phase compensating dielectric lens design with genetic programming

Phase Compensating Dielectric Lens Designwith Genetic Programming

Lei Sun, Evor L. Hines, Roger J. Green, Mark S. Leeson, D. Daciana Iliescu

School of Engineering, Warwick University, Coventry CV4 7AL, United Kingdom

Received 5 April 2006; accepted 15 September 2006

ABSTRACT: This article illustrates a microwave dielectric lens design using genetic pro-

gramming (GP), which, to the best knowledge of the authors, is the first time GP has been

applied to design a microwave dielectric lens. A phase-compensating single layer lens and a

quarter-wave phase-compensating multilayer dielectric lens, which uses four types of materi-

als and a zoned structure, was designed using GP. The dielectric lens shape was designed

using GP strategy, with a random initial shape, to compensate for the phase error of the

multilayer dielectric lens and to achieve the required performance. Standard GP (SGP) and

hierarchical fair competition genetic programming (HFC-GP) was applied to obtain the

solution. The results show that this novel method of microwave lens shape design using GP

is both accurate and stable. In addition, the simulation results of using high frequency simu-

lation software (HFSS) are illustrated and compared. VVC 2007 Wiley Periodicals, Inc. Int J RF and

Microwave CAE 17: 493–504, 2007.

Keywords: genetic programming; Fresnel lens; Fermat’s principle; HFSS

I. INTRODUCTION

This article presents a novel method for phase com-

pensating dielectric lens design using genetic pro-

gramming (GP). Genetic programming is an exten-

sion of genetic algorithms into the area of computer

program induction by evolutionary search [1].

Genetic programming is a rapidly maturing technique

that has been successfully applied in a wide number

of areas, which demonstrate that it is sustainable [2].

The earliest attempt to develop computer programs

by evolution was investigated by Friedberg in the late

1950s [3–5]. As a founder, John Koza understood

and explored the power of program induction by evo-

lution and established the field of GP through exten-

sive demonstration of GP as a domain-independent

method that breeds a population of programs to solve

problems [5]. This article describes a novel method

for single layer and multilayer dielectric microwave

lens design using GP.

GP has been used in many areas, such as circuit

design, robot control [6], wired antenna design [7],

chemistry [8], and financial technical trading rule

determination [8, 9]. To the best knowledge of the

authors, this is the first article concerned with the

application of GP to microwave dielectric lens design

for antenna use.

For lens shape design, Fermat’s principle and ray

tracing are the two most important methods. Besides

these, in the electromagnetics area, the finite-difference

time-domain (FDTD) method, the method of moments

(MoM), finite-element (FE), etc. also can be adopted.

Here, the main research object is to apply GP into

microwave lens shape design firstly, so that Fermat’s

Correspondence to: E.L. Hines; e-mail: [email protected] 10.1002/mmce.20244Published online 16 July 2007 in Wiley InterScience (www.

interscience.wiley.com).

VVC 2007 Wiley Periodicals, Inc.

493

principle is selected as a simple method as the cost func-

tion to establish the GP lens design system.

The rest of article contains the following sections:

II. Microwave lens design using genetic program-

ming; III. Genetic programming simulation results;

IV. GA and HFC-GP based lens comparison V.

HFSS simulation results and comparison, and VI.

Conclusions.

II. MICROWAVE LENS DESIGN USINGGENETIC PROGRAMMING

In this article, we consider two broad strategies for

implementing our GP solution: (1) using standard GP

(SGP) principles, (2) using some advanced GP princi-

ples. We will say more about (1) later when we dis-

cuss the implementation of our system. As far as (2)

is concerned, there are many advanced GP strategies.

Three of these bring (i) in the case of Cartesian

genetic programming (CGP), it is possible to have as

many system outputs as necessary [10]. (ii) Hierarch-

ical fair competition genetic programming (HFC-GP)

sustains the evolutionary search by continuously

incorporating new genetic material into the evolving

pool and keeps lower- and intermediate-level evolu-

tionary processes, going on all the time, rather than

relying only upon ‘‘survival of the fittest’’ [5]. (iii)

Automatically defined functions genetic program-

ming (ADFs GP) [2] uses GP to simultaneously

evolve functions (ADFs) and call programs during

the same run.

The advanced GP strategy we adopt here is based

on the HFC-GP approach. However, in contrast with

the original application of HFC-GP, here we build

two levels into our program: the top level population

pool and the bottom level population pool. The top

level population pool is used to select chromosomes

to run the GP process, and the bottom one is used to

contain the chromosomes with poor fitness values

and additionally generate new chromosomes to com-

plement the top level population pool. For further

details on the HFC-GP technique, Ref. 5 gives a very

clear description. In section III, we will compare the

results of both SGP and HFC-GP.

As a traditional genetic programming application

[1], the microwave lens shape is designed using a

number of functions. An initial population pool is set

up to generate solutions randomly. Using the fitness

function, which evaluates each individual chromo-

some’s value according to its fitness value, all chro-

mosomes in the population pool are sorted. With the

use of genetic operators, including crossover, muta-

tion, and reproduction functions, new chromosomes

are generated and incorporated into the population

pool. The whole cycle will be repeated unless one or

more chromosomes meet the objective, or the system

completes the maximum number of generations. The

schematic outline of the GP MATLAB program is

shown in Figure 1.

Figure 1. Schematic outline of the GP MATLAB pro-

gram. [Color figure can be viewed in the online issue,

which is available at www.interscience.wiley.com.]

TABLE I. Default GP Parameter Values Given

by Koza

Koza

Population size, M 4000

Maximum number of generations to be run, G 51

Probability Pc of crossover 90%

Probability Pr of reproduction 10%

Probability Pm of mutation 0.0%

494 Sun et al.

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

A. Genetic Programming ControlParameter Values Setting

Initial parameter values for the GP application must

be established. Koza indicates that the primary pa-

rameters for controlling a run of genetic program-

ming are the population size, M, and the maximum

number of generations to be run, G [2]. Also, he

gives default parameters values in appendix D of his

book, and selected relevant parameters from Ref. 2

are listed in Table I below.

The simulation results produced using Koza’s val-

ues are shown in section IIIA later.

Figure 3. An example of the GP crossover process. [Color figure can be viewed in the online

issue, which is available at www.interscience.wiley.com.]

Figure 2. An example chromosome generated by GP. [Color figure can be viewed in the online


Microwave Dielectric Lens Design Using GP 495


B. Overview of the Key GeneticProgramming Modules

Crossover, mutation, and fitness functions are three

key functions in the application of a genetic algo-

rithm (GA) and GP. Due to the well known difficulty

in generating suitable tree structure chromosomes,

the function used to generate the initial population is

another very important function. The following para-

graphs describe four functions.

1. Initial Population Function. This function is

used to generate a GP population pool. Each individ-

ual chromosome in the population is composed of the

calculation operators, numerical values and an

unknown parameter, x. The calculation operators con-

sist of ‘‘þ’’ (add), ‘‘�’’ (substract), ‘‘*’’ (multiply), ‘‘/’’

(divide) and ‘‘@’’ (power). Numerical values are

from 0 to 9.

At the beginning of the GP process, 100 chromo-

somes are generated using this function with a maxi-

mum length of 80. An example of a chromosome is

illustrated in Figure 2.

Figure 4. An example of the GP mutation process. [Color figure can be viewed in the online


Figure 5. Illustration of Fermat’s principle [11]. [Color

figure can be viewed in the online issue, which is avail-

able at www.interscience.wiley.com.]

Figure 6. Relative fitness value distribution. [Color fig-

ure can be viewed in the online issue, which is available

at www.interscience.wiley.com.]

496 Sun et al.


The mathematical formula represented by the

chromosome in Figure 2 in string form style is:

4 3 618

xx

148 þ 3

" #ðx�8Þ þ 49ð Þ

14

4

2. Crossover Function. Because each chromosome

is represented by a string, which has a tree-structure,

we can say that a tree-based GP system is adopted in

this research work. During the operation of the cross-

over function, (1) an operator node will be selected

randomly in each of the two parent chromosomes; (2)

then, the two parts of each chromosome will be

exchanged and united into two new chromosomes,

which are the offspring. The crossover process is

illustrated in Figure 3 below, where chromosomes

with higher fitness value are selected with higher

probability.

3. Mutation Function. The mutation function is

used to keep the chromosomes in the population pool

fresh. After several tens of generations chromosomes

in the population pool may become similar. The

mutation function can randomly select some chromo-

somes and choose to change the node at which to cre-

ate a new subtree so that the fitness value changes af-

ter the mutation process is completed. An example of

the GP mutation process is shown in Figure 4 below.

4. Fitness Function. There are a number of meth-

ods that can be used to compensate for the phase

TABLE II. Comparison Between Koza’s Parameter

Values and the Authors’

Koza

The

Authors

Population size, M 4000 100

Maximum number of

generations to be run, G 51 51

Probability Pc of crossover 90% 90%

Probability Pr of reproduction 10% 10%

Probability Pm of mutation 0.0% 1%

Figure 7. Fitness value versus number of generations

graphs for single material lens using SGP design with

Koza’s values. [Color figure can be viewed in the online



graphs for single material lens using SGP design with the

authors’ values. [Color figure can be viewed in the online



graphs for single material lens using HFC-GP design with

Koza’s values. [Color figure can be viewed in the online





graphs for single material lens using HFC-GP design with

the authors’ values. [Color figure can be viewed in the online


TABLE III. Ten Best HFC�GP Results

Running

Time GP Results Generation

Fitness

Value

1 0**�xþx�**x1*4/x2*8@x�xxþ/xx�@x�x1*4/86/�34�*xx�xx 21 999.394

2 0þ@//þx*x6*xþxxx46 40 999.841

3 0/��8@1x1�@4*þ/x56/@1x*4//xþ52�þ76*1x@7�86þ4x 9 999.925

4 0�@þ1�/x/�3/1x1�xxxþþþ9x7�//5��x3x�3xx 28 999.976

5 0��/þ/þþ@89/x/8x/x��þ@xx@x8x�@x@4@þ9þx4x�x@6/5*92�41354/1*4*45 39 999.987

6 0@�@x/6þ6xþ/�9x@þ7x8@/*2x/x�9x@*6x6*4þ/32//4x@53 35 999.551

7 0��*2@**xx7��x3*6xþ@@xx@�9x/*x2@xþ2x@x*9x@@x57 1 999.97

8 0**�þ2*4�53�7x/8þþxx98 2 999.761

9 0///þ3þ7*4/9**xxþ5@7*xx2��þxþx9x*4*@377@/19�x3 6 999.265

10 0�@8@/�xxþx8��3�8xþ*/þx12/x62/*x//x2*x*11þ@xx�6@6x 14 999.934

error. These include, for example, Fermat’s principle

[11], Ray tracing [12], and its finite difference

approach [13]. Of these methods, Fermat’s principle

is the least time consuming. So, Fermat’s principle is

adopted here for fitness value calculation, and Figure

5 below illustrates the principle.

In Figure 5, comV is a standard parameter, and

represents the distance from the source point to the

edge of the lens. Zi is one example of the sampling

points (the authors chose to use 20 sampling points

along the Z coordinate) to investigate the fitness val-

ues of the formula, represented by a chromosome in

string style. Value 1 is the distance between the

source point and the horizontal position of the sam-

pling point along the lens, and value 2 is the lens

thickness at the sampling point.

According to Fermat’s principle, to determine the

best lens shape, the following formulae are applicable:

comV

ko¼ value 1

koþ value 2

kdcomV

ko¼ value 1

koþ value 2

ko=ffiffiffiffier

p

comV ¼ value 1þ ffiffiffiffier

pvalue 2

ð1Þ

where ko is the wavelength in free space, kd is the

wavelength in the dielectric material, and er is the

dielectric material’s relative permittivity.

The aim here is to use GP to generate a lens shape

formula to satisfy eq. (1). Hence, of all the chromo-

somes which are the lens shape formulas represented

as tree style strings, the one which can best meet the

requirements of eq. (1) is the best for lens design.

So, the absolute fitness value of each sampling

point is:

Fitness ¼ comV� value 1� ffiffiffiffier

pvalue 2 ð2Þ

and the total absolute fitness value of each chromo-

some is:

Fitness total ¼Xi

Fitness ðiÞ ð3Þ

Each chromosome in string style represents a for-

mula for the lens shape in 2D. The inputs to the for-

mula are parameters along the Z coordinate, and the

outputs of the formula are the calculation results,

which are the points along the X coordinate. For fit-

ness value calculation, 20 sampling points, selected

evenly along the Z coordinate, are used as the input

parameters for the formula. All these values are com-

pared with the standard marker, which is ‘‘comV’’.

The sum of all their differences is the absolute fitness

value of the chromosome. Therefore, if the sum is 0,

the best lens shape formula is achieved. To avoid the

possibility of negative fitness values, we adopt a rela-

tive fitness value, which is shown below:

498 Sun et al.


Fitness relative ¼ 2�jFitness totalj ð4Þ

For better visualization, the fitness value will be

mapped to the range of (0, 1000):

Fitness relative ¼ 2�jFitness totalj31000 ð5Þ

The relative fitness value distribution is shown in

Figure 6 below.

III. GENETIC PROGRAMMINGSIMULATION RESULTS

A. Results of GP-Based Single LayerLens Design

Due to the convergence characteristic of GP, it was

felt appropriate to tackle the problem with several

runs of a small-population GP, rather than with just

one run of a large-population, as both will often cost

the same in terms of simulation time. The following

method has been applied to produce the simulation

data, instead of using thousands of generations. Each

MATLAB program cycle includes 10 completed GP

cycles. To compare the results of using SGP and

HFC-GP, two different programs have been run using

these two different strategies separately. The fitness

value versus number of generations figures will be

shown later in this section. Meanwhile, to save simu-

lation time and keep individuals in the population

pool activated, in contrast with the default parameter

values given by Koza, the initial population size and

the probability of mutation are set as 100 and 1%,

respectively. Simulations were conducted to deter-

mine whether or not the results obtained with the

results using these new parameters values were

comparable with Koza’s. The initial GP parameter

values adopted in this research work are listed in

Table II.

For further comparison, we use Koza’s parameter

values and the authors’ parameter values in the HFC-

GP MATLAB program. The PC used to simulate the

MATLAB programs has 2.79 GHz CPU and 1.5 GB

of RAM.

Figure 7 below, of fitness value versus number of

generation, shows the simulation results obtained

using SGP and Koza’s parameter values. Figure 8

shows the results using SGP and the authors’ values.

Figure 9 shows the results using HFC-GP and Koza’s

values, and Figure 10 is based on the use of HFC-GP

but using the authors’ parameter values.

There are a number of differences between the

results of Figures 7 and 8. Firstly, not all 10 run

results can be seen in both figures. Only 3 and only 6

run results can be seen in Figures 7 and 8, respec-

tively. So, where are the other results? According to

the simulation data, the other results all are zero, and

hence they cannot be seen in the figures above. Sec-

ondly, the best fitness values for each program run

cycle are significantly different. In Figure 7, the best

one is the first run (around 940), and the worst one,

which is not zero, is the fifth run (around 250). This

shows nearly a 270% difference. In Figure 8, the best

one is the sixth run (around 990), and the worst one,

which is not zero, is the ninth run (around 250). This

Figure 12. 3D plot of the HFC-GP designed single ma-

terial lens. [Color figure can be viewed in the online issue,


Figure 11. Side view of the HFC-GP based design of

single material lens shape. [Color figure can be viewed in

the online issue, which is available at www.interscience.

wiley.com.]



shows nearly a 300% difference. Both sets of results

mean that both programs are very unstable. Finally,

we can see that none of the fitness values have been

improved after a very small number of generation

iterations, which is represented as the flat fitness

value after two or three generation iterations. This

suggests that there are a large number of redundant

chromosomes in the population pool. Comparing Fig-

ures 7, 8, 9, and 10, we can see that the worse simula-

tion result, in Figures 7 and 8, is because of the

redundant chromosomes without using HFC. Hence

HFC is clearly very beneficial.

Comparing Figures 9 and 10, both strategies can

achieve good fitness values, which are typically

nearly 1000, in 40 generation iterations. In other

words, to get a suitable individual chromosome with

good fitness values, the parameter values identified

by the authors in this work are more appropriate than

Koza’s values. To achieve the results using Koza’s

parameter values, it takes about 50 h, whereas our pa-

rameter values only takes around 90 min on a PC,

2.79 GHz CPU, and 1.5 GB of RAM.

In Figure 10, the best fitness value from all the

HFC-GP cycles is used for comparison. Due to the

random nature and dependent probability of the

selection procedure in the GP, the result of every sin-

gle GP cycle is not always the same. The best HFC-

GP result so far is then used in HFSS to simulate

results for the performance of the HFC-GP designed

single material lens. All 10 best GP results are listed

in Table III below.

From this table, the fifth program run gives the

result with the highest fitness value. Using the string

generated by HFC-GP, the 2D and 3D schematics of

the single material microwave lens are illustrated in

Figures 11 and 12, respectively.

The dimensions of the single layer lens as shown

in Table IV.

B. Results of HFC-GP Based MultilayerLens Design

Using the same methodology and the parameter val-

ues given by the authors, as in IIIA, on HFC-GP

based multilayer lens is designed based on the Fres-

nel lens. Reference 14 presents a Genetic Algorithm

(GA) based multilayer dielectric lens, based on the

Fresnel lens.

TABLE IV. 2D HFC-GP Based Design of Single Layer Lens

Z (mm) 0 1 2 3 4 5 6

X (mm) �3.329 �3.32 �3.295 �3.254 �3.196 �3.121 �3.029

Z (mm) 7 8 9 10 11 12 13

X (mm) �2.921 �2.796 �2.654 �2.496 �2.321 �2.13 �1.922

Z (mm) 14 15 16 17 18 19 20

X (mm) �1.697 �1.455 �1.197 �0.922 �0.63 �0.322 0

Figure 14. 3D view of HFC-GP based multilayer lens.

[Color figure can be viewed in the online issue, which is

available at www.interscience.wiley.com.]

Figure 13. 2D view of HFC-GP based multilayer lens.

[Color figure can be viewed in the online issue, which is

available at www.interscience.wiley.com.]

500 Sun et al.


As described in Ref. 14, one of the main advan-

tages of a Frensel lens is that it can be produced at

low cost, and thus we can try to balance the relation-

ship between the cost and the size of the antenna.

One of the main disadvantages of this lens is that it

does not attain the same aperture efficiency as a

shaped lens, since it corrects the phase of the feed

source only at discrete locations over its aperture,

whereas a shaped lens corrects the phase of the feed

source at every location [15]. The solution to this

problem of degradation in gain, due to phase quanti-

sation errors over the Fresnel lens aperture, is to

design the lens to correct the phase at an increased

number of locations [16]. One method to accomplish

this is to use additional grooves in the dielectric ma-

terial [17]. Another method is to use different dielec-

tric constants for the various zones in the lens [18],

which is called a Fresnel-zone plate lens (FZPL). The

cost can be reduced by using the first method, but a

more complex fabrication process will result, and the

lens will be thicker. On the other hand, with the sec-

ond method, a thinner lens can be produced at a

higher cost. Hence by combining the two methods, it

may be possible to balance the cost and fabrication

aspects. In this section, a HFC-GP based multilayer

dielectric lens is designed. The 2D and 3D HFC-GP

based multilayer lens structure is as shown in the fig-

ures below, Figures 13 and 14 respectively.

The dimensions of the multilayer lens are shown

in Table V.

IV. GA AND HFC-GP BASED LENSCOMPARISON

To investigate the shape of HFC-GP based lens

design, a GA-based lens design [14] is employed to

be compared.

Figures 15 and 16 show a comparison of the

results for the GA based lens and HFC-GP based

lens. The 2D single and multilayer lens shapes are ap-

proximate; however, they are not exactly matched.

The reason for this is that in the GA application the

algorithm is applied to individual points (17 points in

total) along the chromosome, and the final 2D lens

shape is generated after all individual points achieve

the best X coordinate with the best fitness value. In

the HFC-GP application, the system provides a 2D

lens shape formula for the whole point of view, with-

out dealing with any individual points. As the authors

TABLE V. 2D HFC-GP Based Multilayer Lens

Z (mm) 0 1 2 3 4 5 6

X (mm) �0.991 �0.935 �0.867 �0.77 �0.647 �0.5 �0.332

Z (mm) 7 8 9 10 11 12 13

X (mm) �0.146 0.054 �0.484 �0.364 �0.242 �0.119 �0.471

Z (mm) 14 15 16 17 18 19 20

X (mm) �0.38 �0.271 �0.118 �0.285 �0.178 �0.069 0.042

Figure 16. Comparison of GA and HFC-GP based multi-

layer lens. [Color figure can be viewed in the online issue,


Figure 15. Comparison of GA and HFC-GP based single

layer lens. [Color figure can be viewed in the online issue,




mentioned in the fitness function for HFC-GP based

lens design, the best solution generated by the HFC-

GP MABLAB program is the chromosome with the

best fitness value for all the fitness values of all sam-

pling points, rather than that of the total fitness values

of each individual point. As a result, there is a differ-

ence between the two sets of results.

Besides using same function parameters values

shown in Table II, both of the simulations of GA and

HFC-GP programs operate on a PC with 2.79 GHz

CPU and 1.5 GB of RAM. The time cost of design a

HFC-GP based single layer lens is 60 min, however,

it costs 20 min to generate a GA based single layer

lens. Also, the cost of design a HFC-GP based multi-

layer lens is 250 min, but it costs 20 min to generate

a GA based multilayer lens.

V. HFSS SIMULATION RESULTSAND COMPARISON

Besides the comparison of the lens shape, in this sec-

tion, an HFSS simulation results comparison of two

lenses are given. Here, a conical horn antenna is

adopted as the source, and the lenses are set at the

top of the horn antenna, to generate lens-horn anten-

nas. The conical horn antenna structure in HFSS is

shown in Figure 17. The lens-horn antenna structure

in HFSS is shown in Figure 18, and, for simplicity,

only the structure of single layer lens horn is illus-

trated in Figure 18.

HFSS simulation results comparisons are shown in

the following figures. For comparison, the antenna

gain patterns (/ ¼ 0 and 90) are provided.

Figure 18. Structure of the single layer lens horn

antenna as used in HFSS. [Color figure can be viewed in


wiley.com.]

Figure 19. Antenna gain comparison for HFC-GP based

single layer lens horn antenna and GA-based single layer

lens horn antenna. [Color figure can be viewed in the online


Figure 17. Structure of the conical horn antenna as used

in HFSS. [Color figure can be viewed in the online issue,


502 Sun et al.


In Figure 19, the red and blue lines are the gain

patterns for the HFC-GP based single layer lens horn

antenna in the plane of / ¼ 0 and 90, respectively;

the green and black lines are the gain patterns for the

GA-based single layer lens horn antenna in the plane

of / ¼ 0 and 90, respectively. It clearly shows that

the difference in the maximum gain at 08 is only 1

dBi, which is not significant. Both of the lens horn

antennas performances have a decrease in the side

lobe gain and back lobe gain; however, the GA based

lens horn antenna is better than the HFC-GP based

single layer lens horn antenna, in this respect. For the

back lobe gain, the gain of the GA based single layer

lens horn antenna is 10 dBi less than that of the HFC-

GP single layer based lens horn antenna, which is

about �7 dBi. For the side lobe gain, the gain of the

GA based single layer lens horn antenna is 12 dBi

less than that of the HFC-GP based single layer lens

horn antenna, which is �19 dBi, in the plane of / ¼0, and nearly the same in the plane of / ¼ 90.

Figure 20 is the comparison between the GP-based

multilayer lens horn antenna and GA-based multilayer

lens horn antenna. The figure shows that in the range

from �508 to 508, the gain pattern in both faces / ¼ 0

and 90 are similar. However, in respects of the back and

side lobes the difference between these performances

can be up to 10 dBi maximum. For example, the gain

power of the back lobe of the HFC-GP based multilayer

lens horn antenna is �18 dBi, which is 10 dBi less than

that of the GA based multilayer lens horn antenna.

All the description above shows that the HFC-GP

simulation system gives a better result than the GA

simulation system.

VI. CONCLUSION

This article discusses the use of HFC-GP to design a

single layer and a multilayer dielectric lens. The sim-

ulation results of the genetic programming design of

dielectric microwave lens shape with tree structure

are reasonable and acceptable.

According to Table III, the HFC-GP simulation

shows its very stable performance characteristic, and all

the best fitness values are between 999 and 1000. All

the best tree structure strings are usually generated

before the 40th generation. The difference between the

best and the worst individual chromosome of the 10

simulation runs is 0.725 (the difference between the

fifth run result and the ninth run result), which is 0.73%.

Compared with the simulation results using two

different methods, the lens shapes produced using

GA and HFC-GP are approximated. However, in the

respect of time consumption, the HFC-GP MATLAB

program takes 60 and 250 min separately for single

and multilayers, rather than the 20 min consumed by

the GA MATLAB program to produce a multilayer

lens shape (operating on the same PC, 2.79 GHz

CPU and 1.5 GB of RAM), and it costs 60 min for an

HFC-GP MATLAB program to generate a single

layer lens, rather than 20 min use of a GA MATLAB

program. This is one aspect for HFC-GP program to

be improved in further research. From the compari-

son of the HFSS simulation results, HFC-GP displays

a better result than GA.

As a conclusion, from this first GP application to

microwave dielectric lens shape design, acceptable

results can be generated using HFC-GP. Compared

with the SGP, HFC-GP application provides a signifi-

cant improvement. For further work, one important as-

pect is to improve the GP MATLAB program’s run-

ning time. Apart from this, other advanced GP techni-

ques could be explored for use in lens shape design to

improve the program efficiency and accuracy.

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MA, 1994.

3. R.M. Friedberg, A learning machine, Part I: IBM,

J Res Dev 2 (1958), 282–287.

4. R.M. Friedberg, B. Dunham, and J.H. North, A learning

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BIOGRAPHIES

Lei Sun received a BSc in Telecommunica-

tions in 2000 from NanKai University, P. R.

China and an MSc in 2002 from the Univer-

sity of York, UK. He received a PhD degree

from the University of Warwick in the area

of antenna design using Artificial Intelligent

Systems. His main research interests are

concerned with Genetic Algorithms, Genetic

Programming, and novel applications in

electromagnetics using the FDTD method. He currently is a mem-

ber of the Intelligent Systems Engineering Laboratory, University

of Warwick.

Evor L. Hines joined the School at the

University of Warwick in 1984. He is an

Associate Professor (Reader) in Electronics.

His main research interest is concerned

with Intelligent Systems and their applica-

tions. Most of the work has focused on

Artificial Neural Networks, Genetic Algo-

rithms, Fuzzy Logic, Neuro-fuzzy Sys-

tems, and Genetic Programming. Typical

application areas include intelligent sensors (e.g. electronic nose); non-

destructive testing of, for example, composite materials; computer

vision, telecommunications; amongst others. He currently leads the

Intelligent Systems Engineering Laboratory. He has co-authored more

than 170 articles.

Roger J. Green became Professor of

Electronic Communication Systems at

Warwick in September 1999, and Head of

the Division of Electrical and Electronic

Engineering in August 2003. He has pub-

lished over 180 papers in the field of

optical communications, optoelectronics,

video and imaging, and has several pat-

ents. Some 46 PhD research students have worked successfully

under his supervision. He leads the Communications and Signal

Processing Research Group, which is the largest in the School of En-

gineering at Warwick. He is also a Senior Member of the IEEE, and

currently serves on two IEEE committees concerned with communi-

cations and signal processing. He is a member of the Smart Optics

Faraday Partnership at Warwick, a member of the UK EPSRC grants

Peer Review College, and an Evaluator for proposals in the EEC

Framework 6 activities.

Mark S. Leeson received a PhD for work

on planar optical modulators from the

University of Cambridge, UK, in 1990

and then worked as a Network Analyst

for a UK bank until 1992. Subsequently,

he held several academic appointments

before joining the University of Warwick

in March 2000 where he is an Associate

Professor. His major research interests are

optical receivers, optical communication systems, communication

protocols, and ad hoc networking. To date he has published over

90 journal or conference papers in these fields. He is a Chartered

Member of the UK Institute of Physics and a Member of the Institute

of Electrical and Electronic Engineers in the USA.

Doina Daciana Iliescu graduated in 1992

from the Polytechnic Institute of Bucha-

rest, Romania, Faculty of Electronics and

Telecommunications, specialization in

Telecommunications and Data Networks.

She received her PhD in Engineering in

1998 from the University of Warwick,

UK, in the field of Optical Engineering.

Since then she has been an Associate Pro-

fessor in the School of Engineering, University of Warwick and a

research member of the Optical Engineering Laboratory.

504 Sun et al.


phase compensating dielectric lens design with genetic programming

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