Application of Support Vector Machines to theAntenna Design

Z. Zheng, X. Chen, K. Huang

School of Electronics and Information Engineering, Sichuan University, Chengdu 610064, China

Received 23 April 2010; accepted 29 July 2010

ABSTRACT: The antenna design is a complicated and time-consuming procedure. This

work explores using support vector machines (SVMs), a statistical learning theory based

on the structural risk minimization principle and has a great generalization capability, as

a fast and accurate tool in the antenna design. As examples, SVMs is used to design a

rectangular patch antenna and a rectangular patch antenna array. Results show, after an

appropriate training, SVMs is able to effectively design antennas with high accuracy.

VC 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE 21:85–90, 2011.

Keywords: support vector machines; antenna design; rectangular patch antenna; rectangular

patch antenna array

I. INTRODUCTION

In the modern society, more and more high-performance,

customized antennas are in demand. The antenna design is

a complicated and difficult procedure. Engineers design

antennas by their intuition, experience, approximate for-

mulae, using full-wave simulation technology [1, 2] or

even utilizing optimization algorithms [3, 4], but all the

methods are time- and labor-intensive. To overcome the

problem, artificial neural networks (ANNs), which is

developed from neurophysiology by morphologically and

computationally mimicking human brains, has been inves-

tigated for the antenna design [5–8]. However, ANNs has

difficulties with generalization, e.g., in many cases it pro-

duces models that can overfit the data, which is a conse-

quence both of the optimization algorithms used for pa-

rameter selection and of the statistical measures used to

select the ‘‘best’’ model [9].

In the past years, the foundations of SVMs have been

developed by Vapnik [10, 11] and are gaining popularity

due to many attractive features. The SVMs formulation

embodies the Structural Risk Minimization principle,

which has been shown to be superior to traditional Empir-

ical Risk Minimization principle, used by ANNs [11].

Hence SVMs has a greater ability to generalize. SVMs

have been widely used to solve classification [12] and

regression problems [9]. And it is also investigated to

apply in the field of microwave device design [13–15],

here it is researched for the antenna design.

SVMs’ great generalization capability provides a

powerful tool for the antenna design. This work investi-

gates the performance of SVMs in the antenna design in

terms of accuracy and efficiency. Examples that use

SVMs to design a rectangular patch antenna and a rectan-

gular patch array will be presented. In the examples, the

FDTD (Finite Difference Time Domain) method is uti-

lized to provide training data and test data for SVMs.

The article is organized as follows. A theoretical over-

view of SVMs is given in Section II. In Section III, we

take a rectangular patch antenna and a rectangular patch

antenna array as examples to estimate whether SVMs can

be effectively applied to the field of the antenna design.

Finally, in Section IV a brief conclusion is given.

II. A THEORETICAL OVERVIEW OF SVMs

The SVMs algorithm is a nonlinear generalization of the

Generalized Portrait algorithm developed in Russia in the

1960s. It is firmly grounded in the framework of statistical

learning theory which characterizes it the properties to

generalize to unseen data [16].

Let us explain the mathematical framework in which

SVMs is defined briefly. Suppose we are given training

data

fðx1; y1Þ…ðx1; y1Þg � v� i; (1)

where i denotes the assembly of real numbers, v denotes

the space of the input patterns (e.g., v ¼ id) and our goal

Correspondence to: X. Chen; e-mail: xingcsc@yahoo.com.cn

VC 2010 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20491Published online 29 November 2010 in Wiley Online Library

(wileyonlinelibrary.com).

85

is to find a function f(x) to approximate the obtained tar-

gets yi for all the training data. The function f(x) can be

described as

f ðxÞ ¼Xl

i¼1

ðai � a�i ÞKhxi; xi þ b; (2)

where x [ v, b [ i and h.,.i denotes the dot product in v.ai, a�i are Lagrange multipliers [17] and K(xi,xj) is theFigure 1 The general procedure to use SVMs.

Figure 2 Rectangular patch antenna.

Figure 3 Performance values of a rectangular patch antenna predicted by SVMs. For (a), (b), and (c), L ¼ 12 mm; for (d), (e), and (f),

W ¼ 18 mm.

86 Zheng, Chen, and Huang

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 1, January 2011

kernel function [18]. We can get definite description of

the function by the methods proposed in previous work

[17, 19].

Loosely speaking, like in the above problem, SVMs

can describe the relationship between the independent and

dependent variables of the training data generated by an

unknown function and use the function f(x) which approx-

imates the unknown function to express the relationship.

The general procedure to use SVMs is illustrated in Fig-

ure 1 [20, 21].

III. APPLICATION OF SVMS TO THE ANTENNA DESIGN

In this section, SVMs is used as a fast and accurate tool

for designing a rectangular patch antenna and a rectangu-

lar patch array. A FDTD code is developed to execute

full-wave simulations to provide training data and test

data for SVMs.

In this work, we use m � SVMs and select Gaussian

kernel function K(x,y) ¼ exp(�kx�yk2/(2r2kernel)). And

here we let m ¼ 0.5, and choose the best SVMs model

among the values of (C, r2kernel) to be (10, 2), (100, 1),

and (1000, 0.5) respectively, so the model is optimal for

all different noise levels under the suppose that the noise

is Gaussian noise [22].

A. The Design of a Rectangular Patch AntennaAs illustrated in Figure 2, a rectangular patch antenna is

made of a rectangular patch with width W and length L,over a ground plane with a substrate thickness h¼ 1.588 mm

and relative dielectric constant er ¼ 4.7. This antenna is

fed from a 50 X coaxial line, whose outer conductor con-

nects with the ground plane while inner conductor pene-

trates the dielectric substrate and connects with the rectan-

gular patch at the feeding point A. The distance from the

feeding point A to one edge of the rectangular patch is

D ¼ 3 mm.

Because for the antenna design, the accurate antenna

analysis is the prerequisite, we firstly explores using

SVMs to predict antennas’ performance values, such as

the resonant frequency, gain and VSWR (voltage standing

wave ratio) as a function of its geometric parameters.

Herein, structural parameters W and L are taken as inputs

to SVMs, while antenna performance values including the

resonant frequency f0, the gain G and the voltage standing

TABLE I Results of Using SVMs to Design a Rectangular Patch Antenna

Objectives of the Antenna Design

Structural Parameters

got by SVMS

Performance Values

Calculated by the FDTD

f0 (GHz) G (dBi) VSWR W (mm) L (mm) f0 (GHz) G (dBi) VSWR

4.8 7.0 1.433 20.20 12.95 4.8375 7.0575 1.699

4.9 7.0 1.433 19.95 12.82 4.881 7.0420 1.705

5.0 7.0 1.433 19.29 12.58 4.9665 7.0055 1.688

5.1 7.0 1.433 18.43 12.32 5.061 6.9714 1.682

5.2 7.0 1.433 17.59 12.06 5.1585 6.9507 1.681

TABLE II Average rms Errors for SVMs Using Different Number of Training Data to Analyze and Design a RectangularPatch Antenna

Number of

Training Data

Average rms Error in the Antenna Analysis Average rms Error in the Antenna Design

f0 G VSWR W L

250 0.01% 0.25% 0.98% 0.65% 0.28%

150 0.02% 0.18% 1.24% 1.02% 0.32%

100 0.03% 0.39% 1.49% 1.24% 0.44%

50 0.24% 0.40% 2.34% 2.71% 1.36%

Figure 4 Rectangular patch antenna array.

TABLE III Value Ranges for Structural Parameters ofthe Rectangular Patch Antenna Array

Structural Parameters Value Range (mm)

D 34–52

D1 3–7

L1 5–7

W 15.3–15.5

W1 4.2–4.4

Application of SVMs to the Antenna Design 87

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

wave ratio VSWR at the resonant frequency f0, are taken

as the outputs. In this example, W and L are in the range

16.0 mm � W � 20.0 mm and 11.0 mm � L � 13.0 mm

respectively. A set of W and L are randomly selected,

then corresponding performance values f0, G, and VSWRare calculated by the FDTD. Some of them are provided

to SVMs as training data, and others are taken as test

data.

After trained by 200 training data, a set of performance

values with L ¼ 12 mm while W varies from 16 to 20

mm and with W ¼ 18 mm while L varies from 11 to 13

mm are predicted by SVMs, as illustrated in Figure 3.

One can observe that the performance values predicted by

SVMs are in good agreement with those obtained by the

full-wave simulation using the FDTD. The average root-

mean-square (rms) error for results of SVMs is less than

1%, which demonstrates that SVMs is able to analyze

antennas with high accuracy.

Then SVMs is used to design the rectangular patch

antenna. Here, performance values f0, G, and VSWR are

regarded as inputs to SVMs, while the outputs are struc-

tural parameters W and L. The objective is to design a

rectangular patch antenna with G ¼ 7.0 dBi, VSWR ¼1.433 (equals to S11 ¼ �15 dB), and resonating at fre-

quencies of 4.8, 4.9, 5.0, 5.1, and 5.2 GHz, respectively.

The number of training data for SVMs is still 200. To

estimate whether the objectives are achieved and the accu-

racy of SVMs, for each set of structural parameters got by

SVMs, the FDTD is carried out to calculate the corre-

sponding performance values. Results are listed in Table

I, and show SVMs achieves all design objectives with

high accuracy.

Usually, better and more robust results may be

obtained from SVMs if more input data sets are supplied

to train SVMs. To further investigate the performance of

SVMs, examples with different number of training data

are presented. In those examples, 100 sets of test data are

randomly generated to measure the accuracy of SVMs in

term of the average rms error. Results are listed in Table

II and shown that more training data is helpful for SVMs

to obtain results with higher accuracy, but SVMs performs

well even with relatively small quantity of training data

such as 50 training data.

B. The Design of a Rectangular Patch ArrayTo further estimate the performance of SVMs, a rectangu-

lar patch array, which has more complicated configuration

and more structural parameters than the rectangular patch

antenna in the previous example, will be designed using

SVMs. As illustrated in Figure 4, the patch array is

printed on a dielectric substrate with relative dielectric

constant er ¼ 2.65 and thickness h ¼ 1.0 mm. In this

example, it works at 5.8 GHz, and structural parameters

{D, D1, L1, W, W1} and performance values {G, VSWR}are considered by SVMs in the antenna design. G and

VSWR are the gain and VSWR at the working frequency

of 5.8 GHz. The value ranges for the structural parameters

are listed in Table III. The method of generating training

data and test data is as the same as that in the previous

example.

Also by measuring 100 sets of test data, Average rms

errors for SVMs to analyze and design the rectangular

patch antenna array are listed in Table IV. We can

observe that the average rms errors are small in most

cases, which demonstrate SVMs still performs well in this

example. But the average rms errors for SVMs to analyze

VSWR and determinate D1 are relatively big. The former

is because this antenna array resonates at or at the vicinity

of its working frequency 5.8 GHz, and so the value of

VSWR changes sharp at 5.8 GHz, which causes relatively

big errors in SVMs’ results. The later is because the per-

formance values {G, VSWR} are relatively insensitive to

D1 in comparison with other structural parameters. There-

fore, besides the number of training data, the accuracy of

TABLE IV Average rms Errors for SVMs to Analyze and Design a Rectangular Patch Antenna Array

Number of

Training Data

Average rms Error in

the Antenna Analysis Average rms Error in the Antenna Design

G VSWR D D1 L1 W W1

200 1.97% 17.87% 4.90% 19.30% 5.41% 0.32% 1.37%

150 2.20% 19.31% 4.91% 19.63% 5.43% 0.32% 1.19%

TABLE V Results of Using SVMs to Design a Rectangular Patch Antenna Array

Objectives of the

Antenna Design Structural Parameters Got by SVMs

Performance Values

Calculated by the FDTD

G (dBi) VSWR D (mm) D1 (mm) L1 (mm) W (mm) W1 (mm) G (dBi) VSWR

14.0 1.433 43.51 5.01 5.79 15.41 4.30 14.03 1.419

14.0 1.222 44.17 5.30 5.99 15.44 4.33 14.05 1.109

14.1 1.433 44.76 4.95 5.80 15.41 4.32 14.12 1.355

14.1 1.222 44.22 5.21 5.99 15.43 4.34 14.07 1.076

88 Zheng, Chen, and Huang

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 1, January 2011

SVMs in the antenna design may be affected by many

factors.

Though some accuracies of the results are not very

good, as shown in Table V, SVMs can still successfully

achieve all given design objectives for this complicated

antenna array, which once more estimates the great

capacity of SVMs in the antenna design.

It needs to be emphasized that the objectives for the

antenna design must be reasonable. In this example, the

gain of the antenna array is about 14 dBi. As shown in

Table VI, results got by SVMs are quite bad when the

objective exceeds the antenna’s performance limit.

IV. CONCLUSION

Nowadays, the antenna design suffers from heavy computa-

tional burden. For example, an antenna design procedure

using optimization algorithms such as the Genetic Algo-

rithm usually invokes thousands of or even more full-wave

simulations [3, 4]. With powerful generalization capability,

SVMs provides a fast and accurate tool in the antenna

design. In this article, an attempt has been made to exploit

the capability of SVMs in the antenna design. Results of

several experiments show SVMs is able to design antennas

with high accuracy after costs moderate full-wave simula-

tions to generate the training and test data sets. This ability

makes SVMs attractive in the antenna design.

ACKNOWLEDGMENT

This work was supported by the National Natural Science

Foundation of China (No. 10876020) and Key Laboratory of

Cognitive Radio (GUET), Ministry of Education, China.

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TABLE VI Results of SVMs to Design an Antenna with Unreasonable Objectives

No.

Objectives of the

Antenna Design Structural Parameters Got by SVMs

Performance Values

Calculated by the

FDTD

VSWR G (dBi) D (mm) D1 (mm) L1 (mm) W (mm) W1 (mm) VSWR G (dBi)

1 1.433 15.00 50.52 0.89 5.26 15.63 4.15 2.687 14.75

2 1.433 16.00 47.89 �3.45 4.64 15.70 2.48 – –

3 1.433 17.00 45.39 0.20 5.97 15.52 1.19 – –

For the Case 2 and 3, the FDTD can not build a valid antenna model.

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

Application of SVMs to the Antenna Design 89

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BIOGRAPHIES

Zhi Zheng was born on March 22,

1985 in Liaoning, China. He received

his B.S. degree in information and

communication engineering in 2008

from Sichuan University. Now he is

working toward his M.S. degree in

electromagnetics and microwave in

Sichuan University. His research is

mainly focused on computational electromagnetics and

antenna design. (king-z-love@163.com)

Xing Chen received the M.S. degree

in radio physics in 1999 and the

Ph.D. degrees in biomedical engi-

neering in 2004, both from Sichuan

University, China. He joined the

teaching staff in 1991, and is now a

Professor in the College of Electron-

ics and Information Engineering of

Sichuan University. His main research interests are in the

fields of antenna design, optimization algorithm, numeri-

cal methods, and parallel computation. He is an IEEE sen-

ior member and a senior member of Chinese Institute of

Electronics. (xingcsc@yahoo.com.cn)

Kama Huang received the M.S.

degree in 1988 and the Ph.D. degree

in 1991 in microwave theory and

technology both from the University

of Electronic Science and Technol-

ogy, China. He has been a professor

of the College of Electronics and In-

formation Engineering of Sichuan

University, China since 1994, and the director of the Col-

lege since 1997. In 1996, 1997, 1999, and 2001, he was a

visiting scientist at the Scientific Research Center ‘‘Vid-

huk’’ in Ukraine, Institute of Biophysics CNR in Italy,

Technical University Vienna in Austria, and Clemson Uni-

versity in USA, respectively. At these institutions, he coop-

erated with the scientists to study the interaction between

electromagnetic fields and complex media in biological

structure and reaction systems. (kmhuang@scu.edu.cn)

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 1, January 2011

90 Zheng, Chen, and Huang