chapter 4 linear antenna array synthesis … · models its processes after the principles of...

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CHAPTER 4

LINEAR ANTENNA ARRAY SYNTHESIS USING

GENETIC ALGORITHM

4.1 INTRODUCTION

Improving antenna characteristics such as directivity, input

impedance, bandwidth, sidelobe level, null depth level and size has always

been the goal of antenna researchers. It is important for the antenna

researchers to have the latest tools to effectively design antennas that meet the

given specifications. Optimization techniques are used to either synthesize an

antenna from given radiation characteristics or simply improve existing

antenna designs. Search routines that utilize numerical methods to provide

radiation properties of antenna generally consume a considerable amount of

time. Therefore, a great deal of work has been devoted to achieving

optimization routines that rapidly and accurately search out an optimum

solution.

Recently, a unique optimization scheme based on genetic

algorithms (GA) (Teruel and Rajo-Iglesias 2006, Stephen 2008, Goldberg and

Holland 1998) has been used to solve a number of electromagnetic problems.

Genetic algorithms, the adaptive heuristic search algorithms, are

programming techniques that mimic biological evolution as a problem-

solving strategy, based on the evolutionary ideas of natural selection and

genetic. Genetic algorithms tend to evolve in an environment in which there is

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a very large set of candidate solutions and in which the search space is uneven

and has many hills and valleys. Genetic algorithms are useful and efficient

when

The search space is large , complex and poorly understood

Domain knowledge is scarce or expert knowledge is difficult to

encode to narrow the search space

No mathematical analysis is available

Traditional search methods fail

The genetic algorithm is a robust, stochastic search method that

models its processes after the principles of natural selection and evolution.

These genetic algorithms are very useful for finding optimum antenna designs

that maximize or minimize certain radiation properties. The GA provides

optimal solutions by successively creating populations that improve over

many generations. Selecting, mating and mutating the previous population

each creates a new generation. This process continues until the population

converges to a single optimal solution. The selection process is based on the

rating of each member relative to the population. This rating is done by fitness

testing of each individual. The fitness value can be the gain, axial ratio, input

impedance, size, sidelobe level or any combination thereof. This fitness value

is received from some type of numerical code that can provide the radiation

characteristics of the antenna. A MATLAB code for SLL reduction and null

depth level is used in this thesis as a fitness function from which the fitness

value is obtained.

This chapter discusses the deployment of Genetic algorithm

optimization method for synthesis of antenna array radiation pattern in

adaptive beamforming. The synthesis problem discussed is to find the weights

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of the Uniform Linear Antenna array elements that are optimum to provide

the radiation pattern with maximum reduction in the sidelobe level (Basak

et al 2008).

4.2 GENETIC ALGORITHM AS AN OPTIMIZATION TOOL

Holland performed much of the foundational work in Genetic

Algorithm during 1960-1970. His goal of understanding the processes of

natural adaptation and designing biologically-inspired artificial systems led to

the formulation of the simple genetic algorithm.

Since its conception, genetic algorithms have enjoyed global use by

many researchers and scientists in many different areas. Although computer

scientists can take much of the credit for the development of GA, areas such

as business, science, and engineering have put the GA to good use.

Researchers, who historically have been obsessed with better and cheaper,

find particular interest in the GA. In Electromagnetics, most all problems are

of a non-differentiable type. Some type of numerical solution is required to

calculate the important characteristics of the problem. When the optimization

is a goal, these problems lend themselves very well to the use of genetic

algorithms. Antenna design is an area of Electromagnetics that has recently

benefited from the use of Gas. Haupt (1994,1995, 2004, 2008), recently

compiled a great deal of work on antenna design using genetic algorithms.

Array thinning and array synthesis are two areas where genetic algorithms

have proven their usefulness. In both of these cases, the GA has provided

results that exceed previous attempts to improve the array designs.

Genetic algorithms are typically implemented using computer

experiments in which an optimization problem is specified. For this problem,

members of a space of candidate solutions, called individuals, are represented

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using abstract representations called chromosomes. GA consists of an

iterative process that evolves a working set of individuals called a population

toward an objective function, or fitness function. Traditionally, solutions are

represented using fixed length strings, especially binary strings, but

alternative encodings have been developed.

The key terms related to genetic algorithm are

Individual – any possible solution

Population – group of all individuals

Search space – all possible solutions to the problem

Chromosome – blueprint for an individual

Trait – possible aspect of an individual

Allele – possible settings of a trait

Locus – the position of a gene on a chromosome

Genome – collection of all chromosomes for an individual

The evolutionary process of a GA is a highly simplified and

stylized simulation of biological version. It starts from a population of

individuals randomly generated according to some probability distribution,

usually uniform and updates this population in steps called generations. In

each generation, multiple individuals are randomly selected from the current

population based upon some application of fitness, bred using crossover, and

modified through mutation to form a new population. The genetic algorithm

operators are defined below

Crossover – exchange of genetic material (substrings) denoting

rules, structural components, features of a machine learning,

search, or optimization problem

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Selection – the application of the fitness criterion to choose

which individuals from a population will go on to reproduce

Reproduction – the propagation of individuals from one

generation to the next

Mutation – the modification of chromosomes for single

individuals

Current GA theory consists of two main approaches – Markov

chain analysis and schema theory. Markov chain analysis is primarily

concerned with characterizing the stochastic dynamics of a GA system, i.e.,

the behavior of the random sampling mechanism of a GA over time. The most

severe limitation of this approach is that while crossover is easy to implement,

its dynamics are difficult to describe mathematically. Markov chain analysis

of simple GAs has therefore been more successful at capturing the behavior of

evolutionary algorithms with selection and mutation only. These include

evolutionary algorithms (EAs) and evolutionary strategies. A schema is a

generalized description or a conceptual system for understanding knowledge-

how knowledge is represented and how it is used. According to this theory,

schemata represent knowledge about concepts: objects and the relationships

they have with other objects, situations, events, sequences of events, actions,

and sequences of actions.

4.2.1 Flow Chart for Simple Genetic Algorithm

The flow of simple Genetic algorithm can be explained through the

flowchart representation as shown in Figure 4.1.

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Initial population (firstgeneration) / best fit and

new population ofremaining generations

Fitness on costfunction evaluationsfor each generation

Selection of best fitindividuals/ rejectionof least fit population

Breed using geneticoperators crossover

and mutation

Generate newoffspring

Figure 4.1 GA reproduction cycle

The cycle is repeated until a termination condition has been reached

such as

1. A solution that satisfies the minimum criteria.

2. Reaching the specified number of generations.

3. Reaching the specified Computation time.

4. Arriving at fitness value and

5. Manual inspection.

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Traditionally, solutions are represented using fixed length strings,

especially binary strings, but alternative encodings have also been developed.

The genetic algorithms provide a directed random search in

complex landscapes. In nature, all living organisms have certain the physical

characteristics or traits, known as the phenome. The phenomes are encoded

into a set of genetic data structure, termed as genome. A particular set of

genetic information is a genotype, and likewise a particular set of physical

characteristics, or traits, is a phenotype. There may, or may not, be a direct

one-to-one mapping of genotypes to phenotypes. These physical

characteristics determine how well suited to its environment a particular

organism is. The suitability of a given organism to its environment is usually

measured as its fitness. Computationally, it is usual to evaluate the fitness of

an organism directly, without considering any kind of phenome.

4.2.2 Encoding

In GAs encoding the solutions of the problem into chromosomes is

a key issue. John Holland, used a single binary bit string. The problem

associated with encoding is that some individuals may correspond to

infeasible or illegal solutions. This may become very severe for constrained

optimization problems and combinatorial optimization problems. Infeasible

solution is one, that chromosome lies outside the feasible region of given

problem. Penalty methods can be used handle infeasible chromosomes. Illegal

solutions have chromosomes that do not represent a solution. Repair

techniques are usually adopted to convert an illegal chromosome to legal one.

Many different methods for encoding the genetic information are in common

use today; Tree encoding, real-valued arrays, permutations, Gray encoding

and so on.

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In general encoding methods can be classified as follows:

i) Binary encoding

ii) Real-number encoding

iii) Integer or literal permutation encoding

4.2.2.1 Binary encoding

Binary encoding (i.e., the bit strings) are the most common

encoding used. Much of existing GAs theories is based on the assumption of

using binary encoding. The binary code doses not preserve the locality of

points in the phenotype space.

4.2.2.2 Real number encoding

Real number encoding performs better than binary encoding for

function optimization and constrained optimizations problems. In real number

encoding, the structure of genotype space is identical to that of the phenotype.

Therefore, it is easy to form effective genetic operators by borrowing useful

techniques from conventional methods.

4.2.2.3 Permutation encoding

Permutation encoding is best used for combinational optimization

problems because it is useful in searching for the best permutation or

combination of items subject to constrain. The encoding schemes can be

better understood while dealing with genetic operators such as cross over and

mutation.

4.3 GENETIC ALGORITHM OPERATORS

There are two basic genetic algorithms operators which are

crossover and mutation. These operators perform a blind search by working

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together to explore – i.e. to investigate new and unknown areas in search

space and exploit- to make use of knowledge of solutions previously found in

search space to help in find better solutions, by creating new variants in the

chromosomes. It is confirmed that mutation operator play the same important

role as that of the crossover

4.3.1 Crossover or Recombination

Crossover plays important role in the design and implementation of

robust evolutionary systems. In most GAs, individuals are represented by

fixed-length strings and crossover operates on pairs of individuals (parents) to

produce new strings (offspring) by exchanging segments from the parents’

strings. Traditionally, the number of crossover points (which determines how

many segments are exchanged) has been fixed at a very low constant value of

1 or 2.

4.3.1.1 Single point crossover

A commonly used method for crossover is called single point

crossover which is shown in Figure 4.2. In this method, a single point

crossover position (called cutpoint) is chosen at random and the parts of two

parents after the crossover position are exchanged to form two offspring

Figure 4.2 Single point crossover

Parent2

Parent1 Offspring1

Offspring2

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4.3.1.2 Multi point crossover

Multi-point crossover is a generalization of single point crossover,

introducing a higher number of cut-points. In this case multi positions are

chosen at random and the segments between them are exchanged as shown in

Figure 4.3.

Figure 4.3 Multi point crossover

4.3.1.3 Uniform crossover

Uniform crossover does not use cut-points, but simply uses a global

parameter to indicate the likelihood that each variable should be exchanged

between two parents as shown in Figure 4.4.

[

Figure 4.4 Uniform crossover

Parent1 Offspring1

Parent2 Offspring2

Parent 1 Offspring 1

Parent 2 Offspring 2

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4.3.2 Mutation

Mutation is a common operator used to help preserve diversity in

the population by finding new points in the search pace to evaluate. When a

chromosome is chosen for mutation, a random change is made to the values of

some locations in the chromosome. A commonly used method for mutation is

called single point mutation. Though, a special mutation types used for varies

problem kinds and encoding methods.

4.3.2.1 Single point mutation

Single gene (chromosome or even individual) is randomly selected

to be mutated and its value is changed depending on the encoding type used,

as shown in Figure 4.5.

[[Figure 4.5 Single point mutation

4.3.2.2 Multi point mutation

Multi genes (chromosomes or even individuals) are randomly

selected to be mutated and there values are changed depending on the

encoding type used which is shown in Figure 4.6.

Figure 4.6 Multi point mutation

OffspringParent

Parent Offspring

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4.3.3 Selection

Selection is the process of determining the number of times a

particular individual is chosen for reproduction and, thus, the number of

offspring that an individual will produce. Selection provides the driving force

in genetic algorithms. With too much force, genetic search will terminate

prematurely. While with too little force, evolutionary progress will be slower

than necessary. Typically, a lower selection pressure is indicated at the start of

genetic search in favor of a wide exploration of the search space, while a

higher selection pressure is recommended at the end to narrow the search

space. In this way, the selection directs the genetic search toward promising

regions in the search space and that will improve the performance of genetic

algorithms. Many selection methods have been proposed, examined and

compared. The most common types are

1) Roulette wheel selection

2) Rank selection

3) Tournament selection

4) Steady state selection

5) Elitism

4.3.3.1 Roulette wheel selection

Roulette wheel selection is most common selection method used in

genetic algorithms for selecting potentially useful individuals (solutions) for

crossover and mutation. In roulette wheel selection which is shown in

Figure 4.7, as in all selection methods, possible solutions are assigned fitness

by the fitness function. This fitness level is used to associate a probability of

selection with each individual. While candidate solutions with a higher fitness

will be less likely to be eliminated, there is still a chance that they may be.

With roulette wheel selection there is a chance some weaker solutions may

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survive the selection process; this is an advantage, as though a solution may

be weak, it may include some component which could prove useful following

the recombination process. The analogy to a roulette wheel can be envisaged

by imagining a roulette wheel in which each candidate solution represents a

pocket on the wheel; the size of the pockets is proportionate to the probability

of selection of the solution. Selecting N individual from the population is

equivalent to playing N games on the roulette wheel, as each candidate is

drawn independently.

Figure 4.7 Roulette wheel selection

4.3.3.2 Rank selection

In ranking selection, as shown in Figure 4.8, the individuals in the

population are sorted from best to worst according to their fitness values.

Each individual in the population is assigned a numerical rank based on

fitness, and selection is based on this ranking rather than differences in

fitness. The advantage of this method is that it can prevent very fit individuals

from gaining dominance early at the expense of less fit ones, which would

reduce the population's genetic diversity and might hinder attempts to find an

acceptable solution. The disadvantage of this method is that it required sorting

the entire population by rank which is a potentially time consuming

procedure.

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(a)

(b)

Figure 4.8 Rank selection effect (a) before ranking (b) after ranking

4.3.3.3 Tournament selection

This method randomly chooses a set of individuals and picks out

the best individual for reproduction. The number of individual in the set is

called the tournament size. A common tournament size is 2, this is called

binary tournament. By adjusting tournament size, the selection pressure can

be made arbitrarily large or small. For example, using large Tournament size

has the effect of increasing the selection pressure, since below average

individuals are less likely to win a tournament while above average

individuals are more likely to win it.

4.3.3.4 Steady state selection

The steady state selection will eliminate the worst of individuals in

each generation. It works as follows; the offspring of the individuals selected

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from each generation go back into the pre-existing population, replacing some

of the less fit members of the previous generation.

4.3.3.5 Elitism

Elitism is an addition to many selection methods that force genetic

algorithms to retain some number of the best individual at each generation. It

improves the selection process and save the best individuals. With elitist

selection, the quality of the best solution in each generation monotonically

increases over time. Without elitist selection, it is possible to lose the best

individuals due to stochastic errors

4.4 GENETIC ALGORITHMS PARAMETERS

One of the more challenging aspects of using genetic algorithms is

to choose the configuration parameter settings. Discussion of GA theory

provides little guidance for proper selection of the settings. The population

size, the mutation rate, and the type of recombination have the largest effect

on search performance. They are used to control the run of a GA. They can

influence the Population and the Reproduction part of the GAs. In traditional

GAs the parameters has fixed values. Some guidelines are used in selecting

these parameter settings are given in the following subsections.

4.4.1 Population Size

The population size is one of the most important parameters that

play a significant role in the performance of the genetic algorithms. The

population size dictates the number of individuals in the population. Larger

population sizes increase the amount of variation present in the initial

population at the expense of requiring more fitness evaluations. It is found

that the best population size is both applications dependent and related to the

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individual size (number of chromosomes within). A good population of

individuals contains a diverse selection of potential building blocks resulting

in better exploration.

If the population loses diversity the population is said to have

“premature convergence” and little exploration is being done. For larger

individuals and challenging optimization problems, larger population sizes are

needed to maintain diversity (higher diversity can also be achieved through

higher mutation rates and uniform crossover) and hence better exploration.

Many researchers suggest population sizes between 25 16 and 100 individual,

while others suggest that it must be very much larger.

4.4.2 Crossover Rate

Crossover rate determines the probability that crossover will occur.

The crossover will generate new individuals in the population by combining

parts of existing individuals. The crossover rate is usually high and

‘application dependent’. Many researchers suggest crossover rate to be

between 0.6 and 0.95.

4.4.3 Mutation Rate

Mutation rate determines the probability that a mutation will occur.

Mutation is employed to give new information to the population (uncover

new chromosomes) and also prevents the population from becoming saturated

with similar chromosomes, simply said to avoid premature convergence.

Large mutation rates increase the probability that good schemata will be

destroyed, but increase population diversity. The best mutation rate is

‘application dependent’. For most applications, mutation rate is between

0.001 and 0.1 while for automated circuit design problems, it is usually

between 0.3 and 0.8.

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Genetic algorithms work on two types of spaces alternatively:

coding space and solution space, or in other words, genotype space and

phenotype space. Genetic operators (crossover and mutation) work on

genotype space, while evolution and selection work on phenotype space. The

selection is the link between chromosomes and the performance of decoded

solutions. The mapping from genotype space to phenotype space has a

considerable influence on the performance of genetic algorithms.

4.5 GENETIC ALGORITHM – A TOOL FOR SYNTHESIS OF

UNIFORM LINEAR ANTENNA ARRAY

In this thesis, two different approaches have been adopted to get a

deeper insight into the application of GA for the synthesis of linear antenna

array,

(1) Using Simple GA

(2) Using GA optimization toolbox in MATLAB

In general, larger population sizes tend to converge to an optimum

solution with fewer generations. This is because the solution space is

sufficiently covered with a larger population size. This luxury may not be

affordable if the fitness function is too time consuming. Also, it may be

redundant to use such a thorough initial search space. A tradeoff occurs

between population size and the time needed to converge to a solution. A

small population converges quickly, but may not find a global optimum. A

large population converges slowly, but more confidently finds the global

optimum. Judicious choices of gene and population size are important, and

dependent on the particular problem. Gene length (number of alleles or bits)

is set relative to how accurate an answer is needed. Larger gene length allows

more quantization levels. The gene length affects the accuracy of the solution.

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It is generally accepted to choose a population size that is 2 to

3 times larger than the total chromosome length (total number of bits). For

instance a chromosome with 4 genes (parameters) that each contains 10

alleles (bits) should have a population of 80 – 120. Larger populations require

more simulations per generation, but in general require fewer generations to

find a solution. This is merely because the initial search space is so large. This

may not be necessary for simple problems. However, more complex

problems, the majority of the initial population may be unacceptable and the

next generation is filled with the very few members that survived the selection

process. In fact, sometimes the initial population is filled with no acceptable

solutions at all. This is remedied by allowing a larger initial population and

thus a more thorough initial search.. Elitist strategy is important in order to

maintain a monotoniously increasing species fitness. The probabilities of

crossover and mutation should be set to 0.6 < pcross < 0.9 and 0.001 < pmut <

0.01. The genetic algorithm itself does not converge down to the exact

solution. It is very good at hunting down the approximate solution but does

not achieve the perfect solution each time. This fact becomes obvious when

multiple runs of the GA turns up many different answers which all lie very

close to one another. Therefore the use of another search routine after the

initial GA search may be necessary. To take advantage of the GA that is

already being used, simply create a new initial population that varies by ~5%

from the previous GA result and rerun the GA. This will force the GA to

significantly reduce its search space. There is no current theory that proves

that the genetic algorithm is capable of achieving the correct solution 100% of

the time. This is because the GA is dependent on the correct setup of the

problem. For instance, if the GA converges quickly to a wrong answer the

population size is too small. These user issues are difficult to quantify into an

exact formula for the setup of the GA. Considering all these criteria, we have

formulated and selected the parameters.

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4.5.1 Problem Formulation

Consider an array of antenna consisting of N number of elements. It

is assumed that the antenna elements are symmetric about the center of the

linear array. The far field array factor of this array with an even number of

isotropic elements (2N) can be expressed as

1( ) 2 cos(2 sin )N

n nnAF a d (4.1)

where an is the amplitude of the nth element is the angle from broadside and

dn is the distance between position of the nth element and the array center. The

objective is to find an appropriate set of required element amplitude an that

achieves interference suppression with maximum sidelobe level reduction.

To find a set of values which produces the desired array pattern, the

algorithm is used to minimize the following cost function given in

Equation (4.2)

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90( )[ ( ) ( )]o dcf W F F (4.2)

where Fo( ) is the pattern obtained using our algorithm and Fd( ) is the

pattern desired. Here it is taken to be the Chebyshev pattern with SLL of

-13dB and W( ) is the weight vector to control the sidelobe level in the cost

function. The value of weight vector is to be selected based on experience and

knowledge. The two different approaches used in our investigation are

detailed as follows.

4.5.2 ULA Synthesis using Simple GA

A continuous GA with a population size 10 and a mutation rate of

0.35 is run for a total of 500 generations using MATLAB and the best result is

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found for every iterations. The cost function is the minimum sidelobe level

for the antenna pattern. Figure 4.9 shows that the antenna array with N = 8

elements has been normalized for a gain of 0dB along the angle 0° and the

maximum relative sidelobe level of -15dB.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-30

-25

-20

-15

-10

-5

0

Figure 4.9 Optimized Radiation pattern with reduced Sidelobe level of

-15dB for N=8 elements

The convergence of the algorithm for maximum reduction in the

relative sidelobe level with N = 8 elements is depicted in Figure 4.10.

0 10 20 30 40 50 60 70 80 90 100-22

-21

-20

-19

-18

-17

-16

-15

-14

-13

-12

generation

Figure 4.10 Convergence of sidelobe level with respect to evolving

generations for N=8 elements

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It starts from -13dB which is the optimized value of chebychev

pattern for the RSLL and after 8 iterations it reaches -18.8dB and after 43

generations it converges to a maximum reduction of -21dB. Figure 4.11

shows the optimized radiation pattern with relative sidelobe level of -15dB

with N=16 and Figure 4.12 shows its convergence curve. The convergence

curve shows that it converges to -19.3dB after 54 generations. Changing the

number of elements causes the contiguous GA to get different optimum

weights. Among N=8, 16, 20, and 24, N=20 performed well and thus selected

as optimized element number. The corresponding array pattern for N=8, 16,

20, and 24 are shown in Figure 4.13. In this, the radiation pattern for N=20

has the best directivity with minimum relative sidelobe level of -14.67dB

below the main beam.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-30

-25

-20

-15

-10

-5

0

Figure 4.11 Optimized Radiation pattern with reduced sidelobe level of

-15 dB for N = 16 elements

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0 10 20 30 40 50 60 70 80 90 100-20

-19

-18

-17

-16

-15

-14

-13

generation



-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-30

-25

-20

-15

-10

-5

0N=24N=20N=16N=8

Figure 4.13 The optimized radiation pattern with reduced sidelobe level

for N=8, 16, 20 and 24

Figure 4.14 shows the convergence of sidelobe level for N=20.

Figure 4.15 shows the optimized radiation pattern with RSLL of -18.7dB with

N=20 elements.

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0 10 20 30 40 50 60 70 80 90 100-20

-19

-18

-17

-16

-15

-14

-13

generation

Figure 4.14 Convergence of sidelobe level with respect to evolvinggenerations for N=20 elements

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-30

-25

-20

-15

-10

-5

0


for number of elements N = 20

Figure 4.16 shows the convergence curve for N=24 elements.

Figure 4.17 shows the optimized radiation pattern with RSLL of -14.97dB

with N=24 elements.

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0 10 20 30 40 50 60 70 80 90 100-19

-18

-17

-16

-15

-14

-13

generation



-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-30

-25

-20

-15

-10

-5

0


for number of elements N= 24

The obtained costs are ranked from best to worst. The most among

suitability criteria is to discard the bottom half and to keep the top half of the

list. But in our program the selection criteria is to discard any chromosome

that has relative sidelobe level less than -15dB. Table 4.1 shows the cost

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function relative to the population that has a SLL less than -15 dB. Among 10

populations only 5 are selected. This limitation speeds up the convergence of

the algorithm. After this natural selection the chromosomes mate to produce

off springs. Mating takes place by pairing the surviving chromosome. Once

paired, the offspring consists of genetic material from both parents.

Table 4.1 Population and Respective Cost Function Values

Index Chromosome (weight vector)Relativesidelobe

level (dB)

1 0.8933 0.9059 0.7956 0.7167 0.5880 0.3725 0.2648 0.2232 -26.9682

2 0.8933 0.7659 0.5982 0.5391 0.9296 0.7216 0.7412 0.2059 -19.0861

4 0.4635 0.7659 0.6338 0.5391 0.9296 0.2594 0.7412 0.2059 -18.2047

7 0.8933 0.7659 0.8712 0.4302 0.9296 0.2594 0.7412 0.2059 -17.5515

5 0.8374 0.4270 0.7824 0.5322 0.5425 0.8238 0.9641 0.3051 -17.5173

Figure 4.18 shows the amplitude excitation for optimized antenna

array as given in Table 4.2.

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

elements

weight vector for N = 20 elements

Figure 4.18 Amplitude distribution for optimized antenna array with

N=20 elements

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Table 4.2 Amplitude excitation values For N=20 elements

corresponding to Figure 4.16

Wn Amplitude excitationW1

W2

W3

W4

W5

W6

W7

W8

W9

W10W11W12W13W14W15W16W17W18W19W20

0.90280.96450.72590.69100.70260.94910.77890.34780.50970.63190.53580.56250.46960.48280.24110.54640.00600.10600.40430.4334

4.5.2.1 Roulette wheel selection

In this thesis, the following parameters are defined; maxgen = 500,

maxfun = 1000 and mincost = -50dB. Population is generated randomly. Then

it is sorted based on its cost - minimum sidelobe level. For choosing mates for

reproduction Roulette wheel selection is used. Each weight vector is assigned

a probability of selection on the basics of either its rank in the sorted

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population or its cost. Rank order selection is the easiest implementation of

roulette wheel selection Figure 4.19 Shows the Roulette wheel selection

probabilities for five parents in the mating pool. The chromosome with low

sidelobe level has higher percent chance of being selected than the

chromosomes with higher sidelobe level. In this case first or the best weight

vector has a 43% chance of being selected.

9% 3%17%

28%

43%

Figure 4.19 Roulette wheel probabilities for five parents in the mating pool

As more generations are added, the percent chance of weight vector

being selected changes. Figure 4.20 shows the Roulette wheel selection for

seven parents in the mating pool.

25%

33%

18%

1%4%

7%

12%

Figure 4.20 Roulette wheel probabilities for seven parents in the mating

pool

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The best weight vector has 33% chance of being selected. The

roulette wheel selection needs to be computed only once, because the number

of parents in the mating pool remains constant from generation to generation.

The Genetic Algorithm has converged well for a variant of options

mentioned above with some trade-offs to have main impact on convergence

speed.

4.5.3 ULA Synthesis using GA Solver Optimization Tool Box

Optimization toolbox with GA-Genetic Algorithm solver in

MATLAB has been used in this thesis to find the amplitude excitations to

achieve minimum sidelobe level of -50 dB. Half the number of elements is

used as the number of variables with the Lower Bound (LB) = 0 and Upper

Bound(UB) = 1. The details of the other parameters set are as follows.

Population size = 20; Selection function = Roulette; Reproduction

(Elite count) = 1; Mutation function = Adaptive feasible; Crossover function

= Single point

A. Case 1

Number of variables = 8; Number of array elements =16;

The experiment has been conducted for 25 times and the best

results are presented here.

Figure 4.21 shows four different plots viz 1) Best fitness 2) Best

individual 3) Score Diversity and 4) Array pattern.

104

0 50 100-60

-40

-20

0

Generation

Best: -48.9263 Mean: -48.8641

1 2 3 4 5 6 7 80

0.5

1

Number of variables (8)

Current Best Individual

-49 -48.8 -48.6 -48.4 -48.20

5

10

15Score Histogram

Score (range)-50 0 50

-60

-40

-20

0

Best f itnessMean fitness

Figure 4.21 Showing (a) best fitness (b) best individual (c) score

histogram and (d) radiation pattern for N = 16 elements

Best result of – 48.9263dB sidelobe level is obtained with a mean

value of -48.8641dB. The number of variables is selected as 8, as the antenna

array consists of even number of elements which is symmetric about the

center. The Score Histogram shows, among 20 of the population, 12

individuals give the best score <-48 dB. It converges to -48dB only after 75

generations.

Figure 4.22 shows that the sidelobe level is reduced to – 36.7213dB

with a mean value of -38.6051dB. The Score Histogram shows 13 individuals

get the score < -36.6 dB. The amplitude excitations of best individuals

are obtained as w1 = 0.9853; w2 = 0.9242; w3 = 0.8215; w4 = 0.6698;

w5 = 0.5218; w6 = 0.3527; w7 = 0.2316; w8 = 0.1406;

105

0 50 100-40

-30

-20

-10

0

Generation

Best: -36.7213 Mean: -36.6051

1 2 3 4 5 6 7 80

0.5

1



-36.8 -36.6 -36.4 -36.2 -360

2

4

6

8Score Histogram


-60

-40

-20

0

Best fitnessMean fitness



The same is tabulated in Table 4.3 for 16 elements. The sidelobe

levels are almost constant for 6 sidelobes and the last one is wider and less

than the remaining. The convergence takes place in 80 generations.

Table 4.3 Amplitude Excitations of a 16 Element Array

W1 W2 W3 W4 W5 W6 W7 W8 W9 W10 W11 W12 W13 W14 W15 W16

0.14 0.23 0.35 0.52 0.67 0.92 0.82 0.98 0.98 0.92 0.82 0.67 0.52 0.35 0.25 0.14

B. Case 2:

Number of variables = 10; Number of array elements = 20;

The experiment is repeated for 10 variables. Figure 4.23 shows that

the sidelobe level is reduced to -31.147dB whereas the mean is -30dB. All the

individuals lie within the range of -30.5dB to -31.5dB. The main beam width

is narrower but the sidelobes are wider.

106

0 50 100-40

-30

-20

-10

0

Generation

Best: -31.1473 Mean: -30.8601

1 2 3 4 5 6 7 8 9 100

0.5

1



-31.5 -31 -30.5 -300

2

4

6Score Histogram


-60

-40

-20

0

Best f itnessMean f itness



A similar case is run and the result is shown in Figure 4.24.

0 50 100-40

-30

-20

-10

Generation

Best: -32.1697 Mean: -31.9476

1 2 3 4 5 6 7 8 9 100

0.5

1

1.5



-32.5 -32 -31.5 -310

2

4

6Score Histogram


-60

-40

-20

0

Best fitnessMean f itness



107

C. Case 3

The experiments are conducted with 22, 42, and 62 elements for 25

runs and their performance are tabulated in Table 4.4 and compared with that

of a table given in Haupt(2004).

Table 4.4 Performance Comparison of Antenna Array with Different

Number of Elements

N 8 16 20 24RSLL (dB) -16.1 -15 -18.7 -14.97Final Convergence (dB) -21.3 -19.3 -19.4 -18.7No. of generations 43 54 31 42

Table 4.5 shows the performance characteristics of five algorithms

for an average of 25 runs with random seed values of the amplitude weights.

Table 4.5 Comparisons of Optimized Sidelobes for Three Different Array

Sizes using Other Algorithms (Haupt 2004) and Genetic

Algorithm

Algorithm

22 Elements 42 Elements 62 ElementsMedianSidelobe

Level(dB)

MedianFunction

Calls

MedianSidelobe

Level (dB)

MedianFunction

Calls

MedianSidelobe

Level (dB)

MedianFunction

Calls

BFGS -30.3 1007 -25.3 2008 -26.6 3016DFP -27.9 1006 -25.2 2011 -26.6 3015Nelder Mead -18.7 956 -17.3 2575 -17.2 3551Steepest descent -24.6 1005 -21.6 2009 -21.8 3013Our proposed work(Genetic Algorithm )

-22.3 830 -20.3 940 -20.9 860

108

Genetic algorithm performs well when compared to Nelder Mead

but poorer when compared to the remaining algorithms. But the function calls

are minimum than all other algorithm. Hence it is cost effective in terms of

computational time. Genetic algorithm shows the best results of median

sidelobe level of -32.04dB with median function calls of 700 when the array

size is 16 elements. Among the three cases the number of elements of the

antenna array with N = 16 performed very well with narrow main beam width

and reduced sidelobe level and minimum number of function calls which cost

less computation time and less complexity.

From our study it is realized that Genetic algorithm has many

variables to control and trade-offs to consider as detailed below

Number of Chromosomes and initial random Population: more

number of chromosomes provide better sampling number,

solution space but at the cost of slow convergence.

Generating the random list, the type of probability distribution

and weighting of the parameters has a significant impact on the

convergence time.

Roulette selection method is employed to decide which

chromosome to discard.

Crossover the chromosome for mating, the chromosome may be

paired from top to bottom randomly best to worst.

Mutation rate is selected to mutate a particular chromosome.

Mutate does not permit the algorithm to get stuck at local

minimum.

109

Stopping Criteria, set in this program are maxgen = 500,

maxfun = 1000 and mincost = -50dB.

In our investigation the Genetic Algorithm has converged well for a

variant of options mentioned above with some trade offs to have main impact

on convergence speed.

4.6 SUMMARY

In this investigation two different approaches of namely (a) Simple

GA and (b) Genetic algorithm Solver in Optimization toolbox of MATLAB

are used to obtain maximum reduction in sidelobe level relative to the main

beam on both sides of 0°. The specialty of the Genetic algorithm is that it can

optimize the large number of discrete parameters. Genetic algorithm is an

intellectual algorithm searches for the optimum element weight of the array

antenna. This investigation demonstrated the different ways to apply Genetic

algorithm such as varying number of elements, to optimize the array pattern.

Adaptive feasible mutation with single point crossover and Roulette selection

showed the performance improvement by reducing the sidelobe level below

-30dB in most of the cases with number of variables as 8 and minimum

function calls when compared to the other methods shown in Table 4.4. The

best result of -48.9dB is obtained for 16 elements proving that this method is

efficient with much of the computation time and complexity are reduced.

Unlike Simple GA (SGA), the Genetic algorithm solver from the optimization

toolbox of MATLAB is used with adaptive feasible mutation, which enables

search in broader space along randomly generated directions to produce new

generations. This improves the performance greatly to achieve the maximum

reduction in sidelobe level with minimum function calls. Experiments proved

the effectiveness of this method.

chapter 4 linear antenna array synthesis … · models its processes after the principles of...

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