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Development of a microwave system for breast tumor detection

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DEVELOPMENT OF A MICROWAVE SYSTEM FOR
BREAST TUMOR DETECTION
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
DEVELOPMENT OF A MICROWAVE SYSTEM FOR BREAST
TUMOR DETECTION
A thesis submitted in partial fulfillment
o f the requirements for the degree o f
Master o f Science Electrical Engineering
By
Mahita Attaluri, B.E. Electronics and Communications
Osmania University, 2003
August 2006
University o f Arkansas
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UMI Number: 1444145
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ABSTRACT
Microwave imaging is evolving as an alternative for the prevalent ionizing, evasive and
expensive methods for breast tumor detection. This method exploits the significant
electrical contrast between breast tumors and healthy breast tissues at microwave
frequencies to detect tumors at a curable stage.
A microwave system based on active microwave imaging was developed to measure the
contrast in the magnitude and phase o f the electromagnetic scattering produced by
illumination o f dielectrically equivalent breast model over 3-18 GHz. The developed
system was calibrated using Multiline TRL calibration algorithm based on Network
analyzer calibration. The calibration using the developed algorithm was validated by
comparing it with that o f an integrated TRL algorithm in a commercial Network
Analyzer. Measurements of a 19mm radius glass sphere with a dielectric constant o f five
scaled accordingly with air to represent the malignant and the normal breast tissue
contrast were taken and calibration was applied using the test system.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
This thesis is approved for
recommendation to the
Graduate Council
Thesis Director:
Dr. Fred Barlow III, Ph.D.
Associate Professor, Electrical Engineering
Dr. Aicha Elshabini, Ph.D.
Distinguished Professor, Electrical Engineering
yj
Dr. Victor Wang, Ph.D.
Research Professor, Electrical Engineering
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THESIS DUPLICATION RELEASE
I hereby authorize the University o f Arkansas Libraries to duplicate this thesis when
needed for research and/or scholarship.
'N L U t - T H L L l
(signature o f student)
Refused__________________________________
(signature o f student)
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ACKNOWLEDGEMENTS
This work has been a significant accomplishment in my life and it would not have been
possible without the support of the people I’m to mention.
Most importantly I would like thank my major advisor Dr. Fred Barlow for his technical
as well as moral support from the inception of this work till this moment. For
transforming me into a researcher, for never discouraging me when I set over optimistic
goals, but for his belief in me that inspired me to eventually achieve them.
I’m thankful to my thesis advisors: Dr. Aicha Elshabini; for being a source of inspiration,
encouragement and kindness in professional and personal life and Dr. Wang, for his
support.
I also want to thank my family for their love and their belief in me. My mother for always
being proud of me, irrespective of my achievements; my father, for teaching me the value
of hard work by his own example; my sister for the inspiration she in herself is. I would
like to share this moment with my parents, sister, grandmother, brother-in-law and my
nephew. They rendered me enormous morale and joy during the whole tenure of my
research.
I thank Kiran, for his belief in me, for constantly encouraging me and being there for me
through every high, low and normal day of research.
I would also like to acknowledge my fellow research students: Brian, for all his help and
for all that I learnt from him; Payam Rashidi, Gokul Talapanuri and Iryna Polyakova; for
automating the system and Shruti Pandalraju and Faisal Magableh for their support.
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INDEX
LIST OF FIGURES................................................................................................................ x
LIST OF TABLES...............................................................................................................xiii
CHAPTER 1
1.1 OBJECTIVE.......................................................................................................1
1.2 THESIS OVERVIEW.......................................................................................2
CHAPTER 2
BREAST TUMOR DETECTION
2.1 PREVALENT METHODS OF BREAST TUMOR DETECTION
5
2.1.1 Mammography
2.1.2 Magnetic Resonance Imaging (MRI)
2.2 IDEAL BREAST SCREENING TOOL FEATURES................................... 7
2.3 MICROWAVE IMAGING................................................................................ 8
2.3.1 Passive Microwave Imaging
2.3.2 Hybrid Microwave Imaging
2.3.3 Active Microwave Imaging
CHAPTER 3
EXPERIMENTAL SET-UP
3.1 ASSUMPTIONS................................................................................................. 11
3.2 ELEMENTARY SYSTEM ..............................................................................11
3.3 MEASURING BACKSCATTER...................................................................12
3.4 PHASE MEASUREMENT.............................................................................. 14
3.4.1 Principle o f operation o f a Mixer as a Phase Detector
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3.4.2 Mathematical proof
3.4.3 Simulation o f a commercial mixer
3.5 SIMULATION OF THE TEST SET-UP....................................................... 18
3.5.1 Guidelines fo r Device Specifications
3.5.2 Amplifier Specifications
3.5.3 Circulator Specifications
3.5.4 Mixer Specifications
3.5.5 Coupler Specifications
3.6 SYSTEM IMPLEMENTATION..................................................................... 25
3.6.1 Switching circuitry in test system
3.6.2 Automation o f the experimental set-up
CHAPTER 4
CALIBRATION
4.1 NEED FOR CALIBRATION.......................................................................... 33
4.2 TEST SYSTEM ERRORS............................................................................... 34
4.2.1 Random Errors
4.2.2 Systematic Errors
4.2.3 Drift Errors
4.3 CALIBRATION................................................................................................ 37
4.3.1 Response Calibration
4.3.2 Vector Calibration
4.4 CALIBRATION METHODS.......................................................................... 38
4.4.1 SOLT
4.4.2 TRL
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4.4.3 TRM
4.4.4 Selected Calibration method — Multiline TRL
CHAPTER 5
Multiline TRL ALGORITHM
5.1 ERROR BOX FORMULATION....................................................................42
5.2 ERROR MODELS WITH STANDARDS....................................................46
5.2.1 Reflect
5.2.2 Thru
5.2.3 Linel
5.2.4 Line2
5.3 MATHEMATICAL ANALYSIS.....................................................................54
5.3.1 Determining propagation constant
5.3.2 Determining error coefficients
5.3.3 Extracting the device parameters from the measurements
5.4 VERIFYING CALIBRATION......................................................................... 69
CHAPTER 6
RESULTS
6.1 CONVERTING MEASUREMENTS TO S-PARAMETERS.................. 73
6.1.1 Magnitude Coversion
6.1.2 Phase Extraction
6.2 EXPERIMENTAL RESULTS....................................................................... 77
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CHAPTER 7
FUTURE WORK
7.1 PROPOSED PROCEDURE FOR CALIBRATION AT DIFFERENT
ANGLES AROUND A CIRCULAR A X IS .................................................80
7.2 EFFECT OF SKIN...........................................................................................84
7.3 TIME DOMAIN GATING............................................................................. 87
REFERENCES.......................................................................................................................89
APPENDIX
MATLAB CODE FOR Multiline TRL ALGORITHM.................................................91
ix
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LIST OF FIGURES
Figure 2.1 Electrical properties o f malignant and normal breast tissue at microwave
frequencies................................................................................................................ 8
Figure 2.2 Illumination o f the breast tissue and scattering in Active Microwave
Imaging.....................................................................................................................10
Figure 3.1 Elementary system for measuring average power of scattering from
dielectrically equivalent tumor and tissue..............................................................12
Figure 3.2 C irculator.............................................................................
12
Figure 3.3 Circulator introduced in the elementary system to measure
backscatter................................................................................................................ 13
Figure 3.4
Mixer operation.................................................................................................. 14
Figure 3.5
Upgraded system with backscatter and phase measurement capabilities.... 16
Figure 3.6
Simulation of a commercial mixer using Microwave Office.......................... 17
Figure 3.7
Experimental setup for measuring magnitude and phase o f scattering from
dielectrically equivalent tumor over 3-18 GHz.................................................19
Figure 3.8
Simulation of the Experimental set-up using Microwave Office.................. 21
Figure 3.9
Devices built into the test system...................................................................... 25
Figure 3.10 Log Periodic Antennas illuminating the breast test model.............................26
Figure 3.11 SPDT Switch Configuration............................................................................... 27
Figure 3.12 SP3T Switch Configuration................................................................................ 27
Figure 3.13 Switching Circuitry in the test system................................................................28
Figure 3.14 The Front panel diagram o f the labview program (or vi) that controls the
digital outputs o f PCI 6503 parallel interface card.........................................31
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Figure 3.15
Front panel diagram o f the labview vi controlling the measurement
process................................................................................................................... 32
Figure 4.1 Reference planes o f measurement before and after calibration....................... 33
Figure 4.2 Signal Flow graph in the forward direction........................................................ 35
Figure 4.3 Signal Flow Graph in the reverse direction........................................................ 35
Figure 4.4 Error box model o f a two port network...............................................................36
Figure 5.1
Error box formulation......................................................................................... 43
Figure 5.2
Cascaded network in ABCR param eters........................................................ 44
Figure 5.3
Interpretation of the experimental system as an Error Box........................... 45
Figure 5.4
Reflect Standard..................................................................................................46
Figure 5.5
Signal Flow graph o f Reflect............................................................................ 47
Figure 5.6
Thru Standard..................................................................................................... 48
Figure 5.7
Signal Flow graph for Thru standard............................................................... 48
Figure 5.8
Line 1 standard.................................................................................................... 50
Figure 5.9
Signal flow graph for Line 1 Standard............................................................. 50
Figure 5.10
Line 2 Standard.................................................................................................... 52
Figure 5.11
Signal flow graph for Line 2 Standard............................................................. 52
Figure 5.12
Extracting device parameters............................................................................ 69
Figure 5.13
Snmagnitude in dB when reflect (short) placed at port 1.............................70
Figure 5.14
Snangle in degrees when reflect (short) placed at port 1..............................70
Figure 5.15
S 22 magnitude in dB when reflect (short) placed at port 2.............................71
Figure 5.16
S22 angle in degrees when reflect (short) placed at port 2.............................71
Figure 5.17 S21 magnitude in dB o f the thru standard.......................................................... 72
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Figure 5.18 S 12 magnitude in dB o f the thru standard............................................................72
Figure 6.1
Two port network representing the test system................................................ 75
Figure 6.2
Reference planes o f measurement for magnitude and phase...........................76
Figure 6.3
Magnitude o f scattering from a 19 mm radius glass sphere o f dielectric
constant 5 ............................................................................................................. 78
Figure 6.3
Phase o f scattering from a 19 mm radius glass sphere o f dielectric
constant 5 ..............................................................................................................79
Figure 7.1
Scattering from a sample at various angles around its circular axis............. 80
Figure 7.2
Error box formulation for case 1.......................................................................81
Figure 7.3
Standards defined for case 1...............................................................................81
Figure 7.4
Error box formulation for case 2 .......................................................................82
Figure 7.5 Thru standards with transmitting antenna at position A and F ...................... 83
Figure 7.6
Bounce diagram o f the breast model with a layer o f skin................................85
Figure 7.7
Reflections in time domain...................................................................................86
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LIST OF TABLES
Table 2.1
Permittivity and conductivity o f normal and malignant breast tissue at
6 GHz..........................................................................................................................9
Table 3.1
The IF voltages for given phase shift using the simulation............................... 18
Table 3.2
Calculating phase shift from IF voltages........................................................... 18
Table 3.3
Amplifier specifications.......................................................................................22
Table 3.4
Circulator specifications....................................................................................... 23
Table 3.5
Mixer specifications...............................................................................................24
Table 3.6
Coupler specifications........................................................................................... 24
Table 3.7
SPDT Switch Specifications.................................................................................27
Table 3.8
SP3T Switch Specifications..................................................................................28
Table 3.9
Relay Specifications............................................................................................. 29
Table 3.10 PCI outputs for measurements............................................................................. 30
Table 7.1
Electrical properties o f skin, normal and malignant breast tissue.................. 84
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CHAPTER 1
1.1 OBJECTIVE
The objective of this work was to develop a safe, low-cost, high accuracy microwave
system for breast tumor detection using microwave imaging. The test system was based
on active microwave imaging where the breast is illuminated with microwaves and the
scattering produced is a function o f permittivity, size, shape and location o f tumors in the
breast, if any. The elementary system comprised o f wide band log periodic antennas,
sourced by a sweep generator illuminating the breast test model and the scattering
produced was measured using a spectrum analyzer. The breast model included a glass
sphere with a dielectric constant o f 5 and air with permittivity o f 1 representing the
dielectric contrast o f the malignant and normal breast tissue. In this model, the skin layer
was not considered and homogeneity o f the breast tissue was assumed. The aim was to
design and build a test system over 3-18 GHz to detect the backscatter and the phase of
the electromagnetic waves in addition to the transmitted waves, so as to increase the
accuracy o f the microwave imaging technique. Two circulators sharing the frequency
band of 3-18 GHz were placed in the system to collect the backscatter from the antennas.
A mixer functioned as a phase detector, where the reference LO signal was coupled from
the source and a dc voltage proportional to the phase o f the transmitted and backscattered
signals was collected at its IF port. However, calibration of the system remained the
greatest challenge in this work. For calibrating the test system an algorithm based on
Multiline TRL calibration method, used for Network Analyzer calibration, was
developed. The calibration from this algorithm was verified by comparing it with that of
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an integrated TRL algorithm in a commercial Network Analyzer. Measurements o f 19
mm radius glass sphere, representing the tumor scaled in dimensions with air, and the
standards: thru, reflect and linel and line 2 were taken with the test system over 3-18
GHz. After converting the measurements into a format compatible with the Multiline
TRL algorithm, calibration was applied to extract the actual scattering from the breast
model.
1.2 THESIS OVERVIEW
This work presents the design, validation and calibration of a microwave system for
breast tumor detection using microwave imaging that is spread over the 6 chapters
outlined below.
•
Chapter 2 provides an introduction to breast tumor detection. The prevalent
methods o f breast tumor detection are discussed and compared with the ideal
breast screening tool as stated by U.S Institute o f Medicine (IOM) [1]. Microwave
Imaging is introduced as an alternative breast screening tool for the contemporary
methods. The different microwave imaging methodologies are also presented.
•
Chapter 3 describes the design,
simulation and
implementation o f the
experimental set-up. It presents the elementary system, where wide band antennas
sourced by a sweep generator illuminate a test model with a dielectric contrast as
that o f the malignant and normal breast tissue and the scattering produced is
measured through a spectrum analyzer. It provides the transition from the
elementary system that measures the transmitted scattering to the upgraded one
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that is capable of measuring backscatter and phase of the scattering. The design
developed is backed by simulations and the guidelines for a safe test system.
•
Chapter 4 introduces calibration. It explains the need for calibration while
presenting the test system errors; random errors, systematic errors and drift errors
that affect a microwave measurement process. Since systematic errors are
significant and predictable in nature, they are studied in detail. The various
calibration methods such as SOLT, TRL, LRM and Multiline TRL used for
eliminating systematic errors are addressed.
•
Chapter 5 establishes Multiline TRL as the choice of the calibration method for
calibrating the test system. It translates the test system into a two port network in
terms o f the systematic errors known as the error box formulation. This chapter
provides the mathematical analysis o f the error box model with Multiline TRL
calibration method which is the basis for the calibration algorithm developed
using Matlab. Also, the calibration algorithm developed is verified using the TRL
algorithm integrated in the commercial network analyzer Agilent E8361 A.
•
Chapter 6 gives the conversion o f the magnitude and phase measurements from
the spectrum analyzer and the mixer respectively in the test system into Sparameters, a format compatible with the Multiline TRL algorithm. It also
presents the calibrated data extracted from the raw measurements o f a glass
sphere o f radius 19 mm and dielectric constant o f 5 and the standards: thru,
reflect, line 1 and line 2.
•
Future work is presented in chapter 7 where a procedure for calibration o f
measurements at angles about a circular axis o f a sample is proposed. It also
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explains the effect o f skin using a simplified breast model, including a skin layer,
analyzed by bounce diagram. It also proposes time domain gating as an additional
feature o f the system which can potentially be used for reducing the effect of the
reflections from the skin.
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CHAPTER 2
BREAST TUMOR DETECTION
2.1 PREVALENT METHODS OF BREAST TUMOR DETECTION
To justify the need for an alternative method o f breast tumor detection the prevalent
breast screening methods are discussed.
2.1.1 Mammography
Mammography undoubtedly is the most widely used breast screening tool. A
mammogram is used to map the densities across the breast. It uses density contrast
between the tumor and the breast tissue to detect tumors.
However it suffers from certain disadvantages.
•
Exposure to radiation
•
Evasive
•
Discomfort
Since it makes use o f the density contrast, it can detect only lesions, but not all lesions
are being detected using a mammogram and not all lesions detected are cancerous. This
increases the false-positive rates associated with the mammograms, substantiating that
upto 25% o f the mammograms lead to inconclusive results. Usually an additional
screening tool such as an ultrasound is required to differentiate between tumors and
lesions. Ultrasound on its own is not a substantial breast screening tool.
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Most importantly there is exposure to ionising radiation. The average radiation dose for
single mammogram is about 0.7 mSv. Though this appears as a small amount, it is
equivalent to the background radiation a person would receive in about 4 months.
As far as the comfort levels of the examination are concerned, it is an uncomfortable
rather painful procedure that requires the breast to be compressed between two metal
plates to obtain an X-ray image o f it.
2.1.2 Magnetic Resonance Imaging (MRI)
Another breast screening that is available today is the MRI technique. This method
detects the water rather hydrogen density across the body and in this case breast.
The disadvantages associated with MRI
•
Expensive
•
Powerful magnetic fields
•
Confinement o f patient during examination
Human body is composed o f 70% o f water, thus the concentration of hydrogen is high.
Each Hydrogen atom contains one single proton. Due to the odd number o f protons
present, an effective spin is induced on each H atom. This spin causes a magnetic
moment to be associated with all hydrogen atoms. A powerful external magnetic can be
used to align the magnetic moments o f numerous H atoms into a magnetic vector. By
applying RF pulses at a frequency that is equivalent to the natural resonance frequency of
the hydrogen atom, decay in the spin on the H atom is caused. By measuring the time
taken for the magnetic vector to decay, the concentration o f hydrogen is determined and
tumors are detected owing to their high water density.
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According to the larmour relationship, the magnetic field required to develop resonance
is proportional to the frequency at which the pulses are applied. Since large magnetic
fields are required to align the magnetic moments associated with the numerous H atoms,
the frequency o f operation falls into the RF region. This is a complicated phenomenon
requiring large magnetic fields and RF sources thereby increasing the cost o f the system
greatly. Also the test equipment is large and may cause discomfort (claustrophobia) to the
patients, since the patients must be placed inside a large magnet.
2.2 IDEAL BREAST SCREENING TOOL FEATURES
As stated by IOM - US Institute of Medicine, an ideal breast screening tool should
possess the following features [1],
• has low health risk
•
is
sensitive to tumors
•
detects breast cancer at a curable stage
•
is non-invasive and simple to perform
•
iscost effective and widely available
•
involves minimal discomfort, so the procedure is acceptable to women
•
provides easy to interpret, objective and consistent results
The prevalent breast screening methods discussed do not satisfy most o f the aspects o f an
ideal breast screening tool. So clearly there is a need for an alternative method o f breast
tumor detection. This is where Microwave Imaging comes into picture.
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2.3 MICROWAVE IMAGING
At microwave frequencies as compared to other cancers, breast tumors have significantly
different electrical properties than healthy breast tissue. Microwave Imaging exploits this
electrical contrast to detect breast tumors.
There have been some studies on the electrical properties o f healthy and malignant breast
tissue at microwave frequencies. Below is a plot for the conductivity and dielectric
constant o f malignant and healthy tissues from various sources as summarizes in the
literature [2].
103
— D e b y e c u rv e
c
CO
A
a
_
A
• C h o u d a ry
A S u ro w ie c
“ '‘‘ ■S'1™ 1 O J o in e s
Inr
Q 101
M alignant
t=t» 10°
N orm al
in r ®
0 1 0 -1
O
A
10
107
A A ^A
10 ®
107
1Q10
Frequency (H z)
Figure 2.1 Electrical properties of malignant and normal breast tissue at microwave
frequencies [2]
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Electrical properties have been measured till 3 GHz, and for higher frequencies
extrapolation is made using a first order Debye dispersion.
j a _ e s- e x jcrs
£r
C0$
-
7
— ------------
1+JCOT
(2 . 1 )
CQ§
From the above expression, the electrical properties at single frequency o f 6 GHz have
been determined based on data from the literature [2],
Table 2.1 Permittivity and conductivity o f normal and malignant breast tissue at 6 GHz
At 6 GHz
Normal breast tissue
Malignant breast tissue
er avg
9.8
50
a avg (S/m)
0.4
7
This data reveals that ratio for the dielectric constants o f breast tissue and tumor at 6 GHz
is about 5:1. This proves the significant electrical contrast between the tumor and healthy
breast tissue at microwave frequencies.
There are various imaging methods for detecting the electrical contrast between the tumor
and tissue. They can be categorized into three types; passive, active and hybrid methods
[3],
2.3.1 Passive Microwave Imaging
The passive methods employ radiometers to measure the temperature across the breast.
Since tumors tend to exhibit greater temperatures as compared to breast tissue, the higher
temperatures measured by the radiometers indicate presence of tunors. It is advantageous
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to use microwaves over IR as tumors are more translucent to microwaves as compared to
IR.
2.3.2 Hybrid Microwave Imaging
Hybrid methods in general and microwave induced acoustic-imaging in particular
illuminate the breast with microwaves. The higher conductivity o f the malignant breast
tissue causes energy to be absorbed in tumors resulting in increased heating o f the breast
in these areas. The increased heating causes the tumors to expand while generating
pressure waves. These waves can be detected by ultrasound transducers.
2.3.3 Active Microwave Imaging
These methods utilize the inverse scattering problem, which employs several microwave
transmitters to illuminate the breast and measure the scattered fields from the breast at
various positions. From the scattered and transmitted electric fields, the physical
properties such as the shape and the location o f the tumor can be obtained.
east
breast
Figure 2.2 Illumination o f the breast tissue and scattering in Active Microwave Imaging
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CHAPTER 3
EXPERIMENTAL SET-UP
3.1 ASSUM PTIONS
The experimental set-up was based on the following assumptions
i)
effect o f skin was neglected
ii)
breast tissue was considered homogenous
The dielectric contrast o f 5:1 o f the malignant and normal breast tissue was simulated by
a breast test model comprising o f a glass sphere o f dielectric constant o f .5 for tumor and
air with a dielectric constant o f 1 representing the breast tissue.
3.2 ELEM EN TA R Y SYSTEM
The elementary system, developed by Faisal Magableh, consisted o f the test model being
illuminated using a wide band transmitting antenna sourced by the sweep generator
HP8340B and the scattering produced was measured by the spectrum analyzer HP8953E
through a wide band receiving antenna. The scattering caused in all directions by
illuminating the breast test model was measured by placing the receiving antenna at the
respective angle around the circular axis about the model except at 0° and a portion of
angles about it, where the backscatter was being received by the transmitting antenna
itself. Backscatter at the transmitting antenna could not be measured with this system as it
did not include a path from the transmitting antenna to the spectrum analyzer.
11
R e p r o d u c e d w ith p e r m is s io n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
,80°
270°
1’ICTRUM I
N A IY 7LR
Figure 3.1 Elementary system for measuring average power o f scattering from
dielectrically equivalent tumor and tissue.
3.3 MEASURING BACKSCATTER
To measure the backscatter being received at the transmitting antenna, a path was
required from the transmitting antenna to the spectrum analyzer. This was created by
placing a circulator between the transmitting antenna and the spectrum analyzer. A
circulator is a three port device that allows signal flow across adjacent ports but the flow
is unidirectional. That is, for an ideal circulator there is a path for energy flow from port 1
to port 2, port 2 to port 3, port 3 to port 1 but there is no path for the energy to flow from
port 1 to port 3, port 3 to port 2 and port 2 to port 1 [4].
CIRCULATOR
Figure 3.2 Circulator
12
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The S-parameters o f an ideal oscillator are as follows
~Slt
S2l
s l2
Sl3~
s 22
_S3l
S22
$21
Si 3 _
'0
=
0
1 0
0
1"
0
1 0
As can be seen in figure 3.3, port 1 o f the circulator received the incident wave from the
sweep generator and directed it to port 2 o f the circulator. The transmitting antenna
placed at port 2 o f the circulator illuminated the tumor and received the backscatter. As
there is very little leakage o f the signal from port 2 to port 1, most o f the backscatter from
the antenna at port 2 o f the circulator flowed to port 3 o f the circulator that was connected
to a switch. The spectrum analyzer through the switch alternately measured the
transmitted and backscattered signals from the receiving and the transmitting antennas
respectively.
270“
l
SWITCH
Figure 3.3 Circulator introduced in the elementary system to measure backscatter
13
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3.4 PHASE MEASUREMENT
Phase of transmitted and backscattered signals is o f significance when the magnitude of
scattering from different sized tumors or tumors at different locations is the same. In such
cases the phase o f the scattered signals can be used to differentiate between the tumors.
3.4.1 Principle o f operation o f a Mixer as a Phase Detector
To detect the phase o f a signal, the basic principle o f a mixer functioning as a phase
detector when the RF and the LO are at the same frequency was utilized.
When the RF and the LO are at the same frequency, the IF is a dc signal with a voltage
proportional to the cosine o f the phase shift between the RF and the LO signals.
Mixer
son
LO
RF
v 2sin(wt + 0 2)
Figure 3.4 Mixer operation
3.4.2 Mathematical proof
The phase detector can be mathematically described as follows [4],
Many non-linear devices can serve as mixers. The nonlinear characteristic o f a device can
be expressed as
I = K (V + v, + v2)”
(3.1)
where n is chosen as 2 for simplicity
/ is the current and V is the bias voltage
14
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v, = Vt sin(®,t + 0 ,) (voltage of signal 1 )
( 3 .2 )
v2 = V2 sin(co2t + 02) (voltage o f signal 2)
(3.3)
Substituting in the equation, we have
/ = K (V + Vy sin(&>^ + 0y)+ V2 sin(ry2/ + 02) ) 2 —» (3.4)
I = K \V 2 + Vy sin 2 (ry,f + (9,) + V2 sin2{co2t + #2) + 2VVl sin(n)^ + #,)
+ 2FF 2 sin(ni2/ + 02) + 2VyV2 s i n ^ f +
(3.5)
)sin(&>2f + #2)]
It can be observed that the output contains direct current and a no o f alternating currents.
The IF current includes the terms Vj and V2.
.'. I IF = 2KV xV2 sin(<y,? + 0y)sin(<»2f + # , ) —> (3-6)
I IF = KV,V2[cos((&>2 -a>2)t + {02 - 0 , ) ) - cos((<y2 + co2)t + {02 + 0y))]
(3.7)
For application as a phase detector (a>2 = coi=co), equation 3.7 becomes
I IF = KVyV2[cos(02 - 6 >,) - c o s ( 2 a>+ (02 + #,))]
(3.8)
On passing the IF signal through a low pass filter the 2co frequency harmonic can be
eliminated. The cut-off frequency o f the LPF is carefully chosen to be smaller than 2co of
frequency o f operation.
The IF voltage after the LPF would be a dc signal as follows
1 ,f ~
k V\V2 c o s ( # 2 - #1)
(3.9)
v,r = ' - f = v y i c o s t e - 0 , )
(3.10)
Therefore the output o f mixer, when the RF and LO ports are at the same frequency is a
dc output proportional to the cosine o f the phase shift between the two signals.
15
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(jI-M-RAIO
r™
1
S O U PLEK
-I
N>
incident
r 1--------------- 1 i n u u eru
»-------------------- -C IR C U L A T O R , *, A v r F N N A
. .
JJ
t r a n s m itte d
Q
A A A A M A ^
^
^
^
p
L ________ T back scatter TU M O R
S W IT C H
M IX H
S P L C 1 K tiV I
A \A I Y /l-K
Figure 3.5 Upgraded system with backscatter and phase measurement capabilities
As can be seen in the figure 3.5, the backscattered or the transmitted signal received at
the switch went to the RF port o f the mixer and coupler was feeding a reference signal
from the sweep generator to the LO o f the mixer. The dc coupled IF port through a low
pass filter was connected to a voltmeter to make the voltage measurements.
3.4.3 Simulation o f a commercial mixer
The functionality of a commercial mixer as a phase detector was simulated using
Microwave Office. When the LO and the RP signals were set to the same frequency, the
output at the IF port through a low pass filter was a dc signal.
16
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
MIXER
ID=MX1
GCONV—6dB
P1DB_IN=10dBm
IP3_IN=20dBm
PLO=10 dBm
NF=OdB
ACVS
ID=V1
Mag=1 V
Ang=90 Deg
RF IN
LPFE
ID=LPFE1
N=10
FP=0.1 GHz
AP=0.1 dB
AS=20dB
0.28V
X I F 0 U 1]}
I t;vi
0.28 V
0.218 V
LO
PORT
P=1
Z=50 Ohm
0.5V
ACVS
ID=V2
Mag=1V
Ang=30 Deg
t[oy]
Figure 3.6 Simulation o f a commercial mixer using Microwave Office
On simulation it was noticed that though the IF voltage is proportional to the cosine o f
the phase shift in the signals, it had a dc offset and could be expressed as
K « , = o + bc os(0 2 - 0 , )
(3.11)
To determine a and b, two sets o f measurements were taken for Vou, with the phase o f the
RF and LO signals set to known values.
17
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Table 3.1 The IF voltages for given phase shift using the simulation
e2
cos( 02-0,)
V
r out
900
30°
0.5
0.28
60°
30u
0 .8 6 6
0.421
From two known phase values and for V] = V 2 = IV, a and b were calculated as a = 0.094
and b = 0.38. Now that a and b coefficients were known, they were applied directly to the
next measurements. From the Vout o f the mixer IF port the phase shift was calculated.
Table 3.2 Calculating phase shift from IF voltages
V
r out
Cos( 92-9,)
( 9 2 ~9\)
0.361
0.707
45°
0.336
0.643
50°
With the simulations proving the mixer’s function as a phase detector, this procedure was
to determine the phase o f the waves in the system.
3.5 SIMULATION OF THE TEST SET-UP
The test system to detect the backscatter and phase in addition to the transmitted signals
was conceived in theory. Due to the cost and design limitations on wide band microwave
devices operating over 3-18 GHz, the system design was dictated by the available
devices. So the system was further complicated to one in figure 3.7. A circulator was
replaced by two o f them, sharing the frequency band o f 3 -18 GHz. Also Amplifiers were
18
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included in the system to improve the SNR o f the measurements. This increased the
number o f switches in the system.
EXPERIM ENTAL SET-U P FOR B R E A ST
TUMOR D E TE C TIO N
1 ■ GHz
1
'j
in c id e n t
/
\
8 1 8 G H z]
AM P
(
TENIURT v v
,J '' ,J —
tr a n s m i tt e d
A /W V i/V *
bad< s c a t t e r T U M O R
1
E
’ ■.
" /
S P Z T S W IT C H
S P 3 T S W IT C H
S P Z T S W IT C H
RF
Figure 3.7 Experimental setup for measuring magnitude and phase o f scattering from
dielectrically equivalent tumor over 3-18 GHz
3.5.1 Guidelines fo r Device Specifications
The choice o f the devices was made taking into consideration various parameters o f each
device and that o f its neighbors in the system. The factors were
•
Minimum detectable input power for each device
•
Maximum power output power o f each device such that it does not harm the
device adjacent to it
•
Maximum power handling capability o f the devices
•
Power dissipation in the devices
•
Heat sinking for devices determined from the power dissipation levels and the
operating temperatures o f the devices
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The following guidelines were used to set some o f the specifications on the devices.
Guideline 1
For the system to be called safe, the maximum power that can be incident on the breast
was set based on the IEEE standards for Maximum permissible exposure (MPE) [5].
Maximum permissible exposure (MPE) for 3-300GHz is a
Power density = 1 0 mW/cm 2
Calculating the maximum safe level values for the incident power on the breast
Area o f the breast = n r 2 = 78.5 cm 2 (Assuming breast as circle o f radius 5 cm)
Power incident on the breast = Power density x Area o f exposure
= 0.785 W
This is the maximum level o f incident power that can be used for the test set up. From the
maximum operating power level for the test system, the gain and 1dB compression points
o f the amplifiers, being the only devices controlling the power levels o f the test system,
were estimated excluding losses in the system.
Guideline 2
The RCS is the Radar cross-section scattering, which is determined from the expression
4T\r 2E 2
RCS = ~ - ^ - ^ 2 rioP;
For a min scattering case o f a depth d = 5 cm, sr = 10.0 - j 1.2 and radius o f the tumor a =
5 mm, RCS= 1.7 x 10‘4 /m 2 based on [6 ].
Thus the minimum scattering is calculated as Esmin = 0.97 V/m
This value set the minimum detectable power level on RF port o f the mixer.
In order to set the other specifications o f the devices and to estimate the losses in the
system, each branch, 3-6 GHz and 6-18 GHz, o f the experimental system was simulated
20
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
in Microwave Office with the properties o f commercial devices that were likely to be
used in building the system. The tumor was simulated by placing an attenuator along
with a phase shifter in the system.
CiC
’IMEI
■ NCOH ■ • ■STATE-1
•asH M
m m .
-4+^
Figure 3.8 Simulation o f the experimental set-up using Microwave Office
3.5.2 Amplifier Specifications
Two amplifiers were required to share the frequency band o f 3-18 GHz. The
specifications of the amplifiers that suited the application are given in table 3.3
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Table 3.3 Amplifier Specifications
Frequency
3-6 GHz
6-18 GHz
PldB
35 dBm min.
35 dBm min.
Gain
20 dB min.
20 dB min.
VSWR
2 :1
2 :1
DC Bias
+12 V @ 2.5 A typ.
+12 V @ 6.5 A typ
Connectors
SMA-F
SMA-F
Since the power from the sweep generator over 3-18 GHz
was limited to 20mW, power
amplifiers were chosen for the test system. As seen from
the specifications, the power
amplifiers require large dc bias power levels while the maximum output ac power o f the
amplifiers is only a small fraction. The bias power that is not converted into ac output is
lost mostly due to dissipation. If there is no proper path for power dissipation away from
the amplifier, it causes heating o f the device, beyond its operating temperatures, that can
deteriorate the performance o f the amplifier or even destroy it. Also the amplifier
parameters such as gain, matching and PldB can drift greatly from the desired
specifications. Proper heat sinking allows the amplifiers to function properly. The power
dissipated is estimated for the given amplifiers.
DC bias power for amplifier 3-6 GHz = 12V x 2.5A = 30 W
RF power max. for amplifier 3-6 GHz = 3.2 W
Power dissipated for amplifier 3-6 GHz = 26.8 W
DC power for amplifier 6-18 GHz = 12V x 6.5A = 78 W
22
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
RF power max. for amplifier 6-18 GHz = 3.2W
Power dissipated for amplifier 6-18 GHz = 74.8 W
Total power dissipated by the amplifiers = 101.6 W
The raise in temperature o f the amplifiers for the power dissipated was calculated using
their thermal resistances and was appropriately heat sinked.
3.5.3 Circulator Specifications
Due to the wide frequency range o f 3-18 GHz, two circulators were required. The
frequency band o f operation on the circulators was selected to be the same as that on the
amplifiers from 3-6 GHz and from 6-18 GHz. This selection simplified the switching
operation during the measurement process. The circulators in the design were placed next
to the amplifiers, thus the maximum power from the amplifier o f 3.2 W (35 dBm)
became the typical power that can be handled by the circulators. Also, the other
specifications o f the circulators were chosen accordingly for the given application.
Table 3.4 Circulator specifications
Frequency
3-6 GHz
6-18 GHz
Isolation
21 dB typ./19 dB min.
15 dB typ./14 dB min.
Insertion loss
0.35 dB typ./0.40 dB min.
0.9 dB typ./l dB min.
VSWR
1.25 typ./l .30 max.
1.45 ty p ./l.50 dB max.
Max Input Power
20 W avg.
20 W avg.
23
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3.5.4 Mixer Specifications
The mixer with the specifications as shown in table 3.5 was chosen for the system. The
conversion loss o f the mixer is a measure o f the power loss due the conversion from RF
to IF. So a minimum available conversion loss was selected in order to maximize phase
resolution. A low pass filter with a pass band o f DC-80 MHz was placed after the mixer
thus eliminating the harmonics from the mixer in the voltage measurements.
Table 3.5 Mixer Specifications
RF
3-18 GHz
LO
3-18 GHz
IF
DC-3 GHz
LO Power
+7 dBm nom.
Conversion loss
6.5 dB
L-R Isolation
35 dB typ./22 dB min.
L-I Isolation
30 dB ty p ./l 6 dB min.
Input PldB
+1 dBm typ.
Input TOIP
+11 dBm typ.
3.5.5 Coupler Specifications
The function o f the coupler was to couple a portion o f the power from the sweep
generator to the mixer for the reference signal and direct the rest o f the signal to the
amplifiers. Since the nominal power available from the sweep generator over 3-18 GHz
was 15dBm and the min power detectable by the mixer was 7 dBm, the coupling was set
to
6
dB.
24
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Table 3.6 Coupler Specifications
Frequency
3-18 GHz
Coupling
6
dB
Insertion loss
0.9 dB
Directivity
12-15 dB
VSWR
1.35
Max Input Power
25 W avg.
3.6 SYSTEM IMPLEMENTATION
In building the system, devices with the specifications that satisfied the guidelines and the
simulation o f the test system were used. The devices as shown in figure 3.9 were
mounted on metal plates attached to the heat sink holding the amplifiers. This helped the
all the devices to be on the same level reducing the coaxial cable lengths and also
providing fair heat sinking for the low power devices.
A m plifiers
c o u p le r
c irc u la to rs
SPD T s w i t c h e
S P3T sw itc h
r e la y
Figure 3.9 Devices built into the test system
25
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G la s s s p h e r e o f dielectric c o n s t a n t 5
j
w
\
/
Lo g p e r i o d i c a n t e n n a s ( 2 GHz-1 SGHz)
Figure 3.10 Log periodic antennas illuminating the breast test model
3.6.1 Switching circuitry in the test system
Since two circulators and amplifiers were used over 3-6 GHz and 6-18 GHz and also the
same mixer was used to detect the voltage proportional to the transmitted as well as the
backscattered waves, three SPDT switches and one SP3T switch were required. The
SPDT switch between the coupler and the amplifiers directed the signals from the source
over 3-6 GHz and 6-18 GHZ to the appropriate amplifiers. The SPDT switch between the
circulators and the transmitting antenna alternately transmitted the amplified signals over
3-6 GHz and 6-18 GHz frequency bands. The transmitted signal at the receiving antenna
and the backscatter from the transmitting antenna collected at the 3-6 GHz and the 6-18
GHz circulators were measured by the spectrum analyzer and the mixer for magnitude
and phase respectively through two switches. An SP3T switch switched the transmitted,
3-6 GHz backscatter or 6-18 GHz backscatter to the SPDT switch which had alternate
26
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paths for the signal to the mixer and the spectrum analyzer. The operation and the
specifications of the SPDT and SP3T switches indicated in the figures 3.12 and 3.13
Single Pole Double Throw switch
Bias: 12 V, 140 mA
C connected to
(+)
NC
Logic 0
NO
Logic 1
Figure 3.11 SPDT switch configuration
Table 3.7 SPDT Switch Specifications
Frequency (GHz)
DC-2.0
2.0-4.0
4.0-8.0
8.0-12.4
12.4-18
VSWR
1 .2 0
1.25
1.30
1.40
1.50
Insertion loss (dB)
0 .2 0
0.25
0.30
0.40
0.50
Isolation (dB)
90
80
80
70
60
Single Pole 3 Throw switch
(C-) ground terminal
C-2
Bias: 15 V, 350 mA
3
1 Input connected to
2
3
0
0
1
0
0
1
'
1
’
Figure 3.12 SP3T switch configuration
27
R e p r o d u c e d w ith p e r m issio n o f th e c o p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Table 3.8 SP3T Switch Specifications
Frequency (GHz)
DC-2.0
2.0-4.0
4.0-8.0
8.0-12.4
12.4-18
VSWR
1 .2 0
1.25
1.30
1.40
1.50
Insertion loss (dB)
0 .2 0
0.25
0.30
0.40
0.50
Isolation (dB)
90
80
80
70
60
In order to automate the switching, a NI PCI 6503 parallel interface digital I/O card was
installed into the PC and the outputs from the card were controlled using Labview. The
available output from the PCI card was 5 V, whereas the bias voltage on the SPDT and
SP3T switches were 12 V and 15 V respectively.
PCI 6503
S
;
SPDT 1
SPDT 2
SPDT 3
Figure 3.13 Switching circuitry in the test system
28
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Since there were two different bias voltages required and 5 V was too low to drive the
switches, two power supplies at the bias voltage levels o f 12 V and 15 V were used and
the supply to the switches was controlled using the relays. These relays were controlled
by the outputs from the PCI card which in turn was controlled using Labview. The relay
needed to have a control voltage o f 5 V and an output voltage o f at least 15 V to operate
with the PCI card as well as the microwave switches at the input and the output ends
respectively. Thus a relay with the following specifications was chosen.
Table 3.9 Relay Specifications
Input control voltage
3.5-32 Vdc
Max. Turn on voltage
3.5 Vdc
Min. Turn off voltage
1.0 Vdc
Typical input current
2.2 mA @ 5 Vdc
Output operating voltage
3-60 Vdc
Load current range
.02-3 Adc
As shown in the figure 3.14, relay 1 controlled the switch SPDT 1 that was used to
measure alternately the phase or magnitude o f the transmitted, 3-6 GHz and 6-18 GHz
backscattered waves through the mixer and the spectrum analyzer respectively. Relay 3
and relay 4 were used to control the SP3T switch that collected the transmitted, 3-6 GHz
backscatter and 6-18 GHz backscatter signals and fed them to SPDT 1. Relay 2
controlled switches SPDT 2 and SPDT 3 that alternately switched between the 3-6 GHz
and 6-18 GHz paths from the coupler to the antenna. Four output lines from the PCI card
29
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
: 47, 45, 43 and 41 controlled the relays. The power supplies at 12 V and 15 V were
connected in parallel with relay inputs and its outputs were connected to the bias pins on
the SPDT and SP3T switches. The control voltage for the relay was supplied by the
output lines from the PCI card. When the control voltage on the relay was high the path
between its input and its output was closed, thus the appropriate bias power was supplied
to the microwave switches. In cases where the switches needed a low on their bias lines,
the control voltage on the relays was cut off. The table 3.10 indicates the states of PCI
output lines to collect magnitude and phase measurements o f the transmitted and the
backscatterd signals over frequency bands 3-6 GHz and 6-18 GHz.
Table 3.10 PCI outputs for measurements
Measurements
47
45
43
41
3-6 GHz Transmitted wave Magnitude
1
1
0
0
3-6 GHz Transmitted wave Phase
1
0
0
0
3-6 GHz Backscattered wave Magnitude
1
1
1
0
3-6 GHz Backscattered wave Phase
1
0
1
0
6-18 GHz Transmitted wave magnitude
0
1
0
0
6-18 GHz Transmitted wave Phase
0
0
0
0
6-18 GHz Backscattered wave Magnitude
0
1
1
0
6-18 GHz Backscattered wave Phase
0
0
1
0
The output lines 47, 45, 43 and 41 from the PCI 6503 card were switched from high to
low based on the table 3.10 to make the measurements using a vi (program) in Labview
developed by Iryna Polyakova as shown in the figure 3.14.
30
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Figure 3.14 The Front panel diagram o f the Labview program (or vi) that controls the
digital outputs o f PCI 6503 parallel interface card
3.6.2 Automation o f the experimental set-up
With the automated switching circuitry and the experimental setup built, the next step
was to further automate the entire measurement process. The automated process was
controlled by a program in Labview, developed by Payam Rashidi and Gokul Talanapuri,
where each measurement device such as the spectrum analyzer, sweep generator,
voltmeter were controlled by their respective drivers in the labview vi. As can be seen in
the figure 3.11, the vi controlled the characteristics o f the generated signal such as the
initial frequency, frequency sweep, the frequency span and the power level. It also
recorded the readings on the spectrum analyzer and the voltmeter and saved the data in
excel files. The figure 3.15 is the front panel diagram of the labview vi controlling the
measurement process.
31
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
I—
Spectrum Analyzer
nmntiiAnlmw Hill ]
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-b7 0-i
■i
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Intial frequency (GHz)
; g oo
rnkmmmm
Wat
'**
sw eep t i m e d '
Sweep Generator VISA session
o 19
5.9E+9
6.0E+9
6.
i
,j
Sweep Generator c
6,lE+<
c -J
Transmitting Antenna
0
Spectrum Analyzer VISA sesston
Spi!Ctrum A u |yzer dup n s A
W" - " - '
& •-* _ *
M /m
. : . , ____________
* **“
v f
‘'
2000
Re
enna +
9000
6000
f°
8000
10000
11520
__J
Frequency (GHz)
f 67 21
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5000
7500
10000
12500
15W0
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17500
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20000
23000
' . '
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iiilll
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■
Receiving Antenna-
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u
t
RKewngAnge (degrees)
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..............
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M n O T i^ M ^ ^ p w
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m easurem ent funrtlan (DC V)
resolution (absolute) (100 OOF 6)
; DC Volts
Jioojnou'
I
range (1 00E+0)
1E H
Figure 3.15 Front panel diagram o f the labview vi controlling the measurement process
The system built and automated, the next step was to calibrate the system.
32
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER 4
CALIBRATION
4.1 NEED FOR CALIBRATION
The reference plane for magnitude and phase measurements o f the transmitted and the
backscattered waves was at the
spectrum analyzer and the voltmeter.
These
measurements included not only the transmitted and backscatter information from the
tumor and tissue but also the impedance mismatches, attenuation, phase shift effects o f
tissue-antenna interface, various devices and cables along the test system. Also noise with
sources internal and external to the test system would affect the measurements.
In other words the test system was adding a degree o f error to the measurements. In order
to get to the measurements that are o f interest, calibration is used to de-embed the error in
any test equipment by measuring standards.
B
-
R eferen ce p la n e s a fte r
c a lib ratio n ,
,
incident /
r"
transm itted
_______
oBmmaam
atTvvw J H
si
back scatt^<TUMOp|
SPZT SWITCH
IB III
wSmtttm
SPZT SWITCH
R e fe re n c e p la n e s b e fo re
c a lib ra tio n
SP3T SWITCH
SPZT SW ITCH
Figure 4.1 Reference planes o f measurement before and after calibration
33
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
4.2 TEST SYSTEM ERRORS
Errors in a microwave test system can be categorized into three types [6 ]:
4.2.1 Random errors: As the name suggests, these errors are unpredictable in nature.
They cannot be eliminated using calibration methods that are to be discussed. There are
two kinds o f random errors that can affect the system. Averaging is the best way to
reduce these errors.
Equipment noise: These errors are characterized by the components in the test system.
The noise floor o f the system depends on that o f the active components present in the test
equipment. Also factors such as poor connector and switch repeatability add to sources of
the random errors. The effect o f these errors can be reduced by increasing the source
power, thus increasing the SNR o f the measurements.
Interference: These errors are introduced by the external environment. It could be from a
nearby microwave source operating at a frequency as that o f the test system or from
reflecting bodies in the testing area. However, if the interference is constant, for example
if the position o f the reflector is fixed, it can be considered as a systematic error.
4.2.2 Systematic errors: If an error occurs repeatedly, it is called a systematic error. A
systematic error is usually due to impedance mismatches, attenuation and phase shift
associated with different media and various components that make up the microwave test
system. Cross talk between the transmission and reflection paths or any constant
interference in the measurement path is also a systematic error. Since these errors are
significant and are also predictable as compared to the random errors, they can be
quantified using calibration techniques. A Signal Flow graph can be used to depict these
34
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
systematic errors [7]. A two port network can be represented by using separate signal
flow graphs for forward and reverse measurements. Six error terms; directivity, source
match, reflection tracking, load match, transmission tracking and isolation, each
indicating a specific source o f error constitutes a signal flow graph. Since there are 12
error terms for a 2 -port network, it is called a
12
term error model.
E xf
1
I
E tf
S 2i
-0
E«
22
2
-LF
*12
DUT
Figure 4.2 Signal Flow graph in the forward direction
E df
•' Directivity
E tf
■' Transmission tracking
E sf
•' Source match
E r f : Reflection tracking
E xf
•' Isolation
E lf
•' Load match
DUX
S 21
-0
s
S 1,
0
22
-s
r
-o
S 12
E xr
Figure 4.3 Signal Flow Graph in the reverse direction
E dr
: Directivity
E Tr
: Transmission tracking
E sr
: Source match
E rr
: Reflection tracking
E xr
: Isolation
E Lr
: Load match
35
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
The 12 term error model is used for test systems that have separate paths for forward and
reverse measurements. For example a 4-sampler VNA uses separate samplers to detect
the incident wave at ports
1
and
2
respectively in addition to the two samplers measuring
the reflected waves at each port. Thus the measurements with source at port one depend
on the sampler at port one and that at port 2 depend on the sampler at port 2. In a 3sampler VNA, a single sampler is switched between ports 1 and 2 to measure the incident
wave. The present test system also uses the same path for forward and reverse
measurements. In these cases the errors in the forward and reverse signal flow graphs of
the
12
term error model overlap and the model can be simplified to an 8 -term error model
often referred to as error box formulation.
DUT
Si 2 B
S 21
Si 1A
••22A
Sn
S 22
s 22 B
S, 1 B
P ort 2
Port 1
Si;
'1 2 A
■221 B
Figure 4.4 Error box model o f a two port network
Relation between 12-term and 8 -term error models
S21aSi2A= E r f
S21bSi2B= E r r
S iia = E d f
S u b -E d r
S22A= E sf ,E lr
S22B= E sr ,E lf
S21a S i 2B= E tf
S21b S i 2A= E tr
The errors are treated as individual error networks at port 1 and port 2 cascaded with
DUT. By mounting standards in place o f the DUT, eight calibration measurements can be
36
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
made to obtain eight equations to solve for eight error terms. Isolation is omitted in error
box formulation. However, isolation can be included by applying correction to measured
data before performing computations using the
8
term error model.
4.2.3 D rift errors: These errors are due to change in the performance o f the test system
with temperature. Since the assumption is that the systematic errors remain constant over
the calibration o f the system and measurement o f the DUT, the deviation o f the test
system performance with temperature is neglected. However for the change in the system
parameters to be negligible, calibration must be performed frequently depending on the
rate at which the drift errors affect the calibration. These errors can be minimized by
providing a test environment with stable ambient temperature. While the active
components such as amplifiers, switches, mixers etc. in the test equipment may be
specified to operate over a wider temperature range, it is advisable to operate them at
room temperature +25° C, ± 5 °C by appropriate heat-sinking thus reducing the need to
recalibrate the system frequently.
4.3 CALIBRATION
The standards determine the accuracy o f calibration and represent it as the quality o f test
system. A system can be calibrated either for magnitude or both magnitude and phase.
Depending on the calibration required there are two types o f calibration.
4.3.1 Response Calibration'. This is the simpler o f the two types o f calibration. In this
calibration not all the errors are corrected. Scalar calibration that is magnitude calibration
37
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
is obtained by normalization o f the DUT measurements with measurements taken using a
standard. For example, transmission calibration o f the DUT is derived from thru standard
normalization; where as for the reflection calibration a known reflect standard
(open/short) can be used. This method only accounts for transmission tracking and
reflection tracking errors in the 12 error model. This method can be used for test systems
that measure only the magnitude and not the phase.
4.3.2 Vector Calibration: For magnitude as well as phase calibration, scalar calibration is
not sufficient. In systems measuring magnitude and phase, it is more appropriate to use
vector calibration that solves for all the 12 errors in a network. However it requires more
standards and measurements from these standards as compared to scalar calibration.
Since the test system needs magnitude and phase calibrations, vector calibration will be
studied in detail. From now on, in this document, vector calibration will be referred to as
calibration.
4.4 CALIBRATION METHODS
Various calibration methods are available depending on the combination o f standards
being used. Some o f the popular methods are as follows [2],
4.4.1 SO LT: It can be used for both 12 term error formulation and error box formulation.
It uses known standards Short, Open, Load and Thru. An ideal Short standard is a perfect
reflector that gives a reflection coefficient o f 1 Z 180°, where as the reflection coefficient
38
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
o f an ideal open would be 1 Z0° and that o f ideal load is zero. A thru is ideally a zero
delay between the two ports, that is the two ports are connected together.
For SOLT calibrations, six reflection measurements are taken by placing short, open and
load standards at each port. The thru provides a transmission and reflection
measurements at each port. Thus 10 measurements are obtained, and if considering the 12
term error formulation, the other two measurements for isolation can be obtained by
terminating either o f the ports and obtaining the transmission measurements.
The quality o f calibration using the SOLT method greatly depends on the accuracy with
which the standards are defined. This is because the properties o f the standards such as
the capacitance o f the open and the inductance o f the short are directly used for error
calculation and thus the deviation in defining the standards introduces the deviation in the
error coefficients and thereby in the calibration.
4.4.2 TRL: Though SOLT is a traditional calibration method, at higher frequencies (>
5 GHz), it becomes difficult to fabricate accurate standards, and a small variation in
defining them results in greater deviation in calibrations as compared to lower frequency
measurements. Thus, it would be convenient to use a method that is less sensitive to the
quality of the standards used for the calibration.
TRL is one such method, while solving for all the 12 error or
8
error model formulations
as SOLT, uses standards that are unknown. TRL calibration uses three standards: Thru,
Reflect and Line. Thru, the same as that defined for SOLT, is a zero delay. A reflect is an
open or a short with high reflection coefficient, but, the reflection coefficient need not be
known. A line is a 90° delay at the center frequency o f the band o f calibration as
39
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
compared to the thru standard. The propagation constant need not be defined and is
calculated from the thru and line measurements. This line is designed such that it varies
from a minimum o f
20
° to a maximum o f 160° electrically over the frequency band o f
calibration. Significant contrast between the thru and the line is required to calculate the
propagation constant o f the line, thus minimum and maximum electrical lengths for the
line are set as a X I 2 (180°) and zero line is electrically equivalent to thru. However, the
electrical delay o f the line needs to be defined accurately as a function o f frequency. Two
measurements are made using the reflect, four measurements each using the Thru and the
Line at both the ports. Using the error box formulation these 10 measurements can be
used to quantify the
8
errors and the other two measurements can be used to determine
the propagation constant o f the line and the reflection coefficient o f the reflect standard.
Inspite of its advantages as compared to SOLT, TRL is limited to higher frequencies.
This is because wavelength increases with the decrease in frequency and thus it requires
very large lines at lower frequencies.
4.4.3 TRM: This is similar to TRL method but just uses a match standard in place o f the
line standard. By introducing the match or rather a load standard, this calibration method
becomes less attractive than TRL for this application. As compared to a line standard
defined for TRL method o f calibration, a match standard is difficult to fabricate in this
case. This method can be used if a suitable termination can be provided for the frequency
band o f calibration. If the thru standard is a non zero line, the method is LRM which uses
the same computations as that used for the TRM.
40
R e p r o d u c e d with p e r m is s io n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
4.4.4 Selected Calibration m ethod - M ultiline TRL: After the discussion o f the various
calibration methods, TRL is an obvious choice for calibrating the system. This decision
was made owing to the 18 GHz higher end o f operation and also due to the absence of
load as one o f the standards for calibration. For the present test system, the antennas are
the microwave source o f energy. A load would require to be an absorbing material for the
electromagnetic waves over the radiation area o f the antennas and prove to be expensive.
However, for wide frequency applications similar to the present o f 3-18 GHz, TRL is not
sufficient as a single line standard is useful only a narrow band o f frequencies. An
enhanced TRL method called the Multiline TRL that uses multiple lines to cover the
required frequency band o f calibration has been adopted.
Multiline TRL has advantages over the TRL method. As this method uses multiple,
redundant line standards, the additional information provided by these standards
overdetermines the error coefficients and thus can be used to minimize the effects o f
random errors that cannot be eliminated through calibration by using linearized error
analysis. In this way, both the accuracy and bandwidth o f the method are improved.
41
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER 5
Multiline TRL ALGORITHM
Multiline TRL has been studied in detail and has been developed for calibration o f the
test system based on the literature [8 ]-[20]. Also the calibration using the developed
algorithm has been compared with that o f the integrated algorithm in the Agilent Vector
Network Analyzer E8361A before applying it to the test system.
The standards were defined: the thru is a zero delay, the Reflect used was a metal plate
made of Aluminum and the line standards followed the same rule as the Line in TRL.
The line cannot have a bandwidth greater than 8:1, that is the when the maximum
electrical length exceeds 160° or the minimum electrical length falls below
20
° and a new
line is required. Also another guideline for Multiline TRL is that the ratio o f the
minimum and the maximum frequencies usable for a line is the same for as that for all the
line standards. For the given frequency range o f 3-18 GHz, two lines were sufficient.
Following these guidelines the lines were formulated as 18.75 mm and 7.5 mm for air as
the tissue medium to cover 3-6 GHz and 6-18 GHz respectively. For a different tissue
medium, these lengths can be accordingly scaled with the phase velocity or rather the
dielectric constant of the media.
5.1 ERROR BOX FORMULATION
With the standards defined, the next step was to define the error model and quantify the
error coefficients. Error box formulation is appropriate for the system as the forward and
reverse paths in the network are identical. However, a 12 term error formulation can be
derived from the error box model.
42
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
DUT
>2 1 A
'12B
S 2i
-o
Si 1 A
22A
sn
?22
J2 2 B
11B
Port 1
Port 2
'12
'12A
S t,
Figure 5.1 Error box formulation
The errors on the either side of the DUT in this model can be treated as individual error
networks as shown above. Since the model now represents a cascaded network with the
error networks and the DUT, it is further convenient to convert the S-parameters into
ABCR parameters. With ABCR parameters, the cascaded network is mathematically
equivalent to multiplying the individual networks. The relation between the ABCR
parameters and the S parameters is as below.
1
R =•
(5.1)
J 2\A
A — {SUS 22
S l2S 2i)
(5.2)
B = S 22
(5.3)
c = -s,
(5.4)
The figure 5.2 is the error box in ABCR parameters. The error box on the right side or at
port
2
is represented by an overbar ass the error coefficients in the reverse signal flow
graph that are in the direction opposite to those in the forward signal flow graph
respectively.
43
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1
o
o
r
7 =
1
1
0
(5.5)
1
0
M = F¥"
Am B'n
Cm 1
DUT
Figure 5.2 Cascaded network in ABCR parameters
where
i
A
i
^2 \A
^\2 A ^2 \a )
( 6 ' l M ‘S '2 2 /(
_ c
SnA
(5.6)
1
22 A
>0
i
X - R x' A
A
Y -R-,
A
l
1
e
° i 2e
0^1 \B ^22B ~
_
2 B ^ 2 IB )
o
22B
(5.7)
1
*^1 IB
The error coefficients o f the test system on either side o f the DUT are defined in the
following way. The transmission path along the sweep generator and the antenna at port 1
through the device or rather tissue and tumor into the antenna at port
2
till the spectrum
analyzer corresponds to the transmission tracking error coefficient in the forward
direction. The mismatch between the antenna at port 1 and the tissue (air) is the source
m atch error. The reflection path through the antenna at port 1 through the circulators to
the spectrum analyzer is the reflection tracking error coefficient. The directivity error is
due to the mismatch at the sweep generator and the coupler. Similarly the error
44
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
coefficients are defined when the sweep generator is switched to port
2
, to make
transmission and reflection measurements at port 2 .
SWEEP
EENERAT
COUPLE
incident /
tfansmitted
0 A/yvW
Iback scatteVTUMOl
5P2T SWITCH
SP2T SWITCH
sP3TswrrcH
SP2T SWITCH
^RF
SPECTRUM !
lo
ANALYZER
Breast tissue and tumor
I
li
ERROR BOX. ^
*-■
ifM 1
Figure 5.3 Interpretation o f the experimental system as an Error Box
45
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Thus there are
8
error coefficients that need to be determined. One measurement using
the Reflect standards and two measurements each using the thru, line 1 and line 2 can be
made at each o f the ports. There are a total o f 14 measurements being made. The
propagation constant o f the line and the error coefficients can be determined separately
due to the number o f lines being used. Since two lines are being used, two sets o f values
for the error coefficients and the propagation constant are available. Guass Markov
theorem was used for the linear error analysis in calculation of the error coefficients. This
will be discussed in detail in section 5.3.1.
5.2 ERROR MODELS WITH STANDARDS
5.2.1 Reflect
An Aluminum plate serves as the reflect standard at both ports 1 and 2.
Figure 5.4 Reflect Standard
46
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
w—
r
'2 1 B
S 21A
R 11
Port 1 ft
rL
r S 11A J1S22A1
_
11B
'21B
>12A
Port 2
Figure 5.5 Signal Flow graph o f Reflect
Therefore the measurements taken before calibration will contain the error coefficients.
Thus the measurements taken using the standards will have the errors embedded in them
as follows. Below is the mathematical interpretation o f the signal flow graphs
^11 —^ 11A +
D
21A12A
rL
(5.8)
i - s 22Ar L
y212? v1 2 5 r L
_ r
.
22 _ °llfi + i _ c r
1
22B
(5.9)
L
where Rn and R 22 are the Si 1 and S22 o f the Reflect standard when placed at port 1 and
port
2
respectively
Since the assumption is that reflect is the same at ports 1 and 2, the same surface o f the
aluminum plate is used as reflect at ports 1 and 2. Since the simulation o f the breast with
tumor model assumes the plane wave propagation, the device is to be place in the far
field region of the antennas. For the antenna with dimension larger than the wavelengths
at operating frequency, far field region can be defined after 2 A. A t the lower end of
frequency 3 GHz, 2 A corresponds to 20 cm. Thus the distance between the antennas and
the device needs to be a maximum o f 20 cm. The Reflect is placed at 20 cm each from
the antenna thus defining the reference planes at
20
cm from each port.
47
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
5.2.2 Thru
The thru standard is a zero delay. Since the reference planes are set at 20 cm from each
antenna, for the thru to be a zero delay the distance between the two antennas is 40 cm
(2 0
cm + 2 0 cm).
Figure 5.6 Thru Standard
The error model with the thru standard is as follows
Port 1
----- w
Q-
1
’ S 11A
1
>12B
-----------w .... ....... ------- r
>kS 22A
------ o
i I S 11B
’r S 22B
S 22B
1^-------------S l 2A
e-
...... i
1
321B
P ort 2
S 21A
Figure 5.7 Signal Flow graph for Thru standard
The raw measurements taken from the test system with thru standard in the place o f the
DUT can be interpreted as follows.
5'tt
' 2 \ A kJ\ 2 AjSy
^ J 22B
Tu ~ S\\ a + 1 _ C C
1
22A
(5.10)
22B
48
R e p r o d u c e d w ith p e r m issio n o f th e c o p y rig h t o w n e r . F u rth er r ep ro d u ctio n p ro h ib ited w ith o u t p e r m issio n .
7 1 _
1 22 ~
^ 2 \B ^ \2 B ^2 2 A
~
o
C
i
J I1S ^
rp
921,4 9122?
-I01
21
_
t>
_
■* 12
10
(5.11)
,2>22S‘->22A
1
(5.12)
11 - 9 22A 922B
*^215*^12,4
1 -9
(5.13)
9
k J 2 2 B iJ>22A
1
where T n, T 22 , T2i and T ]2 are the Sn, S2 2 , S21 and Si 2 parameters o f line
1
standard
It is simpler to use ABCR parameters for the thru or the line models as they form
cascaded networks. The ABCR parameters o f a cascaded system can be found by
multiplying the ABCR parameters o f individual networks.
Mathematically
M ‘ = X T 'Y
(5.14)
where X and Y are the same as defined earlier, M 1 is raw ABCR matrix o f the test system
with Thru and T' is the thru standard ABCR parameters.
1
1
b*.
b*.
= RT
r
1
CT
1
21
s*~
' - ( O L - « )
- s 2T2
1
_
(5.15)
For an ideal thru
0
1
1
0
1
0
0
1
(5.16)
r
=
(5.17)
Since in this test system the propagation medium is air and the thru or lines are defined in
free space (air), the assumption o f ideal standard can be made for the thru and line
standards. However, if the standard is non-ideal, the assumption o f it being ideal
49
R e p r o d u c e d w ith p e r m is s io n o f th e c o p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
introduces an error in the calculation o f the error coefficients. As mentioned earlier these
errors can be reduced by the linear error analysis which will be discussed later.
5.2.3 Line 1
The line 1 is 18.75 mm defined for air and covers frequencies from 3-6 GHz. Thus the
distance between the antennas is that between them for the thru standard in addition to
the 18.75 mm line.
Figure 5.8 Line 1 standard
The signal flow graph for the line is as follows.
e-r'i
S21A
S-I2 B
O
-0
lrS n A
Port 1
i kS 22A
S22B1r
S11B
O ------ 4 -------------- ----------- 4 ---------- --------- 4 ---------S 12 A
e ‘y11
S21B
-0
Port 2
Figure 5.9 Signal flow graph for line 1 Standard
50
R e p r o d u c e d w ith p e r m issio n o f th e c o p y rig h t o w n e r . F u rth er r ep ro d u ctio n p ro h ib ited w ith o u t p e r m issio n .
The measurements made using line 1 contain the error coefficients as follows
T1
_
'L 1 11
_
O
,
c
,
+
p-w >
(5.18)
S 21BS 22Ae
-2/1,
(5.19)
-2 A
<r21.4 c 122? p-?1,
_
—
T 1
22B
*^21 B^\2B^22Ae
1
21
c
+ '1 _ e
1
^22A c 22B
n
_ C
J-‘l 22 ~
T1
c
21A12A
1 S 12AS 22Be
(5.20)
2 / 1,
e p-rh
21 g
12 ^ _________
o
0
--2yl]
^22B^22Ae
e
_
LVn - ,
1
(5.21)
where L i n , LI 22 , L12i and LI 12 are the Su, S22 , S21 and S 12 parameters o f line 1 standard
The cascaded ABCR parameters are defined as follows
M jX = X T JXY
(5.22)
where M jX is the raw ABCR matrix o f the test system with line 1 and T jX is the ABCR
parameters o f the line
T jX = R L1
X ,
B u
-C *
1
1
1
oil
21
/ oil oil
o ilo ilx
W ll °22 _ 12 21 )
oil
_ 22
oil
^11
(5.23)
1
An ideal line has the following S-parameters and ABCR parameters.
-/<,
(5.24)
M =
r*
0
0
(5.25)
e^1
51
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Though linel has been defined for 3-18 GHz, the measurements are taken for the entire
range of frequency o f operation. This is to provide additional information for the error
analysis.
5.2.4 Line2
The line 2 is 7.5 mm defined in air for frequencies from 6-18 GHz. The distance between
the antennas is the thru distance in addition to 7.5 mm line.
Figure 5.10 Line 2 Standard
The signal flow graph for the line 2 is as follows.
O
1A
Port 1
-v !2
S 21A
—►
—
0>12A
S-I2 B
►—
22A
1B
°22B
*-712
-0
S 21B
-0
Port 2
Figure 5.11 Signal flow graph for Line 2 Standard
52
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
The relationship between the raw measurements and the error coefficients is as follows
-
r»
o
rr
- 2yi2
, ^ 2 1 /f°1 2 /* °2 2 g g
T
7
V
l
* u - * u a + Yi Z ov22 /4o22fie
c
(5-26)
-2yl2
0
0
_~2yi2
1 S22BS21Ae
C O O
T 0
—
O
115
22
S22AS22Bs
1
f
9
o
_
12 ~
, ° 2 1 fl° 1 2 g ° 2 2 /ie
21 /?
o
„
(5.28)
-2yi2
12.4 ______
r,
S 22BS 22Ae
1
(5.27)
(5.29)
-2X/2
where L 2 n, L 2 2 2 , L 2 2i and L 2 i2 are the Si i, S22 , S21 and S 12 parameters o f line
2
standard
The cascaded ABCR parameters are as follows
M jl - X T J1Y
(5.30)
where M jl is the raw ABCR matrix of the test system with line 2 and T jl is the ABCR
2
T J1 = R L2
_
r
K>
1
parameters of the line
^L2
1
L2
1
rrZ.2
21
/ oZ.2 QtL2
W ll °22 “
- S
rrZ/2 r i L 2 \
12 ° 2 1 )
“
riL2
^11
1
_
(5.31)
An ideal line 2 has the following S-parameters and ABCR parameters.
e-*2"
0
[s“ ]=
-* 2
(5.32)
Q
-y i2
yv 1 _
0
e
(5.33)
yl2
53
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
It is to be noted that the propagation constant for the line 2 is the same as that o f line 1.
Thus the line standards should be identical as far as the propagation medium is
concerned.
5.3 M A TH EM A TIC A L ANALYSIS
All the standards are defined and the relationship between the measured raw parameters
using the standards and the error coefficients is identified. Now mathematical
computations are made to solve for the unknowns that are the
8
error coefficients, the
propagation constant o f the line standards and the reflection coefficient o f the reflect
standard. These computations are based on known algorithms in literature [8]-[20].
5.3.1 Determining propagation constant
The measured cascade parameters o f the thru and linel standards are
M n =XTn Y
(5.22)
and
, o
1
X
l.
respectively.
I
0
X
o'
1
'1
o
Where T and T ] are assumed ideal as
1
(5.14)
1
M ‘ = X T ‘Y
Eigenvalues and eigenvectors:
For every square matrix G there exists atleast one non zero vector v such that
Gv=/lv
(5.34)
Where v is the eigenvector o f the matrix and A is a scalar value that is called the
eigenvalue o f the matrix.
Now determining the eigenvalues and eigenvectors for the matrix T l]X
54
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
T ijXV = y IV
(5.35)
Determining A , eigenvalues o f the matrix
d e t(r '-'1 - /!/)=
0
=> det
e ^ '-X
0
0
e ' -X
=
0
(5.36)
=>X = e±}1'
Therefore e ±}1' and
are the eigenvalues and eigenvectors respectively o f the
matrix T ijX.
It can be noted that the propagation constant isincluded in theeigenvalues o f the
T iJXmatrix, but T ‘jXis unknown and is being measured as M ‘jX
due to the systematic
errors in the test equipment.
So to determine the eigenvalues and eigenvectors o f the M ‘jXmatrix, the following method
is used.
From (5.14)
(5.37)
Inserting (5.37) in (5.22)
M n = X T Jl(TiY X ~ xM l
(5.38)
M jX(m ' Jr1 X = X T JX(j7' )“'
(5.39)
M iJXX = X T iJX
(5.40)
1
1
1
X
o
(5.41)
X
1
1
0
o '
1
o
o
X
1
o
T in = 7 v ' ( r ) r
-1
'1
1
1
X
1
1
w h e re
55
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
M iJl - M jX{ M ' Y =
mu
mn
m2l
m22
(5.42)
Applying in (5.40) yields
ml2
m21 m2i_
~mn
'A
Ai
_c .
mu ~~A
i
c.
~Ae~*
i
Cxe*'
m2,
m22 Lc iJ
II
m x2
(5.43)
e*
(5.44)
e*
i--i
1
I
mn
o "
Bx V *
1
0
'A
b;
j n 2 1 w22 _ _c >
=>
Bi
1
>i
mu
(5.45)
LC.J
and
~mu
_m2x
mn
m22_
= e*
1
~b ;
(5.46)
i
These have the form o f the eigenvalue and eigenvector definition.
~b ;
From it is evident from g and h that ~ A
5
- c ^-
±^1
and £0 ^ 1 are the eigenvectors and
i
eigenvalues respectively o f the matrix M ij].
Thus it can be noted that T'jl and
have same eigenvalues and the eigenvectors are
related as follows.
Bx T
1 0
'A
A
'O'
~A
_c .
i
1
W
and
.C,_
(5.47)
'B,~
1
U ‘J = X V “
(5.48)
56
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
where
U iJ =
[A.1
b
c,
V ‘J =
1
"1
0
0
1
U lJand
"
V lJ are matrices whose columns are the eigenvectors o f M ‘J] and
T ,J'
respectively.
Since the eigenvalues o f the measured matrix M ,J] also contain the propagation term, it is
used to derive the propagation constant.
From the definition o f eigenvalues, the eigenvalues o f M ,]Xmatrix are
!»,%=■ 1 [ {mu + m 22) ± J ( m n - m 22f + 4m,l \ 2 m
where X{ « 6
2\
(5.49)
and X2 « e rI[ when thru and line standards are ideal. It is advantageous
to use the average o f these values to slightly improve the accuracy in the calculation o f
the propagation constant.
XJ =
1
v
r
X1> + VX*
(5.50)
2J
= e
y ,J —y + Ay 'J
(5.51)
—
\n{Xj )
(5.52)
/,
This clearly demonstrates that error in the propagation constant estimate is minimized by
maximizing the difference between the lengths o f thru and the line.
Similarly a redundant set o f 2^ is derived from thru and line 2 measurements.
57
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
For N lines including the thru standard, N -l sets o f data is collected. These observations
are o f the form
G=y L + er
(5.53)
G includes N -l observations ofy'1
L is N -l pairs o f the length differences between the thru and the other lines
er is N -l the random measurement noise observations in the calculation o f y
If the measurements are equally noisy, averaging could help for an accurate estimate. But
if some measurements are inherently noisier than the other, a weighted average can be
applied with less weight to the nosier measurements for an accurate estimate.
In this method where N -l linearly independent measurements are made using N lines, the
propagation constant is estimated using a weighted-least-squares method. That is, the
sum o f square o f the error er over the range o f frequencies is minimized to determine the
propagation constant.
Minimizing ^ | G - yL\2 and solving for y , we have
Where LHis the Hermitian transpose o f L and W is the weighting matrix.
Guass and Markov found that the inverse o f the measurement noise covariance matrix is
to be used as the weighting matrix for optimum solution [8 ], Using the covariance o f the
noise matrix for the weighting matrix gives a best, linear, unbiased estimator (BLUE) [8 ],
L h V ~ xG
"
l hv
(5.55)
~' l
(5.56)
58
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
where crl is the variance of the composite noise in the individual measurements and
Smn is Kronecker delta.
It is not required to find the <r2
k variance as this term appears in the numerator as well as
the denominator o f the y estimate, and the simplest way to describe Kronecker delta is
for i 5* j
.
for i = j
0
1
(5-57)
Therefore for two lines being used, kronecker delta is a 2 X 2 matrix and so is the
variance inverse matrix.
1
0
0
1
S =
(5.58)
1
N
N
N
'
2
1
3
"
3
1
1
2
N
3
3
(5.59)
(5.60)
L=
Thus the best estimate o f y is determined.
5.3.2 Determining error coefficients
Determining Error box X
M Jl = X T n { r iY X ~ ' M ‘
(5.38)
M j ' ( M i y i X = X T J'( Ti)~'
(5.39)
M iJ' X = X T iJ'
(5.40)
where
59
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
0
X
1
-1
o
1
(5.61)
1
1
1
" i
X
o
m xx
A T 1 = M ji (m ‘Y =
0
o'
*x
1
SO
O
'1
1 _ y y 1^ y ' ) 1 _
I
r
“x
i
y
m X2
(5.62)
m 2X m22
Applying in (5.40)
I
1
<N
s
s'
m 22_
_ m 2X
Ri1
'A
Y>
A
l
'A
t
c,
o *
V *
i
5 >1
l
0
eY
(5.43)
When solving (5.39), it is observed that R t is eliminated since it appears on both the sides
o f the equation. The expressions obtained are in terms o f Bj, Aj and Cj
m xxA x + mxlCx = Axe
(5.63)
mlxA x + m22Cx = Cxe
(5.64)
tnxxB x + m X2 = B xe *
(5.65)
—e
mi\B \ + m 22=
(5.66)
Quadratic equations can be obtained from two sets o f the equations above.
(5 -« )
(5.64)
m 21
+ ( m 22 -
C,i /
■mX2 =
Ax
- { m 22 - m lx) ± J ( m 22 - m xx) 2 + 4mxlm 21
C,
2m 21
(5.65)
(5.66)
B,
0
m- x(Bxf +(m22- m xx\ B x) - m l2 = 0
(m22- m u ) ± J ( m 22- m xx) 2 + 4 m x
(5.67)
(5.68)
(5.69)
(5.70)
2m 2 1
60
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
A
The error coefficients Bj and — solved for using these quadratic equations have two
^1
roots. A decision needs to be made as to which root represents the error coefficients. It
A
can be seen that both Bj and — have equal roots. Assuming them to be equal we have
A
Bl =
=> Ai= Cl Bl => S21aSi2A-SiiaS22A= -S]1aS22A => S 2 IAS12A = 0
6-1
This cannot be true for a practical measurement system, unless it is a unidirectional
system where S )2 is almost negligible. However, this is not applicable to the test system.
A
Thus for a practical test system, the Bi and — values need to be unequal.
Q
A
There are two cases for unequal Bj and —
Case A:
rAA
_ ~ ( m 22 - m u ) + ^ ( m 22 - m n ) 2 + 4m n m 21
v^i/a
(5.71)
2i
~ ( m 22 - m u ) - i l ( m 22 - mu f + 4mi2m 2i
B ]a= ----------------------lm~.
‘ 21
(5.72)
Case B:
r Ax
C
V^i
_ - { m 22 - m n ) - y j ( m 22 - m u ) 2 + 4mn m 2 1
Jb
2
m2l
~ ( m 22- m u ) + ^ ( m 22 - m n f + 4ml2m2l
B \ b = ----------------------- ; ---------------------------2/72,
‘ 21
61
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.73)
(5-74)
A
For appropriate assignment, the two cases for each o f Bj and — are compared based on
the estimates using the propagation constant y
From (63)
determined using BLUE.
m.
c A
V'-'I
(5.75)
e ' —m.
mn
From (65) {B,)est =
e
(5.76)
—m.
The case that is closest to the estimates is used for —L and B,.
C,
Solving for Error box Y , from (5.14)
X = M iT ' ( T iY
(5.77)
Inserting (5.77) in (5.22)
M j] - M ' T ' ( r ' )
(5.78)
y {m
(5.79)
1Y M J] =( t ' Y t j 'Y
Y M jn - T jiXY
(5.80)
1
-1
o'
^ 1 j ’/'* _ "1
0
1_
0
o
yjiI _
1
X1
where
0 "
Y*
0
eA.
M jn = ( m ‘Y M J] =
(5.81)
(5.82)
V2 \
r 22
Applying in (5.77)
i
r 22
A
721
_
_
0
0 '
^12
1
A2 c 2' ru
X1
1
R.
e*
r2
7
~A2
b2
c 2~
1
62
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.83)
When solving (5.83), it is observed that R 2 is eliminated since it appears on both the sides
o f the equation. The expressions obtained are in terms o f B2, A2 and C2
rn A2 +r2lC2 = A2e
(5.84)
rn^2
(5.85)
rn ^ 2 —C2e
f\\B2 + /"21 —B 2e A
(5.86)
rX2B 2 + m 22 = e
(5.87)
Quadratic equations can be obtained from two sets o f the equations above.
(5-84)
(5.85)
V^2 J
+0
r 22 ~
rH.
c2J
- r 21
= 0
A2 __ ~(r 22 ~ r u ) ± J ( r 22- r u ) 2 + 4rl2 r 2
(5.89)
2r.1 2
C,
(5.86)
^12 (^ 2 )
(5.87)
B2 =
"*"(^22
^11X ^ 2 )
(5.88)
(5.90)
r2l
~ ( r22 - ^ i ) ± V ( r 22 - ^ n ) 2 + 4 r 12r 21
2 r.12
(5.91)
Since B 2 and — values need to be unequal, there are again two cases.
C-)
Case A:
An ")
(^22
V ^2 Ja
~ ( r 22
B 2a =
>l( r 22 ^ll) +4^1\ 2 r 2\
2 r,n
(5.92)
~ r n ) ~ VO3 2 - r\ 1f + 4r 12r 21
2 r.12
(5.93)
Case B:
63
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
A ,
^
(^22
fi 1)
y j ( r 22
\ ^C2 J b
D _ _
” 2b ~
ril)
+ ^r!2r2l
^
2r,,12
_ r\ 1 ) + V ( r 22 _ ri 1) + 4^2^21
«
( r 22
^ no
(5.95)
12
vl
For appropriate assignment, the two cases for each o f B 2 and — are compared based on
C2
the estimates using the propagation constant y
From (84)
r
v C 2 j est
determined using BLUE.
r,,
e
-yi\
(5.96)
-r„
From (86) (S,),,, = — -2—
ep —r11
( 5 .97)
A
The case that is closest to the estimates is used for — and By.
C
'-'2
This same procedure is followed for the combination o f thru and line2, thus there are two
A
A
observations o f values for Bj, B 2, — and — .
C
'-'1
^C2
From the 2 observations, the same procedure followed for the propagation constant
A
A
estimate is used to find the best estimates for B h B2, — and — .
C,
C2
B _ h TVBlB
~
h TVBxh
(5.98)
where h is a vector with all elements as 1 and VB 1 is the weighting matrix
64
R e p r o d u c e d w ith p e r m issio n o f th e c o p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
2
+
1
r +2
(5.99)
B .m n
e }i'" —e^'"
12
(5.100)
Y
(5.101)
h TV~lC / A
C / A =h TV~lh
(5.102)
FB , m n
m>„
= (y
V
/
Similarly
where h is a vector with all elements as 1 and Vc is the weighting matrix
2
V,C , m n
+
m=n
e ~r>*
1
.
r + 2e~’
2
(5.103)
I2
(5.104)
V,C , m n
= (v
V
Y
(5.105)
C ,n m 1
A
A
Thus Bi, B 2, — and — best estimates are determined.
Ci
c2
To determine Ai and A 2 , the product Ap =Ai A 2 is determined from thru m easurem ents
and the ratio Ar =Ai /A 2 is determined from the reflect measurements.
From the mathematical interpretation o f the signal flow graphs for the reflect at ports 1
and
2
65
R e p r o d u c e d w ith p e r m is s io n o f th e c o p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
.0 0 5 ) ^
" (1 0 6 )
'
A,
(Ru - B , X l - ( C / A ) 2R „ )
A2
( \ - ( C / A \ R „ l R 22- B 2)
Now considering the thru measurements for Ap
T
C
,
U
S'
S'
2\A
j _
\ 2A
o
1
'f
_
A 22
I 'l
o
,
rp
A
11
21
^ n
12
S'
_
^ 21A
—
11
_
—
22B
S'
S'
215
1
22B
22 A
S'
O .U5
1 R "1" ' 1
W
S'
r.
125
22/1
„ „
^22B^22A
S'12B
- S'22/1 S'2 2 5
S2 1 5 S'12/1
11 — ^S'2 2 5 S 22/1
(Tu - B X T 2 2 - B 2) = T 2,Tu C,C2
Dividing with A 1A 2 on both sides, we have
(T„ - B , i T 22 - B , )
/ /
1
2
t 2, t 12(c i a
\ { c / a )2
Finding A 2 — ApAr , thus to determ ine^/ a choice between the roots is required.
66
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Inserting Ap and Ar in equation, by using the positive root o f Ai
Ru
r tr ia l =
Bx
^v 7 (i-(c/4 « „)
Now considering the difference between the magnitudes o f Ttrial and an estimate o f Y o f
the reflect
Ar =
r.
r
est
trial
r
(5.112)
tr ia l
If A r > V2 then the Ytnal is using the wrong root o f Ai
(5.113)
■■A = ~ 4 Ap A
If A r < V2 then the Y/rial is using the correct root o f Ai
■'■A = ^ A PA
(5.114)
A2 = A p / A j
(5.115)
Since Ai and A 2 are known, Ci and C 2 can be determined
C ,=(C/A),A,
(5.116)
C: = ( C / A ) 2A2
(5.117)
Now determining Rj and R 2
From the thru measurements
M ‘ = X T ‘Y
1
'(^ 11^22 A A n ) T\ |
\ A1
= R 1\
—T-22
1
c.
(5.14)
BA1 " 1
1
. o
0
1
'
r 12
A I7
b
2
C jI
1
67
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(5.118)
. ■ . ^ 2 (C 1C 2 +1) = - L
T
12\
(5.119)
Tn ( ( C I A \ { C I A ) 2AlT 1 + 1)
. s 2iA
(5.121)
c
‘-’ 125
*2
. $21A
0
21 5
1^21
r
- '1 2
c
125
(5.122)
&12A
_ S 2\a _ S\2B
c
c
*J 12 A
J 21 B
. ^2\A
c
125
S UA
c
(5.123)
J 215
( 5 -124)
Using the product and ratio terms, Ri and R 2 are found to be
=±V ^A
(5.125)
The choice o f the root is simple because S 21A is the transmission S-parameter o f error
network. Any transmission S-parameter would is defined as 0 < S 2lA < 1
•••*,
(5-126)
and
R2 = R i I K
<5-127)
Thus all the error coefficients are determined along with the propagation constant. The
reflection coefficient can be determined from either o f the reflect measurements now that
all the error coefficients are known.
68
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
5.3.3 Extracting the device parameters from the measurements
To determine the calibrated DUT measurements from the raw measurements taken, we
can use the ABCR parameters o f the individual error boxes along with that o f the
cascaded network.
Am B
DUT
Figure 5.12 Extracting device parameters
M = X D U TY
(5.128)
D U T = X ~ xMY~'
(5.129)
Thus the calibrated device measurements are obtained, that include only the parameters
o f tissue and tumor, excluding any effects o f the test equipment.
5.4 VERIFYING CALIBRATION
A Multiline TRL calibration algorithm was developed based on the above procedure in
Matlab. In order to verify the accuracy o f algorithm, the calibration produced from this
algorithm was compared with that o f the integrated TRL algorithm in a Network
Analyzer Agilent E8361A. Raw measurements o f the standards were taken using the
Network Analyzer and were entered into the Matlab code. To verify the algorithm the
standards were measured in place o f the device and calibration was applied. The
properties o f the calibrated standards are expected to be close to those o f ideal standards.
69
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Thus when a reflect (short) standard was at port 1, the excepted Sn is a magnitude o f 0
dB and an angle of 180°. The figures 5.13 and 5.14 are the magnitude and angle plots of
Sn with reflect at port 1 with the Matlab algorithm and the PNA calibration.
PN A CAL
CAL (Matlab algorithm)
m
S'
0
■M
x 10*
Figure 5.13 Si (magnitude in dB when reflect (short) placed at port 1
_
i----------------r
PN A CAL
C A L (M a tla b alg o rith m )
—
24
a©
uron unit mm
as
a
aa
i
~~'T... i~...” ...~a£...
Figure 5.14 Si 1angle in degrees when reflect (short) placed at port 1
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er r ep ro d u ctio n p ro h ib ited w ith o u t p e r m issio n .
44..
Reflect was also placed at port 2. The ideal magnitude and phase o f S22 with reflect at
port 2 would be 0 dB and 180°. The following are the plots for the PNA calibration and
the calibration using the Matlab code.
R e fle c t a t p o rt 2
pNACA£
CAL (M atlab a lg o rith m )
no
T3
CM
CM
O
CO
-1
-2
-3 2 .5
3
3 .5
freq u en cy (H z)
x 10
Figure 5.15 S22 magnitude in dB when reflect (short) placed at port 2
S 2 2 a n g le reflect a t port 2
P N A CAL
CAL (M atlab alg o rith m )
F requency(H z)
Figure 5.16 S22 angle in degrees when reflect (short) placed at port
71
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
2
a
Thru standard is also measured and calibrated using both the algorithms. The
transmission parameters S2i and S 12 o f the calibrated thru standard should ideally be OdB.
The figures 5.17 and 5.18 are the S21 and Si2 magnitude in dB o f calibrated thru standard.
thru
PNA CAL
CAL (Matlab algorithm)
QQ
0
t o
...
F r e q u e n c y (H z)
Figure 5.17 S21 magnitude in dB o f the thru standard
Thru
PNA CAL
CAL (Matlab algorithm)
0?*i
oei~
■1
CD ■
TO
cn
r-
CD ;
;
A rw
V
v
'
~
V
r.
J
i
>
, ■.
V V
Frequency(Hz)
Figure 5.18 S 12 magnitude in dB o f the thru standard
72
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
„
,
'•
' 1
CHAPTER 6
RESULTS
6.1 CONVERTING MEASUREMENTS TO S-PARAMETERS
The spectrum analyzer measured the average power o f the transmitted and backscatterd
signals whereas the mixer measured dc voltages that are proportional to the phase o f the
transmitted and backscattered signals. These measurements could not be applied directly
into the Mutiline TRL algorithm as its inputs are o f S-parameter format. In order to make
the measurements compatible with the algorithm they needed to be converted to Sparameters.
6.1.1 M agnitude Conversion
The Scattering matrix relates the voltage waves incident on the ports to those reflected
waves from the ports [4], For a two port network, the S-parameters are defined as
Su =
^ 7
for v ; = 0. S„ =
s a = — 7 for VC =
2
0
for V* = 0.
. Sn = ^
for VC =
0
.
2
where V f and F2+ ai e the incident waves? at poits?
1
and
V[ and V2 are the reflected waves from ports
1
2
respectively,
and
2
respectively.
As can be seen that the reflected and the transmitted or the backscattered waves are
normalized with the incident voltages at either o f the ports. The incident wave was
measured at the sweep generator output, the only point accessible in the test system,
73
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using the spectrum analyzer. Since the transmitted and backscattered waves were also
measured using the same instrument the average power o f the transmitted, backscattered
and the incident waves were all a function o f the same load impedance presented by the
spectrum analyzer. These voltages can be represented as,
K =
(«•>)
K =
( 6 .2 )
r.~ <V1 = 0) =
(6.3)
v ; O V = 0 ) = A — w ,Z t
(6.4)
V{
|(F,* = 0) = ■JPt^ x m r l Z L
(6.5)
,Z L
(6 .6 )
[(»",* = 0) =
where,
Find and Pi„C2 are the incident average power measured for port
Pbackscatteri and Ptransmittedi are the backscattered power at port
port
2
respectively when the wave is incident at port
1
respectively when the wave is incident at port
and
2
respectively
1
and transmitted power at
2
and transmitted power at
1
Pbackscatter2 and Ptransmitted! are the backscattered power at port
port
1
2
ZL is the load impedance o f the spectrum analyzer
In this case S-parameters in terms o f the average powers can be written as,
Su =
(6.7)
V
^m c\
S 21 =
(6.8)
in c I
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
S 22 = \ P^ca„ er2
me
^
^
2
Su =
(6.10)
in c 2
6.1.2 Phase Extraction
From Chapter 3, the dc voltage measured at the IF port o f the mixer was proportional to
the phase o f the transmitted or backscattered wave being measured.
VNe = V refNVNcos(eN)
/V V v -^
( 6 . 11)
ref
■ ■ ■ ■ ■ I m m m w
DUT/
ERROR BOX
STANDARDS
Y
ERROR B
Vi~)
H
H
T
Zl
Figure 6.1 Two port network representing the test system
where
VrejN and Vn is the reference voltage at the mixer LO port and transmitted or
backscattered wave at mixer RF port respectively for each standard and DUT.
VNq is the dc voltage measured at the IF port.
Using network analysis
V
reJN
7 inN
—
7
+ 7
FS = - T < 1 + r “
(6 . 12)
)
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R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
vy N = \ IrpN^inN
z
-~ \ l~p~
r N J K'+
n _ r^/'i,v.I
V v1
(6.13)
1 inN )
where
ZinN and Zs are the input impedance o f the network and the source impedance
respectively.
Vs is the source voltage to the mixer.
V
r reJN V
' N =
(6.14)
where VmN * S UN
Si in and Vs are the Sn o f the network and the source voltage at the reference o f the
mixer, point B, as shown in the figure 6.2.
ENERATOr
COUPLER!
Incident average power
measured
incident
transmitted
WWVVVa /vvvvv
,j A iV W V \ a ^
back scatterTUMOR
SPZTSW rTCH
3
SP2T SWITCH
SP3T SWITCH
s p r rs w r rc H
for th e m ixer 1 R r
Figure 6.2 Reference planes o f measurement for magnitude and phase
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
The phase can be extracted if these parameters are known. Since these values were
already calculated at point A for converting magnitude measurements, the attenuation in
the path from point A to point B was applied to arrive at the required parameters for
phase extraction. Thus the phase o f the scattering was determined from the voltage
measurements. It can be seen in the figure 6.2 that the reference for magnitude
measurements is at point A and that o f phase is at point B. In order to have common
reference plane at point A for both the measurements the phase shift in the path from
point A to point B was accounted for in the extracted phase measurements. The
attenuation and the phase shift in the path from A to B were determined by measuring S2i
o f the path using PNA.
6.2 EXPERIMENTAL RESULTS
Measurements o f a glass sphere o f radius o f 19 mm and dielectric constant o f 5 were
taken using the test system. For calibration, the standards: Thru, Reflect, Line 1 and
Line2 were also measured using the test system. The magnitude and phase measurements
collected from the spectrum analyzer and the voltmeter were first converted into Sparameters, the understandable format for the Multiline TRL algorithm. Calibration was
applied to the raw measurements and the device parameters were extracted. The
magnitude and phase o f the waves scattered from the glass sphere at an angle 180° with
respect to the sphere were plotted.
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transmitted wave magnitude
ri l L i a m t t
1 f Si ■
8
4
3
06
0.8
M
1.4
frequency^)
Figure 6.3 Magnitude o f scattering from a 19 mm radius glass sphere o f dielectric
constant 5
78
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1.8
:xtO
transmittedwaves angle
t|| t
^
It
* 11 )
I® ■
II!
1:3®-
250
phase
in
'§09-
^|®!■
<1 it
wmi
fill
0.2
0.4
0.6
:*
1
1.2
1.6
frequencyJHz)
Figure 6.4 Phase o f scattering from 19 mm radius glass sphere o f dielectric constant 5
79
R e p r o d u c e d w ith p e r m issio n o f th e c o p y rig h t o w n e r . F u rth er r ep ro d u ctio n p ro h ib ited w ith o u t p e r m issio n .
: Vi/I*,
JG
x10
CHAPTER 7
FURURE WORK
7.1 PROPOSED PROCEDURE FOR CALIBRATION AT DIFFERENT ANGLES
AROUND A CIRCULAR AXIS
180° B
Tissue andtumor
270°
45°
Figure 7.1 Scattering from a sample at various angles around its circular axis
When a sample is illuminated using electromagnetic waves, scattering is produced in all
directions. The scattering from the sample at each angle can be measured by placing the
receiving antenna at the respective angle except at 0° where it is collected by the
transmitting antenna as the backscatter. The figure 5.19 indicates 8 angle positions where
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the scattering can be collected. The transmitting and receiving paths are represented by
error boxes. The transmitting antenna along with the transmission path in the test system
will be referred to as the transmitter and the receiving antenna along with the receiving
path in the test system as the receiver. Calibration at different angles is studied by
considering two cases.
Case 1:
When the transmitter is at position A and the receiver at B, i.e 180° with respect to A, the
calibration is the same as mentioned earlier.
DUT
X1
Y1
Figure 7.2 Error box formulation for case 1
The standards are defined as discussed in the previous sections.
180° B
180“ B
180° B
180° B
180° B
o
Yi
t Yi
\
v
•
DUT
Thru
Line 1
Line 2
J
X,
x,
0° A
Tluu
0° A
0° A
Line 1
0° A
0° A
Reflect
Line 2
DUT
Figure 7.3 Standards defined for case 1
81
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By applying the measurements o f the standards in the multiline TRL algorithm, the error
boxes X { and Y{ are determined. The calibrated device parameters with the receiver at
180° are extracted from the raw measurements.
DUTcaseX= X ; 'M Y \'
However the calibration obtained in the above case cannot be applied to any other
positions of the receiver. This is because the waves follow a different path indicated by
the different error boxes Y in each case. Also, as mentioned in chapter 4, any constant
source o f interference becomes a systematic error and is included in the error model. This
interference need not be the same at each angle. For example, a metal object placed
opposite to the 90° position of the receiving antenna will have greater interference at this
angle as compared to other receiver positions. So the system needs to be calibrated for
each angle.
Case 2:
This is when the receiver moves to position C, whereas the transmitter remains at
position A.
DUT
Figure 7.4 Error box formulation for case 2
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In this case the error box X ) determined for Case 1 can be used as the as the position of
the transmitter is the same in both the cases. However the position o f the receiver in this
case has changed and the error box F, cannot be applied for calibration in this case.
The calibration for these positions o f the antennas is different from the case with antennas
opposite to each other. This is because as seen in the figure, the transmitting antenna is
not focused at the receiving antenna, so the energy received at the antenna at position C is
a fraction o f the energy transmitted by antenna at position A which negates the definition
o f thru.
|
225°
225° C
o
c
Y,
X'
Thru
X2
X
45°
0° A
0° A
Figure 7.5 Thru standards with transmitting antenna at positions A and F
To obtain X\ and Y2 , the transmitting antenna is moved to position F, while not changing
the position o f receiving antenna from position C. Since the position o f the receiving
antenna does not change the error box F2 remains the same, whereas the error box
associated with the transmitting antenna changes to X 2 from Xj. By applying calibration
with the standards measured through the antennas at F and C positions, the error boxes X 2
83
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
and Y2 are determined. Now the devices parameters for the case 2 are obtained using
error box X\ obtained from case 1 and error box Y2 .
DUTcme2 = X^MY~2
Similarly calibration is performed at different angles.
7.2 E F FE C T O F SKIN
In this work the effect o f skin was neglected, but the skin has significant electrical
properties, when in conjunction with breast tissue, causing reflections that cannot be
assumed to be negligible. The electrical properties o f skin, healthy tissue and tumor are as
shown in table 7.1 [3].
Table 7.1 Electrical properties o f skin, normal and malignant breast tissue
Dielectric constant
Conductivity(S/m)
Skin
36
4
Tissue
9
0.4
Tumor
50
4
To get a simplified estimate o f the effect o f skin a planar model o f the breast along with a
layer o f skin was considered. This was analyzed using bounce diagram where the model
was assumed to be slabs o f skin, tissue and tumor. As shown in the figure 7.1, A, B and C
are the air and skin, skin and tissue and tissue and tumor interfaces respectively.
Since the reference planes were shifted to interface A after calibration, the wave is
assumed to be incident at A. At A due the dielectric contrast between the air and skin
media, a part o f the incident wave is transmitted and a part o f it reflected. The transmitted
84
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wave travels through the skin and reaches the interface B after time ti which is a function
o f permittivity and thickness o f the skin.
ft
1
t=Q
S\
T
j
-T1 T2 IT
3 t,
Vp
T ) ,/
TiTzRsIT
4tl
A A /V W
2tj+2tj
■TiTsTsfTz
lz ^
6t
I
1
I
1
1
1
1
1
1
(
1
I
i
Breast tissue e=9.8
|
Tumor e=50
|
Skin e=36
Calibration reference plane
1
1
1
I
I
1
1
1
1
1
1
1
1
i
I
I
I
1
i
i
1
I
\
I
I
i
A
B
A ir-Skin
interface
S k in -T issu e
interface
1
1
I
i
1
I
»
\
1
t
1
(
*
T issu e-T u m o r
interface
Figure 7.6 Bounce diagram o f the breast model with a layer o f skin
Then the wave suffers a reflection due to the mismatch between the skin and tissue
properties and reaches interface A back at 2ti, where as the transmitted wave travels
through the tissue and reaches the tumor after time ti+t 2 where t2 depends on the tissue
properties and the location o f the tumor. At the tumor tissue interface C, due to the
significant contrast between the tumor and the tissue, a part o f the wave is transmitted
and the rest is reflected back that reaches the interface A at 2ti+2t2. This cycle repeats in
itself till the wave is completely attenuated.
85
R e p r o d u c e d w ith p e r m issio n o f th e c o p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Since interface A is the reference plane o f measurement, the reflections collected at A are
interpreted in time domain.
i
i
Information of interest
Figure 7.7 Reflections in time domain
As can be observed there are reflections at time instants ti, 2ti, 4fi, 6 t i
and 2ti+2t2,
4ti+4t2................ The only reflections o f interest are at multiples o f time 2ti+2t2 since
these include reflections due to the dielectric contrast between the tumor and the tissue.
Since t2 is a function o f the permittivity o f the breast tissue and the location o f the tumor,
it remains unknown. Thus, the information o f interest by itself cannot be extracted.
However, the unwanted reflections can be eliminated if the instant at which they occur is
known. The unwanted reflections occur at multiples o f time tj, and if the permittivity and
the thickness o f the skin are known, ti can be determined and thus reflections can be
eliminated. But t2 is unknown so the information o f interest at 2ti+2t2 can occur at any
instant from 2ti to any instant time or may overlap with the multiples of ti. Thus, only
the first reflection due to the skin at fi can be eliminated as it lies before 2fi.
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
7.3 TIME DOMAIN GATING
The measurements from the breast test model using the test system are a function o f
frequency. In cases similar to that addressed in the previous sections, interpretation o f the
device characteristics with time domain is useful. By analyzing the magnitude, duration,
and shape o f the reflected waveform, the mismatches in a device can be determined.
Time domain gating becomes an appealing feature, where further calibration cannot be
applied to shift the reference planes. However the time delay for shifting the reference
planes should be known precisely. Thus if the time delay o f an electromagnetic wave in
the skin layer can be determined, time domain gating can be potentially used to shift the
reference planes at the air-skin interface to the skin-tissue interface. There are three steps
involved in Time domain gating: Converting frequency domain to time domain,
Extracting samples o f interest and converting the time domain data o f interest back to the
frequency domain.
The data collected in the frequency domain can be converted to time domain using
inverse Fourier transform. The discrete inverse fourier transform function is,
j
AM
2 mkn
N
where
Hn are the sampled points in frequency domain and
N is the sampling rate
The sampling rate needs to be fixed carefully such that important information is not lost
and the complexity o f the conversion algorithm is not increased.
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er r ep ro d u ctio n p ro h ib ited w ith o u t p e r m issio n .
The samples o f interest are extracted from the time domain series and converted back to
frequency domain using Fourier Transform.
N -l
2m kn
H. = Z
k=0
Now the frequency domain data contains only the information o f interest.
88
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
R EFER EN C ES
[1] Mammography and Beyond: Developing Techniques fo r the Early Detection o f
Breast Cancer. Washington, DC: Inst. Med., Nat. Academy Press, 2000.
[2] X. Li and S.C. Hagness, “A confocal microwave imaging algorithm for breast cancer
detection,” IEEE Microwave Wireless Components Lett., vol. 11, pp. 130-132, Mar.
2001 .
[3] E. C. Fear, S. C. Hagness, P. M. Meaney, M. Okoniewski, and M. A.Stuchly,
“Enhancing breast tumor detection with near field imaging,” IEEE Microwave Mag.,
vol. 3, pp. 8-56, Mar 2002.
[4] David M. Pozar, “ Microwave Engineering,” Second edition, John Wiley & sons,inc.
[5] C95.1b-IEEE Standard for Safety Levels with Respect to Human Exposure to Radio
Frequency Electromagnetic Fields
[6] “Applying Error Correction to Network Analyzer Measurements”, Helwett Packard
Application Note 1287-3
[7] Wartenberg, Scott A, “RF measurements of die and packages” .
[8] DeGroot, D.C. (NIST, Boulder, CO, USA); Jargon, J.A.; Marks, R.B, “Multiline
TRL revealed," 60th ARFTG Conference Digest, Fall 2002, p 131-55
[9] R. B. Marks, “A multiline method o f network analyzer calibration,” IEEE
Transactions on Microwave Theory and Techniques, vol. 39, no. 7, pp. 1205-1215,
July, 1991.
[10] R. B. Marks, “Formulations o f the basic vector network analyzer error model
Including switch terms,” 50th ARFTG Conference Digest, pp. 115-126, Dec., 1997.
[11] G. F. Engen and C. A. Hoer, “Thru-reflect-line: An improved technique for
calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microwave
Theory Tech., vol. MTT-27, pp. 987-993, Dec. 1979.
[12] R. A. Soares, P. Gouzien, P. Legaud, and G. Follot, “A unified mathematical
approach to two-port calibration techniques and someapplications,” IEEE Trans.
Microwave Theory Tech., vol. 37, pp. 1669-1674, NOV. 1989.
[13] G. F. Engen, C. A. Hoer, and R. A. Speciate, “The application o f Thru-ShortDelay’ to the calibration o f the dust six-port,” in 1978, IEEE MTT-S Int. Symp.
Dig., pp. 184-185.
[14]
J. Verspecht, F. Verbeyst, and M. Vanden Bossche, “Large-signal network
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analysis: Going beyond S-parameters,” presented at ARFTG Short Course on
Measurements and Metrology for RF Telecommunications, Broomfield, CO, Nov.
29, 2000.
[15] S. Vandenberghe, D. Schreurs, G. Carchon, B. Nauwelaers, W. De Raedt, \Sparameter Reciprocity Relations, Normalization, and TLR Error Box Completion,"
Internal Journal RF and mm-wave CAE, John Wiley & Sons, Inc.,vol. 12, no. 5,
2002, pp. 418-427.
[16] Juroshek, J.R., “Correcting for systematic errors in one-port calibration-standards
Differential measurements,” 62nd ARFTG Microwave Measurements Conference,
4-5 December 2003, Boulder, Colorado.
[17] J.A. Jargon and R.B. Marks, "Two-tier multiline TRL for calibration o f low-cost
network analyzers," 46th ARFTG Conference, Scottsdale, AZ, 1995.
[18] G. F. Engen, “A least squares solution for use in the six-port measurement
technique,” IEEE Trans. Microwave Theory Tech., vol. MTT-28, pp. 1473-1477,
Dec. 1980.
[19] D. Williams, “De-embedding and unterminating microwave fixtures with nonlinear
least squares,” IEEE Trans. Microwave Theory Tech.,vol. 38, pp. 787-791, June
1990.
[20] R. A.Speciale, “ A Generalization o f the TSD Network-Analyzer Calibration
Procedure, Covering n-Port Scattering-Parameter Measurements, Affected by
Leakage Errors,” IEEE Trans. Microwave Theory Tech., vol.. MTT-25, NO. 12,
December 1977
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APPENDIX
MATLAB CODE FOR MULTILINE TRL ALGORITHM
y=501;
y2=501;
c=3 *10A8;
yi=i;
Z0=50;
e l =1.0005;
e2=0;
fo r k= l:l:y 2
kl= k;
freq(k) =data(kl, 1);
TI l(k)= (10A(data(kl, 2)/20)) *(cos(data(kl ,3) *pi/l 80)+(i*sin(data(kl ,3) *p
i/180)));
T21(k)=(10A(data(kl,4)/20)) *(cos(data(kl,5) *pi/l 80)+(i*sin(data(kl ,5) *p
i/180)));
T12(k)=(l0A(data(kl,6)/20) ) *(cos(data(kl, 7)*pi/l80)+(i*sin(data(kl, 7) *p
i/180)));
T22(k)=(l 0A(data(kl, 8)/20)) *(cos(data(kl,9) *pi/l 80)+(i*sin(data(kl, 9) *p
i/180)));
L lll( k ) = ( l 0A(data(kl, 10)/20)) *(cos(data(kl, 11) *pi/180)+(i *sin(data(kl,
1 l)* p i/l 80)));
L I 21 (k)=(l 0A(data(kl, 12)/20))*(cos(data(kl, 13)*pi/l 80)+(i*sin(data(kl,
13)*pi/180)));
L I 12(k)=(10A(data(kl, 14)/20)) *(cos(data(kl, 15) *pi/180)+(i*sin(data(kl,
15)*pi/180)));
L 1 2 2 (k )= (l 0A(d a ta (k l ,1 6 )/2 0 ))* (c o s(d a ta (k l,1 7 )* p i/1 8 0 )+ (i* sin (d a ta (k l,
17)*pi/180)));
L21 l(k)= (10A(data(kl,18)/20)) *(cos(data(kl,19) *pi/l 80)+(i*sin(data(kl,
19)*pi/180)));
91
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
L 221(k)-(10A(data(kl,20)720)) *(cos(data(kl ,21)*pi/180)+(i*sin(data(kl,
21)*pi/180)));
L 212(k)= (l 0A(data(kl ,22)/20))*(cos(data(kl ,23)*pi/l 80)+(i*sin(data(kl,
23)*pi/180)));
L222(k)=(10A(data(kl ,24)/20))*(cos(data(kl,25)*pi7180)+(i*sin(data(kl,
25)*pi/180)));
R l l (k)=( 10A(data(kl ,26)720)) *(cos(data(kl,27) *pi/180)+(i*sin(data(kl,2
7)*pi/180)));
R22(k)=(10A(data(kl, 28)720)) *(cos(data(kl,29) *pi/l 80)+(i *sin(data(kl, 2
9)*pi/180)));
S 11D 1(k)=(l 0A(data(kl ,30)/20))*(cos(data(kl ,3 l)* p i/l 80)+(i*sin(data(k
l,31)*pi/180)));
S21D 1 ( k ) - ( l 0A(data(kl ,32)720)) *(cos(data(kl ,33) *pi/l 80)+(i*sin(data(k
l,33)*pi/180)));
S12D l(k)= (10A(data(k1,34)720)) *(cos(data(kl,35) *pi/180)+(i*sin(data(k
l,35)*pi/180)));
S22D1 (k)=(l 0A(data(kl, 36)720)) *(cos(data(kl,3 7) *pi/180)+(i *sin(data(k
l,37)*pi/180)));
S I lD2(k)=(10A(data(kl ,38)720))*(cos(data(kl ,39)*pi/l 80)+(i*sin(data(k
l,39)*pi/180)));
S21D2(k)=(10A(data(kl,40)/20))*(cos(data(kl,41)*pi/180)+(i*sin(data(k
l,41)*pi/180)));
51 2D2(k)=(l 0A(data(kl, 42)720)) *(cos(data(kl,43) *pi/l 80)+(i *sin(data(k
l,43)*pi/180)));
52 2D 2 (k)=(10A(data(kl ,44)/20))*(cos (data(kl ,45)*pi/180)+(i*sin(data (k
l,45)*pi/180)));
T i l C(k)=(10A(data(kl, 46)720)) *(cos(data(kl, 4 7) *pi/l 80)+(i *sin(data(kl,
47)*pi/180)));
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
T21 C(k)—(10A(data(kl, 48)/20)) *(cos(data(kl ,49) *pi/180)+(i *sin(data(kl,
49)*pi/180)));
T12C(k)=(10A(data(kl,50)/20))*(cos(data(kl ,5 l)* p i/l 80)+(i*sin(data(kl,
51)*pi/180)));
T22C(k)=(10A(data(kl, 52)720)) *(cos(data(kl ,53) *pi/180)+(i *sin(data(kl,
53)*pi/180)));
L I 1 lC (k )—(10A(data(kl,54)/20))*(cos(data(kl,55)*pi/180)+(i*sin(data(k
l,55)*pi/180)));
L121C(k)=(l 0A(data(kl, 56)/20)) *(cos(data(kl, 5 7) *pi/l 80)+(i *sin(data(k
1,57) *pi/180)));
L I 12C(k)=(10A(data(kl ,58)720)) *(cos(data(kl ,59)*pi/l 80)+(i*sin(data(k
l,59)*pi/180)));
L122C(k) =(10A(data(kl, 60)720)) *(cos(data(kl, 61) *pi/l 80)+(i*sin(data(k
1,61) *pi/l 80)));
L211 C(k)—(10A(data(kl, 62)720)) *(cos(data(kl, 63) *pi/l 80)+(i *sin(data(k
l,63)*pi7180)));
L221C(k)—(10A(data(kl, 64)720)) *(cos(data(kl, 65) *pi7180)+(i *sin(data(k
1,65) *pi/l 80)));
L212C(k)=(10A(data(kl, 66)720)) *(cos(data(kl, 67) *pi/180)+(i *sin(data(k
l,67)*pi/180)));
L222C(k)=(10A(data(kl, 68)720)) *(cos(data(kl, 69) *pi/l 80)+(i *sin(data(k
l,69)*pi/180)));
R 111 C (k)= (l0A(data(kl, 70)720)) *(cos(data(kl, 71) *pi/l80)+(i*sin(data(k
l,71)*pi7180)));
R121 C (k)= (l0A(data(kl, 72)/20))*(cos(data(kl,73)*pi/180)+(i*sin(data(k
l,73)*pi/180)));
R112C(k)=(10A(data(kl, 74)720)) *(cos(data(kl, 75) *pi/l80)+(i*sin(data(k
1,75) *pi/180)));
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R122C(k)=(]0A(data(kl, 76)720)) *(cos(data(kl, 77) *pi/l80)+(i*sin(data(k
l,77)*pi/180)));
L21 JC(k)=(l 0A(datafkl, 78)720)) *(cos(data(kl, 79) *pi/180)+(i*sin(data(k
l,79)*pi/180)));
L221C(k)=(10A(data(kI,80)/20))*(cos(data(kl,81)*pi7180)+(i*sin(data(k
l,81)*pi/180)));
L 212C(k)=(l 0A(data(kl ,82)720)) *(cos(data(kl,83) *pi/180)+(i*sin(data(k
l,83)*pi/180)));
L222C(k)=(1 0A(data(kl, 84)720)) *(cos(data(kl, 85) *pi/180)+(i *sin(data(k
l,85)*pi/180)));
ISOF(k)=(10A(data(kl, 86)720)) *(cos(data(kl,83) *pi/180)+(i*sin(data(kl,
87)*pi/180)));
ISOR(k)=(10A(data(kl, 88)720)) *(cos(data(kl,85) *pi/180)+(i*sin(data(kl,
89)*pi/180)));
% APPL YINGISOLA TJON
TF(k)=T22r(k);
T R (k )-T ll r(k);
TICF(k)=TF(k)/(1- (IS0F(k)/T21r(k)));
TICR (k)=TR (k)7(1- (ISOR (k)/T12r(k)));
%0APPLYING SW ITCH TERM CORRECTION
TIC12(k)=T12r(k) -ISOR (k);
TIC21 (k)=T21r(k)-ISOF(k);
LlIC12(k) =L112r(k)-IS0R(k);
L1IC21 (k) =L121 r(k)-ISOF(k);
L2IC12(k) =L212r(k) -ISOR (k);
L21C21 (k) =L221r(k)-lSOF(k);
DT(k)=1-TIC12(k) *TIC21 (k) *TlCF(k) *TICR(k);
D L l(k)= l-L lIC 12(k) *LlIC21(k) *TICF(k) *TICR(k);
DL2(k)=l-L2IC12(k) *L2IC21 (k) *TICF(k) *TICR(k);
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T ll(k )-(T llr(k)-T IC 1 2 (k) *TIC21(k) *TICF(k))/DT(k);
T12(k)=(TIC12(k)-Tl lr(k) *TIC12(k) *TICR(k))/DT(k);
T21 (k)=(T!C21 (k)-T22r(k) *TIC21 (k) *TICF(k))/DT(k);
T22(k)=(T22r(k)-TIC12(k) *TIC21(k) *TICR(k))/DT(k);
L I 1I (k)=(L11 lr(k)-LllC 12(k) *LlIC21(k) *TICF(k))/DLl(k);
L112(k)=(LlIC12(k)-Ll llr (k ) *LlIC12(k) *TICR(k))/DLl (k);
L I 21 (k)=(LIIC21 (k)-Ll 22r(k) *LlIC21(k) *TICF(k))/DLl(k);
LJ22(k)=(L122r(k)-LlIC12(k) *L1IC21 (k) *TICR(k))/DLl(k);
L211 (k)=(L21 lr(k)-L2IC12(k) *L2IC21(k) *TICF(k))/DL2(k);
L212(k)=(L2IC12(k)-L211r(k) *L2IC12(k) *TICR(k))/DL2(k);
L221 (k)=(L2IC21 (k)-L222r(k) *L2IC21 (k) *TICF(k))/DL2(k);
L222(k)=(L222r(k)-L2IC12(k) *L2IC21 (k) *TICR(k))/DL2(k);
% CONVERTING S TO ABCR PARAMETERS
D e lL l(k )= L llI(k ) *L122(k)-Ll 12(k) *L121(k);
ALl(k)= -D elLl(k);
RJLl(k)-l/L121(k);
BL1 (k)=LJ 11 (k);
C L l(k)-(-l)*L122(k);
DelL2(k) =L211 (k) *L222(k)-L212(k) *L221 (k);
AL2(k)=-DelL2(k);
RL2(k)=l/L221(k);
BL2(k)=L21J(k);
CL2(k)=(-l)*L222(k);
DelT(k)=Tl 1 (k) *T22(k)-T12(k) *T2J (k);
AT(k)=-DelT(k);
RT(k)=l/T21 (k);
B T (k)= T ll(k);
CT(k)=(-l)*T22(k);
%LINES DELA Y IN METERS
ll(k)= 20*10A(-3);
l2(k)=(30*10A(-3));
%o CALCULATING r
Rt(:, :,k) =RT(k) *[A T(k),BT(k);CT(k), 1];
R d l (:, :,k)=RLl(k) *[ALl(k),BLl(k);CLl(k), 1];
R d 2 (:,k ) =RL2(k) *[AL2(k),BL2(k);CL2(k), 1];
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M(:, :,k) =Rdl (:, :,k) *inv(Rt(:, :,k));
0 ( : , k ) = R d 2 (:,k ) *inv(R t(:,k));
N(:, :,k)=inv(Rt(:, :,k)) *Rdl ( : , k);
Q (:,k )= in v (R t(:,k )) *R d2(:,k);
q ll(k )= Q (l,l,k );
ql2(k)= Q (l,2,k);
q21(k)=Q(2,l,k);
q22(k)=Q(2,2, k);
q(k)=sqrt(((q22(k)-qll(k))A2)+(4*ql2(k)*q21(k)));
n ll(k ) =N(1,1, k);
nl2(k) =N(1,2, k);
n21(k)=N(2,l,k);
n22 (k) =N(2,2, k);
n(k)=sqrt(((n22(k)-nl l(k))A2)+(4*nl2(k)*n21(k)));
m l l(k) =M(1,1, k);
m l2(k) =M(1,2, k);
m21(k)=M(2,l,k);
m22(k) =M(2,2,k);
m (k)=sqrt(((m 22(k)-m ll(k))A2)+(4*ml2(k)*m21(k)));
o ll(k )= 0 (l,l,k );
ol2(k)= 0(l,2,k);
o21(k)= 0(2,l,k);
o22(k)=0(2,2,k);
o(k) =sqrt(((o22(k)-ol 1 (k))A2)+(4 *ol2(k) *o21 (k)));
[v l ( : , k), d l ( : , k)] = eig(M (:,k));
E ll(k )= d l(l,l,k );
E12(k) =dl (2,2,k);
[v2(:,:,k),d2(:,:,k)]=eig(0(:<:,k));
E21(k) =d2( l,l,k);
E22(k)=d2(2,2,k);
E R l(k)= (E 12(k)+ (l/E l 1 (k)))/2;
rldl(k)=-log(ERl(k));
rl(k)= rldl(k)/ll(k);
ER2(k) =(E22(k)+(l/E21(k)))/2;
r2dl (k)=-log(ER2 (k));
r2(k) =r2dl(k)/l2(k);
G(:, :,k)=[rldl(k);r2dl(k)];
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
L(:,:,k)=[ll(k);l2(k)J;
LH (
k)=(conj(L
k)))
V(:,:,k)=[2/3, -1/3; -1/3,2/3];
Q1 (:, :,k) =LH(:, :,k) * V(:, :,k) *G(:, :,k);
Q2(:,:,k)=LH(:,:,k)*V(:,:,k)*L(:,:,k);
r(k)=Ql(:,:,k)/Q2(:,:,k);
alpha(k) =20*log(real(r(k)))/log(10);
beta(k) =imag((r(k) *1 *c)/(2 *pi *freq(k)));
ereff(k) =-((r(k) *1 *c)/(2 *pi *freq(k)))A2;
alpha 1 (k) =20 *log(real(rl (k)))/log(10);
betal (k)=imag((rl (k) *1 *c)/(2*pi*freq(k)));
ereffl(k)= -((rl(k) *1 *c)/(2*pi*freq(k)))A2;
alpha2 (k) =2 0 *log(real (r2(k)))/log(10);
beta(k)=imag((r2(k) *1 *c)/(2 *pi*freq(k)));
ereff2(k)=-((r2(k) *1 *c)/(2*pi*freq(k)))A2;
% CALCULATING ERROR COEFFICIENTS
B l a l (k)—((m l 1 (k)-m22(k))+m(k))/(2 *m21 (k));
B lb l(k )= ((m ll (k)-m22(k))-m(k))/(2 *m21 (k));
C l A l a i (k)= l/(((m l 1 (k)-m22(k))-m(k))/(2 *m21 (k)));
CIA lb l(k )= l/(((m l 1 (k)-m22(k))+m(k))/(2 *m21(k)));
B la2(k)= ((ol 1 (k)-o22(k))+o(k))/(2 *o21 (k));
B lb2(k)= ((ol 1 (k)-o22(k))-o(k))/(2 *o21(k));
C lA la 2 (k)= l/(((o l 1 (k)-o22(k))-o(k))/(2 *o21 (k)));
CIA 1b2(k)= l/(((ol 1 (k)-o22(k))+o(k))/(2 *o21 (k)J);
B estl (k)=(m 12 (k))/(exp(r(k) *11(k))-ml 1(k));
ClAestl(k)=m21(k)/(exp(-r(k)*ll(k))-m22(k));
B estl 1 (k)=(ol2(k))/(exp (r(k) *l2(k))-ol 1 (k));
C lA estl 1 (k)=o21(k)/(exp(-r(k) *l2(k))-o22(k));
B ael (k) =abs(Bestl (k)-B la l (k));
B bel (k)=abs(Bestl (k)-B lbl(k));
i f B ael (k)<Bbel (k)
B ll(k )= B la l(k );
else
B ll(k )= B lb l(k );
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end;
C lA a e l (k) = abs(C lAestl (k)-C !A la l (k));
C l A b e l (k) = abs(C lAestl (k )-C lA lb l (k)J;
i f C lA a e l (k)<Cl.Abel (k)
C lA ll(k )= C lA la l(k );
else
C lA ll(k )= C lA lb l(k );
end;
B ae2(k)-abs(B estl I(k)-Bla2(k));
Bbe2(k) =abs (Bestl 1 (k)-Blb2(k));
i f Bae2(k)<Bbe2(k)
B12(k) =Bla2(k);
else
B12(k)=Blb2(k);
end;
ClAae2(k) = abs(C lAestl 1 (k fC IA la2(k));
CIA be2(k) = abs(C lAestl 1 (k)-ClAlb2(k));
i f ClAae2(k)<ClAbe2(k)
C lA12(k)= C lAla2(k);
else
C lA12(k)= C lAlb2(k);
end;
B 2al(k)= ((nl I(k)-n22(k))+n(k))/(2*nl2(k));
B 2bl (k)=((nl 1 (k)-n22(k))-n(k))/(2 *nl 2 (k));
C lA 2 a l (k) = l/(((nl 1 (k)-n22(k))-n(k))/(2 *nl2(k)));
C lA 2 b l(k )= l/(((n ll (k)-n22(k))+n(k))/(2 *nl2(k)));
B2a2(k)=((ql 1 (k)-q22(k)) +q(k))/(2 *ql2(k));
B2b2(k)=((qll(k)-q22(k))-q(k))/(2*ql2(k));
C lA 2a2(k)= l/(((ql 1 (k)-q22(k))-q(k))/(2 *qJ2(k))j;
C lA 2b2(k)= l/(((ql 1 (k)-q22(k))+q(k))/(2 *ql 2 (k)));
Best2(k) =n21 (k)/(exp(-r(k) *11(k))-n ll (k));
C lAest2(k)= nl2 (k)/(exp (f k ) *ll(k))-n22(k));
Best22 (k)=q21 (k)/(exp(-r(k) *l2(k))-qll (k));
C lAest22(k)= ql2 (k)/(exp(r(k) *l2(k))-q22 (k))
Bael2(k)=abs(Best2(k)-B2al(k));
Bbel2(k)=abs(Best2(k)-B2bl(k));
i f Bael2(k)< Bbel2(k)
B21(k)=B2al(k);
else
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B 21 (k) =B2b 1 (k);
end;
C lA a el 2(k)=abs(ClAest2(k)-ClA2al(k));
C lAbel2(k)= abs(C lAest2(k)-C lA2bl(k));
i f C lA a e l 2(k)<Cl A b e l 2(k)
C lA21(k)= C lA2al(k);
else
C lA 21(k)= C lA 2bl(k);
end;
Bae22(k) =abs(Best22(k)-B2a2(k));
Bbe22(k)=abs(Best22(k)-B2b2(k));
i f Bae22(k)<Bbe22(k)
B22(k)=B2a2(k);
else
B22(k)=B2b2(k);
end;
ClAae22(k) =abs(ClAest22(k)-ClA2a2(k));
C 1Abe22(k)=abs(C 1Aest22(k)-C 1A2b2(k));
i f ClAae22(k)<ClAbe22(k)
ClA22(k)=ClA2a2(k);
else
ClA22(k)=ClA2b2(k);
end;
B l e ( : ,k ) =[B11 (k);Bl 2 (k)];
B2e(:, :,k) =[B21 (k);B22(k)];
CIA le(:, :,k) =[C1A 11 (k);ClA12(k)];
ClA2e(:, :,k)=[ClA21 (k);ClA22(k)];
h(:,:,k)=[l ;1];
h T (:,k)= (co n j (h (:, :,k)))';
V b ll(k )=(abs (exp (-r(k) *11 (k)))A2+(l/abs(exp(r(k) *11(k)))A2) +2 *((abs(exp(-r(k) *11(k))))A2))/((abs(exp(-r(k) *11(k))
exp(r(k) *11(k))))A2);
Vb22(k)=(abs(exp(-r(k)*ll(k)))A2+(l/abs(exp(r(k) *11 (k)))A2) +2 *((abs(exp(-r(k) *11(k))))A2))/((abs(exp(-r(k) *11(kf)
exp(r(k) *11(k))))A2);
Vbl 2(k)=((exp(-r(k) *11 (k)) *conj(exp(-r(k) *ll(k))))+(exp(r(k) *11(k)) *conj(exp(-r(k) *l2(k)))))/((exp(-r(k) *11(k))exp(r(k) *11(k))) *conj(exp(-r(k) *l2(k))-exp(r(k) *l2(k))));
Vb21 (k) =conj(Vbl2(k));
Vb(:,:,k)=[Vbll(k), Vbl2(k);Vb21(k), Vb22(k)];
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R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Vcl 1 (k)-(abs(exp(-r(k) *11 (k)))A2+(l/abs(exp(r(k) *11(k)))A2)+(2/((abs(exp(-r(k) *11 (k))))A2)))/((abs(exp(-r(k) *11 (k))exp(r(k)*ll(k))))A2);
Vc22(k)—(abs(exp(-r(k)*ll(k)))A2+(l/abs(exp(r(k) *11 (k)))A2)+(2/((abs(exp(-r(k) *11(k))))A2)))/((abs(exp(-r(k) *11(k))exp(r(k) *ll(k))))A2);
Vcl 2(k)=((l/(exp(-r(k) *11(k))*conj(exp(-r(k)*ll (k)))))+(l/(exp(r(k) *11(k)) *conj(exp(-r(k) *12(k))))))/((exp(-r(k) *11 (k))exp(r(k) *11 (k))) *conj(exp(-r(k) *12(k))-exp(r(k) *12(k))));
Vc21(k) -c o n j (Vcl 2(k));
V c (:,k ) -[ V c l 1 (k), Vcl2(k);Vc21(k), Vc22(k)];
X I (:, :,k) =hT(:, :,k)*Vb(:, :,k) *Ble(:, :,k);
X2(:, :,k) =hT(:, :,k)*Vb(:, :,k) *h(:, :,k);
B1 (k) =X1 (:, :,k)/X2(:, :,k);
X3(:, :,k) =hT(:, :,k)*Vb(:, :,k) *B2e(:, :,k);
B2(k) = X 3(:,k)/X 2(:, :,k);
X4(:,:,k)=hT(:, :,k)*Vc(:, :,k) *C lAle(:, :,k);
X5(:, :,k) =hT(:, :,k) * Vb(:, :,k) *h(:, :,k);
CIA 1 (k) =X4(:, :,k)/X5(:,:,k);
X6(:, :,k)~hT(:, :,k)*Vb(:, :,k) *ClA2e(:, :,k);
ClA2(k) =X6(:, :,k)/X5(:, :,k);
Ap(k)=-(Bl(k) *(B2(k))-Bl(k) *T22(k)-B2(k) *T1 l(k)+ D elT(k))/(lC lA l(k) *T1 l(k)-C lA 2(k) *T22(k)+ClAl(k) *ClA2(k) *DelT(k));
Ar(k) =((R11 (k)-Bl (k)) *(1-R22(k) *(ClA2(k))))/((R22(k)-B2(k)) *(1-
(Rll(k)*(ClAl(k)))));
Ttrial(k)-(R11(k)-B 1(k))/(sqrt(Ap(k) *Ar(k)) *(1 -R l 1(k) *(C1A 1 (k))));
Test=-1;
Tval (k)=abs(((Test/abs(Test))-(Ttrial (k)/abs (Ttrial (k)))));
ifTval(k)>sqrt(2)
A 1(k) =-sqrt(Ap(k) *Ar(k));
else
A l(k) =sqrt(Ap(k) *Ar(k));
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
end;
A2(k)=Al (k)/A r(k);
C l (k) =A l(k)* C lA l(k);
C2(k)=A2(k)*ClA2(k);
Rl(k)= l/(((T21(k) *T12(k))A0.25) *sqrt(l+ C !A l(k) *ClA2(k) *Al(k) *A2(k))
);
R2(k)=Rl(k) *sqrt(T12 (k)/T2 l(k));
S12B(k)=1/R2(k);
S21A(k)= l/Rl(k);
S llA (k)= B l(k);
S I lB(k)=B2(k);
S22A(k)=-Cl(k);
S22B(k)=-C2(k);
S I 2A (k)=(A l(k)+ (SI 1A (k) *S2 2A (k)))/S21A (k);
S21B(k)=(A2(k)+(S1 lB (k) *S22B(k)))/S12B(k);
% EXTRA CT1NG D U T PARAMETERS FRO M ERROR BOXES
X (:,:,k)= R l(k)*[A l(k),B l(k);C l(k),l];
Ybar(:, :,k) =R2(k) *[A2(k), C2(k);B2(k), 1];
DelD(k) =S11D1 (k) *S22D1 (k)-Sl 1D1 (k) *S21D1 (k);
AD(k)=-DelD(k);
RD (k)= l/S21D l (k);
BD (k)=Sl lD l(k);
CD(k)=(-l)*S22Dl(k);
Me(:, :,k) =RD(k) *[AD(k),BD(k);CD(k), 1];
D(:, :,k) =inv(X(:, :,k)) *Me(:, :,k) *inv(Ybar(:, :,k));
S21d(k)=1/D(2,2, k);
S22d(k) =-D(2, l,k) *S21d(k);
SI ld(k) =D(l,2,k) *S21d(k);
Del(k) =-D(l, l,k) *S21d(k);
S12d(k)= (Slld(k) *S22d(k)-Del(k))/S21d(k);
Sllr(k)= 20*log(abs(Tl 1 (k)));
S21r(k) =20 *log(abs(T21 (k)j);
S12r(k)=20*log(abs(T12(k)));
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R e p r o d u c e d w ith p e r m issio n o f th e c o p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
S22r(k)=20*log(abs(T22(k)));
S I la(k) =angle(Sl ld(k));
i f sign(Sl la(k))= = -l
S I langflc) =-Sl la(k);
else
S I lang(k) = Slla(k);
end;
S22a(k)=angle(S22d(k));
i f sign(S22a(k))==-l
S22ang(k) =-S22a(k);
else
S22ang(k) =S22a(k);
end;
end;
% SMOOTHING THE MEASUREMENTS
fo r t= (yl+ l)/2 :l:y2 -(yl-l)/2
b=0;
fo r p = t-(y l-l)/2 :l:t+ (y l-l)/2
a= Slld(p);
b=a+b;
end;
S I l(t)= b /y l;
end;
fo r t= l:l:(y l-l)/2
S ll(t)= S ll((y l+ l)/2 );
end;
fo r t= (y2-(yl-l)/2)+ l:l:y2
S I 1 (t) =S11 (y2-(yl-l)/2);
end;
fo r t= (yl+ J)/2:l:y2-(y]-l)/2
b=0;
fo r p = t-(y l-l)/2 :l:t+ (y l-l)/2
a=S21d(p);
b=a+b;
end;
S21(t)=b/yl;
end;
fo r t= l:l:(y l-l)/2
S21 (t)=S2J((yl +1)/2);
end;
fo r t= (y2-(yl-l)/2)+ l:l:y2
S21(t)=S21(y2-(yl-l)/2);
end;
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
fo r t= (yl+ J)/2:l:y2 -(yl-l)/2
b=0;
fo r p = i-(y l-l)/2 :l:t+ (y l-l)/2
a=Sl2d(p);
b=a+b;
end;
S12(t)=b/yl;
end;
fo r t= l:l:(y l-l)/2
S12(t)=S12((yl+l)/2);
end;
fo r t= (y2-(yl-l)/2)+ 1:1 :y2
S12(t)= Sl2(y2-(yl-l)/2);
end;
fo r t= (yl+ l)/2 :l:y 2 -(y l-l)/2
b=0;
fo r p = t-(y l-l)/2 :l:t+ (y l-l)/2
a=S22d(p);
b=a+b;
end;
S22(t)=b/yl;
end;
fo r t= l:l:(y l-l)/2
S22(t)=S22((yl+l)/2);
end;
fo r t= (y2-(yl-l)/2)+ l:l:y2
S22(t) =S22(y2-(yl-l)/2);
end;
fo r t= (yl+ l)/2 :l:y 2 -(y l-l)/2
b=0;
fo r p = t-(y l-l)/2 :l:i+ (y l-l)/2
a= Sl lD l(p );
b=a+b;
end;
S I lD (t)= b/yl;
end;
fo r t= l:1 :(yl-l)/2
S I lD (t)= Sl lD ((y l +l)/2);
end;
fo r t= (y2-(yl-l)/2)+ 1:1 :y2
S llD (t) =S1 lD (y2-(yl-l)/2);
end;
103
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
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