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VC 2011 Wiley Periodicals, Inc.

IMPROVED MEASUREMENT OFCOMPLEX PERMITTIVITY USINGARTIFICIAL NEURAL NETWORKSWITH SCALED INPUTS

Azhar Hasan and Andrew F. PetersonSchool of Electrical and Computer Engineering, Georgia Institute ofTechnology, Atlanta, GA 30332-0250; Corresponding author:peterson@ece.gatech.edu

Received 19 November 2010

ABSTRACT: A procedure is described to enhance the accuracy of

microwave measurements of the complex permittivity of a dissipativemedium. Monopole probe measurements are used in conjunction withtwo real-valued neural networks, which are integrated together to

reconstruct the complex permittivity from the measured reflectioncoefficients. The approach is tested over the frequency range from 2.5 to

5 GHz, for the real part of the permittivity in the range 3–10 and theimaginary part in the range 0– 0.5. The performance of the network isalso demonstrated for a reduced frequency range from 3.5 to 5 GHz.

Less than 4% error was observed in the presence of white Gaussiannoise with an SNR of 10dB. VC 2011 Wiley Periodicals, Inc. Microwave

Opt Technol Lett 53:2139–2142, 2011; View this article online at

wileyonlinelibrary.com. DOI 10.1002/mop.26221

Key words: neural network; complex permittivity; measurement;monopole; reflection coefficient

1. INTRODUCTION

The dielectric characterization of materials in the microwave

frequency range uses a variety of resonant and nonresonant

methods [1]. A particularly simple technique uses a monopole

probe immersed in the material under investigation; the input

impedance of the probe provides sufficient information to

extract the complex permittivity (er). In one approach, the nor-

malized impedance of the probe is expressed as a rational func-

tion of order three and the coefficients of the function are deter-

mined based on the profile of a standard medium [2]. A more

elaborate model for the impedance of a monopole antenna in a

half space [3] can be used in conjunction with various in-situ

procedures for solving the associated nonlinear inverse problem.

In one approach, the dielectric values are calculated by compar-

ing the measured and calculated values of the input impedance

using an iterative two-dimensional complex zero finding routine

[4]; in another, a least square fit is used to match the measured

and calculated input impedance [5]; and in a third, an analysis

of resonant peaks determined from the analytical model is used

with Prony’s method [6].

An alternate approach uses neural networks with a back-

propagation algorithm to reconstruct the permittivity profile of

lossless stratified medium from complex reflection coefficients

[7]. For a dissipative medium, the complex-valued nature of the

permittivity makes the problem incompatible with conventional

neural networks, which are designed to process real-valued data.

One possible approach is to use a complex valued back propaga-

tion neural network, which might result in better accuracy, but

that is considered much more complicated to implement [8].

Another possible approach is to split the network into two net-

works, one dealing with the real part and the other dealing with

the imaginary part (with both using real valued input data). One

such approach, for complex permittivity measurement, used the

finite difference time domain technique to generate training data

[9]. However, the use of two networks leads to a loss of correla-

tion between the real and imaginary parts of the complex input

data [10]. As there is usually considerable correlation between

the real and imaginary parts of the complex data [11], such a

loss reduces the accuracy of the results.

This article proposes the use of two back propagation net-

works, integrated together to reconstruct the complex permittiv-

ity of dissipative media. Instead of separating the real and imagi-

nary parts of the input data, the phase and magnitude of the

complex reflection coefficients are used as inputs to each of the

two back propagation networks, one of which solves for the real

part of the complex permittivity, whereas the other solves for the

imaginary part. Scaling is incorporated to improve the perform-

ance in the presence of noise. To demonstrate the procedure, the

analytical model of [3] will be used to train and test the neural

networks. The analytical model is briefly discussed in Section 2,

and the neural network setup is described in Section 3. The simu-

lation results and analysis are presented in Section 4.

2. THE MONOPOLE ANTENNA MODEL

The monopole antenna is widely used as a sensor probe for the

in-situ characterization of soil moisture content [6], snow and

soil wetness [5], and the measurement of electrical properties of

materials [2]. For a monopole probe immersed in a nonmagnetic

dielectric medium, the input impedance (and reflection coeffi-

cient) measured with a vector network analyzer (VNA) depends

on the permittivity of the medium. The input impedance of the

probe is related to the reflection coefficient through Eq. (1).

C ¼ Zin � Zo

Zin þ Zo

(1)

In Eq. (1), C is the complex reflection coefficient, Zin is the

input impedance of the monopole probe, and Zo is the character-

istic impedance of the coaxial cable connected to the probe,

which in this case is 50 X. The input impedance Zin is computed

using an expression developed by Wu [3], given by

Z�in ¼xlo

j4kðSþ CUÞ (2)

where

k � bþ ja ¼ x½lð�0 � j�00Þ�1=2(3)

The functions S, C, and U, along with the approximations in-

herent in these expressions, are described in detail in AppendixA of [4]. These functions along with k depend on the measure-

ment frequency, the geometrical dimensions of the monopole

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 9, September 2011 2139

probe (height and radius), and the electrical properties of the

medium (er ¼ er0 - jer

00).The monopole geometry is shown in Figure 1, where h is the

height and a is the radius of the probe, and d is the diameter of

the ground plane. The probe is connected to a VNA and is

immersed in the dielectric medium of relative permittivity er.

The reflection coefficient is measured using the VNA.

The finite diameter d of the ground plane can be neglected

for diameters greater than 4\k [6], enabling the analytical model

of Wu [3] to be used for the input impedance and reflection

coefficient calculations. The validity of the model depends upon

the ratio of monopole length to wavelength (h/k), but the model

is not limited to small values of h/k. Approximations made in

the model limit its range of validity to (a/h) < 1, (a/k) < 1, bh> 1 and (r/xe) 1. The current has a maximum value at the

base, and vanishes at the end of the antenna. For short monopole

antennas (h k/10), the current distribution is approximately

linear, while for longer antennas (h � k/4) the current distribu-

tion is approximately sinusoidal.

In this research we use a 10-mm long monopole probe,

which corresponds to the first resonant frequency of �7 GHz in

air. The validity of the model for a monopole probe of given

length (height) is limited to a specific frequency range depend-

ing upon the ratio h/k. For a 10-mm long probe, values of er

ranging from 3 to 10 correspond to resonant frequencies ranging

from 4.5 to 2.76 GHz, whereas er ranging from 3 to 6 corre-

sponds to frequencies from 4.5 to 3.5 GHz. The phase and mag-

nitude response to a change in complex permittivity is more pro-

nounced close to the monopole resonance, and therefore our use

of this model is limited to the frequency range from 2.5 to 5

GHz for the real part of complex permittivity ranging from 3 to

10. The loss tangent of the medium also limits the validity of

the model. It was observed that the analytical model only works

well for values of tand 0.2 and consequently we restricted the

values of er00 to the range from 0 to 0.5 when the smallest value

of er0 is 3.0.

The analytical model is used in this research is to generate the

training data for the target range of permittivity values to train the

back propagation neural network, presented in the next section.

3. NEURAL NETWORK MODEL WITH SCALED INPUTS

The nonlinear inverse problem of reconstructing the permittivity

profile of a nondissipative media (er00= 0) from measured com-

plex reflection coefficient data (C) can be solved using a real-

valued neural network using the Levenberg Marquardt back

propagation algorithm [7]. In the lossy case, the imaginary part

of er constitutes an important part of the information and cannot

be ignored. For complex-valued data, one approach is to use

complex-valued neural networks (CVNN), which can handle

inputs, outputs and weights, all as complex numbers. The

approach proposed in this article is much simpler to implement

than the CVNN, and produces almost the same level of accu-

racy. The new methodology uses the input data in polar repre-

sentation (magnitude and phase) instead of Cartesian representa-

tion (real and imaginary) and allows the use of real-valued

artificial neural networks for solving the complex permittivity

profile without any loss of information. The magnitude and

phase of C are cascaded in each vector of the input data matrix

D as shown in Eq. (4).

D ¼ jCjffC

� �(4)

For dissipative media, the magnitude and phase of C both

depend on the imaginary part of er. However it was observed

that the change in phase is more pronounced than the change in

magnitude, as shown in Figure 2.

This model works accurately for noise free data [12], how-

ever it was observed to be sensitive to the presence of noise.

Measurement noise was simulated by adding white Gaussian

noise, and as the noise level increased a larger error was

observed in the estimated values of er00. To compensate for the

change in |C| in response to a change in er00, the data matrix D

was modified to Dsc by scaling the magnitude of reflection coef-

ficient by a factor N as shown in Eq. (5).

Figure 2 Input vectors corresponding to different values of complex

permittivity er ¼ er0 - jer

00. [Color figure can be viewed in the online

issue, which is available at wileyonlinelibrary.com]

Figure 1 A monopole probe of height h and radius a immersed in a

dissipative dielectric medium with permittivity er ¼ er0 � jer

00, and con-

nected to VNA. [Color figure can be viewed in the online issue, which

is available at wileyonlinelibrary.com]

2140 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 9, September 2011 DOI 10.1002/mop

Dsc ¼N � jCjffC

� �(5)

An appropriate value of scaling factor was observed to be N� 6. Several vectors from Dsc corresponding to different values

of complex er are shown in Figure 3.

Two separate networks are used using the complex reflection

coefficient as the input in the form described in Eq. (5). Each

input vector represents the reflection coefficient for one value of

er, computed using the analytical model described in Section 1.

An appropriate number of neurons was determined through sim-

ulations and comparison, and it was found that a network with

one hidden layer having 25 neurons gives superior performance

in terms of mean square error. Despite having the same input

training data and neuron layer architecture, the networks differ

in their weight matrices. They are tracking different outputs and

thus the performance function will respond differently to the

same weights for tracking the output and minimizing the error.

The setup comprises two networks, identical in architecture, but

having different performance functions, weight matrices and out-

puts. The two outputs are combined together to reconstruct the

complex permittivity as shown in Figure 4. In the input vector,

|Cfn| and < Cfn indicates the magnitude and phase of the input

reflection coefficient computed/measured for frequency fn. The

frequency range of interest is divided into n steps.

The complex reflection coefficient training data were gener-

ated for 21 frequency steps of 130 MHz each from 2.5 to 5

GHz. The real part of complex permittivity (er’) was swept from

3.0 to 10.0 with an increment of 0.05, whereas the imaginary

part (er00) was swept across 0–0.5 with a step size of 0.005. The

setup resulted in a training matrix Dsc of dimension 42 �14241, which is easily handled by a common desktop computer.

Measurement noise was simulated by adding white Gaussian

noise with an SNR of 10 dB to the training data and the same

proportion of noise was induced in the test data.

4. SIMULATION RESULTS AND ANALYSIS

The computed input vectors corresponding to different values of

complex permittivity are shown in Figure 3. The network was

trained and tested with one hidden layer with 20, 25, 30, and 35

neurons, and it was observed that a network using one hidden

layer with 25 neurons gave the best results. The network was

then tested using test data corresponding to 231 different values

of er, which were distinct from the data used for training the net-

work. The network output values were found to be quite accurate

and the recorded percent error was less than 3.5% (Table 1). To

confirm that the network has converged to the global minima and

not local minima, we repeated the simulation multiple times and

observed the results for consistency, as suggested by [13].

The frequency range was determined by the resonant fre-

quency of the monopole corresponding to the er values of the

medium. It was observed that frequency range could be further

reduced, if the range of complex permittivity values is also

Figure 3 Scaled input vectors corresponding to different values of

complex permittivity er ¼ er0-jer

00. [Color figure can be viewed in the

online issue, which is available at wileyonlinelibrary.com]

Figure 4 Neural network model comprising two networks using one hidden layer each, with #1 solving for er0 and #2 reconstructing er

00 using the Lev-

enberg Marquardt back-propagation algorithm. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com]

TABLE 1 Comparison of Actual Values of er 5 er0 - jer

00 versusTwo Sets of Estimated Values and the Respective PercentError for Frequency Range 2.5–5 GHz

Actual Values

NN Output

(Set1) % Error

NN Output

(Set2) % Error

3.01 � j0.015 3.08 � j0.05 0.84 2.98 � j0.023 0.59

4.59 � j0.36 4.67 � j0.51 3.21 4.56 � j0.41 1.26

5.27 � j0.213 5.32 � j0.21 0.78 5.21 � j0.25 1.37

5.9 � j0.5 5.69 � j0.45 1.49 5.91 � j0.45 0.87

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 9, September 2011 2141

reduced. The same setup was used over a frequency range from

3.5 to 5 GHz by limiting the sweep range of er’ from 3 to 6. The

input vectors for 4 different er values are shown in Figure 5.

The back-propagation based neural network model was

trained with the input data arranged in a matrix Dsc as in Eq.

(5). The training data matrix is generated using the same step

size for frequency and er00, whereas the incremental step is

reduced to 0.025 for er’, resulting in a training data matrix with

dimensions 38 � 12221. The reduced frequency range network

was tested with 231 test vectors and the observed percent error

was less than 4% in the presence of white Gaussian noise (10-

dB SNR). The results for two sets of four randomly selected

values, incorporating white Gaussian noise with an SNR of 10

dB to simulate measurement noise, are tabulated in Table 2.

5. CONCLUSIONS

A procedure is described to enhance the accuracy of complex

permittivity measurements. Two real-valued neural networks are

integrated together to reconstruct the complex permittivity from

the measured reflection coefficients. The approach is tested over

the frequency range from 2.5 to 5 GHz, for the real part of the

permittivity in the range 3–10 and the imaginary part in the

range 0–0.5. The performance of the network is also demon-

strated for a reduced frequency range from 3.5–5GHz. Less than

4% error was observed in the presence of white Gaussian noise

with an SNR of 10 dB.

The accuracy of the neural network output is obviously de-

pendent upon the accuracy of the model used to train the net-

work, as well as the resolution of the training data. Some a pri-

ori knowledge of the medium is required, which can be obtained

from an analytical model, numerical simulations, or actual meas-

urements. The approach should be successful for any range of

the real and imaginary parts of the complex permittivity as long

as the size of data set remains within the user’s computing

capability.

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VC 2011 Wiley Periodicals, Inc.

A DUAL-BAND SHORTED MONOPOLEANTENNA FOR WLAN-BANDAPPLICATION

Wen-Chung Liu and Yang DaiDepartment of Aeronautical Engineering, National FormosaUniversity, 64 Wenhua Road, Huwei, Yunlin 632, Taiwan, Republicof China; Corresponding author: wencliu@nfu.edu.tw

Received 19 November 2010

ABSTRACT: A novel printed monopole antenna for dual-bandoperation is presented. The antenna comprises a radiator having a stripand two side-loaded patches, and is fed by a microstrip feedline. By

folding top one of the side-loaded patches on the other side of thesubstrate as well as connecting the two patches by a short pin, good

antenna performances including dual wide bandwidths of 590 MHz and1.12 GHz, high antenna gains of �2.5 and 3.6 dBi, and monopole-likeradiation patterns over the two operating bands, respectively, are

obtained. The measured results show good agreement with the simulated

Figure 5 Scaled input vectors corresponding to different values of

complex permittivity er ¼er0-jer

00 for reduced frequency range. [Color fig-

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

wileyonlinelibrary.com]

TABLE 2 Comparison of Actual Values of er 5 er0 2 jer

00 vsTwo Sets of Estimated Values and the Respective PercentError for Frequency Range 3.5–5 GHz

Actual Values

NN Output

(Set1) % Error

NN Output

(Set2) % Error

3.01 � j0.015 2.98 � j0.037 2.87 3.04 � j0.026 1.36

4.59 � j0.36 4.67 � j0.51 3.86 4.62 � j0.35 0.78

5.27 � j0.213 5.32 � j0.21 0.97 5.24 � j0.19 0.75

5.9 � j0.5 5.69 � j0.49 3.51 5.86 � j0.37 2.29

2142 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 53, No. 9, September 2011 DOI 10.1002/mop