26 Chapter Two: Biogeography Based Optimization There are three main types of problems that can be handled by the classical optimization Techniques which are: single variable functions, multivariable functions with no constraints, multivariable functions with both equality and inequality constraints [12]. for problems with equality constraints the Lagrange multiplier method can be used. If the problem has inequality constraints, the Kuhn-Tucker conditions can be used to identify the optimum solution. These methods lead to a set of nonlinear simultaneous equations that may be difficult to solve. The other classical methods of optimization include [12] : Linear programming: studies the case in which the objective function f is linear and the set of design variable space is specified using only linear equalities and inequalities. Integer programming: studies linear programs in which some or all variables are constrained to take on integer values. Quadratic programming: allows the objective function to have quadratic terms, while the set of design variables must be specified with linear equalities and inequalities. Nonlinear programming: studies the general case in which the objective function or the constraints or both contain nonlinear parts. Stochastic programming: studies the case in which some of the constraints depend on random variables. Dynamic programming: studies the case in which the optimization strategy is based on splitting the problem into smaller sub-problems. Combinatorial optimization: is concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete one. Infinite-dimensional optimization: studies the case when the set of feasible solutions is a subset of an infinite-dimensional space, such as a space of functions. Constraint satisfaction: studies the case in which the objective function f is constant (this is used in artificial intelligence, particularly in automated reasoning) 17

27 Chapter Two: Biogeography Based Optimization Advanced Optimization Techniques In recent years, some optimization methods that are conceptually different from the traditional mathematical programming techniques have been developed. These methods are labeled as modern or nontraditional methods of optimization. Most of these methods are based on certain characteristics and behavior of biological, molecular, swarm of insects, and neurobiological systems. these methods have been developed only in recent years and are emerging as popular methods for the solution of complex engineering problems. Most require only the function values (and not the derivatives). The genetic algorithms: are based on the principles of natural genetics and natural selection. And it is a general method for solving search for solutions problems as many other evolutions inspire techniques and its simple form it works by choosing some candidates solution and then systematically perform the mutation process until finding the best fitness. [17, 12] Simulated annealing: is based on the simulation of thermal annealing of critically heated solids. Both genetic algorithms and simulated annealing are stochastic methods that can find the global minimum with a high probability and are naturally applicable for the solution of discrete optimization problems [21, 12]. The particle swarm optimization: is based on the behavior of a colony of living things, such as a swarm of insects, a flock of birds, or a school of fish, just like other algorithms it tries to improve candidate solutions iteratively with regard to a given measure of quality, works by putting a population of candidates solution as particles and move them around in the search space according a formula of position and velocity and this is supposed to move the swarm into a better solution in the search space till getting to the best one [19, 12]. Ant colony optimization: is based on the cooperative behavior of real ant colonies, which are able to find the shortest path from their nest to a food source and in many practical systems, the objective function, constraints, and the design data are known only in vague and linguistic terms. Fuzzy optimization methods have been developed for solving such problems [20, 12]. 18

28 Chapter Two: Biogeography Based Optimization Neural-network-based methods: in this methods the problem is modeled as a network consisting of several neurons, and the network is trained suitably to solve the optimization problem efficiently [12] Biogeography based optimization Biogeography based optimization (BBO) is an evolutionary algorithm (EA) that optimizes a function by stochastically and iteratively improving candidate solutions with regard to a given measure of quality or fitness function, since it does not make any assumptions about the problem, it can be applied to a wide class of problems [13]. It is typically used to optimize multidimensional real-valued functions and It does not require the function to be differentiable therefore it can be used on discontinuous functions. Like many EAs, BBO was motivated by a natural process; in particular, BBO was motivated by biogeography, which is the study of the distribution of biological species through time and space, it has been introduced by Dan Simon in 2008 [13]. Generally, Mathematical models of biogeography describe how species migrate from one island to another, how new species arise, and how species become extinct. The term island here is used descriptively rather than literally. That is, an island is any habitat that is geographically isolated from other habitats. So, Geographical areas that are well suited as residences for biological species are said to have a high habitat suitability index HIS. and the variables that characterize habitability are called suitability index variables SIVs, SIVs can be considered the independent variables of the habitat, and HSI can be considered the dependent variable [14]. Candidate solutions of a problem are represented by an array of integers as: Habitat = [SIV 1, SIV 2, SIV 3,..., SIV N ]. (2.2) The Habitat suitability index (HSI) can be also referred to as The value of the fitness function since it is directly proportional to it [14,15], and it is found by evaluating the fitness function: Fitness (Habitat) = HSI = f (SIV1, SIV2, SIV3,..., SIVN). (2.3) 19

29 Chapter Two: Biogeography Based Optimization (A) (B) Figure. 2.1: illustrates a model of species abundance in a single habitat where (a) illustrate the simplest migration model where (E=I) BBO Main Operators The BBO algorithm is based on two main operators Migration and Mutation Migration Operator In BBO algorithm each habitat (Hi) is a solution candidate for the optimization problem and the position of each habitat (Hi) is an n-dimensional search space represented by (SIVs) which is an n-dimensional vector, and the quality of each habitat is measured by the HSI which is directly proportional to the fitness function value. This algorithm uses Migration operator as a powerful tool to share information between habitats in the solution space. The Migration operator shares information between habitats based on immigration and emigration rates, probabilistically. Each habitat has its own immigration λi and emigration rates μi which are the functions of species in the habitat. For a given habitat, the immigration λi rate is inversely proportional to the HSI (fitness) value, while the emigration μi rate is directly proportional to HSI value. The habitats with high immigration rates (poor solutions) are more likely to accept information from the other habitats with high HSI values, while the habitats with low immigration rates (good solutions) share their information with other poor habitats with a high probability [14,15,16]. The immigration and emigration rates are calculated for each habitat as follows: 20

30 Chapter Two: Biogeography Based Optimization μ i = E ( K S max ) (2.4) λ i = I (1 K S max ) (2.5) I: Maximum possible immigration rate E: Maximum possible emigration rate K: Number of species in the ith habitat Smax : Maximum number of species Habitats with a high HSI value tend to have a large number of species, while those with a low HSI have a small number of species. from Figure 1, it can be concluded that the habitat with few species (poor solution, low HSI) like S1, has a low emigration rate and a high immigration rate. This means that the habitat with low HSI tends to take information about the good habitats with the high probability, while the probability of sharing its information for other habitats is relatively low. On the other hand, the habitat which has more species (good solution, high HSI) like S2, has a low immigration rate and a high emigration rate. Such habitats with high HSI values share their information with the other habitats with a high probability. By utilizing this mechanism [14,15,16], the Migration Operator of the BBO algorithm can achieve adequate exploitation ability between the habitats in the search space. For each variable of a given solution (Hi), the immigration λi rate decides whether or not to immigrate. If the immigration condition is satisfied, the migration procedure occurs between the immigrating and emigrating habitats as follows: H i (SIV) H i (SIV) This last equation explains that one of the variables of ith habitat is replaced by a variable of jth habitat. Which are respectively the immigrating and emigrating habitats, it is worth mentioning that the emigrating habitat (Hj) is selected based on the emigration rates μ, and the probability of selecting jth habitat as emigrating habitat is calculated as follow: P(emigration from H j ) = NP: Population size. μ j N i=1 μ i, for j = 1,2,3,.., N P (2.6) 21

31 Chapter Two: Biogeography Based Optimization Figure 2.2: The migration procedure of the BBO algorithm. Here, the roulette wheel selection technique is used to select the emigrating habitat Mutation Operator In most cases, it is possible that a meta-heuristic algorithm is trapped to the local optimum by lapse of the iteration. In order to escape from the local traps in the search space, the BBO algorithm utilizes a Mutation Operator. Which is a probabilistic operator that modifies a habitat s SIV randomly based on mutation rate (pmutate) [3], which is related to the habitat s probability. The mutation rate (pmutate) for each habitat is calculated as follows: mmax : user-defined parameter. Pmax : max{ Pi } pmutate = m max ( 1 P i P max ) (2.7) Pi = probability of the number of each species Based on this equation a variable of each habitat mutates randomly in search space with a given probability. For a better explanation, the mutation operator of the BBO algorithm 22

32 Chapter Two: Biogeography Based Optimization Figure 2.3: The mutation procedure of the BBO algorithm can be described as in Figure 2.3. However, in most studies, for simplicity, the probability of performing mutation operator for the all habitats is set to Another feature of the BBO algorithm is that the elite habitats with high HSI values are selected to keep and transfer from previous generation to the current one. Therefore, the Keeprate parameter is defined for this purpose. For Example, 20% of habitats with high HSI values are selected to keep in each generation. It means that the 20% of elite habitats from the previous population are transferred to the current generation and combined with new habitats (Keeprate=0.2). Finally, the habitats with high HSI values are selected from the combined population of habitats to form a new population. Flowchart of main BBO Algorithm The following figure is a flowchart that describe the simplified algorithm of the BBO, the algorithm starts by generating random habitats and evaluating the cost function then the migration rates are calculated to be used in formulating the migration operator after that the mutation operator is applied to save the algorithm from trapping in a local minima, at the end the habitats with high HSI would be conserved to pass to the next generation,the same steps will be repeated until the termination criterion will be satisfied 23

33 Chapter Two: Biogeography Based Optimization Initialization Generation of Random habitats and evaluation of cost function of each habitat Migration Rates Calculations Calculate λ i and µ i for each habitat based on HSI values and migration model i=1 For K=1 - No - Rand < λ i? Select the emigrating habitat (H j ) based on the emigration rates (µ i ) Replace the selected variable of the Immigrating habitat (H i ) by a variable of emigrating habitat (H j ) K=k+1 - No - Rand < pmutation? i=i+1 Replace the selected variable by the randomly selected variable in the search space K < n? Evaluate the cost function of habitat i i < N H? Combine the habitat with high HIS values from previous generation and select the N H Habitats with high HIS values from combined population - No - Termination criterion is satisfied? End Figure 2.4: the flowchart of the main BBO Algorithm 24

34 Chapter Two: Biogeography Based Optimization Algorithm Testing with Benchmark Functions: Ackley Function. Global minimum : f(0, 0) = 0 d i=1 f(x) = a exp ( b 1 d x d i=1 i 2 ) exp ( 1 cos(cx d i) ) + a + e (2.7) Best Cost = 0 Figure 2.5: convergence to the global minima after 12 iterations Sum Squares Function Global minimum: f(0, 0) = 0 Best Cost = 0 d 2 f(x) = i=1 ix i (2.8) 25

35 Chapter Two: Biogeography Based Optimization Figure 2.6: convergence to the global minima 15 iterations Levy Function Global minimum : f(1,., 1) = 0 f(x) = sin 2 d (πω 1 ) + i=1 (ω i 1) 2 [ sin 2 (πω i + 1)] + (ω d 1) 2 [1 + sin 2 (2πω d )] Where ω i = 1 + x i 1,for all i = 1,, d while d is the dimension and our results 4 below are simulated with d=10. Best Cost = e-32 (2.9) Figure 2.7: convergence to the global minima after 152 iterations 26

36 Chapter Two: Biogeography Based Optimization Sphere Function Global minimum : f(0,., 0) = 0 d 2 f(x) = i=1 x i (2.10) While d is the dimension and for our simulation test we took d=20. Best Cost = 3.333e-16 Figure 2.8: convergence to the global minima after 225 iterations Conclusion In this chapter, we gave a small overview about optimization techniques and how they evolved throughout history while outlining the most know ones. After that, we presented a one of the new nature-inspired global optimization algorithm that is motivated by biogeography which is the study of the distribution of biological species through time and space and it describes how species migrate from one island to another, how new species arise, and how species become extinct. We also made a simulation with Matlab software to test the algorithm with certain benchmark functions, and in the next chapter we will use this algorithm in our sidelobes reduction problem. 27

37 Chapter Three Discussion & Results

38 Chapter Three: Discussion and Results Discussion and Results In this chapter, the biogeography based optimization Algorithm explained in the previous chapter is used to minimize our objective function that is formulated to reduce sidelobes level of multiple array factors of different shapes, and in each shape our goal is to obtain an optimal set excitations amplitudes that provide a radiation pattern with maximum sidelobes level reduction. 3.1 Problem Formulation The problem formulation which is the objective function to be optimized follows from the previous chapter and it will be the sum of differences between our Array Factor and its corresponding mask. Shown in the figure bellow an example of a Mask, Noting that the values of AF and AFmin and the intervals of each one of them should be modified to adapt to the specific array factor of the 2D shape we want to optimize for, this adjustment must be suitable to the sidelobes level (SLL) and the Directivity (Dir) of the uniform array factor of that specific shape. Figure 3.1: the shape of the mask used in the optimization 28

39 Chapter Three: Discussion and Results Fitness function: fitness = N 1 AF(θ) d AF(θ) p N (3.1) AF(θ) d : The desired array factor which is represented by the mask shown above. AF(θ) p : Our proposed array factor of our chosen 2D shape. N Number of point we would have, throughout our work we chose a 0.01 spacing for θ and since it is bounded 180 < θ < 180 we would have N = BBO Algorithm Parameters During the simulation in this chapter the following BBO parameters has been used: Number of population: 60, KeepRate=0.2, Number of iterations =100, Maximum migration rates E=1 and I=1, Mutation probability = Rectangular Planar Array The treated array in this part is going to be the Rectangular Planar array, the array is optimized to get the maximum SLL reduction using the mask explained above, the amplitudes excitations are varied for 7 x7 planar array and 10 x 10 array from 0 to 1. For each case we will deal with the uniform and the non-uniform array, the first one is obtained by setting the amplitudes excitations to 1 and the second will represent the results of our work which is obtained by getting the amplitudes excitations that will ensure the best reduction of SLL. 29

40 Chapter Three: Discussion and Results Case 1 We have used a 49 elements laying along the x-y plane (07X07 planar array) By setting the parameters presented above, we see from figure (3.2) that the side lobes level and the Directivity of the uniform array are dB, 18.34dB respectively, and after optimizing the values obtained are SLL= dB, Directivity=17.14dB. Case 2 Figure 3.2: pattern of 7 x 7 Planar Rectangular Array optimization From figure (3.3) by varying the amplitude, the side lobes level and the Directivity of the uniform array are dB to 21.70dB respectively, and after optimizing the values obtained are SLL= dB, Directivity=20.27dB. Figure 3.3: pattern of 10 x 10 Planar Rectangular Array optimization 30

41 Chapter Three: Discussion and Results Result Discussion Optimization SLL using our mask, the figures above show the pattern of the array factor when optimizing with 49 elements and 100 elements, for the 7 x 7 array the uniform one has a ratio DIR/SLL of , we have optimized it to , for the 10 x 10 array the uniform one has a ratio DIR/SLL , we have optimized it to and although we have obtained a better SLL for both cases but we had a decrease in directivity. 3.4 Circular array In this part a circular array would be treated that has the parameters bellow: r = N (4 π) (3.1), φn = 2π n N (3.2) Where r the radius of the circle is, N is the number of elements and φn is the angular position of the n th element. And the element spacing= λ/2, λ=1. The array is optimized to get the maximum SLL reduction, the amplitudes excitations are varied for 50 elements and 100 elements from 0 to 1. Case 1 We have used a 50 elements circular array, the results obtained are shown in Figure (3.4), for the uniform array we have got SLL = dB, and Directivity=17.42dB and after optimizing we have a got a reduction of sidelobes level to dB, and a decrease in Directivity to 16.35dB. Figure 3.4: pattern of a 50 element circular array optimization 31

42 Chapter Three: Discussion and Results Case 2 A 100 elements circular array is used for this case, from Figure (3.5) above, we have successfully reduce the sidelobes level from dB to dB and for the Directivity is 20.34dB for the uniform and 19.09dB after optimizing. Figure (3.5): pattern of the. Circular array with optimization Result Discussion Optimization SLL using our mask, the figures show the pattern of the array factor when optimizing with 50 elements and 100 elements, for the first case of 50 element circular array, the uniform one has a ratio DIR/SLL of , we have optimized it to , for the 50 element circular array, the uniform one has a ratio DIR/SLL , we have optimized it to and although we have obtained a better SLL for both cases but we had a decrease in directivity 32

43 Chapter Three: Discussion and Results 3.5 Concentric Circular array We are using a concentric circular array consisting a number 7 rings; each ring hold a specific number of element hold a specific number of element, with a spacing between element = λ/2, λ=1. Case 1 We have used a 92 elements concentric circular array consisting of 5 rings, each ring hold a specific number of element Nm= [ ] Figure (3.6): pattern of 92 element Concentric Circular array with optimization The results obtained are shown in Figure (3.6), for the uniform array we have got SLL = dB, and Directivity=16.23dB and after optimizing we have a got a reduction of sidelobes level to -34.2dB, and a decrease in Directivity to 15.29dB. 33

44 Chapter Three: Discussion and Results Case 2 We have used a 168 elements concentric circular array consisting of 7 rings, with: Nm= [ ]. Figure 3.7: pattern of 168 element Concentric Circular array with optimization For the second case the results obtained are shown in Figure (3.8), for the uniform array we have got SLL = -15 db, and Directivity=18.95dB and after optimizing we have a got a reduction of sidelobes level to dB, and a decrease in Directivity to 16.95dB. Result Discussion Optimization SLL using our mask, the figures show the pattern of the array factor when optimizing with 92 elements spread over five rings and 168 elements spread over 6 rings, for the first case of 92 element circular array, the uniform one has a ratio DIR/SLL of 1.082, we have optimized it to.4486, for the 168 element concentric circular array, the uniform one has a ratio DIR/SLL 1.25 and we have optimized it to and although we have obtained a better SLL for both cases but we had a decrease in directivity. 3.6 The hexagonal array In this part we treat a hexagonal array, same as above we used two different size of element array to see the difference resulting in the directivity, the number of side lobes obtained and the better reduction where it will occur. For both cases we optimize for better sidelobes reduction. 34