2007 IEE ANN and IFS paper related to antenna.pdf

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007 659
A Hybrid Method Based on Combining Artificial
Neural Network and Fuzzy Inference System
for Simultaneous Computation of Resonant
Frequencies of Rectangular, Circular, and
Triangular Microstrip Antennas
Kerim Guney and Nurcan Sarikaya
Abstract—A hybrid method based on a combination of artifi-
cial neural network (ANN) and fuzzy inference system (FIS) is
presented to calculate simultaneously the resonant frequencies of
various microstrip antennas (MSAs) of regular geometries. The
ANN is trained with the Bayesian regulation algorithm. An algo-
rithm that integrates least square method and backpropagation
algorithm is used to identify the parameters of FIS. The resonant
frequency results of the proposed hybrid method for the rectan-
gular, circular, and triangular MSAs are in very good agreement
with the experimental results available in the literature.
Index Terms—Fuzzy inference system, microstrip antenna
(MSA), neural network, resonant frequency.
I. INTRODUCTION
MICROSTRIP antennas (MSAs) are used in a broad
range of applications from communication systems to
biomedical systems, primarily due to their simplicity, con-
formability, low manufacturing cost, light weight, low profile,
reproducibility, reliability, and ease in fabrication and integra-
tion with solid-state devices [1]–[3]. These attractive features
have recently increased the application of MSAs and stimulated
greater effort to investigate their performance. The patch of
MSA may be of any geometrical shape. The rectangular, cir-
cular, and triangular patches are the basic and most commonly
used MSAs. These patches can be used for the simplest and the
most demanding applications.
In MSA designs, it is important to determine the resonant fre-
quencies of the antenna accurately because MSAs have narrow
bandwidths and can only operate effectively in the vicinity of the
resonant frequency. So a model to determine the resonant fre-
quency is helpful in antenna designs. Several methods [1]–[41],
varying in accuracy and computational effort, have been pro-
posed and used to calculate the resonant frequency of the rect-
angular, circular, and triangular MSAs. These methods can be
Manuscript received March 1, 2006; revised October 20, 2006.
K. Guney is with the Department of Electronic Engineering, Faculty of Engi-
neering, Erciyes University, 38039 Kayseri, Turkey (e-mail: kguney@erciyes.
edu.tr).
N. Sarikaya is with the Department of Aircraft Electrical and Electronics,
Civil Aviation School, Erciyes University, 38039 Kayseri, Turkey (e-mail:
nurcanb@erciyes.edu.tr).
Digital Object Identifier 10.1109/TAP.2007.891566
broadly classified into two categories: analytical and numer-
ical methods. The analytical methods, based on some funda-
mental simplifying physical assumptions regarding the radia-
tion mechanism of antennas, are the most useful for practical
design as well as providing a good intuitive explanation of the
operation of MSAs. However, these methods are not suitable for
many structures, in particular, if the thickness of the substrate is
not very thin. The numerical methods provide accurate results
but usually require tremendous computational effort and numer-
ical procedures, resulting in roundoff errors, and may also need
final experimental adjustment to the theoretical results. They
suffer from a lack of computational efficiency, which in prac-
tice can restrict their usefulness due to high computational time
and costs. In general, the numerical methods are based on an
electromagnetic boundary problem, which leads to expression
as an integral equation, using proper Green functions, either in
the spectral domain (the SDA method) or directly in the space
domain, using moment methods. Without any initial assump-
tion, the choice of test functions and the path integration appears
to be more critical during the final, numerical solution. The nu-
merical methods also suffer from the fact that any change in
the geometry (patch shape, feeding method, addition of a cover
layer, etc.) requires the development of a new solution.
In our previous works [37]–[40], the effective side length and
effective patch radius expressions obtained from the genetic al-
gorithm (GA) and the tabu search algorithm (TSA) have been
presented for calculating the resonant frequencies of the cir-
cular and triangular MSAs. GA and TSA were used to deter-
mine optimally the unknown coefficient values of the models
chosen for the effective side length and effective patch radius
expressions. We also proposed fuzzy inference system (FIS) for
computing the resonant frequencies of rectangular MSAs [41].
The optimum design parameters of the FIS were determined by
using the classical, modified, and improved tabu search algo-
rithms. It was shown in [37]–[41] that the results of the methods
based on GA, TSA, and FIS are better than those of the conven-
tional analytical and numerical methods.
During the last decade, ANN models have been increasingly
used in the design of antennas, microwave devices, and cir-
cuits [42], [43] due to their ability and adaptability to learn,
generalizability, smaller information requirement, fast real-time
operation, and ease of implementation features [44], [45]. A
0018-926X/$25.00 © 2007 IEEE

660 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007
neural network model can be developed by learning from mea-
sured/simulated antenna data, through a training process. The
aim of the training process is to minimize the training error be-
tween the target output and the actual output of the ANN. The
trained ANN model can be used during antenna design to pro-
vide instant answers to the task it learned. ANN in this paper is
used to model the relationship between the parameters of MSAs
and the measured resonant frequency results.
In order to improve the performance of ANN, in this paper,
a hybrid method based on combining a trained ANN with the
adaptive-network-based fuzzy inference system (ANFIS) is pre-
sented. The ANFIS [46], [47] can be considered as a class of
adaptive networks which are functionally equivalent to FISs.
Usually, the transformation of human knowledge into a fuzzy
system (in the form of rules and membership functions) does
not give exactly the target response. So, the parameters of the
FIS should be determined optimally. The main aim of ANFIS is
to optimize the parameters of the equivalent FIS by applying a
learning algorithm using input–output data sets. In the proposed
hybrid method, first, the resonant frequencies are computed by
using ANN, and then the inaccuracies in the ANN computation
are corrected by the ANFIS.
Due to their attractive features, ANN [48]–[54] and ANFIS
[55], [56] were used in computing the resonant frequencies
of various MSAs. The bandwidth and input resistance of the
MSAs were also computed by using ANN [57]–[59] and ANFIS
[60]–[62]. In general, in the literature, each different parameter
of each different MSA was computed by using a different indi-
vidual ANN or ANFIS model. However, a single neural model
trained with the backpropagation (BP), delta-bar-delta (DBD),
and extended delta-bar-delta (EDBD) algorithms was presented
by Guney et al. [52] for simultaneously calculating the resonant
frequencies of the rectangular, circular, and triangular MSAs.
The performance of this single neural model was improved in
[63] by using a parallel tabu search (PTS) algorithm for the
training process. The results of single neural models [52], [63]
are not in very good agreement with the experimental results
available in the literature [4], [5], [8], [12], [13], [16], [19],
[22], [23], [30], [31] for the rectangular, circular, and triangular
MSAs. For this reason, in this paper, a single hybrid method
based on a combination of ANN with ANFIS is presented
to calculate simultaneously and accurately the resonant fre-
quencies of the rectangular, circular, and triangular MSAs. To
calculate the resonant frequencies by using this single hybrid
method, the equivalent patch dimensions for the circular and
triangular MSAs are obtained by equating the patch areas
of the circular and triangular MSAs to that of an equivalent
rectangular microstrip antenna. The resonant frequency results
calculated by using the single hybrid method are in very good
agreement with the experimental results [4], [5], [8], [12], [13],
[16], [19], [22], [23], [30], [31].
In this paper, the next section briefly describes the resonant
frequencycomputationoftherectangular,circular,andtriangular
MSAs. The basic principles of ANN and ANFIS are presented in
thefollowingsection.Subsequently,theapplicationofthehybrid
method to the resonant frequency computation is explained. The
resultsarethenpresentedandconclusionsaremade.
Fig. 1. Geometry of rectangular MSA.
II. RESONANT FREQUENCY OF MICROSTRIP ANTENNAS
Fig. 1 illustrates a rectangular patch of width and length
over a ground plane with a substrate of thickness and a
relative dielectric constant . The resonant frequency of
the antenna can be calculated from [1]–[3]
(1)
where is the effective relative dielectric constant for the patch,
is the velocity of electromagnetic waves in free space, and
take integer values, and and are the effective dimen-
sions. To compute the resonant frequency of a rectangular patch
antenna driven at its fundamental mode, (1) is written as
(2)
The effective length can be defined as follows:
(3)
where is the edge extension.
The resonant frequency of a circular MSA for the
mode is expressed as [1]–[3]
(4)
where is the th zero of the derivative of the Bessel func-
tion of order and is the circular patch radius. The domi-
nant mode is , for which .
To account for the fringing fields, there have been a number of
suggestions in the literature [1]–[3]. Most of the suggestions are
about replacing the patch radius by an effective value and
leaving the substrate dielectric constant unchanged.
For a triangular MSA, the resonant frequencies obtained
from the cavity model with perfect magnetic walls are given by
[1]–[3]
(5)

GUNEY AND SARIKAYA: HYBRID METHOD FOR SIMULTANEOUS COMPUTATION OF RESONANT FREQUENCIES 661
Fig. 2. Diagram for equating the patch area of the triangular and circular MSAs
with the rectangular MSA.
where is the length of a side of the triangle. To account for
the fringing fields, the side length should be replaced by an
effective value .
A survey of the literature [1]–[41] clearly shows that the
resonant frequencies of the rectangular, circular, and triangular
MSAs are determined by and the dimensions of the
patch ( and for the rectangular MSA, for the circular
MSA, and for the triangular MSA). To calculate the resonant
frequencies of the rectangular, circular, and triangular MSAs
by using the single hybrid method, the areas of the circular
and triangular patches are equated to that of the rectangular
MSA. The following formulas are then used for the equivalent
dimensions of the circular and triangular patches with reference
to Fig. 2
and for the circular MSA (6)
and for the triangular MSA (7)
where is the height of the triangular patch. It is evident from
(6) and (7) that multiplying by is equal to the area of the
corresponding patch.
In the calculation of the resonant frequencies by using the
single hybrid method, first the equivalent values of and for
the circular and triangular MSAs should be obtained by using
(6) and (7). The resonant frequencies of the rectangular, circular,
and triangular MSAs are then determined by ,
and . The fundamental modes for the rectangular and circular
MSAs are ( and ) and ,
respectively. These modes are widely used in MSA applications.
III. ARTIFICIAL NEURAL NETWORKS (ANNS)
In the course of developing an ANN model, the architec-
ture of the neural network and the learning algorithm are the
two most important factors. ANNs have many structures and
architectures [42]–[45]. The class of ANN and/or architecture
Fig. 3. Architecture of ANFIS.
selected for a particular model implementation depends on the
problem to be solved. After several experiments using different
architectures coupled with different training algorithms, in this
paper, the multilayered perceptron (MLP) neural network archi-
tecture [44], [45] is used to compute the resonant frequencies of
MSAs.
The MLPs have a simple layer structure in which succes-
sive layers of neurons are fully interconnected, with connection
weights controlling the strength of the connections. MLPs can
be trained using many different learning algorithms [42]–[45].
In this paper, the Bayesian regulation (BR) algorithm [64] is
used to train the MLPs.
IV. ADAPTIVE-NETWORK-BASED FUZZY INFERENCE
SYSTEM (ANFIS)
The FIS is a popular computing framework based on the
concepts of fuzzy set theory, fuzzy if-then rules, and fuzzy
reasoning [47]. Among many FIS models, the Sugeno fuzzy
model is the most widely applied one for its high interpretability
and computational efficiency and built-in optimal and adaptive
techniques. The Sugeno fuzzy model provides a systematic
approach to generate fuzzy rules from a set of input–output
data pairs.
The ANFIS is a class of adaptive networks that are function-
ally equivalent to FISs [46], [47]. The ANFIS architecture con-
sists of fuzzy layer, product layer, normalized layer, defuzzy
layer, and summation layer. A typical architecture of ANFIS
is depicted in Fig. 3, in which a circle indicates a fixed node,
whereas a square indicates an adaptive node. For simplicity,
it was assumed that the FIS has two inputs and and one
output . The ANFIS used in this paper implements a first-order
Sugeno fuzzy model. For this model, a typical rule set with two
fuzzy if-then rules can be expressed as
Rule If is and is then
(8a)
Rule If is and is then
(8b)
where and are the fuzzy sets in the antecedent and
and are the design parameters that are determined during the
training process. As in Fig. 3, the ANFIS consists of five layers.

Fig. 4. Hybrid model for resonant frequency calculation of rectangular, cir-
cular, and triangular MSAs.
TABLE I
RESONANT FREQUENCIES OF RECTANGULAR MSAS
FOR TM (m = 1 AND n = 0) MODE
Layer 1: Each node in the first layer employs a node function
given by
(9a)
(9b)
where and can adopt any fuzzy membership
function (MF). In this paper, the following generalized bell MF
is used:
bell (10)
where is the parameter set that changes the shapes of
the MF. Parameters in this layer are referred to as the premise
parameters.
TABLE II
RESONANT FREQUENCIES OF CIRCULAR MSAS
FOR TM (m = n = 1) MODE
TABLE III
RESONANT FREQUENCIES OF TRIANGULAR MSAS FOR VARIOUS MODES
Layer 2: Each node in this layer calculates the firing strength
of a rule via multiplication
(11)
Layer 3: The th node in this layer calculates the ratio of the
th rule’s firing strength to the sum of all rules’ firing strengths
(12)
where is referred to as the normalized firing strengths.

TABLE IV
COMPARISON OF THE HYBRID METHOD AND THE SINGLE NEURAL MODELS FOR RESONANT FREQUENCIES OF MSAS
AND THE SUM OF THE ABSOLUTE ERRORS BETWEEN THE THEORETICAL AND EXPERIMENTAL RESULTS

Layer 4: In this layer, each node has the following function:
(13)
where is the output of layer 3 and is the parameter
set. Parameters in this layer are referred to as the consequent
parameters.
Layer 5: The single node in this layer computes the overall
output as the summation of all incoming signals, which is ex-
pressed as
(14)
It is seen from the ANFIS architecture that when the values of
the premise parameters are fixed, the output of the ANFIS can
be expressed as
(15)
Substituting (12) into (15) yields
(16)
Substituting the fuzzy if-then rules into (16), it becomes
(17)
After rearrangement, the output can be written as a linear com-
bination of the consequent parameters
(18)
The optimal values of the consequent parameters can be
found by using the least square method (LSM). When the
premise parameters are not fixed, the search space becomes
larger and the convergence of training becomes slower. The
hybrid learning (HL) algorithm [46], [47] combining LSM
and BP algorithms can be used to solve this problem. It was
shown in [46] and [47] that the HL algorithm is highly efficient
in training the ANFIS. This algorithm converges much faster
since it reduces the dimension of the search space of the BP
algorithm. During the learning process, the premise parameters
in layer 1 and the consequent parameters in layer 4 are tuned
until the desired response of the FIS is achieved.
The HL algorithm has a two-step process. First, the conse-
quent parameters are identified using LSM when the values of
the premise parameters are fixed. Then, the consequent parame-
ters are held fixed while the error is propagated from the output
end to the input end, and the premise parameters are updated by
the BP algorithm.
V. APPLICATION OF HYBRID METHOD TO THE CALCULATION
OF THE RESONANT FREQUENCY
The hybrid model proposed in this paper for calculating the
resonant frequency of the rectangular, circular, and triangular
MSAs is shown in Fig. 4. This hybrid model is based on com-
bining ANN and ANFIS. For the hybrid model, the inputs are
, and and the output is the measured resonant
frequencies . In the hybrid model, first, the resonant fre-
quencies are computed by using ANN, and then the inaccura-
cies in the ANN computation are corrected by the ANFIS.
ANN and ANFIS models are a kind of black box models
whose accuracy depends on the training data sets. A good col-
lection of the training data, i.e., data that are well distributed,
sufficient, and accurately simulated, is the basic requirement to
obtain an accurate model. There are two types of data genera-
tors for antenna applications: the measurement and simulation.
The selection of a data generator depends on the application
and availability of the data generator. The training and test data
sets used in this paper have been obtained from previous experi-
mental works [4], [5], [8], [12], [13], [16], [19], [22], [23], [30],
[31] and are given in Tables I–III for the rectangular, circular,
and triangular MSAs, respectively. A total of 68 data sets are
listed in Tables I–III. Fifty-four data sets were used to train the
hybrid model, and the remaining 14 data sets, marked with an
asterisk in Tables I–III, were used for testing. The equivalent
values of and for the circular and triangular MSAs were
calculated by using (6) and (7). The input and output data sets
were scaled between zero and one before training.
Training the ANN with the use of a learning algorithm to
calculate the resonant frequencies of MSAs involves presenting
them sequentially with different sets ( , and ) and
corresponding measured resonant frequencies . First, the
input vectors ( , and ) are presented to the input
neurons and output vector is computed. ANN output is then
compared to the known output of the training data sets and errors
are computed. Error derivatives are then calculated and summed
up for each weight until all the training sets have been presented
to the network. These error derivatives are then used to update
the weights for neurons in the model. Training proceeds until
errors are lower than prescribed values. The trained ANN es-
tablishes the relationship between the parameters of the MSAs
and the measured resonant frequency results.
Currently, there is no deterministic approach that can opti-
mally determine the number of hidden layers and the number of
neurons. A common practice is to take a trial and error approach
that adjusts the hidden layers to strike a balance between mem-
orization and generalization. After several trials, it was found
in this paper that a two hidden layered network achieved the
task with high accuracy. The most suitable network configura-
tion found was 6 6 12 1. It means that the number of neu-
rons was 6, 6, 12, and 1 for the input layer, the first and second
hidden layers, and the output layer, respectively. The tangent
sigmoid function was used in the hidden layers. The linear ac-
tivation function was used in the output layer. The number of
epoch is 381. Initial weights of the neural model were set up
randomly.
Training an ANFIS by using the HL algorithm for computing
the resonant frequency involves presenting it sequentially with
ANN output values and corresponding measured values
. Differences between the target output and the
actual output of the ANFIS are evaluated by the HL algorithm.
The adaptation is carried out after the presentation of each data

TABLE V
RESONANT FREQUENCIES OBTAINED FROM THE CONVENTIONAL METHODS FOR RECTANGULAR MSAS AND THE SUM
OF THE ABSOLUTE ERRORS BETWEEN THE EXPERIMENTAL RESULTS AND THE THEORETICAL RESULTS
TABLE VI
RESONANT FREQUENCIES OBTAINED FROM THE CONVENTIONAL METHODS AND THE METHODS BASED ON GA AND TSA FOR CIRCULAR MSAS
AND THE SUM OF THE ABSOLUTE ERRORS BETWEEN THE EXPERIMENTAL RESULTS AND THE THEORETICAL RESULTS

TABLE VII
RESONANT FREQUENCIES OBTAINED FROM THE CONVENTIONAL METHODS AND THE METHODS BASED ON GA AND TSA FOR TRIANGULAR MSAS
AND THE SUM OF THE ABSOLUTE ERRORS BETWEEN THE EXPERIMENTAL RESULTS AND THE THEORETICAL RESULTS
set until the calculation accuracy of the ANFIS is deemed sat-
isfactory according to some criterion (for example, when the
error between and the actual output for all the training set
falls below a given threshold) or when the maximum allowable
number of epoch is reached.
Selection of training parameters for ANFIS mostly depends
on experience besides the type of problem at hand. In this paper,
the number of epoch is 164 for training ANFIS. The number of
MFs for the input is 11. The number of rules is 11. The type of
MF for the input is the generalized bell. It is apparent from (10)
that the generalized bell MF is specified by three parameters.
Therefore, the ANFIS used here contains a total of 55 fitting
parameters, of which 33 are the premise param-
eters and 22 are the consequent parameters.
VI. RESULTS AND COMPARISIONS
The resonant frequencies calculated by using the hybrid
method proposed in this paper for the rectangular, circular, and
triangular MSAs are listed in Table IV. For comparison, the
results obtained by using the single neural model proposed by
Guney et al. [52] and the improved single neural model pro-
posed by Sagiroglu and Kalinli [63] are also given in Table IV.
, and in Table IV represent, respec-
tively, the values calculated by using the single neural model
trained with EDBD, DBD, and BP algorithms and by using
the improved single neural model trained with PTS algorithm.
The sum of the absolute errors between the theoretical and
experimental results for every method is also listed in Table IV.
As is seen from Table IV, the results of hybrid method show
better agreement with the experimental results as compared to
the results of the previous single neural models [52], [63]. The
very good agreement between the measured values and our com-
puted resonant frequency values supports the validity of the hy-
brid method. It needs to be emphasized that better results may
be obtained from the hybrid method either by choosing different
training and test data sets from those used in this paper or by
supplying more input data set values for training. Better results
can also be obtained by using a different individual neural or
ANFIS model for each different MSA.
In order to make a further comparison, the resonant fre-
quency results of conventional methods [1], [2], [5]–[9], [11],
[15], [17]–[21], [23]–[29], [31]–[34], [36] and the methods
based on GA [37], [38] and TSA [39], [40] for the rectangular,
circular, and triangular MSAs are given in Tables V–VII. The
sum of the absolute errors between the experimental results
and the theoretical results in Tables V–VII for every method
is also given in the last rows of Tables V–VII. It is clear from
Tables IV–VII that the results of the hybrid method are better
than those predicted by other scientists. It should be noted that
the conventional methods and the methods based on GA and
TSA were used to compute the resonant frequencies of each
different MSA. However, the hybrid method is valid for the
resonant frequency computation of all three different types
of MSAs, including the rectangular, circular, and triangular
MSAs.
It is well known that the accuracy of a properly trained
ANN and ANFIS depends on the accuracy and the effective
representation of the data used for training. In this paper, the
hybrid model is trained and tested with the experimental data
taken from previous experimental works. It is apparent from
Tables V–VII that the theoretical resonant frequency results
of the conventional methods and those based on GA and TSA
are not in very good agreement with the experimental results.
For this reason, the theoretical data sets obtained from these
methods are not used in this paper. Only the measured data set
is used for training and testing the hybrid model.
VII. CONCLUSION
The hybrid method is presented to accurately and simulta-
neously compute the resonant frequencies of the rectangular,
circular, and triangular MSAs. This hybrid method is based
on the combination of ANN and ANFIS. The ANN is trained
with BR algorithm. The optimal values for the premise and
consequent parameters of ANFIS are obtained by the HL
algorithm. The results of the hybrid method are in very good

agreement with the measurements, and better accuracy with
respect to the single neural models, the conventional methods,
and the methods based on GA and TSA is obtained. The
main advantage of the method proposed here is that the single
hybrid method is used to simultaneously calculate the resonant
frequencies of all three different types of MSAs, including the
rectangular, circular, and triangular MSAs. The hybrid method
offers an accurate and efficient alternative to previous methods
for the calculation of the resonant frequency. This method is
not limited to the resonant frequency computation of MSAs. It
can easily be applied to other antenna and microwave circuit
problems. The high-speed real-time computation feature of the
hybrid method recommends its use in antenna computer-aided
design programs. We expect that the hybrid method will
find a wide application area in antenna and electromagnetic
engineering.
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Kerim Guney was born in Isparta, Turkey, on
February 28, 1962. He received the B.S. degree from
Erciyes University, Kayseri, in 1983, the M.S. degree
from Istanbul Technical University, Turkey, in 1988,
and the Ph.D. degree from Erciyes University, in
1991, all in electronic engineering.
From 1991 to 1995, he was an Assistant Professor
and now is a Professor in the Department of Elec-
tronic Engineering, Erciyes University, where he is
working in the areas of optimization techniques (the
genetic, the tabu search, the differential evolution, the
ant colony optimization, and the clonal selection algorithms), fuzzy inference
systems, neural networks, their applications to antennas, microstrip and horn
antennas, antenna pattern synthesis, and target tracking.
Nurcan Sarikaya was born in Kayseri, Turkey, on
October 4, 1978. She received the B.S. and M.S. de-
grees from Erciyes University, Kayseri, in 2001 and
2003, respectively, both in electronic engineering.
Currently, she is pursuing the Ph.D. degree at the
Department of Aircraft Electrical and Electronics of
the Civil Aviation School, Erciyes University.
She is currently a Research Assistant in the Depart-
ment of Electrical and Electronics Institute of Science
and Technology, Erciyes University. Her current re-
search activities include neural networks, fuzzy infer-
ence systems, and their applications to antennas.

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