A MICROWAVE TOMOGRAPHY SYSTEM FOR FORESTRY

APPLICATIONS

By

John James Doroshewitz

A THESIS
Submitted to

for the degree of

Michigan State University

in partial fulfillment of the requirements

Electrical Engineering – Master of Science

2019

ABSTRACT

A MICROWAVE TOMOGRAPHY SYSTEM FOR FORESTRY

APPLICATIONS

By

John James Doroshewitz

A healthy urban forest is important for both human health and environmental
quality. There is currently a lack of technology that can analyze living trees in
their natural environment without destroying them.

This thesis presents a microwave tomography imaging system that is in-
tended for use in several forestry applications. Recently, there has been an
increased demand for imaging systems that do not use ionizing radiation but
produce comparable accuracy and sensitivity to traditional imaging techniques.
Microwave tomography is an emerging medical imaging research area that has
rarely been used for anything besides human tissue imaging. The system de-
scribed in this thesis was specifically designed for in vivo measurement of tree
trunks and is one of the first of its kind.

The system forms an image by using frequency domain sampling over a wide
bandwidth (1-5 GHz) and a ring of antennas. Microstrip monopole antennas
specifically designed for use in microwave tomography systems are used and
their design is presented. The system also uses a microwave switch matrix to
switch between active pairs of antennas. The design of this switch matrix and
the controlling software is also presented. Lastly, this thesis discusses the use of
time reversal image reconstruction and the signal processing involved to create
an image. Measured results of simple targets and preliminary measurements of
tree trunk samples are both included.

ACKNOWLEDGMENTS

The first of many thanks has to go to my entire family. As the 13th member of
my family to receive a Bachelor’s degree from Michigan State and the second
to receive a Master’s, my destiny as a Spartan was written from birth. My
parents, Jeri and John, and both my sisters, Jessica and Jackie, have been every
kind of support that I could ask for during this process. Our family identifies
as Spartans and it is truly a privilege to walk out of here with a second degree
from the school I have always loved.

Of course, none of this is possible without the support, guidance, and pushing
for excellence from my advisor, Dr. Jeffrey Nanzer. You care deeply about
providing your students with the best opportunity and education possible. From
the student design competition at APS as an undergrad up until the last day of
graduate school you always pushed me to be the best engineer, scientist, and
student that I could be.

I would also like to thank the rest of the faculty in the EMRG. Thank you
to Dr. Ed Rothwell for first introducing me to electromagnetics and sparking
my interest in this topic (big loops=bad, small loops=good) as well as for
your guidance throughout both undergraduate and graduate school, you were
always a trusted person to come to for advice on any topic. Thank you to
Dr. Prem Chahal, the first EMRG professor that I did research under. Your
experimental approach pushed my creativity and problem solving skills essential
to research. And lastly, a thank you to Dr. Shanker Balasubramaniam, who was
always available to ask theoretical EM questions or lecture me about something
computational (although you are still disappointed in my first 280 exam score).
A big thank you to Chris Oakley for his help with the creation of the switch
matrix, to Vinny Gjokaj and Brian Wright for their help in the antenna fabri-

iii

cation, and to Jason Merlo for his advice on the entire system design. Without
you guys, the tomography system would still not be functional. Also, thank
you to Saptarshi Mukherjee whose time reversal code and knowledge of image
reconstruction was essential to this project.

This work was supported in part by the MSU Forestry Department, which
gave me a unique opportunity to research a non-traditional area of electromag-
netics. Thank you to Dr. David Macfarlane and Dr. Emily Huff for their support
and advice on the forestry end of this work, and a huge thank you to Sam Clark
of the Kellogg Experimental Forest who cut and prepared the wood samples for
measurement. His work and willingness to drive the samples to East Lansing
were vital to the success of this research. Thank you Dan Brown for allowing
us to use his oven in the Forestry Wood Shop.

Lastly, I would like to thank the students of the EMRG. Without your friend-

ship and support, this work would have never been possible.

TABLE OF CONTENTS

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1.1 Introduction .

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2.1 Overview .
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2.2 Antenna Design .
2.3 Switch Matrix Design . .
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LIST OF FIGURES .
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CHAPTER 1 INTRODUCTION .
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1.1.1 Forestry Motivation .
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1.1.2 Wood Imaging Review .
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1.1.3 Microwave Tomography Review .
1.1.4 Contributions of This Work .
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CHAPTER 2 TOMOGRAPHY SYSTEM DESIGN .
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3.4 Energy Calculation and Reconstruction .
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CHAPTER 3 SIGNAL PROCESSING .
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4.3.2.1 Air Gap Images
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CHAPTER 4 MEASURED RESULTS .
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CHAPTER 5 CONCLUSION .
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4.1 Introduction .
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4.4 Review .

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APPENDICES .
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PYTHON CODE TO TAKE DATA OFF HP8753D
APPENDIX A
NETWORK ANALYZER .
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SWITCH MATRIX SCHEMATIC AND PART
APPENDIX B
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NUCLEO CODE FOR CONTROLLING SWITCH
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APPENDIX E MATLAB CODE TO READ IN DATA .
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MATLAB CODE TO BACK PROPAGATE
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LIST OF FIGURES

1.1 Proposed mockup of the microwave tomography system intended to

nondestructively evaluate tree trunks.

2.1 Realized microwave tomography system.
2.2 Ring of antennas used to perform microwave tomography on samples

under testing.

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2.3 Copper patterning of the monopole imaging antenna. Top copper

(conductor) is in blue and bottom copper (ground plane) is in orange. 13

2.4 Fabricated antenna design on Rogers 3006. The conducting (left)

and ground (right) planes are both shown.

2.5 Simulated and measured S11 for the tomography measurement an-

tennas.

2.6 Simulated (a) and measured (b) antenna patterns of the tomography

measurement antennas.

2.7 Front side of the switch matrix.
2.8 Output ports on the rear of the switch matrix.
2.9 Schematic of switch tree to create a 2x16 switch matrix con-
necting Tx and Rx to any pair of ports 1-16. Switches labeled
"A" are HMC270AMS8GE switches and those labeled "B" are
HMC322ALP4E switches.

2.10 The Nucleo F411RE used to control the microwave switch matrix.
2.11 Inside of the switch matrix.
2.12 The Hewlett-Packard 8753D network analyzer and its controlling

computer.

3.1 Test setup of a brass cylinder being imaged by the microwave to-

mography system.

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vii

3.2 Measured frequency data in dB (top), zero padded and conjugate
mirrored frequency data in dB (middle), and time domain pulse
after the IFFT processing (bottom).

3.3 Time domain waveform of the measured empty environment.
3.4 Time domain waveform of the measured cylinder environment.
3.5 Time domain waveform of the measured empty environment sub-

tracted from the measured cylinder environment.

3.6 Location of each of the antennas in the ring.
3.7 Sub-optimal reconstructed images of antennas 1 (a), 2 (b), 3 (c), and

4 (d) transmitting.

3.8 Sub-optimal reconstructed images of antennas 5 (a), 6 (b), 7 (c), and

8 (d) transmitting.

3.9 Sub-optimal reconstructed images of antennas 9 (a), 10 (b), 11 (c),

and 12 (d) transmitting.

3.10 Sub-optimal reconstructed images of antennas 13 (a), 14 (b), 15 (c),

and 16 (d) transmitting.

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3.11 Total reconstructed image obtained from averaging the total energy

in all 16 of the previous images, producing the brass cylinder shape. 44

4.1 Imaging results of a brass cylinder using pairwise measurement

technique. (a) Measurement setup. (b) Reconstruction.

4.2 Imaging results of a plastic cylinder (
r = 3.8) using pairwise
measurement technique. (a) Measurement setup. (b) Reconstruc-
tion.

4.3 Imaging results of a brass cylinder using the entire measurement

system. (a) Measurement setup. (b) Reconstruction.

4.4 Imaging results of a plastic cylinder (
r = 3.8) using the entire
measurement system. (a) Measurement setup. (b) Reconstruction.
4.5 Imaging results of a plastic square prism (
r = 3.7) using the entire
measurement system. (a) Measurement setup. (b) Reconstruction.

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viii

4.6 Imaging results of a wood block using the entire measurement sys-

tem. (a) Measurement setup. (b) Reconstruction.

4.7 Imaging results of a 1" diameter hole in a green basswood sample.
The solid portion of the trunk (left (a)) was subtracted from the
portion with the hole (right (a)). (b) Reconstruction.

4.8 Imaging results of a solid, green basswood sample. Two solid por-
tions of the trunk (a) were subtracted from each other and only noise
remains in the reconstructed image (b).

4.9 Imaging results of a 2" diameter hole in a green sugar maple sample.
The solid portion of the trunk (left (a)) was subtracted from the
portion with the hole (right (a)). (b) Reconstruction.

4.10 Imaging results of a solid, green sugar maple sample. Two solid
portions of the trunk (a) were subtracted from each other and only
noise remains in the reconstructed image (b).

4.11 Imaging results of a 2" diameter hole in a completely dried basswood
sample. The solid portion of the trunk (left (a)) was subtracted from
the portion with the hole (right (a)). (b) Reconstruction.

4.12 Imaging results of a 2" diameter hole in a completely dried sugar
maple sample. The solid portion of the trunk (left (a)) was subtracted
from the portion with the hole (right (a)). (b) Reconstruction.

4.13 Imaging results of a solid, fully dried basswood sample. Two solid
portions of the trunk (a) were subtracted from each other and only
noise remains in the reconstructed image (b).

4.14 Imaging results of a bolt added to a 1" hole in a completely dried

basswood sample (a). (b) Reconstruction.

4.15 Imaging results of a bolt added to a 1" hole in a completely dried

sugar maple sample (a). (b) Reconstruction.

4.16 Imaging results of a nail in a completely dried basswood sample.
The solid portion of the trunk (right (a)) was subtracted from the
portion with the nail (left (a)). (b) Reconstruction.

4.17 Imaging results of nails in a completely dried sugar maple sample.
The solid portion of the trunk (left (a)) was subtracted from the
portion with nails (right (a)). (b) Reconstruction.

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ix

4.18 Imaging results of a horizontal nail (illustrated by the red line) in a
completely dried basswood sample. The solid portion of the trunk
(right (a)) was subtracted from the portion with the nail (left (a)).
(b) Reconstruction.

B.1 Switch matrix schematic.

74
87

x

CHAPTER 1

INTRODUCTION

1.1 Introduction

A microwave tomography system designed for imaging tree trunks using
non-ionizing radiation is presented in this work. The realization of this system,
including the hardware design and the signal processing is described in detail.
Measured results for simple targets such as cylinders are presented as well as
more complex targets, specifically tree trunk samples with simulated health
issues in them. Tree health is the primary motivation for this work, discussed
in Section 1.1.1.

1.1.1 Forestry Motivation
The primary focus of this work is creating a tomography system to image the
inside of tree trunks without having to do any destructive evaluation to any
components of the tree. As urban population continues to increase, the degra-
dation of urban forests raises a concern regarding the impacts on human health
and the overall quality of the environment.
In 2012, Nowak and Greenfield
wrote about the positive benefits of preserving a healthy urban tree canopy and
its ability to mitigate some of the concerns [1]. Benefits of a healthy urban forest
can yield positive effects on not only the environment, but also a community’s
economy due to the increase in energy efficiency [2] and its overall societal
well-being by adding recreational opportunities and positive emotional effects
[3].

Preserving a healthy urban forest requires an effective and efficient technique
to evaluate the risk posed by each individual tree. The largest risk presented

1

from trees in an urban area is stem and branch breakage or even whole-tree
failure and toppling, often caused by wind and ice storms [4]. Currently, there
are few options to properly evaluate the risk presented by a living, standing
tree. Dahle et. al identifies several defects such as decay, cavities, cracks,
splits, among others as indicators of trees at risk to fall. These characteristics
are more qualitative and very difficult to translate into a reliable risk analysis
procedure. They also discussed quantitative measurements such as aspect ratio
(the ratio of branch diameter to stem diameter, with a large ratio being more at
risk) or the angle of attachment of branches to trunk. Despite being numerical
values, these measurements, again, are difficult to translate into a risk analysis
[5]. Other than these parameters, there are no other accepted ways to determine
the risk of failure of a living tree without destructively sampling the tree in
some form.

1.1.2 Wood Imaging Review
In 2003, Bucur published a detailed review of the ability to image wood struc-
tures at a high resolution [6]. He organizes the imaging techniques into five
main categories:
ionizing radiation, thermal techniques, ultrasonic, nuclear
magnetic resonance (NMR) imaging, and microwave techniques. Research
done in this area and pros and cons of each technique is discussed:

Ionizing Radiation Ionizing radiation, such as x-ray or γ-ray radiation is one
of the more traditional imaging techniques used in biological imaging.
The use of x-ray radiation as an imaging technique was famously first
discovered by Röntgen in 1895. The technology and its development
continued through the 20th century as the most prominent medical imaging
technique [7]. X-ray or γ-ray imaging has two major drawbacks: cost of
system and health risks. Since the imaging system described in this work

2

is designed for the imaging of trees, cost is a major issue. The end-user of
these products typically do not have huge budgets for imaging the inside
of trees. Although there is a benefit of adding this imaging system, there
is not a dire need for sophisticated imaging systems as compared to other
places that would utilize x-ray imaging, i.e. hospitals. The second, more
important, drawback is the fact that this radiation is ionizing. Using x-ray
for imaging of living things ionizes the tissue which presents a major health
risk [8]. Clearly, this goes against the goal of non-destructively evaluating
the sample.
Despite these drawbacks, x-ray and γ-ray imaging have been used in
previous application to image tree samples. Most work has been in the
field of computed tomography (CT), which is preferable to conventional
radiography for various reasons such as the lack of need for film, receiving
data in near-real-time, improved calibration procedures, among numerous
others [9]. In 1983, Onoe et. al created a portable x-ray CT scanner that
could be used for measuring the annual rings inside of a living tree without
destructively evaluating it (besides the unavoidable radiation exposure
when using this imaging method) [10]. In 1989, Wagner et. al created a
CT scanner that was able to scan logs for internal defects as they came into
sawmills [11]. Rust also used x-ray CT to estimate the xylem, the tissue
that transports water and minerals to the rest of the tree, of living trees
[12].

Thermal Techniques Thermography is a term for techniques used that evalu-
ate the sample using its surface temperature. Complex relationships are
then drawn between the temperature distribution and the potential presence
of different materials, defects, etc. Thermography is not an extensively
studied field, largely due to these relationships. It relies on a largely exter-

3

nal physical property to predict internal properties and defects. Although
this is a more quantitative measurement than things like decay and cavi-
ties discussed in the previous section, it still lacks concrete relationships
between external and internal characteristics and relies on guesswork. In
addition, thermography is a generally slow process and some of the most
interesting results obtainable from this method need rapid capture times,
which is difficult to achieve.
Some early work has been done in thermography to characterize living
trees and their defects. Catena has published several works on using
thermography to detect hidden cavities inside of the trunks of trees [13, 14].
Although this requires very large cavities to provide useful results, it can
be very useful as these are the trees most at risk to fall and cause damage.
Wyckhuyse et. al were able to show signs of wood decay and rotten
wood using thermography by estimating moisture content of the samples.
This is a step in the right direction towards characterizing the trees but in
published literature it seems to be the maximum achievable result using
this technique. Although it is useful for some low-level preliminary tests,
there is still further demand for more accurate and robust systems.

Ultrasonic Ultrasonic imaging is similar to ionizing radiation imaging as it re-
turns cross-sectional images by using transmission or reflection data from
all directions surrounding the sample. In contrast to ionizing radiation,
however, ultrasound uses high frequency pressure, or sound, waves for
its characterization. The advantage of these pressure waves is they are
completely safe and non-destructive to biological tissue when they are
administered at low intensities. Ultrasound, however, does require direct
contact between its transceivers and the sample in order to properly trans-
mit the sound waves. In a forestry application this is not a huge issue as the

4

outside of the tree can easily be accessed and patient comfort is not a con-
cern. These factors, as well as the generally low price of equipment [15],
point towards ultrasound being a promising imaging technique. However
in comparison with x-ray CT, ultrasound CT is inferior in both sensitivity
and ability to specify inconsistencies [15].
In tree imaging, ultrasound has been used to test wood samples in various
ways. In 1994, Biagi et. al used ultrasound to characterize wood poles
and timber as it entered a sawmill [16]. Comino et. al performed a
similar ultrasonic analysis but instead tested living trees in their natural
state in order to detect internal decay and defects [17]. Socco et. al has
showed over the course of several papers the feasibility of using ultrasonic
tomography for imaging of living trees and has shown that there is ability to
detect inconsistencies and decay, but there are limitations to its resolution
and accuracy [18–20]. These fundamental limitations are the major reason
why ultrasound is not the best option for tree imaging moving forward in
research.

Nuclear Magnetic Resonance NMR has proven to be one of the most accurate
and powerful characterization techniques developed to date. It uses a strong
magnetic field to align the magnetic polarizations of the sample’s nuclei
and then applies a weak, oscillating magnetic field to the same sample.
The response of the aligned nuclei is then measured and the sample is
reconstructed from these measured fields [21]. This technique is very
involved but it produces extremely accurate results on a relatively real-
time basis. In addition, this magnetic resonance does not have any lasting
effects on the nuclei so it is not harmful in any way to living tissue. The
biggest deterrent to NMR is the cost of the technology. Since there is
a demand in forestry applications for measurement equipment to be low-

5

cost, a consumer product using NMR is not feasible considering current
technology.
Despite the high cost, there have been attempts to characterize trees using
NMR in field tests.
In 1997, Pearce et. al observed water contents in
sycamore trees and used this information to be able to diagnose several
conditions such as decayed wood, sooty bark disease, and various species
of invasive fungi growing in the tree [22]. Merela et. al used NMR to
diagnose wounded branch structures in beech trees by identifying areas
with decreased moisture content as the tree cuts off water supply to dam-
aged tissue [23]. This research proves to be a promising technique to give
accurate and valuable diagnoses to trees but will continue to be unfeasible
in most applications due to its high price.

Microwave techniques Microwave tomography is the focus of this work and
its benefits and current uses will be discussed in detail Section 1.1.3. It has
been studied very little for use in forestry applications prior to this work.
There has only been one group that has published work for microwave
applications in imaging tree. Pastorino et. al [24] and Boero et. al
[25] both have worked on the imaging of the trunks of living trees using
microwave tomography. They were able to show simulated and measured
results using this technique. This is, to the author’s knowledge, the first
and only use of microwave tomography for forestry applications to date.

1.1.3 Microwave Tomography Review
Microwave tomography is an emerging imaging technique that is proving to be
useful for analyzing complex structures, specifically those that contain dielectric
inconsistencies inside of them. Using low to medium power frequencies in the

6

radio or microwave bands provides non-ionizing radiation that is a safe alternate
to x-ray or γ-ray imaging. In addition, radio frequency components are cheaper
and widely available in comparison to the ionizing radiation or NMR hardware.
In terms of sensitivity and accuracy, early data suggests that microwave imaging
is better than any of the other proposed methods [26]. Although thermography
and ultrasound are cheap and portable methods, the advantages proposed by
microwave tomography while still remaining cheap and portable makes it a vital
area of research for future biological imaging advancements.

Since microwave tomography is a cheap, safe alternate to traditional imaging
techniques, it has become of particular interest to the medical imaging field
[27]. Specifically, researchers have attempted to create microwave tomography
systems for breast imaging when screening for tumors or other signs leading
to breast cancer [28]. Currently, the process involves yearly (or even more
frequently in certain cases) x-ray examinations of the breasts, a process called
mammography. Researchers have focused much of the effort on this specific
image because it is a routine x-ray and it exposes tissue that is already at risk of
developing cancer to more of the dangerous ionizing radiation. Whereas many
other x-ray examinations are done only when necessary, a routine examination
such as a mammogram screening is a logical candidate to be replaced by
microwave tomography.

There has been many different uses of microwave tomography for imaging
breast tissue. An interesting technique that combines microwave and ultra-
sound methods called thermoacoustic imaging. The thermoacoustic effect is a
phenomenon that causes molecules to vibrate when exposed to heat or other
microwave radiation [29]. This vibration, in turn, can be used as an ultrasonic
signal to image a medium that was excited by microwave energy. This was of
particular interest in the late 90s when microwave imaging was still a relatively

7

new topic. Kruger et. al [30, 31], Wang et. al [32], and Ku et. al [33],
among many others, presented systems and measurements that demonstrate
thermoacoustic imaging for breast tissue. There are also examples of using
actual microwave tomography for imaging. Meaney et. al [34, 35], Souvorov,
et. al [36], and Hagness et. al [37, 38] are some examples of researchers who
created microwave simulations and systems to perform such measurements,
among many others.

Microwave techniques are proving to be very promising in the area of medical
imaging. One drawback, however, is many of the current methods proposed
are computationally expensive. These systems rely on non-linear inversion
methods to do their image reconstruction [39–41]. These techniques require
large computing power, long computation times, or often both to successfully
invert the problem and reconstruct an image. Although this is much less of a
problem in a medical imaging scenario, where hospitals can store large clusters
of computers in house to perform the expensive computations, this becomes
a much bigger issue for forestry applications. In order to do in vivo imaging
of tree trunks, a computationally cheap solution that can still provide near-real
-time images must be created.

1.1.4 Contributions of This Work
In this thesis, a frequency domain sampling microwave tomography system
that utilizes time reversal, a less computationally expensive method than a full
inverse problem, to perform its reconstruction is presented. This system uses
specifically designed planar monopole antennas to perform its data collection,
a microwave switch matrix to handle the switching between antennas, and a
network analyzer to do its data collection. The system is designed for the
intent to use in forestry applications. Unlike the methods that are described in

8

Section 1.1.2, this uses non-ionizing radiation and cheap components to create a
reasonable solution for the forestry experts that would potentially be interested
in this final product.
In comparison to prior work, the tomography system
presented here is a safer, cheaper, and more mobile alternative to image tree
trunks.

Figure 1.1 Proposed mockup of the microwave tomography system intended to
nondestructively evaluate tree trunks.

The design process for the all hardware and software components included in
this system, from why each specific method was used to fabrication and testing,
is presented in the following chapters. Chapter 2 will cover the hardware
design of the system. The design and fabrication of the monopole antennas, the
creation of the switch matrix and its programming, and the transceiver used and
its motivation are all discussed in this chapter. Chapter 3 will cover the software
design of the system. The process of converting data from the frequency domain
to the time domain, the time reversal simulation, and the image reconstruction
process are included in this chapter. Chapter 4 will present many different
measured results from the system. Measured results of both simple targets as
well as preliminary measurements of actual tree trunks with simulated defects in
them are presented to show the feasibility of this system as a low-cost, portable,
near-real-time imaging system for in vivo measurement of tree trunks.

9

CHAPTER 2

TOMOGRAPHY SYSTEM DESIGN

2.1 Overview

This chapter will describe the microwave tomography system used for imag-
ing of wood samples, seen in Fig. 2.1. A ring of antennas, seen in Fig. 2.2,
is created that surround the sample under test (SUT) to collect data that will
eventually be used to create an image, which is discussed in Chapter 3.
In
this work, the ring consists of 16 equidistantly spaced antennas that act as
both a transmitter and receiver to measure the scattering off of the SUT. This
ring is surrounded by microwave absorbing material to block the outside RF
interference that could be present in the environment. Although this would
not be available in the field testing environment, it helps give the best possible
results for a first prototype of a system. Control of these antennas and the data
collection is done through a computer connected to both a network analyzer and
a microwave switch matrix. The antenna design and the design of the switch
matrix will be described in this chapter while the code for the control of the
network analyzer and the switch matrix can be seen in Appendices A and C,
respectively.

10

Figure 2.1 Realized microwave tomography system.

Figure 2.2 Ring of antennas used to perform microwave tomography on samples
under testing.

11

2.2 Antenna Design

A major component of all microwave systems is the antenna used in the
system. As electronic systems become increasingly more digital with the
development of high speed digitizers, antennas will remain to be necessary for
any wireless system, as a transition from a guided wave to a unbounded wave
cannot be avoided until possibly very far in futuristic systems. In microwave
tomography systems, it is popular to use antennas that are directional in their
pattern in order to collect information from inside the ring and avoid any
interference or noise from outside the ring. Due to space constraints in the
ring of antennas, it is common to use directional planar antennas as opposed to
a simple horn antenna. The most commonly used directional antenna design
used in tomography is the Vivaldi, or tapered slot, antenna. This is a planar
wideband antenna that creates a highly directive radiation pattern. This antenna
was first designed by Gibson in 1979 [42] and improved by Gazit in 1988 [43].
Medical imaging using microwave tomography have utilized Vivaldi antennas
in many different applications [44, 45]. Other directive planar antenna designs,
such as the patch antenna [46, 47] or more complex designs such as the folded
quasi self complementary antenna (FQSCA) [48] have been utilized in [49, 50].
The antennas used for this design are wideband planar monopole antennas
often used for imaging. In contrast to what is discussed above, these antennas
display more of a omnidirectional antenna pattern in the x-y plane. Non-
directive antennas are less typical in the microwave tomography research field
due to the collection of data from areas not in the center of the ring. In addition,
there is increased coupling between the antennas if neighboring antennas have
overlapping beams. Despite these drawbacks, prior research has been conducted
using non-directive antennas. The most popular omnidirectional antenna used
in all wireless applications, microwave tomography included, is the dipole

12

antenna [51] or its equivalent; a monopole antenna above a ground plane
reflection [52, 53].

An omnidirectional radiation pattern was chosen for this application due to
the use of time reversal imaging, discussed in detail in Chapter 3. This imaging
process assumes that the antennas radiate in an omnidirectional pattern.
In
order to mimic this sitatuion as closely as possible, the pattern was chosen as
such. This particular antenna design was specifically published for wideband
imaging in circular imaging systems [54]. This antenna consists of a semi-
circular patch antenna fed by a microstrip transmission line over a semi-circle
of the same size on the rear side that acts as a partial ground plane, which can
be seen in Fig. 2.3. This antenna acts as a monopole that creates a dipole-like
antenna with its image above the ground plane.

Figure 2.3 Copper patterning of the monopole imaging antenna. Top copper
(conductor) is in blue and bottom copper (ground plane) is in orange.

The antenna design was modified and optimized using Ansys HFSS. Once
final dimensions were optimized, 16 of the antennas were fabricated and tested.
In order to limit the physical size of the antennas, they were fabricated on Rogers

13

3006 dielectric board, which has a moderately high dielectric constant (
r =
6.15). A board height of 1.27 mm was chosen and end-launch SMA connectors
were added to the edge of the antenna feed line. In order to compensate for the
gap of the connector being larger than the thickness of the substrate, copper tape
was added to increase the stability of the connection. The copper material was
cut on a CNC milling machine and the excess copper removed, leaving only the
pattern behind. An image of the front and rear of the fabricated antennas can
be seen in Fig. 2.4.

Figure 2.4 Fabricated antenna design on Rogers 3006. The conducting (left)
and ground (right) planes are both shown.

The simulated and measured S11 and antenna patterns can be seen in Figs.
2.5 and 2.6, respectively. Nearly the entire bandwidth of 1 to 5 GHz measured
below -10 dB, besides the lowest frequencies and a small frequency range
between 3 and 3.5 GHz. The resonance ship upwards around 1 GHz accounts
for the lack of gain at that frequency in the pattern. Besides this difference
in the 1 GHz pattern, the antenna trends well with simulation and shows the
dipole-like characteristics that were targeted in this design.

14

Figure 2.5 Simulated and measured S11 for the tomography measurement an-
tennas.

15

11.522.533.544.55Frequency (GHz)-50-45-40-35-30-25-20-15-10-50S11MeasuredSimulated-10 dB(a)

(b)

Figure 2.6 Simulated (a) and measured (b) antenna patterns of the tomography
measurement antennas.

16

0°90°180°90°-30-25-20-15-10-505101 GHz2 GHz3 GHz4 GHz5 GHz0°90°180°90°-20-15-10-5051 GHz2 GHz3 GHz4 GHz5 GHz2.3 Switch Matrix Design

Another important piece of this system design is the microwave switch
matrix. This switch matrix consists of 22 switches, 4 single pole eight throw
(SP8T) switches and 18 single pole double throw (SPDT) switches, that can
quickly connect any of the 16 antennas to any of the other remaining 15. On
one side of the housing box has two ports, one for transmit and one for receive,
as seen in Fig. 2.7, and the other side has the 16 ports for each of the outputs
to the antennas, seen in Fig. 2.8. The switch tree that connects these two sides
can be seen in Fig. 2.9. Each signal route passes through three switches from
transmit to antenna and three from antenna to receive. In order to help mitigate
some of this insertion loss from the three switches, a Mini-Circuits ZX60-V82+
amplifier was added to the transmit stage before the signal is passed through
any switch. Since the switching would be happening many times and to create
the most robust switch matrix as possible, high speed solid state switches were
selected. The SP8T switches used are HMC322ALP4E and the SPDT switches
used are HMC270AMS8GE.

These switches have many DC voltage control lines that are used to control
the logic. To properly handle these voltages, a PCB was designed, fabricated,
and populated with surface mount devices. The schematic and part numbers
of this PCB can be referenced in Appendix B. This circuit board is controlled
through a Nucleo F411RE development board from STMicroelectronics, seen
in Fig. 2.10. The Nucleo F411RE runs C++ code on the Mbed platform.
The Nucleo is connected to a controlling laptop through a USB hub, allowing
for easy control of the connected ports of the switch matrix. The computer
connects with the Nucleo through a serial connection and simply accepts two
hexadecimal characters, first one for transmit connection and second one for
receive connection, as its input for control. The input hexadecimal characters

17

Figure 2.7 Front side of the switch matrix.

0-9 correspond with ports 1-10 and characters A-F correspond with ports 11-16,
all respectively. The C++ code installed on the Nucleo can be seen in Appendix
C. The complex arrangement of the inside of this system can be seen in Fig.
2.11.

Overall, there are 240 measurements that are recorded per scan, 16 antennas
paired with each of the other 15. In order to do a complete scan of a sample,
the system takes approximately 18.5 minutes, providing near-real-time imaging
of the sample.

18

Figure 2.8 Output ports on the rear of the switch matrix.

19

Figure 2.9 Schematic of switch tree to create a 2x16 switch matrix connecting Tx
and Rx to any pair of ports 1-16. Switches labeled "A" are HMC270AMS8GE
switches and those labeled "B" are HMC322ALP4E switches.

Figure 2.10 The Nucleo F411RE used to control the microwave switch matrix.

20

Figure 2.11 Inside of the switch matrix.

21

2.4 Transceiver

For this research, a Hewlett-Packard 8753D network analyzer, seen in Fig.
2.12, is used. A network analyzer measures the scattering parameters of a one
or two port network. This analyzer also has the ability to collect the complex
scattering parameters of the network it is measuring. This network analyzer is
connected to the switch matrix discussed in the previous section that connects
it to each of the 16 antennas in the ring. This network analyzer measures S21
for each pair of antennas, recording Sij (i = [1,16]; j = [1,16]; i (cid:44) j) for the
SUT. Data is collected from 1 to 5 GHz over 1601 points, yielding a step size of
2.5 MHz. The decision to use these sampling parameters and its consequences
will be further discussed in Chapter 3.

Figure 2.12 The Hewlett-Packard 8753D network analyzer and its controlling
computer.

Using a network analyzer for the measurement is an important factor in
future iterations of this system. The goal of the final product is to have a

22

low-cost, portable measurement system for use in field testing of tree trunks.
Using a frequency domain measurement of scattering parameters with a network
analyzer allows for the implementation of a software defined radio (SDR). Using
an SDR to sweep across a wide bandwidth as an alternative to an expensive
network analyzer can provide a cheap, portable alternative to the current system
design. Although SDRs are generally instantaneously narrowband, they have
a wide operational bandwidth so the wide frequency sweep can be done in a
similar manner to the network analyzer.

SDRs have only recently become an area of research in electrical engineer-
ing. In 2006, Svensson introduced the idea of using generic hardware that can
be programmed in software to manage radio standards across a wide bandwidth
[55]. In 2009, Welch et. al explored the possibility of using SDRs as an afford-
able alternate for radio signal processing instrumentation but acknowledged its
shortcomings in the cleanliness of the signal [56]. Regardless, they described
SDRs as a cheap and fairly reliable alternative to expensive instrumentation,
especially in classroom or academic research areas where budgets are generally
tight. More recent research has involved creating special receivers that help
create a more reliable receiver across a wide bandwidth [57–59], which will
be key to this particular application.
It was not until 2016 until Helaly and
Adnani explored the possibility of using a software defined radio for spectrum
analysis in [60] and [61]. Also in 2016, Marimuthu et. al used an SDR to
do basic medical imaging using a wideband sweep similar to the future system
proposed in this work [62]. Stancombe is continuing to advance this SDR
imaging technology to develop a portable head imaging system in 2019 [63].
In the near future, SDR technology and research will be expanded greatly and
its use as a reliable alternative to expensive testing equipment will become a
feasible possibility for the tomography system.

23

CHAPTER 3

SIGNAL PROCESSING

3.1 Overview

A major component of the microwave tomography system is the processing of
the measured signals to create a reconstructed image. There are three primary
sections of signal processing involved in converting measured transmission
coefficients in the frequency domain to a coherent reconstructed 2D image:
frequency to time domain conversion, time reversal simulation, and energy
calculation and reconstruction. Each of these three procedures will be described
in detail in this chapter. The MATLAB code for initializing the system setup,
preparing the measured data from the frequency to the time domain, and the
back propagation and energy calculation can be seen in Appendices D, E, and
F, respectively.
3.2 Frequency Domain Motivation

The microwave tomography system collects data in the frequency domain in
terms of the transmission coefficients Sij (i = [1,16]; j = [1,16]; i (cid:44) j). This
creates a 16 by 16 scattering parameter matrix that contains all the transmission
coefficients of each possible pair of antennas.
In this case, the reflection
coefficients, which are found on the diagonal of the scattering parameter matrix,
are set to 0. Although reflection coefficients could possibly be used, this is not
traditional in time reversal simulations which will be discussed in the following
section.

In order to prepare this data for this time reversal simulation and reconstruc-
tion, time domain data must be obtained. Although data could also be obtained

24

in the time domain thus avoiding this step, this requires precise timing equip-
ment and a high speed oscilloscope. In addition, the use of an microwave switch
matrix for this system would also add complexity to the analysis. Individual
line length and signal paths are not uniform through this switch matrix and
each channel would require individual characterization. As an alternative, the
data is collected on a network analyzer in steady state and transformed into the
time domain. The steady state measurement also generally provides a higher
signal-to-noise ratio (SNR).

3.2.1 Frequency to Time Domain Conversion
The frequency domain information that is obtained from the network analyzer
or SDR must be converted from the frequency domain to the time domain. This
section will follow the code included in Appendix E. The previously mentioned
16 by 16 scattering parameter matrix is measured over 1601 frequency points
from 1 to 5 GHz, with a frequency step size of 2.5 MHz. In order to obtain
strictly real time domain waveforms like those that would be obtained from
an oscilloscope, frequency data must be conjugate matched across the zero
frequency axis.

First, zeros are added to each S-parameter from 0 to 1 GHz at the same
frequency sampling rate used for the collected data. Once the frequency domain
is populated from 0 to 5 GHz, this information is reflected over the DC axis
and conjugated. This creates frequency domain information from -5 to 5 GHz,
containing 4001 points, 2000 points on each side of the DC axis and one point
for the DC frequency. From this point, an inverse fast Fourier transform (IFFT)
is applied to the data, creating the strictly real time domain waveform. Since
the frequency domain waveform was taken across a wide bandwidth, this leads
to a narrowband pulse in the time domain.

25

The last step is to cut the information down to reduce computation time. This
process produces a 4001 time step pulse, the vast majority of which are very
close to zero. It has been tested and shown that using only 250 time steps where
the actual pulses occur in the waveform provides a nearly if not completely
identical reconstruction as using the entire waveform and it drastically cuts
down on computation time and resources. After the creation of these time
domain pulses from the frequency sampled data, the data is ready for the time
reversal simulation which will be described in the next section.

In order to create the reconstructed images, a subtraction of these time
waveforms is necessary. The scattered field can be calculated using these
waveforms by using the equation

E scat = Etot − Einc

(3.1)
where E scat, Etot, and Einc represent the scattered, total, and incident
electric fields, respectively. The total field uses the measured data from the
environment with the object being tested inside of it and the incident field is a
measurement of the empty field. Subtracting these two fields leaves only the
scattered field in the simulation. This, in theory, will create an image of the
scatterer with the additional environmental noise subtracted from the empty
field calibration step.
3.3 Time Reversal Simulation

For this imaging application, the imaging technique selected is time rever-
sal. This theory is well documented as a localization technique for imaging
environments in radar applications. The following sections will discuss both
its theory and uses in many different microwave and ultrasonic applications
followed by its implementation in the microwave tomography system discussed

26

in this thesis.

3.3.1 History
The first mentioning of time reversal in literature as a electrical engineering
concept was in 1957 from Bogert of Bell Labs [64]. It was originally presented
as a concept to mitigate phase distortion in communication signals to increase
quality of transmission. This technique was used to create digital filters with
zero phase shift added to the signal. A similar idea was presented in 1965 using
the concept of time reversal as a digital filter [65]. For about 30 years from this
publication, the concept of time reversal was used solely for this purpose. It
was not used as an imaging technique until much later.

Time reversal technique evolved in the late 1980s and early 1990s as a method
for focusing fields to locate targets in an environment as opposed to strictly a
filtering technique for phase distortion [66–68]. These works, however, used
ultrasonic waves to perform their imaging. It was not until 2004 when Lerosey
et al. demonstrated the first use of time reversal focusing using electromagnetic
radiation [69]. They used wideband microwave radiation at a central frequency
of 2.45 GHz to focus convergence of the waves back to the initial source.
Since then, time reversal of electromagnetic fields converging to their source
has become the main focus of research. Although there is still some recent
use of time reversal at microwave frequencies for delay spread compression of
communication signals (e.g. [70]) as it was originally introduced by Bogert,
its use as a back propagation of waves to their source has proven to be a much
more useful tool in modern research.

The major use of time reversal in recent research has used this back propa-
gation of electromagnetic waves for radar applications and medical imaging
problems. For radar applications, the use of time reversal has been demon-

27

strated for the imaging of targets in cluttered environments [71], as well as for
obscured targets [72] such as through-wall [73] or buried [74] targets. Mi-
crowave imaging is an important development due to its use of non-ionizing
radiation. Consequently, time reversal has been investigated for its use in breast
imaging for cancerous tumors [75–77].

Using it for an imaging application is the focus of this thesis due to its
relatively easy implementation as well as its low computation cost. As will be
discussed in the following section, the low computation cost of this technique
is vital for the final system since it provides near-real-time imaging and it limits
monetary costs.

3.3.2 Theory and Implementation
In this work, time reversal is implemented to the real time domain waveforms
to construct an image of the testing environment. The process and theory of
electromagnetic time reversal described in this section is based off the develop-
ment in [78]. Assuming that the imaging environment is lossless and passive,
commonly done for air environments, the scalar wave equation can be written
as

φ(r, t) = 0,

In this scenario, the term φ(cid:0)r, t(cid:1) represents a solution of the

(3.2)
where c is the speed of light in a vacuum and n is the refractive index in
the medium.
scalar wave equation. Since there is a double partial time derivative applied
to this solution, that means that an equivalent solution to this wave equation is
φ(r,−t). Considering that φ(r, t) represents a diverging wave from the trans-
mitting sources, then φ(r,−t) is a wave doing the opposite, converging on the
transmitting sources. The time reversal operator is indicated as T in this section.

(cid:18)

∇2 − n2(r)
c2

2
∂
∂t2

(cid:19)

28

The diverging and converging scenarios can be considered equal in equation
form as φ(r, t) = T φ(r,−t) [69]. Since these two solutions equivalently satisfy
the scalar wave equation, their simulation in reversed time is a valid physical
representation of the measurement scenario.

In order to construct this image, the waveforms are back propagated through
a 2D Finite Difference Time Domain (FDTD) simulation [79]. This simulation
has been adapted from the work that was done in [80]. The simulation is based
upon the solving of the governing equations of electromagnetics, Maxwell’s
equations:

∇ · E =
ρ


∇ · H = 0
∇ × E = −jωµH
∇ × H = J + jω
E

(3.3)

where E and H are the electric and magnetic fields, respectively, J is the electric
current source, ρ is the electric charge density, ω is the angular frequency, and

 and µ are the permittivity and permeability, respectively.

In order to numerically simulate this environment, these equations must
be discretely solved in time. The time domain version of Equation 3.3 can
be obtained by replacing the jω in the third and fourth equation with a time
derivative. Maxwell’s equations can be solved discretely in time by creating a
mesh of Yee cells [81] where the time step is related to time taken for the speed
of light to cross one cell. This time step, however is different from the time step
derived from the IFFT performed in 3.2.1, which is based upon the 10 GHz
bandwidth of the transformation. Consequently, an interpolation of the pulsed
waveforms obtained is required to map the IFFT time steps to the FDTD time
steps. The numerical model for this simulation uses a TMz mode, where Hz,

29

Ex, and Ey are vanishing to zero due to the lack of a transverse magnetic field
component [80]. Each cell is created to be a square for simplicity. The length
of the side of each square is selected to exactly satsify the Courant condition
for stabiltiy c∆t ≤ ∆x√
[82]. It is a common rule of thumb to select a mesh size
2
of ∆x = ∆y = λ/10 in order to sufficiently mesh the simulation domain, for the
reconstructed images presented in the results chapter a mesh size of λ/40 was
selected to ensure a well-meshed simulation.

FDTD simulations are used to simulate electromagnetic waves in a domain
that can essentially be considered an infinite open space. Since this is clearly
computationally impossible, the domain must be bounded by effective boundary
conditions. These boundary conditions are necessary to avoid reflections off of
the edge of the computational domain, which introduces non-physical energy
back into the simulation, skewing results. Although many different boundary
conditions have been researched and utilized in electromagnetic simulations
[83–86], the Mur absorbing boundary condtions (ABC) [83] was selected for
this case. These conditions are designed to avoid reflections off of the boundary
interface. Although Mur ABCs can provide inaccuracies when waves hit the
boundary at an oblique angle or when the source is close to the boundary [78],
these situations are avoidable in the current simulation environment. ABCs
are the simplest to implement and standard in simulations so more robust
boundary conditions [84–86] were deemed unnecessary. Standard Mur ABCs
were imposed on each boundary of the simulation environment.
3.4 Energy Calculation and Reconstruction

There are two primary methods for reconstructing an image from the mea-
sured results. The first method involves the entropy of the system, using the
maximum of the inverse of the normalized variance of the fields. This technique

30

is known as the inverse varimax norm [75, 87]. Since entropy is colloquially
known as "randomness", the electric field is the most focused (least random) at
the time instant when the entropy is at a local minimum. At this time instant, a
plot of the electric field is concentrated at the scatterers present in the imaging
environment. The second method is using the time integrated energy to localize
the target [80].

Once the frequency domain information is properly transformed and in-
terpolated into the time domain and the time reversal and FDTD simulation
properly prepared, the last step left is to simulate and reconstruct the image.
As described before, time reversal simulation back propagates waves from the
receivers and they converge onto the transmitters in a antenna system. During
the measurement phase, each antenna takes a turn as a transmitter and the other
15 antennas act as receivers, filling the 16 by 16 matrix of each antenna pair.
The received signals for one transmitting case is placed at the location of each
receiver and the waves are back propagated through the simulation environment
for the duration of the time waveforms. As these waves propagate through the
environment, the energy at each of the FDTD cells is calculated with each
passing time step. The total integrated energy (E) for each cell is calculated as
a running sum per time step:

E(x, y) =

z (x, y, ti)∆t
E2

(3.4)
where N is the total number of time steps, Ez(x, y, ti) is the electric field in the
z direction at the (x, y) location at time ti, and ∆t is the time step.

This simulation is done for each transmitting energy case, giving the to-
tal integrated energy at each cell from E1 to E16. Each of these provides a
sub-optimal image of the environment. In order to create a full reconstructed

N


i=1

31

image, the 16 energy matrices are averaged to create one reconstructed image
of the total average time integrated energy of the FDTD simulation environ-
ment. Overall, the time it takes for this information to process and to create a
reconstructed system is about 60-65 minutes. Examples of this reconstruction
will be presented and discussed in the next chapter.
3.5 Reconstruction Example

In this section, an example of the signal processing procedure will be outlined
from start to finish. The waveforms as they are being processed are included
to help better visualize the steps taken to turn measured frequency domain data
into a fully processed image. This section will outline the signal processing
steps to image a brass cylinder using the full system. This image of the test
setup can be seen in Fig. 3.1. This section will cover only the signal processing,
the hardware aspect and image quality will be discussed in Section 4.3.1.

3.5.1 Frequency to Time Domain Example
Once the frequency domain information is sampled by the network analyzer,
the first step is to prepare it for the time reversal simulation. The procedure
to do this conversion follows Section 3.2.1 and uses the code that is included
in Appendix E. The code takes an input of measured frequency domain data
and creates time domain pulses, each waveform can be seen in Fig. 3.2. The
time domain waveforms of the measured empty (Einc) field, the measured field
with the brass cylinder (Etot), and the resulting subtracted field (E scat) can be
seen in Figs. 3.3, 3.4, and 3.5, respectively. These waveforms represent the
received signals at antennas 2 through 16 while antenna 1 is transmitting, which
is why the first line is set to exactly zero. Note that these waveforms are spaced
vertically for visualization purposes only and the y-axis is not scaled to any true

32

Figure 3.1 Test setup of a brass cylinder being imaged by the microwave to-
mography system.

value.

There are two notable aspects of these waveforms that can be used as sanity
checks for proper performance of the measurement as well as the frequency
to time domain conversion. The first aspect is the extremely large response of
antennas 2 and 16 in Figs. 3.3 and 3.4. This is largely due to the fact that
antennas 2 and 16 are direct neighbors of antenna 1. The mutual coupling be-

33

Figure 3.2 Measured frequency data in dB (top), zero padded and conjugate
mirrored frequency data in dB (middle), and time domain pulse after the IFFT
processing (bottom).

tween these antennas is much higher than the coupling between any of the other
antennas. This is a good indication that the measurement was taken correctly
since mutual coupling between neighboring antennas is always expected, espe-
cially while using planar non-directional antennas. The second sanity check is
the moving wavefront of each antenna pair. The arrival time of the pulses starts
with the neighboring antennas while antenna 9 receives the pulse last. This is
expected due to the physical placement of the antennas. The varying flight time
helps confirm that the inverse Fourier transform was performed correctly. After
the subtraction, only the information from the added cylinder remains, as seen
in Fig. 3.5. The mutual coupling is brought down significantly and only small

34

11.522.533.544.55Frequency (GHz)-40-200S21 (dB)-5-4-3-2-1012345Frequency (GHz)-40-200S21 (dB)050100150200250300350400Time (ns)-0.100.1Figure 3.3 Time domain waveform of the measured empty environment.

ripples in the waveforms remain.

35

01020304050Time (ns)-0.0500.050.10.150.20.250.30.350.4Figure 3.4 Time domain waveform of the measured cylinder environment.

36

01020304050Time (ns)-0.0500.050.10.150.20.250.30.350.4Figure 3.5 Time domain waveform of the measured empty environment sub-
tracted from the measured cylinder environment.

37

01020304050Time (ns)00.050.10.150.20.250.30.353.5.2 Time Reversal and Image Reconstruction
Once the wavefroms are prepared, they are ready to process from time domain
pulses to a reconstructed image. This is achieved by the code seen in Appendix
F. This code steps through the process of time reversal from the measured
results. During this process, the energy is being calculated at each cell in the
simulation environment using Equation 3.4. This is done for every transmitting
case, which produces a sub-optimal image each time. The sub-optimal images
can be seen in Figs. 3.7, 3.8, 3.9, and 3.10. The position of each antenna
number is labeled in Fig. 3.6. These locations will be used throughout this
entire thesis. Notice that in these images there is no energy focused around
the transmitting antenna. Also, depending on the position of the transmitting
antenna there is a large amount of energy focused around the antennas that are
in direct reflective paths of the cylinder, specifically in (c) and (d) of Fig. 3.10.
Regardless, each of these images produces a clear location for the cylinder, so
doing it 16 times provides a sufficient amount of information for the final image.
The final image can be seen in Fig. 3.11. This is done by averaging the 16
sub-optimal images created from each transmitting case, creating one image of
the final environment. It is clear to locate the cylinder and the energy focused
around particular antennas is greatly reduced by this averaging, which helps
clean up the image and reduce unwanted noise. The impacts of this image and
its comparison to an optical image will be discussed in Section 4.3.1.

38

Figure 3.6 Location of each of the antennas in the ring.

39

(a) Antenna 1 transmitting.

(b) Antenna 2 transmitting.

(c) Antenna 3 transmitting.

(d) Antenna 4 transmitting.

Figure 3.7 Sub-optimal reconstructed images of antennas 1 (a), 2 (b), 3 (c), and
4 (d) transmitting.

40

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15(a) Antenna 5 transmitting.

(b) Antenna 6 transmitting.

(c) Antenna 7 transmitting.

(d) Antenna 8 transmitting.

Figure 3.8 Sub-optimal reconstructed images of antennas 5 (a), 6 (b), 7 (c), and
8 (d) transmitting.

41

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15(a) Antenna 9 transmitting.

(b) Antenna 10 transmitting.

(c) Antenna 11 transmitting.

(d) Antenna 12 transmitting.

Figure 3.9 Sub-optimal reconstructed images of antennas 9 (a), 10 (b), 11 (c),
and 12 (d) transmitting.

42

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15(a) Antenna 13 transmitting.

(b) Antenna 14 transmitting.

(c) Antenna 15 transmitting.

(d) Antenna 16 transmitting.

Figure 3.10 Sub-optimal reconstructed images of antennas 13 (a), 14 (b), 15
(c), and 16 (d) transmitting.

43

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15Figure 3.11 Total reconstructed image obtained from averaging the total energy
in all 16 of the previous images, producing the brass cylinder shape.

44

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-15CHAPTER 4

MEASURED RESULTS

4.1 Introduction

In this chapter, the measured results of the microwave tomography system
are presented. There are several variables that were tested using two separate
methods. The first method was done using only two antennas and physically
moving them to each pair of positions, collecting the frequency data per pair.
This is described and the reconstructed images are included in Section 4.2.
Simple cylindrical targets are imaged in this section. The other measurements
that are presented are all using the system described in Chapter 2, using the
switch matrix and the 16 antennas simultaneously. It should be noted in all of
these images that there is a large amount of energy focused around the antennas.
This is a expected result and will show up in all reconstructed images as there
will always be energy around the antennas.
4.2 Pairwise Measurements

The pairwise measurement was performed by moving the pair of antennas
connected to ports 1 and 2 of the network analyzer and recording S21 at each
location, filling the 16 by 16 S-parameter matrix one-by-one. Clearly, this
method is extremely time consuming and not a feasible way to execute in
vivo measurements of tree trunks but it was an important step to verify the
functionality of both the antennas and the imaging code before the switch matrix
was fully constructed. A full 240 measurement matrix took between 6 and 7
hours to conduct using this method. The images shown were reconstructed
using 500 points of the pulse instead of 250 (see Section 3.2.1 for more on

45

this), so the reconstruction time was about 90 minutes. Thus, the overall time
from the start of measurement to reconstructed image was approximately 8
to 9 hours. This is not accounting for the calibration step of measuring an
empty environment that represents the incident field that is subtracted from the
scattered field (Equation 3.1) since this only is taken once and then used for all
the scattered measurements. If this is factored in, another 6 to 7 hours should
be added to the timing.

The first, and potentially easiest, case that was tested was using a brass
cylinder. Since metal is very conductive it is often considered a close to real-
world example of a perfect electrical conductor (PEC). This cylinder being very
reflective is the best chance to successfully reconstruct an image. The optical
image and its reconstructed microwave image can be seen in Fig. 4.1. This
cylinder has a height of 6" and a dimeter of 1.25". In this image, the energy
information had to be normalized per transmitting case, which explains the
intensity of the color axis. The normalization was likely required due to the
inconsistency of the pairwise measurements, the requirement to move antennas
to each location by hand had a lot of variability and inconsistent energy transfer
due to antenna positioning, orientation, etc.

The next case that was tested was a plastic cylinder. This is closer to a real
world scenario since the signals pass through the object being tested instead
of going around them like they do a PEC. The image and its reconstructed
microwave image can be seen in Fig. 4.2. This cylinder has a height of 12" and
a diameter of 5". This cylinder is black Acetron GP, which has an dielectric
constant 
r = 3.8 and a dissipation factor tanδ = 0.005. Again, there is
normalization of the energy for the same reasons as listed for the PEC case.

In both of the images, there is a good correlation between the optical image
and the microwave reconstructed image. There is some shadowing in the

46

reconstructed picture but for the first run, it pointed to a successfully operating
system. Since these measurements took so long to take a single set, these were
the only results attained. This merely confirmed the operation of the network
analyzer and the antennas in the system and after this the switch matrix was
added.

47

(a)

(b)

Figure 4.1 Imaging results of a brass cylinder using pairwise measurement
technique. (a) Measurement setup. (b) Reconstruction.

48

(a)

(b)

Figure 4.2 Imaging results of a plastic cylinder (
r = 3.8) using pairwise
measurement technique. (a) Measurement setup. (b) Reconstruction.

49

4.3 System Measurements

In this section, the entire system was used for measurements of scatterers.
This uses the automated switch matrix (Section 2.3) to take all 240 measure-
ments without the need for human interaction between the sampling, which
drastically cuts down on the measurement time. All 16 antennas are in the
system simultaneously, as in Fig. 2.2. This is more reflective of the physical
system that will be used in the final product. The cut of the measurement time
went from 6 to 7 hours for the pairwise measurements to about 18.5 minutes, a
huge improvement that allows many measurements to be taken in the course of
a day.

4.3.1 Simple Scatterers
In this section, only simple, homogeneous, scatterers are used, similar to those
seen in Section 4.2. These measurements, again, were used to validate the
operation of the system and the reconstruction code. For these experiments,
a 250 point pulse was used again, so this reconstruction time took about 45
minutes, for a total reconstruction time of about 60-65 minutes from beginning
of measurement to reconstructed image. This is a nearly 90% decrease in
total time as compared to the pairwise measurements, showing the value of the
switch matrix for this system. For any feasible commercial imaging system it
is absolutely essential.

The same brass (pseudo-PEC) and dielectric plastic cylinders were measured
as a starting case. The optical images and reconstructed microwave images of
the brass and plastic cylinders can be seen in Figs. 4.3 and 4.4, respectively.
The color axis for this image does not use the averaging that was used in the
pairwise measurements, which is a more accurate simulation representation of

50

the measurement scenario.

51

(a)

(b)

Figure 4.3 Imaging results of a brass cylinder using the entire measurement
system. (a) Measurement setup. (b) Reconstruction.

52

(a)

(b)

Figure 4.4 Imaging results of a plastic cylinder (
r = 3.8) using the entire
measurement system. (a) Measurement setup. (b) Reconstruction.

53

After the functionality of the system was confirmed by measuring the cylin-
ders, more complex structures were tested to push the abilities of the system.
Fig. 4.5 shows the optical image and reconstructed microwave image of a plastic
square prism. This prism is made out of Delrin, an electrically similar material
to the cylinder measured above (
r = 3.7). The square base of the object is 4"
by 4". The sharper corners can cause a more difficult imaging problem but the
shape seems to appear in the microwave image.

Fig. 4.6 is an optical image and reconstructed microwave image of a wooden
block. This block is a painted 4x4 from any typical hardware store (3.5" by
3.5"). The dielectric constant of this type of wood varies since it is generally
made from a combination of multiple wood species, but it typically varies from
1.25-5. This ability of this system to image a wood structure is an important
step towards the final system goal which is to image tree trunks.

Again, both of these more complex used the same physical parameters as
above so the total imaging time was approximately 60-65 minutes. This il-
lustrates an important aspect of the system. Since time reversal is not an
inverse problem, the time from measurement to final image is always the same.
Whereas an inverse problem would have trouble with a more complex structure
compared to a cylinder and a more inconsistent material than the solid plastic or
brass samples, these images take the exact same amount of time and computing
power as the very simple sample targets. For a portable, low-cost imaging
system this characteristic is extremely important.

54

(a)

(b)

Figure 4.5 Imaging results of a plastic square prism (
r = 3.7) using the entire
measurement system. (a) Measurement setup. (b) Reconstruction.

55

(a)

(b)

Figure 4.6 Imaging results of a wood block using the entire measurement system.
(a) Measurement setup. (b) Reconstruction.

56

4.3.2 Tree Images
This section presents images of tree trunk samples with various types of simu-
lated abnormalities on their inside. The two types of damage that are simulated
are drilled holes, which are simulating unhealthy trees that are at risk of falling
down in a wind or ice storm due to decay in their trunks, and metal inclusions,
which are a threat to the lumber industry because they can damage saw blades
and halt production when these blades break. Two species of trees were tested
in this section, a basswood tree, which is less dense dicot tree and a sugar maple,
which is a dense dicot tree species.

In this section, a different subtract step was taken in the signal processing.
Whereas in the simple target case an empty environment was used as an inci-
dent field calibration step, other sections of the tree itself were considered the
"incident field". Although this is a more complex measurement setup, it is not
unrealistic for a real world system. Since decay in trees rarely spans the entire
trunk, measurements could be taken at several heights of the trunk and each pair
could be subtracted from the other. The subtraction used and optical images of
both samples are presented with each presented result. Due to the addition of
a second measurement, the time to create these images rises to approximately
80-85 minutes from start of measurement to reconstructed image.

4.3.2.1 Air Gap Images
Fig. 4.7 shows the imaging result of a basswood sample with a 1" diameter
hole subtracted from a solid sample. The hole appears in the microwave
reconstructed image as a small dielectric inconsistency. Although this is not a
completely conclusive result and the inconsistency is hard to distinguish from
the noise, it is a step in the right direction. For comparison, two portions of
solid basswood logs were subtracted from each other. This result can be seen

57

in Fig. 4.8. As is seen in this reconstructed image, two solid portions of logs
produce only noise and no definite inconsistencies can be recognized. Note that
all the optical images seen in these two figures were taken after the logs had
been completely dried, which explains the cracks and splits seen in the logs.
The measurement was taken when the logs were freshly green, so each of the
sections were completely solid with no cracks or splits other than the drilled
hole. Also, the solid log in Fig. 4.8 has a nail added to it, which was used for
the metal inclusions test. Again, this picture was taken much later and the data
used for the microwave imaging was a measurement of the solid log with no
nail included.

58

(a)

(b)

Figure 4.7 Imaging results of a 1" diameter hole in a green basswood sample.
The solid portion of the trunk (left (a)) was subtracted from the portion with
the hole (right (a)). (b) Reconstruction.

59

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.3012345610-16(a)

(b)

Figure 4.8 Imaging results of a solid, green basswood sample. Two solid
portions of the trunk (a) were subtracted from each other and only noise remains
in the reconstructed image (b).

60

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.3012345610-16Similar results were obtained for a sugar maple sample. The imaging results
for a 2" diameter hole and a solid log (again the nails in the optical image were
added after the microwave measurement was taken) can be seen in Figs. 4.9 and
4.10, respectively. Again, the results are not conclusive, but there is a definite
contrast between the hole gap image and the solid log image. This points to a
feasible idea for a future system.

61

(a)

(b)

Figure 4.9 Imaging results of a 2" diameter hole in a green sugar maple sample.
The solid portion of the trunk (left (a)) was subtracted from the portion with
the hole (right (a)). (b) Reconstruction.

62

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.511.522.510-16(a)

(b)

Figure 4.10 Imaging results of a solid, green sugar maple sample. Two solid
portions of the trunk (a) were subtracted from each other and only noise remains
in the reconstructed image (b).

63

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.511.522.510-16These images were all obtained using measurements of freshly cut, green
tree trunk samples. These samples were measured to contain 50-50% water
by weight, clearly a much different dielectric consistency than pure wood.
Measurements of air gaps were more successfully obtained after completely
drying the samples and measuring only the wood samples. This demonstrates a
functioning system and feasible results for using this as a tree imaging system.
The following images will present images using completely dried wood.

Figs. 4.11 and 4.12 show the imaging results of holes in a basswood and a
sugar maple sample, respectively. As is seen in these images, the results are
much more conclusive and the holes are much more evident in the reconstructed
image. It is suspected that the penetration of the waves into the sample are to
blame for this drastic difference between the green and dry measurements. The
addition of water to the trees introduces much more loss into the system and
the waves are likely not reaching from one antenna to another in a direct path.
This is an issue that will need to be further investigated. Fig. 4.13 presents
two solid logs subtracted from each other, in which noise only remains, again
demonstrating the feasibility of the measurement technique.

64

(a)

(b)

Figure 4.11 Imaging results of a 2" diameter hole in a completely dried basswood
sample. The solid portion of the trunk (left (a)) was subtracted from the portion
with the hole (right (a)). (b) Reconstruction.

65

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.20.40.60.811.21.41.610-16(a)

(b)

Figure 4.12 Imaging results of a 2" diameter hole in a completely dried sugar
maple sample. The solid portion of the trunk (left (a)) was subtracted from the
portion with the hole (right (a)). (b) Reconstruction.

66

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.30123456710-17(a)

(b)

Figure 4.13 Imaging results of a solid, fully dried basswood sample. Two solid
portions of the trunk (a) were subtracted from each other and only noise remains
in the reconstructed image (b).

67

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.20.40.60.811.21.41.610-164.3.2.2 Metal Inclusion Images
This section will show images of logs that have metal inclusions embedded in
their trunks. Figs. 4.14 and 4.15 show the results of a bolt added to a basswood
and a maple trunk sample, respectively. The pictures seen in these figures have
1" holes drilled in them. Measurements of these logs (completely dried) were
taken with the log and then again with a bolt added to these 1" holes. These two
measurements were subtracted from each other and the images were obtained.
Although this measurement situation is not necessarily feasible in a real world
scenario, it is an important first step to confirming the use of this system to
image metal inclusions.

68

(a)

(b)

Figure 4.14 Imaging results of a bolt added to a 1" hole in a completely dried
basswood sample (a). (b) Reconstruction.

69

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-16(a)

(b)

Figure 4.15 Imaging results of a bolt added to a 1" hole in a completely dried
sugar maple sample (a). (b) Reconstruction.

70

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.511.510-16Figs. 4.16 and 4.17 show the imaging results of nails added to a sample
subtracted from a solid sample of basswood and sugar maple trees, respectively.
Again, these samples were completely dried in order to properly image the
metal inclusions. Unlike the above bolt images, these images were obtained
by subtracting solid portions from the nail-added portions, which is a more
realistic testing scenario. Fig. 4.18 shows the imaging result of a nail added
horizontally to the outside of a basswood log subtracted from a solid sample
of basswood log. The orientation of this nail is a more lifelike scenario, but
the quality of the image is drastically less clear than the vertical nails. This is
likely due to the size of the nail and the very little energy that is actually being
reflected off of it. This issue and the sensitivity will need to be explored further.

71

(a)

(b)

Figure 4.16 Imaging results of a nail in a completely dried basswood sample.
The solid portion of the trunk (right (a)) was subtracted from the portion with
the nail (left (a)). (b) Reconstruction.

72

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.10.20.30.40.50.60.70.80.9110-16(a)

(b)

Figure 4.17 Imaging results of nails in a completely dried sugar maple sample.
The solid portion of the trunk (left (a)) was subtracted from the portion with
nails (right (a)). (b) Reconstruction.

73

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.511.522.510-17(a)

(b)

Figure 4.18 Imaging results of a horizontal nail (illustrated by the red line) in
a completely dried basswood sample. The solid portion of the trunk (right (a))
was subtracted from the portion with the nail (left (a)). (b) Reconstruction.

74

-0.3-0.2-0.100.10.20.3-0.3-0.2-0.100.10.20.300.511.522.533.544.5510-174.4 Review

In this chapter, the use of the microwave tomography system was success-
fully demonstrated in many different scenarios, including tree trunk samples.
The system is very successful at imaging simple targets such as homogeneous
cylinders and rectangular prisms. While the system does not produce ideal
results, there is evidence pointing toward the use of this system for the imaging
of tree trunks. Usable results are not attainable unless the logs are completely
dried of their water content. This is likely due to the lack of penetration of the
waves into the sample due to the increase of lossy water within the trunk. More
power could help mitigate these issues, the use of a very lossy switch matrix
and the power limitations of the network analyzer could be contributing to these
issues.

75

CHAPTER 5
CONCLUSION

5.1 Overview

A microwave tomography system that is intended for use in several forestry
applications was presented. This system consists of a frequency domain sam-
pling transceiver, a 2-to-16 port switch matrix, and a ring of 1-5 GHz planar
monopole antennas that are used to image an environment using microwave
tomography. The system collects the transmission coefficient between each
pair of antennas in the ring. After the data is collected, the wideband frequency
domain data is transformed into time domain pulses for each transmit-receive
pair. Once the time domain pulses are obtained, they are back propagated in free
space using a 2D FDTD simulation, a process known as time reversal. At each
time step of the time reversal simulation, the energy per mesh cell is calculated
from the electric field. At the end of this process, the time integrated energy per
mesh cell is averaged for each transmitting case. The energy focuses around
scatterers in the imaging environment, providing a 2D tomographic image.

This system was used to image both simple targets such as cylinders and
rectangular prisms made of brass, plastic, and wood to demonstrate the imaging
capabilities of the system. These images were very conclusive and the location
of the objects in the ring were easily discernible. The system was also used
to image more complex targets, specifically samples of tree trunks of various
species of wood. These results were less conclusive and required an extra
subtraction step but there was evidence pointing towards the feasibility of using
it for forestry applications. The images were much more successful using
completely dried wood as compared to freshly cut green wood. The penetration

76

of the antennas is a major concern, the lossy switch matrix and limitations of
the network analyzer are possible reasons for this issue, which will need to be
further explored.
5.2 Cost Analysis

The price and complexity of this system was consistently the motivation
behind the decisions made during this process. Since this system was designed
for forestry applications, the cost and ease of use of the system was of the utmost
importance. The system requires cheap hardware and software that requires as
little computing power as possible.

The system hardware cost about $5,000, not considering the transceiver. The
largest hardware cost is the network analyzer and the cables that connect it to
the antennas as well as the transceiver. This cost about $4,500 in total for all
the components, with the other $500 accounting for the price of the antenna
substrate as well as other small miscellaneous costs. This is a reasonable cost
for a final product that would be used by natural parks, power companies, etc.,
however, this is ignoring the price of the transceiver. The transceiver used
in this study is a large, heavy network analyzer that costs thousands to tens
of thousands of dollars and is not portable, which is vital to performing in
vivo measurements of trees. This is where the use of a software defined radio,
discussed in Section 2.4, becomes important. Software defined radios are much
cheaper and portable, making them a good alternate to the current system and
a good option for the next prototype.

In terms of the software, the system was able to create an image in about an
hour using the low computational method of time reversal. All the simulations
were run on a single core of an Intel Core i5-2500T CPU that runs at 2.30 GHz.
The computer had 8 GB of RAM to do the calculations. This is a relatively

77

low computational cost and can easily be found in an inexpensive portable
laptop, another major selling point of the prototype. The low cost of signal
processing in comparison with other imaging systems makes it very attractive
for its intended application.
5.3 Future Work

Section 5.2 outlines the future work required for this system before it is able
to perform in vivo measurements of tree trunk. The first goal is to make the
system portable. The substitution of an SDR for the network analyzer will
greatly improve the portability of the system. Since the processing power is
low enough to be reasonably done on a laptop, it is not of great importance but
vectorizing the code could cut down on processing requirements and help reduce
the time from measurement to final image. The last requirement of the system
to create a fully portable system is to create a housing system for the antennas
on the ring. Currently, the ring rests on a stationary stage that has slots for the
antennas. A better, portable ring will need to be designed for the next prototype
of the system. This will add a slight cost to the final system but it should
not be anywhere on the magnitude of the switch matrix, keeping the cost in a
reasonable price range. With these few additions and optimizations, a portable,
low cost, near-real-time microwave tomography system can be created.

78

APPENDICES

79

APPENDIX A

PYTHON CODE TO TAKE DATA OFF HP8753D NETWORK

ANALYZER

# tomography_capture . py
#
# U t i l i t y t o c o n t r o l

s w i t c h m a t r i x and c a p t u r e

measurements

t o be used t o

t r e e / s u b j e c t .

images of

# r e c o n s t r u c t
#
# Ver 1 . 0
# Updated 1 / 1 0 / 2 0 1 8
#
# Change Log
#

im port

sys

and i n t e r r u p t h a n d l i n g

im port

s i g n a l

c t r l −c ) h a n d l i n g

im port

t r a c e b a c k

p r i n t i n g
time

im port

between samples

im port d a t e t i m e

save

f o l d e r

80

# Tracebacks

# S i g i n t

(

# Traceback

# S l e e p i n g

# Naming

im port

s e r i a l

m a t r i x communication

im port
p o r t

s e r i a l . t o o l s . l i s t _ p o r t s
l i s t i n g

im port hp8735_na as na
GPIB communication

# Switch

# S e r i a l

# HP−8735−NA

from i t e r t o o l s

im por t p e r m u t a t i o n s

#

Combination of p o r t s

t o s w i t c h

c o n s t a n t s

# P o r t
PORT = ’COM4’
BAUD = ’ 115200 ’

# Sampling c o n s t a n t s
SAMPLE_DELAY = 10e−3

# seconds

# Text
c l a s s

f o r m a t t i n g
c o l o r :

PURPLE = ’ \ 0 3 3 [ 9 5m’
CYAN = ’ \ 0 3 3 [ 9 6m’
DARKCYAN = ’ \ 0 3 3 [ 3 6m’
BLUE = ’ \ 0 3 3 [ 9 4m’
GREEN = ’ \ 0 3 3 [ 9 2m’
YELLOW = ’ \ 0 3 3 [ 9 3m’
RED = ’ \ 0 3 3 [ 9 1m’
BOLD = ’ \ 0 3 3 [ 1m’

81

UNDERLINE = ’ \ 0 3 3 [ 4m’
END = ’ \ 0 3 3 [ 0m’

c l a s s

tomography_capture ( o b j e c t ) :

" " "
C l a s s
" " "

t o manage tomography_capture

def _ _ i n i t _ _ ( s e l f , port , baud ) :

s u p e r ( tomography_capture ,
s e l f . p o r t = p o r t
s e l f . baud = baud

s e l f ) . _ _ i n i t _ _ ( )

# I n i t i a l i z e
s e l f . s e r = s e l f . o p e n _ s e r i a l ( )

s e r i a l p o r t o b j e c t

# I n i t i a l i z e network a n a l y z e r ,

r e t u r n s NA

o b j e c t

s e l f . ena = na . i n i t _ 8 7 3 5 d ( )

# Bind s i g n a l h a n d l e r
s i g n a l . s i g n a l ( s i g n a l . SIGINT ,

s e l f .

t o g r a c e f u l l y shutdown

e x i t _ h a n d l e r )

def

e x i t _ h a n d l e r ( s e l f ,
frame ) :
p r i n t ( ’ C l o s i n g s e r i a l p o r t . . . ’ )

sig ,

82

s e l f . s e r . c l o s e ( )
p r i n t ( ’ E x i t i n g tomography_capture . . . ’ )
e x i t ( )

def o p e n _ s e r i a l ( s e l f ) :

a l l

# L i s t
p o r t s = s e r i a l . t o o l s . l i s t _ p o r t s . comports ( )

a v a i l a b l e p o r t s

p r i n t ( " A v a i l a b l e
p r i n t ( ’− ’ ∗ 60)
f o r

s e r i a l p o r t s " )

i , p o r t
i f p o r t . name i s None :

i n enumerate ( p o r t s ) :

p r i n t ( ’ { : } .
d e v i c e ) )

e l s e :

{ : } ’ . f or ma t ( i +1 , p o r t .

p r i n t ( ’ { : } .

{ : } ’ . f or ma t ( i +1 , p o r t .

name ) )
p r i n t ( ’− ’ ∗ 60)
p r i n t ( ’ ’ )

# Open p o r t
t r y :

# p r i n t ( c o l o r .GREEN)
p r i n t ( ’ Using :

" { : } " @ { : } baud ’ . f or ma t (

PORT, BAUD) )

p r i n t ( ’ ’ )
# p r i n t ( c o l o r .END)

83

s e r = s e r i a l . S e r i a l (PORT, BAUD,

t i m e o u t

=1)
s e r i a l . s e r i a l u t i l . S e r i a l E x c e p t i o n as

e x c e p t

e :

p r i n t ( e )
p r i n t ( ’ E x i t i n g . . . ’ )
e x i t ( )

e x c e p t E x c e p t i o n :

p r i n t ( " Unknown e x c e p t i o n i n o p e n _ s e r i a l

" )

p r i n t ( ’− ’ ∗60)
t r a c e b a c k . p r i n t _ e x c ( f i l e = sys . s t d o u t )
p r i n t ( ’− ’ ∗60)
e x i t ( )

r e t u r n s e r

def

c o l l e c t _ d a t a ( s e l f , d e l a y =100e −3) :
a r r a y [0−F ]
# Generate hex v a l u e
h e x _ a r r a y = [ f or mat ( x ,

’ 01X’ )

f o r x i n range

( 0 , 16) ]

# Get network a n a l y i z e r
f r e q _ a r r a y = na . g e t _ f r e q s ( s e l f . ena )

smapling f r e q u e n c i e s

# Get c u r r e n t

time f o r naming save

f o l d e r

84

t i m e _ s t r = d a t e t i m e . d a t e t i m e . now ( ) . s t r f t i m e

( "%Y−%m−%d−%H%M%S " )

# I t e r a t e over a n t e n n a rx / t x p a i r s
f o r p a i r

i n p e r m u t a t i o n s ( hex_array , 2) :

p r i n t ( " Sampling p o r t s :

" + s t r ( p a i r ) )

# S e l e c t p o r t s
s e l f . s e r . w r i t e ( ( ’ { : } { : } ’ . f or ma t (∗ p a i r ) ) .

encode ( ’ u t f −8 ’ ) )

# S e t t l i n g time d e l a y
time . s l e e p ( d e l a y )

polReal , polImag = na . record_measurement

( s e l f . ena )

na . save_measurement ( polReal , polImag ,

f r e q _ a r r a y , p a i r ,

t i m e _ s t r )

def main ( ) :

t c = tomography_capture (PORT, BAUD)
t c . c o l l e c t _ d a t a ( d e l a y =SAMPLE_DELAY)

i f __name__ == " __main__ " :

main ( )

85

[style

86

APPENDIX B

SWITCH MATRIX SCHEMATIC AND PART NUMBERS

Figure B.1 Switch matrix schematic.

87

B.1 Part Numbers

R1-R29 ESR03EZPF1002

Z1-Z29 CMDZ5L1

I1-I8 SN74HCT04DRG4

P1 MAX764

C1 TPSD107K020R0085

C2-C3 T491A104M035AT

C4 T510X686K025ATE045

D1 1N5817

L1 CDRH8D58

H1-H2 SSW-119-01-T-D

H3-H5 1-796692-2

88

APPENDIX C

NUCLEO CODE FOR CONTROLLING SWITCH MATRIX

# i n c l u d e "mbed . h "

void r e s e t _ s w i t c h e s ( ) ;
void s e t _ t x ( ) ;
void s e t _ r x ( ) ;

S e r i a l pc ( SERIAL_TX , SERIAL_RX ) ;

/ / TX and RX 2−way s w i t c h e s
D i g i t a l O u t
D i g i t a l O u t

tx_sw ( PD_2 ) ;
rx_sw ( PC_2 ) ;

/ / TX 8−way lower h a l f
D i g i t a l O u t
D i g i t a l O u t
D i g i t a l O u t

s w i t c h
tx_low_A ( PC_10 ) ;
tx_low_B ( PC_11 ) ;
tx_low_C ( PC_12 ) ;

/ / TX 8−way upper h a l f
D i g i t a l O u t
D i g i t a l O u t

s w i t c h
tx_high_A ( PA_13 ) ;
tx_high_B ( PA_14 ) ;

89

D i g i t a l O u t

tx_high_C ( PA_15 ) ;

/ / RX 8−way lower h a l f
D i g i t a l O u t
D i g i t a l O u t
D i g i t a l O u t

s w i t c h
rx_low_A ( PC_13 ) ;
rx_low_B ( PC_14 ) ;
rx_low_C ( PC_15 ) ;

/ / RX 8−way upper h a l f
D i g i t a l O u t
D i g i t a l O u t
D i g i t a l O u t

s w i t c h
rx_high_A ( PA_0 ) ;
rx_high_B ( PA_1 ) ;
rx_high_C ( PA_4 ) ;

/ / Antenna p o r t s
D i g i t a l O u t p o r t 0 1 ( PC_3 ) ;
D i g i t a l O u t p o r t 0 2 ( PC_9 ) ;
D i g i t a l O u t p o r t 0 3 ( PC_8 ) ;
D i g i t a l O u t p o r t 0 4 ( PB_8 ) ;
D i g i t a l O u t p o r t 0 5 ( PC_6 ) ;
D i g i t a l O u t p o r t 0 6 ( PB_9 ) ;
D i g i t a l O u t p o r t 0 7 ( PC_5 ) ;
D i g i t a l O u t p o r t 0 8 ( PA_5 ) ;
D i g i t a l O u t p o r t 0 9 ( PA_12 ) ;
D i g i t a l O u t p o r t 1 0 ( PA_6 ) ;
D i g i t a l O u t p o r t 1 1 ( PA_11 ) ;

90

D i g i t a l O u t p o r t 1 2 ( PA_7 ) ;
D i g i t a l O u t p o r t 1 3 ( PB_12 ) ;
D i g i t a l O u t p o r t 1 4 ( PB_6 ) ;
D i g i t a l O u t p o r t 1 5 ( PC_7 ) ;
D i g i t a l O u t p o r t 1 6 ( PA_9 ) ;

c h a r b u f f [ 4 ] ;
t x _ a n t ;
c h a r
c h a r
r x _ a n t ;

i n t main ( )
{

/ / Set baud r a t e of UART
pc . baud (115200) ;

/ / r e s e t _ s w i t c h e s ( ) ;

while ( 1 )
{

pc . p r i n t f ( " Waiting . . . . \ r \ n " ) ;

pc . g e t s ( buff , 2) ;
t x _ a n t = b u f f [ 0 ] ;

/ / pc . p r i n t f ( " Got

t h i s

f a r \ r \ n " ) ;

91

( t x _ a n t == ’ r ’ )

r e s e t _ s w i t c h e s ( ) ;

pc . p r i n t f ( "Now I am h e r e \ r \ n " ) ;
c o n t i n u e ;

i f
{

}

pc . g e t s ( buff , 2) ;
r x _ a n t = b u f f [ 0 ] ;

( r x _ a n t == ’ r ’ )

r e s e t _ s w i t c h e s ( ) ;

pc . p r i n t f ( " Woooow . \ r \ n " ) ;
c o n t i n u e ;

i f
{

}

pc . p r i n t f ( " S e t t i n g s w i t c h e s . . . . \ r \ n " ) ;

92

r e s e t _ s w i t c h e s ( ) ;

s e t _ t x ( ) ;

s e t _ r x ( ) ;

}

}

void s e t _ t x ( )
{

s w i t c h ( t x _ a n t )
{

c a s e

c a s e

’ 0 ’ :
break ;
’ 1 ’ :

tx_low_A = 0 ;
break ;
’ 2 ’ :

tx_low_B = 0 ;
break ;
’ 3 ’ :

c a s e

c a s e

93

tx_low_A = 0 ;
tx_low_B = 0 ;
break ;
’ 4 ’ :

tx_low_C = 0 ;
break ;
’ 5 ’ :

tx_low_A = 0 ;
tx_low_C = 0 ;
break ;
’ 6 ’ :

tx_low_B = 0 ;
tx_low_C = 0 ;
break ;
’ 7 ’ :

tx_low_A = 0 ;
tx_low_B = 0 ;
tx_low_C = 0 ;
break ;
’ 8 ’ :

tx_sw = 0 ;
break ;
’ 9 ’ :

tx_sw = 0 ;
tx_high_A = 0 ;
break ;
’A’ :

c a s e

c a s e

c a s e

c a s e

c a s e

c a s e

c a s e

94

c a s e

c a s e

c a s e

c a s e

c a s e

tx_sw = 0 ;
tx_high_B = 0 ;
break ;
’B ’ :

tx_sw = 0 ;
tx_high_A = 0 ;
tx_high_B = 0 ;
break ;
’C ’ :

tx_sw = 0 ;
tx_high_C = 0 ;
break ;
’D’ :

tx_sw = 0 ;
tx_high_A = 0 ;
tx_high_C = 0 ;
break ;
’E ’ :

tx_sw = 0 ;
tx_high_B = 0 ;
tx_high_C = 0 ;
break ;
’F ’ :

tx_sw = 0 ;
tx_high_A = 0 ;
tx_high_B = 0 ;
tx_high_C = 0 ;

95

break ;

d e f a u l t :

break ;

}

}

void s e t _ r x ( )
{

s w i t c h ( r x _ a n t )
{

c a s e

c a s e

c a s e

c a s e

c a s e

’ 0 ’ :
p o r t 0 1 = 0 ;
break ;
’ 1 ’ :
p o r t 0 2 = 0 ;
rx_low_A = 0 ;
break ;
’ 2 ’ :
p o r t 0 3 = 0 ;
rx_low_B = 0 ;
break ;
’ 3 ’ :
p o r t 0 4 = 0 ;
rx_low_A = 0 ;
rx_low_B = 0 ;
break ;
’ 4 ’ :

96

c a s e

c a s e

c a s e

c a s e

c a s e

p o r t 0 5 = 0 ;
rx_low_C = 0 ;
break ;
’ 5 ’ :
p o r t 0 6 = 0 ;
rx_low_A = 0 ;
rx_low_C = 0 ;
break ;
’ 6 ’ :
p o r t 0 7 = 0 ;
rx_low_B = 0 ;
rx_low_C = 0 ;
break ;
’ 7 ’ :
p o r t 0 8 = 0 ;
rx_low_A = 0 ;
rx_low_B = 0 ;
rx_low_C = 0 ;
break ;
’ 8 ’ :
p o r t 0 9 = 0 ;
rx_sw = 0 ;
break ;
’ 9 ’ :
p o r t 1 0 = 0 ;
rx_sw = 0 ;
rx_high_A = 0 ;

97

c a s e

c a s e

c a s e

c a s e

c a s e

break ;
’A’ :
p o r t 1 1 = 0 ;
rx_sw = 0 ;
rx_high_B = 0 ;
break ;
’B ’ :
p o r t 1 2 = 0 ;
rx_sw = 0 ;
rx_high_A = 0 ;
rx_high_B = 0 ;
break ;
’C ’ :
p o r t 1 3 = 0 ;
rx_sw = 0 ;
rx_high_C = 0 ;
break ;
’D’ :
p o r t 1 4 = 0 ;
rx_sw = 0 ;
rx_high_A = 0 ;
rx_high_C = 0 ;
break ;
’E ’ :
p o r t 1 5 = 0 ;
rx_sw = 0 ;
rx_high_B = 0 ;

98

c a s e

rx_high_C = 0 ;
break ;
’F ’ :
p o r t 1 6 = 0 ;
rx_sw = 0 ;
rx_high_A = 0 ;
rx_high_B = 0 ;
rx_high_C = 0 ;
break ;

d e f a u l t :

break ;

}

}

void r e s e t _ s w i t c h e s ( )
{

tx_sw = 1 ;
rx_sw = 1 ;

tx_low_A = 1 ;
tx_low_B = 1 ;
tx_low_C = 1 ;

tx_high_A = 1 ;
tx_high_B = 1 ;

99

tx_high_C = 1 ;

rx_low_A = 1 ;
rx_low_B = 1 ;
rx_low_C = 1 ;

rx_high_A = 1 ;
rx_high_B = 1 ;
rx_high_C = 1 ;

p o r t 0 1 = 1 ;
p o r t 0 2 = 1 ;
p o r t 0 3 = 1 ;

p o r t 0 4 = 1 ;
p o r t 0 5 = 1 ;
p o r t 0 6 = 1 ;
p o r t 0 7 = 1 ;
p o r t 0 8 = 1 ;
p o r t 0 9 = 1 ;
p o r t 1 0 = 1 ;
p o r t 1 1 = 1 ;
p o r t 1 2 = 1 ;

100

p o r t 1 3 = 1 ;
p o r t 1 4 = 1 ;
p o r t 1 5 = 1 ;
p o r t 1 6 = 1 ;

}

101

APPENDIX D

MATLAB CODE TO PLACE ANTENNA LOCATIONS

a l l
a l l

c l o s e
c l e a r
c l c

c i r c l e _ d = . 5 0 0 ;

p a i r ends

%d i s t a n c e between o p p o s i t e

c i r c l e _ r = c i r c l e _ d / 2 ;
antenna_n =16;

%r a d i u s
%number of a n t e n n a s

a n t e n n a _ l o c a t i o n = z e r o s ( antenna_n , 2 ) ;

f o r

i _ a n t e n n a =0: antenna_n −1
a n t e n n a _ l o c a t i o n ( i _ a n t e n n a +1 ,1) = c i r c l e _ r ∗ cos ( p i

/2+2∗ p i ∗ i _ a n t e n n a / antenna_n ) ;

a n t e n n a _ l o c a t i o n ( i _ a n t e n n a +1 ,2) = c i r c l e _ r ∗ s i n ( p i

/2+2∗ p i ∗ i _ a n t e n n a / antenna_n ) ;

end

save v a r s / a n t e n n a _ l o c a t i o n . mat a n t e n n a _ l o c a t i o n

102

APPENDIX E

MATLAB CODE TO READ IN DATA

a l l

c l o s e
c l c

% s p a r a m e t e r m a t r a c i e s

a r e m a t e r i a l ( freq , rx , t x )

where S_21 i s

t x =1 rx =2

%%

empty_zeros = z e r o s ( 2 0 0 1 , 1 6 , 1 6 ) ;
b l o c k _ z e r o s = z e r o s ( 2 0 0 1 , 1 6 , 1 6 ) ;

f o r

i _ r x =1:16
f o r

i _ t x =1:16
i f

i _ t x ~= i _ r x
name= ’ v a r s / empty / sample_%d_%d . csv ’ ;
s t r = s p r i n t f ( name , i _ t x , i _ r x ) ;
d a t a = c s v r e a d ( s t r ) ;
empty_zeros ( 4 0 1 : 2 0 0 1 , i_rx , i _ t x ) = d a t a

( : , 2 ) +1 j ∗ d a t a ( : , 3 ) ;

name= ’ v a r s / PEC / sample_%d_%d . csv ’ ;

103

s t r = s p r i n t f ( name , i _ t x , i _ r x ) ;
d a t a = c s v r e a d ( s t r ) ;
b l o c k _ z e r o s ( 4 0 1 : 2 0 0 1 , i_rx , i _ t x ) = d a t a

( : , 2 ) +1 j ∗ d a t a ( : , 3 ) ;

e l s e
end

end
d i s p ( i _ r x )

end

%%

e m p t y _ f u l l = z e r o s ( 4 0 0 1 , 1 6 , 1 6 ) ;
e m p t y _ f u l l ( 2 0 0 1 : 4 0 0 1 , : , : ) = empty_zeros ;
b l o c k _ f u l l = z e r o s ( 4 0 0 1 , 1 6 , 1 6 ) ;
b l o c k _ f u l l ( 2 0 0 1 : 4 0 0 1 , : , : ) = b l o c k _ z e r o s ;

window= z e r o s ( 4 0 0 1 , 1 ) ;
window ( 2 0 0 1 : 4 0 0 1 ) =ones ( 1 , 2 0 0 1 ) ;

f o r

i _ f r e q =1:2000
i _ r x =1:16
f o r
f o r

i _ t x =1:16
e m p t y _ f u l l (2001− i _ f r e q , i_rx , i _ t x ) = c o n j (

e m p t y _ f u l l (2000+ i _ f r e q , i_rx , i _ t x ) ) ;

b l o c k _ f u l l (2001− i _ f r e q , i_rx , i _ t x ) = c o n j (

104

b l o c k _ f u l l (2000+ i _ f r e q , i_rx , i _ t x ) ) ;

window(2001− i _ f r e q ) =window (2000+ i _ f r e q ) ;

end

end

end

empty_time= z e r o s ( 4 0 0 1 , 1 6 , 1 6 ) ;
empty_TR_full = z e r o s ( 4 0 0 1 , 1 6 , 1 6 ) ;
empty_TR_pulse= z e r o s ( 2 5 0 , 1 6 , 1 6 ) ;

b l o c k _ t i m e = z e r o s ( 4 0 0 1 , 1 6 , 1 6 ) ;
b l o c k _ T R _ f u l l = z e r o s ( 4 0 0 1 , 1 6 , 1 6 ) ;
block_TR_pulse = z e r o s ( 2 5 0 , 1 6 , 1 6 ) ;

f o r

i _ r x =1:16
f o r

i _ t x =1:16
empty_time ( : , i_rx , i _ t x ) = i f f t s h i f t ( i f f t (
e m p t y _ f u l l ( : , i_rx , i _ t x ) . ∗ window ( : ) , ’
symmetric ’ ) ) ;

empty_TR_full ( : , i_rx , i _ t x ) = f l i p ( empty_time

( : , i_rx , i _ t x ) ) ;

empty_TR_pulse ( : , i_rx , i _ t x ) = empty_TR_full

( 1 45 0 : 16 9 9 , i_rx , i _ t x ) ;

105

b l o c k _ t i m e ( : , i_rx , i _ t x ) = i f f t s h i f t ( i f f t (
b l o c k _ f u l l ( : , i_rx , i _ t x ) . ∗ window ( : ) , ’
symmetric ’ ) ) ;

b l o c k _ T R _ f u l l ( : , i_rx , i _ t x ) = f l i p ( b l o c k _ t i m e

( : , i_rx , i _ t x ) ) ;

block_TR_pulse ( : , i_rx , i _ t x ) = b l o c k _ T R _ f u l l

( 1 45 0 : 16 9 9 , i_rx , i _ t x ) ;

end

end

save v a r s / empty_TR . mat empty_TR_pulse
save v a r s / block_TR . mat block_TR_pulse

f i g u r e ;

f o r

i _ r x =1:16
p l o t ( l i n s p a c e ( 0 , 4 9 9 , 2 5 0 ) / 1 0 , ( f l i p ( empty_TR_pulse

( : , i_rx , 1 ) ) ) + i _ r x ∗ . 0 2 , ’ LineWidth ’ , 1 . 2 5 )

hold on

end
x l a b e l ( ’ Time ( ns ) ’ )
s e t ( gca , ’ F o n t S i z e ’ , 2 4 )

f i g u r e ;

106

f o r
%

i _ r x =1:16

p l o t ( abs ( block_TR_pulse ( : , i_rx , 8 ) ) + i _ r x ∗ . 0 1 )

p l o t ( l i n s p a c e ( 0 , 4 9 9 , 2 5 0 ) / 1 0 , ( f l i p ( block_TR_pulse

( : , i_rx , 1 ) ) ) + i _ r x ∗ . 0 2 , ’ LineWidth ’ , 1 . 2 5 )

hold on

end
x l a b e l ( ’ Time ( ns ) ’ )
s e t ( gca , ’ F o n t S i z e ’ , 2 4 )

f i g u r e ;

f o r
%

i _ r x =1:16

p l o t ( abs ( block_TR_pulse ( : , i_rx , 8 ) )−abs (

empty_TR_pulse ( : , i_rx , 8 ) ) + i _ r x ∗ . 0 1 )

p l o t ( l i n s p a c e ( 0 , 4 9 9 , 2 5 0 ) / 1 0 , ( f l i p ( block_TR_pulse

( : , i_rx , 1 ) )− f l i p ( empty_TR_pulse ( : , i_rx , 1 ) ) ) +
i _ r x ∗ . 0 2 , ’ LineWidth ’ , 1 . 2 5 )

hold on

end
x l a b e l ( ’ Time ( ns ) ’ )
s e t ( gca , ’ F o n t S i z e ’ , 2 4 )

107

MATLAB CODE TO BACK PROPAGATE THE MEASURED DATA

APPENDIX F

a l l
a l l

c l o s e
c l e a r
c l c

%% Load d a t a and l o c a t i o n s
% l o a d t i m e _ m a t r i x . mat

l o a d v a r s / a n t e n n a _ l o c a t i o n . mat

l o a d v a r s / empty_TR . mat
l o a d v a r s / block_TR . mat

t i c

%% Medium P a r a m e t e r s
c=3e8 ;
f_c =3e9 ;

f r e q u e n c y

mu0=4∗ p i ∗1e −7;
e p s i l o n 0 =8.85 e −12;
e p s i l o n _ r =1;

108

%speed of
%c e n t e r

l i g h t

%mu naught
%e p s i l o n naught

%% FDTD Mesh P a r a m e t e r s
kk =4;
d e l t a _ p = ( ( c / f_c ) / 1 0 ) / kk ;

based on c e n t e r

f r e q u e n c y

%s c a l i n g f a c t o r

%g r i d s i z e

d e l t a _ t = ( ( d e l t a _ p ) / ( c∗ s q r t ( 2 ) ) ) ;

%sampling time

t i m e s t e p s = f l o o r ( ( 1 / 1 0 e9 ) / ( d e l t a _ t ) ∗250) ;

t s _ i f f t = ( 0 : 2 4 9 ) ∗1/10 e9 ;
t s _ f d t d = ( 0 : t i m e s t e p s −1)∗ d e l t a _ t ;

i n p u t = z e r o s ( t i m e s t e p s , 1 6 , 1 6 ) ;

i _ r x =1:16
f o r

f o r

%

i _ t x =1:16
i n p u t ( : , i_rx , i _ t x ) = i n t e r p 1 ( t s _ i f f t

,

block_TR_pulse ( : , i_rx , i _ t x )−
empty_TR_pulse ( : , i_rx , i _ t x ) , t s _ f d t d ) ;
i n p u t ( : , i_rx , i _ t x ) = i n t e r p 1 ( t s _ i f f t

,

empty_TR_pulse ( : , i_rx , i _ t x ) , t s _ f d t d ) ;

end

end

% t i m e s t e p s =8485;

109

%% S p a t i a l Grid
xlim = . 6 0 0 ; %.330
ylim = . 6 0 0 ;

x r e f =−xlim : d e l t a _ p : xlim ;
y r e f =−ylim : d e l t a _ p : ylim ;

i f mod ( l e n g t h ( x r e f ) , 2 ) ~=1

xlim=xlim+ d e l t a _ p ;
x r e f ( l e n g t h ( x r e f ) +1)=xlim ;

end

i f mod ( l e n g t h ( y r e f ) , 2 ) ~=1

ylim=ylim+ d e l t a _ p ;
y r e f ( l e n g t h ( y r e f ) +1)=ylim ;

end

xn= l e n g t h ( x r e f ) ;
yn= l e n g t h ( y r e f ) ;

%% F i n d i n g p o s i t i o n s of each a n t e n n a

antenna_n =16;

a n t e n n a _ g r i d _ x = z e r o s ( antenna_n , 1 ) ;

110

a n t e n n a _ g r i d _ y = z e r o s ( antenna_n , 1 ) ;
antenna_x = z e r o s ( antenna_n , 1 ) ;
antenna_y = z e r o s ( antenna_n , 1 ) ;

f o r

im =1: antenna_n
[ ~ , a n t e n n a _ g r i d _ x ( im ) ]= min ( abs ( xref −

a n t e n n a _ l o c a t i o n ( im , 1 ) ) ) ;

[ ~ , a n t e n n a _ g r i d _ y ( im ) ]= min ( abs ( xref −

a n t e n n a _ l o c a t i o n ( im , 2 ) ) ) ;

antenna_x ( im ) = a n t e n n a _ g r i d _ x ( im ) ∗ d e l t a _ p −xlim ;
antenna_y ( im ) = a n t e n n a _ g r i d _ y ( im ) ∗ d e l t a _ p −ylim ;

end

%% FDTD i n i t i a l i z a t i o n
sumador= z e r o s ( yn , xn ) ;
Ez_new2D= z e r o s ( yn , xn ) ;
Ez_old2D= z e r o s ( yn , xn ) ;
Hx_new2D= z e r o s ( yn −1 , xn ) ;
Hx_old2D= z e r o s ( yn −1 , xn ) ;
Hxm_new2D= z e r o s ( 2 , xn ) ;
Hxm_old2D= z e r o s ( 2 , xn ) ;
Hy_new2D= z e r o s ( yn , xn −1) ;
Hy_old2D= z e r o s ( yn , xn −1) ;
Hym_new2D= z e r o s ( yn , 2 ) ;
Hym_old2D= z e r o s ( yn , 2 ) ;
er_m=ones ( yn , xn ) ;

111

sigma_m= z e r o s ( yn , xn ) ;
i j =1;

%% C o e f f i c i e n t s
coef1 = ( ( c∗ d e l t a _ t )− d e l t a _ p ) / ( ( c∗ d e l t a _ t ) + d e l t a _ p ) ;

f o r Maxwell ’ s E q u a t i o n s

%%C o e f i c i e n t of MUR boundary c o n d i t i o n

coef2 =( d e l t a _ t / ( mu0∗ d e l t a _ p ) ) ;

%

%C o e f i c i e n t of

c o r r e c t i o n eq .

f o r H f i e l d

coef3 =( d e l t a _ t . / ( e p s i l o n 0 ∗er_m∗ d e l t a _ p ) ) ;
c o e f i n c =( d e l t a _ t . / ( e p s i l o n 0 ∗er_m ) ) ;
c o e f f 3 =(1−sigma_m . ∗ d e l t a _ t . / ( 2 ∗ er_m ) ) . / ( 1 + sigma_m . ∗

d e l t a _ t . / ( 2 ∗ er_m ) ) ;

c o e f f 4 = 1 . / ( 1 + sigma_m . ∗ d e l t a _ t . / ( 2 ∗ er_m ) ) . ∗ ( d e l t a _ t

. / ( e p s i l o n 0 ∗er_m . ∗ d e l t a _ p ) ) ;

A= [ ] ;
%% FDTD S i m u l a t i o n s
f o r

i _ a n t e n n a =1: antenna_n

c l c
i _ a n t e n n a
t o c

sumador= z e r o s ( yn , xn ) ;
Ez_new2D= z e r o s ( yn , xn ) ;
Ez_old2D= z e r o s ( yn , xn ) ;

112

Hx_new2D= z e r o s ( yn −1 , xn ) ;
Hx_old2D= z e r o s ( yn −1 , xn ) ;
Hxm_new2D= z e r o s ( 2 , xn ) ;
Hxm_old2D= z e r o s ( 2 , xn ) ;
Hy_new2D= z e r o s ( yn , xn −1) ;
Hy_old2D= z e r o s ( yn , xn −1) ;
Hym_new2D= z e r o s ( yn , 2 ) ;
Hym_old2D= z e r o s ( yn , 2 ) ;
er_m=ones ( yn , xn ) ;
sigma_m= z e r o s ( yn , xn ) ;
i j =1;
f o r n =1: t i m e s t e p s

%

t _ r e f ( n ) =n∗ d e l t a _ t ;

%% Updating H F i e l d s
Hy_new2D ( ( 1 : ( yn −1) ) , ( 1 : ( xn −1) ) ) =Hy_old2D ( ( 1 : ( yn

−1) ) , ( 1 : ( xn −1) ) ) + coef2 ∗( Ez_new2D ( ( 1 : ( yn −1) )
, ( 2 : ( xn ) ) )−Ez_new2D ( ( 1 : ( yn −1) ) , ( 1 : ( xn −1) ) ) ) ;
%%Update of

t h e Hy f i e l d s no b o u n d a r i e s

Hx_new2D ( ( 1 : ( yn −1) ) , ( 1 : ( xn −1) ) ) =Hx_old2D ( ( 1 : ( yn

−1) ) , ( 1 : ( xn −1) ) )−coef2 ∗( Ez_new2D ( ( 2 : ( yn ) )
, ( 1 : ( xn −1) ) )−Ez_new2D ( ( 1 : ( yn −1) ) , ( 1 : ( xn −1) ) ) )
; %%Update of

t h e Hx f i e l d s no b o u n d a r i e s

Hym_new2D ( ( 1 : yn ) , 1 ) =Hy_old2D ( ( 1 : yn ) , 1 ) + coef1 ∗(

Hy_new2D ( ( 1 : yn ) , 1 )−Hym_old2D ( ( 1 : yn ) , 1 ) ) ;

113

%%Hy ABCs f o r

l e f t

and r i g h t

Hym_new2D ( ( 1 : yn ) , 2 ) =Hy_old2D ( ( 1 : yn ) , xn −1)+ coef1
∗( Hy_new2D ( ( 1 : yn ) , xn −1)−Hym_old2D ( ( 1 : yn ) , 2 ) ) ;

%%Hy ABCs f o r

top and bottom

Hxm_new2D ( 1 , ( 1 : xn ) ) =Hx_old2D ( 1 , ( 1 : xn ) ) + coef1 ∗(

Hx_new2D ( 1 , ( 1 : xn ) )−Hxm_old2D ( 1 , ( 1 : xn ) ) ) ;

%%Hx ABCs f o r

l e f t

and r i g h t

Hxm_new2D ( 2 , ( 1 : xn ) ) =Hx_old2D ( yn − 1 , ( 1 : xn ) ) + coef1
∗( Hx_new2D ( yn − 1 , ( 1 : xn ) )−Hxm_old2D ( 2 , ( 1 : xn ) ) ) ;

%%Hx ABCs f o r

top and bottom

%%Old f i e l d s become

found a t each i t e r a t i o n

t h e

f i e l d s

Hy_old2D=Hy_new2D ;
j u s t
Hx_old2D=Hx_new2D ;
Hym_old2D=Hym_new2D ;
Hxm_old2D=Hxm_new2D ;

%% Updating E f i e l d s

found u s i n g one of

l ==1 && j ~=1 && j ~=yn

% e l s e i f
% The E f i e l d i s
l =1;
j j = 2 : ( yn −1) ;
Ez_new2D ( j j , l ) =Ez_old2D ( j j , l ) + coef3 ( j j , l ) . ∗ (
Hy_new2D ( j j , l )−Hym_new2D ( j j , 1 ) )−coef3 ( j j , l )
. ∗ ( Hx_new2D ( j j , l )−Hx_new2D ( j j −1 , l ) ) ;

t h e ABCs

114

%e l s e i f
% The E f i e l d i s

l ==xn && j ~=1 && j ~=yn

found u s i n g one of

t h e boundary

c o n d i t i o n s

l =xn ;
j j = 2 : ( yn −1) ;
Ez_new2D ( j j , l ) =Ez_old2D ( j j , l ) + coef3 ( j j , l ) . ∗ (

Hym_new2D ( j j , 2 )−Hy_new2D ( j j , l −1) )−coef3 ( j j , l )
. ∗ ( Hx_new2D ( j j , l )−Hx_new2D ( j j −1 , l ) ) ;

%e l s e i f
% The E f i e l d i s

j ==1 && l ~=1 && l ~=xn

found u s i n g t h e ABC f o r

t h e

p l a c e s around t h e b o u n d a r i e s

j =1;
l l = 2 : ( xn −1) ;
Ez_new2D ( j , l l ) =Ez_old2D ( j , l l ) + coef3 ( j , l l ) . ∗ (

Hy_new2D ( j , l l )−Hy_new2D ( j , l l −1) )−coef3 ( j , l l )
. ∗ ( Hx_new2D ( j , l l )−Hxm_new2D ( 1 , l l ) ) ;

% e l s e i f
% The E f i e l d i s

j ==yn && l ~=1 && l ~=xn

found u s i n g t h e ABC f o r

t h e

p l a c e s around t h e b o u n d a r i e s

j =yn ;
l l = 2 : ( xn −1) ;
Ez_new2D ( j , l l ) =Ez_old2D ( j , l l ) + coef3 ( j , l l ) . ∗ (

Hy_new2D ( j , l l )−Hy_new2D ( j , l l −1) )−coef3 ( j , l l )
. ∗ ( Hxm_new2D ( 2 , l l )−Hx_new2D ( j −1 , l l ) ) ;

115

%e l s e i f
% The E f i e l d i s

l ==1 && j ==1

found u s i n g t h e ABC f o r

t h e

p l a c e s around t h e b o u n d a r i e s

l =1;
j =1;
Ez_new2D ( j , l ) =Ez_old2D ( j , l ) + coef3 ( j , l ) ∗( Hy_new2D
( j , l )−Hym_new2D ( j , 1 ) )−coef3 ( j , l ) ∗( Hx_new2D ( j ,
l )−Hxm_new2D ( 1 , l ) ) ;

%e l s e i f
% The E f i e l d i s

l ==xn && j ==yn

found u s i n g t h e ABC f o r

t h e

p l a c e s around t h e b o u n d a r i e s

l =xn ;
j =yn ;
Ez_new2D ( j , l ) =Ez_old2D ( j , l ) + coef3 ( j , l ) ∗(

Hym_new2D ( j , 2 )−Hy_new2D ( j , l −1) )−coef3 ( j , l ) ∗(
Hxm_new2D ( 2 , l )−Hx_new2D ( j −1 , l ) ) ;

%e l s e i f
% The E f i e l d i s

l ==1 && j ==yn

found u s i n g t h e ABC f o r

t h e

p l a c e s around t h e b o u n d a r i e s

l =1;
j =yn ;
Ez_new2D ( j , l ) =Ez_old2D ( j , l ) + coef3 ( j , l ) ∗( Hy_new2D

( j , l )−Hym_new2D ( j , 1 ) )−coef3 ( j , l ) ∗(Hxm_new2D

116

( 2 , l )−Hx_new2D ( j −1 , l ) ) ;

%e l s e i f
% The E f i e l d i s

l ==xn && j ==1

found u s i n g t h e ABC f o r

t h e

p l a c e s around t h e b o u n d a r i e s

l =xn ;
j =1;
Ez_new2D ( j , l ) =Ez_old2D ( j , l ) + coef3 ( j , l ) ∗(

Hym_new2D ( j , 2 )−Hy_new2D ( j , l −1) )−coef3 ( j , l ) ∗(
Hx_new2D ( j , l )−Hxm_new2D ( 1 , l ) ) ;

%e l s e
% The e l e c t r i c

t h e o t h e r

f i e l d i s

found f o r

a l l
c o m p u t a t i o n a l domain .

t h a t

i n t h e
l o c a t i o n s

These
a r e not around t h e

p l a c e s
a r e
t h e
b o u n d a r i e s .
j j = 2 : ( yn −1) ;
l l = 2 : ( xn −1) ;
Ez_new2D ( j j , l l ) =Ez_old2D ( j j , l l ) + coef3 ( j j , l l ) . ∗ (
Hy_new2D ( j j , l l )−Hy_new2D ( j j , l l −1) )−coef3 ( j j ,
l l ) . ∗ ( Hx_new2D ( j j , l l )−Hx_new2D ( j j −1 , l l ) ) ;

%% P l a c i n g F i e l d s
i f ( n<= l e n g t h ( i n p u t ) −16)
l l = a n t e n n a _ g r i d _ x ;

a t R e c e i v e r s

l = l e n g t h ( l l ) ;

f o r

i =1: l

117

j = a n t e n n a _ g r i d _ y ;

i f

i ~= i _ a n t e n n a
Ez_new2D ( j ( i ) , l l ( i ) ) =( Ez_old2D ( j ( i ) ,

l l ( i ) ) + coef3 ( j ( i ) , l l ( i ) ) ∗(
Hy_new2D ( j ( i ) , l l ( i ) )−Hy_new2D ( j ( i
) , l l ( i ) −1) )−coef3 ( j ( i ) , l l ( i ) ) ∗(
Hx_new2D ( j ( i ) , l l ( i ) )−Hx_new2D ( j ( i
) −1 , l l ( i ) ) ) )− 5∗ c o e f i n c ( j ( i ) , l l ( i
) ) ∗ i n p u t ( n , i , i _ a n t e n n a ) ;

end

end

end

%% L o c a l i z a t i o n t e c h n i q u e s

% I n t e g r a t e d Energy

Ez_old2D=Ez_new2D ;
i f n >1000
sumador = ( ( Ez_old2D ) .^2∗ d e l t a _ t ) +sumador ;
end
e n e r g i a { i _ a n t e n n a }= sumador ;
% Entropy
e n t 2 ( n ) =(

( sum ( sum ( ( Ez_new2D . ^ 2 ) ) ) ^2) / ( sum ( sum (

Ez_new2D . ^ 4 ) ) ) ) ;

e n t r o p y { i _ a n t e n n a }= e n t 2 ;
p =( Ez_new2D . ^ 2 ) . / ( sum ( sum ( Ez_new2D . ^ 2 ) ) ) ;

==0) =1e −13;

p ( p

118

e n t 3 ( n ) = − sum ( sum ( p . ∗ log ( p ) ) ) / ( max ( max ( p ) ) ) ;

%
%
%

%

i f ( ( mod ( n , 1 0 ) ==0) )

n

end

D i s p l a y I t e r a t i v e E l e c t r i c F i e l d s

i f ( ( mod ( n , 1 0 0 0 ) ==0) )

%
%
%
%
%
%
% %

uu= r e a l ( Ez_old2D ) ;
imagesc ( yref , xref , uu ) ;
colormap ( hsv ( 1 2 8 ) )
hold on
p l o t ( a n t e n n a _ g r i d _ x , a n t e n n a _ g r i d _ y , ’ ow ’ )
p l o t ( antenna_x , antenna_y , ’ ow ’ )

c i r c l e ( [ 0 , 0 ] , r _ t , 1 0 0 0 , ’w− ’) ; % T a r g e t

i t e r a t i o n =n

c o l o r b a r
frame = g e t f r a m e ;

end

i f n >8480

i t e r a t i o n =n

end

%
%

%
%
%

a t

e n t r o p y minima

% % F i e l d s
% i f ( n==3475 | | n ==5038)
A( i j , : , : ) =Ez_new2D ;
%

119

i j = i j +1;

%
% end

end

% End of

i t e r a t i o n s

% %% Focusing Images
f i g u r e ; p l o t ( e n t 2 / max ( e n t 2 ) ) % Entropy p l o t
t i t l e ( ’ Entropy P l o t ’ )
% f i g u r e ;

imagesc ( xref , yref , e n e r g i a { i _ a n t e n n a })%

Energy p l o t

% c o l o r b a r
% colormap j e t
% c a x i s ( [ 0 1e −15])
% hold on
% % p l o t ( source_pos_x , source_pos_y , ’ ok ’ ) ; p l o t (

sensor_x , sensor_y , ’+ k ’ ) ; p l o t ( 0 , 0 , ’ + k ’ ) ; % T a r g e t

f i e l d s

% t i t l e ( ’ Time I n t e g r a t e d Energy ’ )
% P l o t of
% f i e l d 1 ( : , : ) =A ( 1 , : , : ) ;
% f i g u r e ;
% t i t l e ( ’ F i e l d a t Entropy Minima Time ’ )
% hold on
% % p l o t ( source_pos_x , source_pos_y , ’ ok ’ ) ; p l o t (

imagesc ( xref , yref , ( f i e l d 1 ) )

sensor_x , sensor_y , ’+ k ’ ) ; p l o t ( 0 , 0 , ’ + k ’ ) ; % T a r g e t

120

end

%%
energy = z e r o s ( s i z e ( e n e r g i a { i _ a n t e n n a }) ) ;
f o r
%

im =1: antenna_n

energy = energy + e n e r g i a {im } ( : , : ) / max ( max ( e n e r g i a

{im } ( : , : ) ) ) ;

energy = energy + e n e r g i a {im } ( : , : ) ;

end
energy = energy / antenna_n ;
% f i g u r e ;
% p l o t ( e n t 2 / max ( e n t 2 ) )

imagesc ( xref , yref , energy )

f i g u r e ;
hold on
% p l o t ( a n t e n n a _ g r i d _ x , a n t e n n a _ g r i d _ y )
p l o t ( antenna_x , antenna_y , ’ or ’ , ’ MarkerSize ’ , 1 5 )
c o l o r b a r
colormap j e t
a x i s
a x i s
% c a x i s ( [ 0 . 9 e −16]) ;
c a x i s ( [ 0 100e −22])

e q u a l
([ −.35 . 3 5 −.35 . 3 5 ] )

t o c

121

BIBLIOGRAPHY

122

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