A Low-Power Radar Imaging System, 2 Edition By Gregory Louis Charvat

A Low-Power Radar Imaging System, 2nd Edition
By
Gregory Louis Charvat
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Electrical and Computer Engineering
2007
ABSTRACT
A Low-Power Radar Imaging System, 2nd Edition
By
Gregory Louis Charvat
A near real-time radar-based imaging system is developed in this dissertation.
This system uses the combination of a spatially diverse antenna array, a high sensitivity range-gated frequency-modulated continuous wave (FMCW) radar system,
and an airborne synthetic aperture radar (SAR) imaging algorithm to produce near
real-time high resolution imagery of what is behind a dielectric wall. This system
is capable of detecting and providing accurate imagery of target scenes made up
of objects as small as 6 inch tall metallic rods and cylinders behind a 4 inch thick
dielectric slab. A study is conducted of through-dielectric slab imaging by the development of a 2D model of a dielectric slab and cylinder. The SAR imaging algorithm
is developed and tested on this model for a variety of simulated imaging scenarios
and the results are then used to develop an unusually high sensitivity range-gated
FMCW radar architecture. An S-band rail SAR imaging system is developed using
this architecture and used to image through two different dielectric slabs as well as
free-space. All results are in agreement with the simulations. It is found that freespace target scenes could be imaged using low transmit power, as low as 5 picowatts.
From this result it was decided to develop an X-band front end which mounts directly
on to the S-band rail SAR so that objects as small as groups of pushpins and aircraft
models in free-space could be imaged. These results are compared to previous Xband direct conversion FMCW rail SAR work. It was found that groups of pushpins
and models could be imaged at transmit powers as low as 10 nanowatts. A spatially
diverse S-band antenna array will be shown to be developed for use with the S-band
radar; thereby providing the ability for near real-time SAR imaging of objects behind
dielectric slabs with the same performance characteristics of the S-band rail SAR.
The research presented in this dissertation will show that near real-time radar imaging through lossy-dielectric slabs is accomplished when using a highly sensitive radar
system located at a stand-off range from the slab using a free-space SAR imaging
algorithm.
Copyright by
Gregory Louis Charvat
2007
For my parents, Dave and Rita Charvat
v
ACKNOWLEDGMENTS
I would like to acknowledge my parents Dave and Rita Charvat, who encouraged
me in my interest of electronics at an early age. I would like to acknowledge my
grandparents, Bud and Jane Charvat and Girard and Therese Nefcy (now deceased)
for their thoughts and support.
I would like to thank my Junior High School shop teacher Kerry Pytel, who taught
me that large projects are feasible if I put my mind to it. I would like to acknowledge
the Amateur Radio community and my parents for suggesting it to me. Without this
interest I would not have been able to gain an intuitive understanding of RF design
principles at an early age. I credit Ardis Herrold for introducing me to research in
general by sponsoring me as a competitor in the International Science and Engineering
Fair in High School. And my friend Dr. Daniel Fleisch, who hired me as an intern at
Aeroflex Lintek in Powell Ohio when I was a Junior in High School. It was from Dan
that I realized the Ph.D. was the career path for me.
I acknowledge Dr. Charles MacCluer who helped me to acquire my first independent study at Michigan State University, which began my working relationship with
the Electrical and Computer Engineering Department.
A special thanks to Dr. Chris Coleman who introduced me to the Electromagnetics
Group and encouraged the EM group to hire me as a graduate research assistant.
Chris was also responsible for introducing me to the field of SAR imaging and has
continued to help me with this over the years.
Most importantly I would like to acknowledge my advisor, Dr. Leo Kempel. Thank
vi
you for all of your help and support on this research and other projects that we
developed over the years. I would like to thank Dr. Kempel, Dr. Ed Rothwell, and
Dr. Shanker Balasubramaniam for introducing me to the study of electromagnetic
theory and causing me to become genuinely interested.
I acknowledge Dr. Eric Mokole, head of the Surveillance Technology Branch, Radar
Division, at the Naval Research Laboratory in Washington D.C. for initiating the
working relationship and providing the funding for this research effort.
I would like to thank the Dow Chemical Company for sponsoring my graduate
assistantship for no fewer than two years and introducing me to the patent process.
I would like acknowledge the Air Force Research Laboratory, who has also been an
important sponsor of my work.
I would like to acknowledge the Michigan State University Electromagnetics Research Group for providing outstanding facilities, equipment, and assistance on all of
the projects that I have developed at MSU. It has been an honor to work with so
many talented people, many of whom will be lifetime friends.
vii
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . xxii
CHAPTER 1
Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1
Previous work in through-lossy-dielectric slab imaging radar system
design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Previous work in through-lossy-dielectric slab imaging algorithms . .
4
1.3
Previous work in through-dielectric slab theoretical imaging models .
5
1.4
Previous work in FMCW radar design . . . . . . . . . . . . . . . . .
5
1.5
Previous work in spatially diverse antenna array design . . . . . . . .
6
1.6
Discussion of research to be presented . . . . . . . . . . . . . . . . . .
7
CHAPTER 2
Synthetic Aperture Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1
Data collection geometry . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
The Range Migration Algorithm (RMA) . .
2.2.1 Simulation of a single point scatterer
2.2.2 Cross range Fourier transform . . . .
2.2.3 Matched filter . . . . . . . . . . . . .
2.2.4 Stolt interpolation . . . . . . . . . .
2.2.5 Inverse Fourier transform in to image
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2.3
Simulation of multiple point scatterers . . . . . . . . . . . . . . . . .
12
CHAPTER 3
Scattering from a Perfect Electric Conducting Cylinder . . . . . . . . . . . . .
3.1 T M z scattering solution to a 2D PEC cylinder . . . . . . . . . . . . .
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3.2
Echo width of a 2D PEC cylinder . . . . . . . . . . . . . . . . . . . .
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3.3
Range profiles of a 2D PEC cylinder . . . . . . . . . . . . . . . . . .
31
3.4
SAR image of a the 2D PEC cylinder model . . . . . . . . . . . . . .
32
viii
CHAPTER 4
A Dielectric Sheet Model Using Wave Matrices . . . . . . . . . . . . . . . . .
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4.1
Reflection and transmission at a dielectric boundary . . .
4.1.1 Incident, reflected and transmitted electric fields .
4.1.2 Incident, reflected and transmitted magnetic fields
4.1.3 Boundary conditions at an air-dielectric interface
4.1.4 Transmission and reflection coefficients . . . . . .
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4.2
Wave matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
4.3
A dielectric sheet model based on wave matrices . . . . . . . . . . . .
56
4.4
Simulated range profiles of the dielectric sheet model . . . . . . . . .
63
4.5
Simulated SAR image of the dielectric slab model . . . . . . . . . . .
65
CHAPTER 5
Simulation of a through-dielectric slab radar image . . . . . . . . . . . . . . .
80
5.1
Simulated range profiles of a 2D PEC cylinder behind a dielectric slab
80
5.2
Simulation of a through-dielectric slab radar image . . . . . . . . . .
82
5.3
Simulation of a through-dielectric slab radar image using coherent
background subtraction . . . . . . . . . . . . . . . . . . . . . . . . . .
87
CHAPTER 6
Radar System Design Requirements and Theoretical Analysis . . . . . . . . .
97
6.1
Flash and through-lossy-dielectric slab attenuation
6.2
The necessity of a range gate
6.3
In-line attenuator approximation . . . . . . . . . . . . . . . . . . . . 99
6.3.1 Comparison of through-lossy-dielectric slab and free-space
range profile results . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3.2 Lack of multi-bounce in simulated range profile results . . . . 100
6.3.3 Comparison of lossly and lossless dielectric range profiles . . . 100
6.3.4 In-line attenuator approximation applied to theoretical range
profile results . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4
Using the RMA for through-dielectric slab imaging . . . . . . . . . .
6.4.1 Comparison of simulated free-space and through-dielectric slab
SAR imagery . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Comparison of simulated free-space and through-losslessdielectric slab SAR imagery . . . . . . . . . . . . . . . . . . .
6.4.3 Simulated offset through-lossy-dielectric slab imagery . . . . .
6.4.4 Offset through-lossless-dielectric slab imagery . . . . . . . . .
6.4.5 Summary of using the RMA for through-dielectric slab imaging
6.5
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97
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Summary of the general design requirements . . . . . . . . . . . . . . 104
ix
CHAPTER 7
High Sensitivity Range-Gated FMCW Radar Architecture . . . . . . . . . . . 122
7.1
Noise bandwidth and receiver sensitivity . . . . . . . . . . . . . . . . 122
7.2
Time domain range-gate and receiver sensitivity . . . . . . . . . . . .
7.2.1 Example of a 40 nS time domain range-gate in a pulsed IF radar
system ranging a target through a dielectric slab . . . . . . . .
7.2.2 Example of a 20 nS time domain range-gate in a pulsed IF radar
system ranging a target through a dielectric slab . . . . . . . .
7.2.3 Time domain range-gates and their limitations . . . . . . . . .
123
124
125
125
7.3
High sensitivity range-gated FMCW radar architecture . . . . . . . . 126
7.3.1 FMCW radar . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.3.2 High sensitivity range-gated FMCW radar system design theory 127
7.4
High sensitivity range-gated FMCW radar architecture conclusions and
advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
CHAPTER 8
S-Band Through-Dielectric Slab Rail SAR Imaging System . . . . . . . . . . . 142
8.1
Radar control and data acquisition . . . . . . . . . . . . . . . . . . . 142
8.2
S-band transmitter signal chain . . . . . . . . . . . . . . . . . . . . . 144
8.3
S-band receiver signal chain . . . . . . . . . . . . . . . . . . . . . . . 145
8.4
Calibration and background subtraction . . . . . . . . . . . . . . . . 147
8.5
S-band SAR data processing . . . . . . . . . . . . . . . . . . . . . . . 148
CHAPTER 9
Through-Dielectric and Free-Space S-Band Rail SAR Imaging Results . . . . . 165
9.1
Free-space imaging results . . . . . . . . . . . . . . . . . . . . . . . . 165
9.1.1 Comparison of measured and simulated free-space cylinder imagery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
9.1.2 Free-space SAR imagery of various targets . . . . . . . . . . . 166
9.2
Low power free-space imaging results . . . . . . . . . . . . . . . . . . 172
9.2.1 Comparison of measured and theoretical low power free-space
cylinder imagery . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.2.2 Low power free-space SAR imagery of various targets . . . . . 173
9.3
Through-lossy-dielectric slab imaging results . . . . . . . . . . . . . . 178
9.3.1 Comparison of measured and simulated through-lossy-dielectric
slab imagery of cylinders . . . . . . . . . . . . . . . . . . . . . 178
9.3.2 Through-lossy-dielectric slab SAR imagery of various targets . 179
9.4
Through an unknown lossy-dielectric slab imaging results . . . . . . . 187
9.5
Low-power through an unknown lossy-dielectric slab imaging results . 191
x
9.6
Discussion of S-band rail SAR imaging results . . . . . . . . . . . . . 194
CHAPTER 10
X-Band Rail SAR Imaging System . . . . . . . . . . . . . . . . . . . . . . . . 195
10.1 High sensitivity range-gated FMCW X-band radar front end . . . . . 195
10.2 Free-space X-band imaging . . . . . . . . . . . . . . . . . . . . . . . . 202
10.3 Low power free-space X-band imaging . . . . . . . . . . . . . . . . . . 207
10.4 Comparison of high sensitivity range-gated FMCW to a typical FMCW
radar imaging system . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
10.5 Discussion of X-band rail SAR imaging results . . . . . . . . . . . . . 213
CHAPTER 11
High Speed SAR Imaging Array . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11.1 SAR on an array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11.2 Array implementation . . . . . . . . . . .
11.2.1 Array antenna spacing and physical
11.2.2 Array implementation and interface
11.2.3 Processing and control software . .
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11.3 Array measured data . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.1 Comparison of free-space simulations and measured data . . .
11.3.2 Free-space imagery . . . . . . . . . . . . . . . . . . . . . . . .
11.3.3 Comparison of through-dielectric slab simulations and measured data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3.4 Through-dielectric slab imagery . . . . . . . . . . . . . . . . .
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11.4 Discussion of the high speed SAR imaging array . . . . . . . . . . . . 252
CHAPTER 12
Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
APPENDIX A
MATLAB code for simulating multiple point scatterers . . . . . . . . . . . . . 257
APPENDIX B
MATLAB code for opening SAR data to be processed by the RMA . . . . . . 261
APPENDIX C
RMA MATLAB code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
APPENDIX D
MATLAB code for calculating PEC cylinder echo width and range profiles . . 273
xi
APPENDIX E
MATLAB code for simulating SAR data of the 2D PEC cylinder model . . . . 278
APPENDIX F
MATLAB code for calculated a simulated range profile of a lossy-dielectric slab 281
APPENDIX G
MATLAB code for simulating rail SAR data of the dielectric slab model . . . 285
APPENDIX H
MATLAB code for acquiring a simulated range profile of a PEC cylinder on the
opposite side of a dielectric slab . . . . . . . . . . . . . . . . . . . . . . . . . . 290
APPENDIX I
MATLAB code for simulating SAR data of a PEC cylinder on the opposite side
of a dielectric slab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
APPENDIX J
MATLAB code for opening measured calibration and SAR data . . . . . . . . 301
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
xii
LIST OF TABLES
Table 3.1
Summary of range profile results. . . . . . . . . . . . . . . . . . .
33
Table 4.1
Parameters and specifications used for simulating range profiles of
the dielectric slab. . . . . . . . . . . . . . . . . . . . . . . . . . .
68
A simulated SAR image of the dielectric slab was calculated using
these parameters and specifications. . . . . . . . . . . . . . . . . .
68
The substitutions shown here were used to simulate range profiles
of a cylinder behind a dielectric slab. . . . . . . . . . . . . . . . .
86
The substitutions shown here were used to simulate SAR image
data of a cylinder behind a dielectric slab. . . . . . . . . . . . . .
86
Table 4.2
Table 5.1
Table 5.2
Table 8.1
S-band modular component list. . . . . . . . . . . . . . . . . . . . 155
Table 8.2
S-band modular component list (continued). . . . . . . . . . . . . 156
Table 10.1 X-band front end modular components list. . . . . . . . . . . . . 199
Table 11.1 High speed SAR imaging array modular component list. . . . . . 230
Table 11.2 32 bit hex word for communicating with the switch matrix (1 of 2). 232
Table 11.3 32 bit hex word for communicating with the switch matrix (2 of 2). 232
Table 11.4 High speed SAR array hex look-up table (1 of 2). . . . . . . . . . 232
Table 11.5 High speed SAR array hex look-up table continued (2 of 2). . . . 233
xiii
LIST OF FIGURES
Figure 2.1
Rail SAR data collection geometry. . . . . . . . . . . . . . . . . .
14
Figure 2.2
Real values of the SAR data matrix for a single point scatterer. .
15
Figure 2.3
Phase of the SAR data matrix for a single point scatterer. . . . .
16
Figure 2.4
Magnitude of simulated point scatterer after downrange DFT,
showing the wave-front curvature of the point scatterer. . . . . . .
17
Figure 2.5
Magnitude after the cross range DFT. . . . . . . . . . . . . . . .
18
Figure 2.6
Phase after the cross range DFT. . . . . . . . . . . . . . . . . . .
19
Figure 2.7
Phase after the matched filter. . . . . . . . . . . . . . . . . . . . .
20
Figure 2.8
Magnitude after 2D DFT of matched filtered data. . . . . . . . .
21
Figure 2.9
Phase after Stolt interpolation. . . . . . . . . . . . . . . . . . . .
22
Figure 2.10 A subsection of the phase after Stolt interpolation. . . . . . . . .
23
Figure 2.11 SAR image of a simulated point scatterer. . . . . . . . . . . . . .
24
Figure 2.12 SAR image three simulated point scatterers. . . . . . . . . . . . .
Figure 3.1 T M z incident plane wave on the PEC cylinder geometry. . . . . .
Figure 3.2 2D T M z incident plane wave on the PEC cylinder geometry. . . .
Figure 3.3 Bistatic echo width of a 2D PEC cylinder with T M z plane wave
25
Figure 3.4
Figure 3.6
Figure 3.7
Figure 3.8
Figure 3.9
Figure 4.1
34
incident electric field observed at a distance of 20 ft from the cylinder. 35
Range profile of a 23.62 inch radius cylinder with T M z plane wave
incidence, showing the downrange time delay of the front face of
the cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 3.5
33
Real value of the range profile of a 23.62 inch radius cylinder with
T M z plane wave incidence. . . . . . . . . . . . . . . . . . . . . .
Range profile of a 1.96 inch radius cylinder with T M z plane wave
36
37
incidence, showing the downrange time delay of the front face of
the cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
Real value of the range profile of a 1.96 inch radius cylinder with
T M z plane wave incidence. . . . . . . . . . . . . . . . . . . . . .
39
SAR image of the 2D cylinder model with a radius of a = 3 inches,
located 30 ft downrange. . . . . . . . . . . . . . . . . . . . . . . .
40
SAR image of the 2D cylinder model with a radius of a = 6 inches,
located 30 ft downrange. . . . . . . . . . . . . . . . . . . . . . . .
41
Incident, reflected and transmitted fields from an air-dielectric interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
xiv
Figure 4.2
Incident, reflected and transmitted fields from an air-dielectric interface represented by wave amplitude coefficients. . . . . . . . .
71
Figure 4.3
The wave matrix geometry for multiple dielectric layers. . . . . .
72
Figure 4.4
The air-dielectric-air geometry for oblique plane wave incidence. .
73
Figure 4.5
Conductivity of the lossy-dielectric slab model, where r = 5.
74
Figure 4.6
Geometry for simulated range profile data of a lossy-dielectric slab. 75
Figure 4.7
Range profile of a 3.94 inch thick simulated slab at incidence angle
φi = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
Range profile of a 3.94 inch thick simulated slab at incidence angle
φi = π
6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
Simulated SAR imaging geometry of slab only. . . . . . . . . . . .
78
Figure 4.10 SAR image of a 4 inch thick lossy-dielectric slab model. . . . . . .
79
Figure 5.1
Geometry of simulated range profile data. . . . . . . . . . . . . .
89
Figure 5.2
Range profile of a lossy-dielectric slab in front a 3 inch radius cylinder at normal incidence. . . . . . . . . . . . . . . . . . . . . . . .
90
Range profile of a lossy-dielectric slab in front a 3 inch radius cylinder at an oblique incidence. . . . . . . . . . . . . . . . . . . . . .
91
Figure 5.4
Simulated SAR imaging geometry. . . . . . . . . . . . . . . . . .
92
Figure 5.5
SAR image of a simulated target scene made up of a 3 inch radius
cylinder target behind a 4 inch thick lossy-dielectric slab. . . . . .
93
SAR image of a simulated target scene made up of a 6 inch radius
cylinder target behind a 4 inch thick lossy-dielectric slab. . . . . .
94
SAR image of a simulated target scene made up of a 3 inch radius
cylinder target behind a 4 inch thick lossy-dielectric slab using
background subtraction. . . . . . . . . . . . . . . . . . . . . . . .
95
SAR image of a simulated target scene made up of a 6 inch radius
cylinder target behind a 4 inch thick lossy-dielectric slab using
background subtraction. . . . . . . . . . . . . . . . . . . . . . . .
96
Figure 4.8
Figure 4.9
Figure 5.3
Figure 5.6
Figure 5.7
Figure 5.8
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Figure 6.1
Simulated range profiles of a cylinder with radius a = 3 inches in
free-space (a) and behind a 4 inch thick lossy-dielectric slab (b). . 106
Figure 6.2
Simulated range profiles of a cylinder with radius a = 6 inches in
free-space (a) and behind a 4 inch thick lossy-dielectric slab (b). . 107
Figure 6.3
Simulated range profiles of a cylinder with radius a = 3 inches in
free-space (a) and behind a 4 inch thick lossless-dielectric slab (b). 108
Figure 6.4
Simulated range profiles of a cylinder with radius a = 3 inches in
free-space (a) and behind a 12 inch thick lossy-dielectric slab (b).
xv
109
Figure 6.5
Simulated range profiles of a cylinder with radius a = 3 inches in
free-space (a) and behind a 12 inch thick lossless-dielectric slab (b). 110
Figure 6.6
Simulated SAR imagery of a 2D cylinder with radius a = 3 inches
in free-space (a) and behind a 4 inch thick lossy-dielectric slab
using background subtraction (b). . . . . . . . . . . . . . . . . . . 111
Figure 6.7
Simulated SAR imagery of a 2D cylinder with radius a = 6 inches
in free-space (a) and behind a 4 inch thick lossy-dielectric slab
using background subtraction (b). . . . . . . . . . . . . . . . . . . 112
Figure 6.8
Simulated SAR imagery of a 2D cylinder with radius a = 3 inches
in free-space (a) and behind a 4 inch thick lossless-dielectric slab
using background subtraction (b). . . . . . . . . . . . . . . . . . . 113
Figure 6.9
The 2D PEC cylinder is offset in cross range from the center of
the rail to show the theoretical effects of an offset target behind a
dielectric slab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Figure 6.10 Simulated SAR imagery of a 2D cylinder offset by approximately
2 feet with radius a = 3 inches in free-space (a), behind a 4 inch
thick lossy-dielectric slab using background subtraction (b). . . . 115
Figure 6.11 Simulated SAR imagery of a 2D cylinder offset by approximately
2 feet with radius a = 3 inches in free-space (a), behind a 4 inch
thick lossless-dielectric slab using background subtraction (b). . . 116
Figure 6.12 Simulated SAR imagery of a 2D cylinder with radius a = 3 inches
in free-space (a) and behind a 12 inch thick lossy-dielectric slab
using background subtraction (b). . . . . . . . . . . . . . . . . . . 117
Figure 6.13 Simulated SAR imagery of a 2D cylinder with radius a = 3 inches
in free-space (a) and behind a 12 inch thick lossless-dielectric slab
using background subtraction (b). . . . . . . . . . . . . . . . . . . 118
Figure 6.14 Simulated SAR imagery of a 2D cylinder offset by approximately
2 feet with radius a = 3 inches in free-space (a), behind a 12 inch
thick lossy-dielectric slab using background subtraction (b). . . . 119
Figure 6.15 Simulated SAR imagery of a 2D cylinder offset by approximately
2 feet with radius a = 3 inches in free-space (a), behind a 12 inch
thick lossless-dielectric slab using background subtraction (b). . . 120
Figure 6.16 A simple attenuation model for the lossy-dielectric slab for use in
determining system specifications. . . . . . . . . . . . . . . . . . . 121
Figure 7.1
Block diagram of a typical coherent pulsed IF radar system where
the IF bandwidth must be wide enough to capture the returned
pulse from the target scene. . . . . . . . . . . . . . . . . . . . . . 136
Figure 7.2
Side view of a typical through-dielectric slab imaging geometry. . 137
xvi
Figure 7.3
Example of a 40nS range-gate where the slab is located at d1 =
20 feet down range, target is located d3 = 60 nS down range and
mulipath (from slab to radar and back again) is shown 80 nS down
range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Figure 7.4
Example of a 20nS range-gate where the slab is located at d1 =
10 feet down range, target is located d3 = 30 nS down range and
mulipath (from slab to radar and back again) is shown 40 nS down
range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Figure 7.5
A direct conversion FMCW radar system. . . . . . . . . . . . . . 140
Figure 7.6
Simplified block diagram of the high sensitivity range-gated
FMCW radar system. . . . . . . . . . . . . . . . . . . . . . . . . 141
Figure 8.1
Block diagram of the IF. . . . . . . . . . . . . . . . . . . . . . . . 149
Figure 8.2
Attenuator chassis. . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Figure 8.3
Power splitter and delay line. . . . . . . . . . . . . . . . . . . . . 151
Figure 8.4
S-band transmitter front end. . . . . . . . . . . . . . . . . . . . . 152
Figure 8.5
S-band receiver front end. . . . . . . . . . . . . . . . . . . . . . . 153
Figure 8.6
High level block diagram of the PC connected to motion control
and data acquisition/triggering. . . . . . . . . . . . . . . . . . . . 154
Figure 8.7
Schematic of IF filter switch matrix XTAL Filter Mux1 and XTAL
Filter Mux2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Figure 8.8
Schematic of double balanced mixer module MXR3. . . . . . . . . 158
Figure 8.9
Schematic of the video amplifier VideoAmp1. . . . . . . . . . . . 159
Figure 8.10 Schematic of the Ramp Generator. . . . . . . . . . . . . . . . . . 160
Figure 8.11 The S-band rail SAR imaging system. . . . . . . . . . . . . . . . 161
Figure 8.12 Data conditioning hardware, motion control and power supplies. . 161
Figure 8.13 From the top down: Hewlett Packard HP3325A Synthesizer/Function Generator BFO, the power splitter chassis and the
Weinschel Engineering 430A Sweep Oscillator with the 432A RF
Unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Figure 8.14 The S-band transmitter front end.
. . . . . . . . . . . . . . . . . 163
Figure 8.15 The S-band receiver front end. . . . . . . . . . . . . . . . . . . . . 163
Figure 8.16 Radar IF (a), inside of the radar IF (b). . . . . . . . . . . . . . . 164
Figure 8.17 ATTN1, ATTN2 and ATTN3 mounted on the attenuator chassis.
164
Figure 9.1
Free-space SAR imaging geometry. . . . . . . . . . . . . . . . . . 167
Figure 9.2
SAR imagery of an a = 3 inch radius cylinder in free-space; simulated (a), measured (b). . . . . . . . . . . . . . . . . . . . . . . . 168
xvii
Figure 9.3
SAR imagery of an a = 6 inch radius cylinder in free-space; simulated (a), measured (b). . . . . . . . . . . . . . . . . . . . . . . . 169
Figure 9.4
Measured free-space SAR imagery of an a = 4.3 inch radius sphere
(a), a group of 6 inch tall 3/8 inch diameter carriage bolts in a
block ‘S’ pattern (b). . . . . . . . . . . . . . . . . . . . . . . . . . 170
Figure 9.5
Picture of 6 inch tall 3/8 inch diameter carriage bolts (a), target
scene of carriage bolts in a block ‘S’ pattern (b). . . . . . . . . . . 171
Figure 9.6
Simulated SAR image of an a = 3 inch radius cylinder in free-space
(a), measured SAR image of an a = 3 inch radius cylinder using a
transmit power of 100 picowatts (b). . . . . . . . . . . . . . . . . 174
Figure 9.7
Theoretical SAR image of an a = 6 inch radius cylinder in freespace (a), measured SAR image of an a = 6 inch radius cylinder
using a transmit power of 100 picowatts (b). . . . . . . . . . . . . 175
Figure 9.8
SAR image an a = 4.3 inch radius sphere in free-space using 100
picowatts of transmit power. . . . . . . . . . . . . . . . . . . . . . 176
Figure 9.9
SAR image of a group of carriage bolts in free-space using 10
nanowatts of transmit power. . . . . . . . . . . . . . . . . . . . . 176
Figure 9.10 SAR image of a group of carriage bolts in free-space using 100
picowatts of transmit power. . . . . . . . . . . . . . . . . . . . . . 177
Figure 9.11 SAR image of a group of carriage bolts in free-space using 5 picowatts of transmit power. . . . . . . . . . . . . . . . . . . . . . . . . 177
Figure 9.12 The 4 inch thick lossy-dielectric slab. . . . . . . . . . . . . . . . . 180
Figure 9.13 Through-lossy-dielectric slab image scene where the S-band rail
SAR is located 29.5 feet from the slab. . . . . . . . . . . . . . . . 181
Figure 9.14 Through-slab rail SAR imaging geometry. . . . . . . . . . . . . . 182
Figure 9.15 Theoretical SAR image of an a = 3 inch radius cylinder behind a
4 inch thick lossy-dielectric slab (a), measured SAR image of an
a = 3 inch radius cylinder behind a 4 inch thick lossy-dielectric
slab (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Figure 9.16 Theoretical SAR image of an a = 6 inch radius cylinder behind a
4 inch thick lossy-dielectric slab (a), measured SAR image of an
a = 6 inch radius cylinder behind a 4 inch thick lossy-dielectric
slab (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Figure 9.17 Sphere with a radius of a = 4.3 inches imaged behind a 4 inch thick
lossy-dielectric slab. . . . . . . . . . . . . . . . . . . . . . . . . . 185
Figure 9.18 12 oz aluminum beverage cans in a block ‘S’ configuration imaged
behind a 4 inch thick lossy-dielectric slab. . . . . . . . . . . . . . 185
xviii
Figure 9.19 Diagonal row of three 6 inch tall 3/8 inch diameter carriage bolts
imaged behind a 4 inch thick lossy-dielectric slab. . . . . . . . . . 186
Figure 9.20 Experimental setup for imaging through an unknown lossydielectric slab (a), a close up view of the unknown lossy-dielectric
slab (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Figure 9.21 A radar image of a cylinder with a radius of a = 3 inches behind
the unknown dielectric slab. . . . . . . . . . . . . . . . . . . . . . 189
Figure 9.22 A radar image of a cylinder with a radius of a = 6 inches behind
the unknown dielectric slab. . . . . . . . . . . . . . . . . . . . . . 189
Figure 9.23 A radar image of a sphere with a radius of a = 4.3 inches behind
the unknown dielectric slab. . . . . . . . . . . . . . . . . . . . . . 190
Figure 9.24 A target scene of carriage bolts in a block ‘S’ configuration imaged
behind the unknown dielectric slab. . . . . . . . . . . . . . . . . . 190
Figure 9.25 A cylinder with a radius of a = 3 inches radar imaged behind an
unknown slab using 10 nanowatts of transmit power. . . . . . . . 192
Figure 9.26 A cylinder with a radius of a = 6 inches radar imaged behind an
unknown slab using 10 nanowatts of transmit power. . . . . . . . 192
Figure 9.27 A sphere with a radius of a = 4.3 inches radar imaged behind an
uknown slab using 10 nano-watts of transmit power. . . . . . . . 193
Figure 9.28 A target scene of carriage bolts in a block ‘S’ configuration radar
imaged behind an unknown slab using 10 nanowatts of transmit
power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Figure 10.1 Block diagram of the X-band front end. . . . . . . . . . . . . . . 198
Figure 10.2 Schematic of the module that contains AMP7 and FL14. . . . . . 200
Figure 10.3 X-band front end (a), X-band front end in operation with adjustable transmit attenuator ATTN5 in line (b). . . . . . . . . . . 201
Figure 10.4 The X-band rail SAR and target scene. . . . . . . . . . . . . . . . 204
Figure 10.5 X-band rail SAR image of a 1:32 scale F14 model. . . . . . . . . . 204
Figure 10.6 One pushpin (a), image scene of ‘GO STATE’ in pushpins (b). . . 205
Figure 10.7 X-band rail SAR image of a group of pushpins. . . . . . . . . . . 206
Figure 10.8 Zoomed-out X-band rail SAR image of a group of pushpins. . . . 206
Figure 10.9 X-band rail SAR image of a group of pushpins using 100 nanowatts
of transmit power. . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Figure 10.10X-band rail SAR image of a group of pushpins using 10 nanowatts
of transmit power. . . . . . . . . . . . . . . . . . . . . . . . . . . 208
xix
Figure 10.11X-band rail SAR image of a group of pushpins fading into the noise
using 1 nanowatt of transmit power. . . . . . . . . . . . . . . . . 209
Figure 10.12SAR image of a 1:32 scale model F14 using a direct conversion
FMCW radar system. . . . . . . . . . . . . . . . . . . . . . . . . 211
Figure 10.13SAR image of a group of pushpins using a direct conversion FMCW
radar system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Figure 10.14Zoomed out SAR image of a group of pushpins using a direct conversion FMCW radar system showing the presence of significant
clutter outside the target scene of interest. . . . . . . . . . . . . . 212
Figure 11.1 A radar device electronically switched across an array of evenly
spaced antenna elements is equivalent to a rail SAR. . . . . . . . 216
Figure 11.2 When bi-static transmit and receive elements are close together
(relative to target range) spaced apart by x it is equivalent to a
single mono-static element at the mid point x/2. . . . . . . . . . 217
Figure 11.3 Side view of a simple bi-static antenna array using 48 transmit and
48 receive elements for a total of 96 elements. . . . . . . . . . . . 218
Figure 11.4 A more advanced bi-static antenna array producing 12 mono-static
phase centers at the expense of only 2 receive elements and 6 transmit elements (units in inches). . . . . . . . . . . . . . . . . . . . . 219
Figure 11.5 The high speed SAR imaging array physical layout (all units are
in inches), antenna combinations and phase center locations (1 of 2).225
Figure 11.6 The high speed SAR imaging array physical layout (all units are
in inches), antenna combinations and phase center locations (2 of 2).226
Figure 11.7 The overall radar system block diagram with modifications required to control the antenna array. . . . . . . . . . . . . . . . . . 227
Figure 11.8 Block diagram of the transmitter switch matrix. . . . . . . . . . . 228
Figure 11.9 Block diagram of the receiver switch matrix. . . . . . . . . . . . . 229
Figure 11.10Layout of the LTSA used for both the transmit and receive elements
in the high speed SAR array (units in inches). . . . . . . . . . . . 231
Figure 11.11A receiver LTSA antenna element with LNA mounted on the antenna (a), close up view of LNA mounted at the end of the LTSA
(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Figure 11.12The front of the array showing the vertically polarized LTSA antennas (a), the array at an angle showing the ’V’ of the LTSA
elements pointing out towards the target scene (b). . . . . . . . . 235
Figure 11.13The back side of the array showing the LTSA antenna feeds running
to the switch matrix box mounted on the back of the array. . . . 236
Figure 11.14The near real-time S-band radar imaging system. . . . . . . . . . 237
xx
Figure 11.15SAR imagery of a a = 3 inch radius cylinder in free space: simulated (a), acquired from the high speed array (b). . . . . . . . . . 240
Figure 11.16SAR imagery of a a = 6 inch radius cylinder in free space: simulated (a), acquired from the high speed array (b). . . . . . . . . . 241
Figure 11.17SAR image of an a = 4.3 inch radius sphere in free-space. . . . . . 243
Figure 11.18SAR image of a group of 6 inch long 3/8 inch diameter carriage
bolts in free-space. . . . . . . . . . . . . . . . . . . . . . . . . . . 243
Figure 11.19Image of a 12 oz aluminum beverage can in free-space. . . . . . . 244
Figure 11.20Image of a group of 3 inch tall nails in free-space. . . . . . . . . . 244
Figure 11.21Image of a group of 2 inch tall nails in free-space. . . . . . . . . . 245
Figure 11.22Image of a group of 1.25 inch tall nails in free-space. . . . . . . . 245
Figure 11.23Imagery of an a = 3 inch radius cylinder behind a d = 4 inch thick
lossy-dielectric slab: simulated (a), acquired from the high speed
array (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Figure 11.24Imagery of an a = 6 inch radius cylinder behind a d = 4 inch thick
lossy-dielectric slab: simulated (a), acquired from the high speed
array (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Figure 11.25SAR image of a 12 oz aluminum beverage can through a d = 4
inch thick lossy-dielectric slab. . . . . . . . . . . . . . . . . . . . . 250
Figure 11.26SAR image of a 6 inch tall 3/8 inch diameter carriage bolt through
a d = 4 inch thick lossy-dielectric slab. . . . . . . . . . . . . . . . 250
Figure 11.27SAR image of an a = 4.3 inch radius sphere through a d = 4 inch
thick lossy-dielectric slab. . . . . . . . . . . . . . . . . . . . . . . 251
xxi
KEY TO SYMBOLS AND ABBREVIATIONS
BFO: Beat Frequency Oscillator
BPF: Band Pass Filter
CW: Continuous Wave
DIO: Digital I/O
DFT: Discrete Fourier Transform
FM: Frequency Modulated
FMCW: Frequency Modulated Continuous Wave
HPF: High Pass Filter
IDFT: Inverse Discrete Fourier Transform
I/O: Input/Output
IF: Intermediate Frequency
IFP: Image Formation Processor
LO: Local Oscillator
LPF: Low Pass Filter
LSB: Least Significant Bit
LNA: Low Noise Amplifier
xxii
MDS: Minimum Detectable Signal
PEC: Perfect Electric Conductor
RF: Radio Frequency
RCS: Radar Cross Section
RMA: Range Migration Algorithm
SAR: Synthetic Aperture Radar
TEM: Transverse Electromagnetic
VCO: Voltage Controlled Oscillator
xxiii
CHAPTER 1
INTRODUCTION AND BACKGROUND
A near real-time through-dielectric slab imaging system is developed in this dissertation. This system uses the combination of a high speed antenna array, a high sensitivity range-gated FMCW radar system, and an airborne synthetic aperture radar
(SAR) imaging algorithm to produce near real-time high resolution imagery of what
is behind a dielectric slab. This system is capable of detecting and providing accurate
imagery of target scenes made up of objects as small as 6 inch tall rods and small
cylinders behind a 4 inch thick lossy dielectric slab at a stand-off range of 20 to 30
feet.
A number of research topics are developed in order to accomplish this. In Chapter
2 the Range Migration SAR algorithm is developed and tested on simulated point
scatterers. In Chapters 3 through 5, a 2D through-lossy-dielectric slab SAR imaging
model is developed which simulates a cylinder on the opposite side of a lossy dielectric
slab. The SAR algorithm is tested on this model, with various wall thicknesses,
cylinder diameters, and cylinder locations. These results are explored in detail in
Chapter 6. Based on this analysis an unusually sensitive range-gated FMCW radar
system architecture is developed in Chapter 7. Using this radar architecture, an Sband rail SAR imaging system is developed in Chapter 8. In Chapter 9, through-lossydielectric slab and free space imagery are then acquired using the S-band rail SAR,
where target scenes made up of objects as small as bolts and soda cans are imaged.
Results were found to be in agreement with the theoretical model. One of the more
interesting results from this chapter was the fact that the rail SAR was capable of
imaging free space targets using extremely low amounts of transmit power, down to 5
picowatts. Due to this result, in Chapter 10 it was decided to develop an X-band front
1
end based on the high sensitivity range-gated FMCW radar architecture, and image
target scenes made up of small objects such as pushpins to compare this to previous
work in [60]. It was found that the high sensitivity radar architecture operating at
X-band was in agreement with previous results, and that the new system is capable of
imaging targets as small as pushpins by using only 10 nanowatts of transmit power.
Due to the results of the S-band rail SAR, it was decided to develop a high speed
array to image through-dielectric slabs in near real-time. The development of this
array, with free space and through-lossy-dielectric slab imaging results is presented
in Chapter 11. It was found that the array imaging results were in agreement with
the theoretical model. Future work and conclusions are discussed in Chapter 12.
1.1
Previous work in through-lossy-dielectric slab imaging radar system
design
The research area of through-dielectric slab imaging is very diverse. Numerous approaches have been considered such as the use of millimeter wave radiometers [1],
[2], [3]. Others are using UWB impulse radar at X-band to image through-dielectric
slabs [4], [5], [6]. Still yet another approach is to use random noise radar [7]. Some
researchers choose to focus on algorithm development, and use a vector network analyzer (VNA) as the radar sensor with bi-static transmit and receive antennas mounted
on an XY scanner directly up against the lossy dielectric slab [8]. One of the most
impressive systems for imaging through a dielectric slab is in [9], which is a 2.2 meter
long 4 element bi-static antenna array connected to a 450 MHz to 2 GHz stepped
frequency CW radar transmitting 50 milliwatts of power. This same system was then
modified to operate as a distributed network for imaging through-dielectric slabs in
[10]. Another interesting approach is the use of a 900 MHz through-dielectric slab
doppler radar [11], designed operate as a through slab motion detector. A two receiver channel 2.4 GHz CW radar was developed as a doppler direction-of-arrival
2
system for tracking indoor movers in [14]. This machine tracks targets behind dielectric slabs by the doppler shift of their motion. Research has been conducted on
a robot mounted pulsed-doppler radar motion sensing and ranging system operating
at 5.8 GHz [12], [13], designed to maneuver around a building and detect motion
and range through-dielectric slabs and map out locations of that motion. The Electromagnetics Research Group at Michigan State University has a history of imaging
targets through thin lossy dielectric slabs with UWB impulse radar [15].
The majority of current through-lossy-dielectric slab research has been focused on
UWB impulse radar operating in the 1 to 3 GHz frequency range [16], [17], [18], [19],
[20] and [21]. UWB impulse radar technology is promising because the reflection off
of the dielectric slab can be time gated (or range-gated) out of the data, protecting
data acquisition equipment from being saturated and allowing for maximum system
dynamic range to be applied to the range bins behind the dielectric slab. UWB impulse radar by itself is not very sensitive, and requires the use of coherent integration
to achieve useful signal to noise ratios when attempting to image objects behind dielectric slabs. It is for this reason that there has been much effort in the area of
optimizing coherent integration methods so that more coherent integration occurs at
range bins behind the slab causing a reduction in CPU cycles when processing [22].
Other radar systems for imaging through-dielectric slabs exist that are not presented in publication. These can be found by conducting a simple internet search.
These will not be cited in this dissertation for archival purposes.
The radar systems developed for through-lossy-dielectric slab imaging in this dissertation are different from the majority of previous work in that they are not UWB
impulse radar systems, CW doppler, stepped frequency, or pulsed doppler. The
through-dielectric imaging systems developed in this dissertation are FMCW, using
a modified FMCW radar architecture that allows for hardware range gating without the use of time domain solid state switches or an increased IF bandwidth. The
3
approach taken here is to develop the most sensitive, greatest dynamic range radar
which is fast and range-gated. It will be shown that a high sensitivity range-gated
FMCW radar architecture is developed which achieves this goal.
1.2
Previous work in through-lossy-dielectric slab imaging algorithms
There are a number of interesting through-lossy-dielectric slab algorithms that have
been developed, such as Space Time Focusing, tested on a 3D ray tracing model of a
5 story building in [28]. In this work 3D positions of simulated targets were located
using an array of numerous radar sensors placed around the theoretical building. A
number of microwave tomography and inverse scattering methods have been used to
image cylinders behind lossy-dielectric slabs [29], [30], [31], [32], and dielectric barrier
surfaces behind lossy-dielectric slabs [33]. Most of the through-lossy-dielectric slab
radar systems that have been developed use some sort of beam forming algorithm
[16], [17]. Much of the focus of beam forming algorithm research has been to develop
methods to counter the effects of the dielectric slab on a radar system with antennas
placed directly on the slab [23], [24], [25], [26], [27]. The effects of Snell’s law severely
affect the beam forming algorithms by changing the phase centers of the array elements of these radar systems. This dissertation is different than this previous work in
that the radar systems developed here are designed to be placed at stand-off ranges of
20 to 30 feet or greater from the dielectric slab. It will be shown that placing the radar
systems at this stand-off range does not require special treatment or de-convolution
of the dielectric slab in order to process accurate radar imagery.
The through-dielectric slab imaging radar systems presented in this dissertation
will use SAR imaging algorithms such as those in [34], [35], [36], rather than beamforming techniques used in most other through-slab work. More specifically, the radar
systems presented in this dissertation use the Range Migration Algorithm, which is
an airborne SAR imaging algorithm from [37]. SAR imaging is a specific type of
4
beamforming, and the RMA is one of the more advanced SAR algorithms in use
presently.
1.3
Previous work in through-dielectric slab theoretical imaging models
In the research field of through-dielectric imaging numerous approaches have been
developed to model a lossy dielectric slab in a through-dielectric slab imaging scenario.
These include the use of finite difference time domain (FDTD) [38], [41], geometric
optics (GO) [39] [40], modeling the slab as a transmission line [4], and the use of
electric field integral equation (EFIE) solved using the method of moments (MoM)
for a lossy current sheet which simulates the wall of a wood packaging crate [15].
Some general purpose dielectric models were considered for this research for use in
modeling the dielectric slab. One approach considered was the use of the EFIE and
MoM to model a dielectric slab, by developing a thick version of the models in [42],
[43] and [44] for modeling arbitrarily shaped dielectric cylinders. Some unpublished
work was conducted. The integral equation solution to planar dielectrics [45] (similar
to the work done by [46], and [46]) was considered for some time, and resulted in the
publication of [48]. The Universal Theory Diffraction (UTD) was also explored for
modeling a dielectric [49].
Based on the geometry of the problem it was decided use the plane wave approximation to model the dielectric slab. Wave matrix theory from [63] is used to model
the slab. The solution to a 2D PEC cylinder from [61] is used to model a cylinder
behind the slab.
1.4
Previous work in FMCW radar design
For decades FMCW radar has been used primarily for aircraft radio altimeters where
the FMCW chirp would range the distance from aircraft to ground [50], [51], [52]. It
was not until the late 1980’s and early 90’s that advancements in low cost high speed
5
signal processors allowed for the application of FMCW to more sophisticated applications, where the video signal of the FMCW radar could be digitized and analyzed
using Fourier analysis on a single chip digital signal processor (DSP) [53]. These more
advanced signal processing techniques have led to the use of FMCW radar systems in
many applications such as rail SAR imaging [60] and automotive obstacle avoidance
[55], [56], [57]. Previous FMCW rail SAR imaging systems such as [60] use a direct
conversion FMCW radar architecture. The FMCW radar systems developed in this
dissertation are different from most previous work in that they use a hardware rangegate without electronic high speed RF switches. This unusual range-gate provides
for a high sensitivity receiver without the use of coherent integration or reduction
of chirp rate. This modified FMCW radar architecture will be referred to as high
sensitivity range-gated FMCW.
Imagery from both a direct conversion X-band FMCW rail SAR imaging system in
[60] and an X-band rail SAR utilizing the high sensitivity range-gated FMCW radar
architecture developed in this dissertation will be presented for comparison purposes.
This will show that the high sensitivity range-gated FMCW radar architecture is of
superior performance compared to direct conversion FMCW radar systems.
1.5
Previous work in spatially diverse antenna array design
The array developed for this dissertation is a large switched antenna array, where
various combinations of transmit and receive elements are switched on to the transmitter and receiver to provide spatially diverse radar data for the imaging algorithm.
Many of the previous through-dielectric slab UWB impulse radar systems [4], [5],
[6], [16], [17], [18], [19], [20] and [21], use some type of switched antenna array. In
this setup the transmitter or receiver are switched onto different transmit and receive
elements, acquiring data across different combinations of elements. This multi-static
data acquisition approach allows for the development of spatially diverse radar data,
6
facilitating the use of various radar imaging algorithms and beam forming using the
unique advantages of UWB impulse radar used on an antenna array [54]. These arrays were developed to be very small, designed to be portable or hand held. The
array developed in this dissertation is different in that it is significantly larger than
the hand held systems. This array was developed to be 88 inches long, because of
this, it produces higher cross range resolution than most existing systems using UWB
impulse or even that of other FMCW radar devices used in switched antenna arrays
[55], [56], [57].
1.6
Discussion of research to be presented
The research that will be presented in this dissertation will show that throughdielectric slab radar imaging is entirely feasible by combining the use of airborne
SAR imaging algorithms, an unusualy high sensitivity range-gated FMCW radar design, and a large high speed array to produce near real-time radar imagery of what
is behind an unknown lossy dielectric slab.
7
CHAPTER 2
SYNTHETIC APERTURE RADAR
Synthetic Aperture Radar (SAR) is often used as an airborne imaging sensor [58]
producing nearly photographic radar imagery of a target scene. SAR Imaging can
also be done on a small scale as was shown in [59]. In the laboratory environment,
extremely high resolution SAR imagery can be achieved on a small scale using a wide
bandwidth radar system mounted on a precision linear rail [60]. In this dissertation,
a small rail SAR much like [60] will be developed to SAR image through a dielectric
slab. The small aperture data collection geometry used in this dissertation for both
simulated and measured data is shown in Section 2.1. The Range Migration Algorithm
(RMA) is the SAR imaging algorithm that will be used for all radar imaging in this
dissertation. This algorithm is discussed in Section 2.2. The RMA will be tested and
proven to be effective for imaging simulated SAR data of ideal point scatterers in
Section 2.3.
2.1
Data collection geometry
The rail SAR imaging geometry used for both theoretical and measured data is shown
in Figure 2.1. In this data collection geometry a small radar device is mounted on an
8 ft long automated linear rail. The radar has a small wide beamwidth horn antenna
that is directed out towards an unknown target scene parallel to the rail. The radar
begins its journey at the end of the rail near the drive motor. The drive motor moves
the radar to a location, stops, then the radar acquires a range profile. After the range
profile is complete the drive motor moves the radar to the next known location and
the radar acquires another range profile. This process is repeated at regular intervals
along the rail until the radar device reaches the end of the rail. The range profiles
8
over each rail position produce a 2D data matrix in the form of
s xn , ω(t) ,
where s xn , ω(t) is the 2D frequency domain range profile data matrix, xn is the
nth cross range position of the radar on the automated rail, and
BW
ω(t) = 2π cr t + fc −
2
(2.1)
is the instantaneous radial frequency of the received chirp waveform. In this cr is
the chirp rate of the linear frequency modulated (FM) chirped radar system, fc is
the radar center frequency, and BW is the chirp bandwidth. The SAR rail length is
L, where in the case for all simulations and some measurements L = 8 ft, and the
aperture spacing is ∆x = 2 inches between range profiles across the rail.
It is important to note that a linear FM radar system exactly like [60] will be
assumed for all cases, both theoretical and experimental. As such, the instantaneous
received frequency of the radar system is a function of chirp rate and time. The chirp
rate for a linear FM radar system is
cr =
BW
,
T
(2.2)
where T is the chirp length and BW = fstop −fstart is the transmit bandwidth from
the transmit start frequency to the transmit stop frequency in Hz. In the example
presented in this chapter T = 10 milliseconds. Thus, the chirp rate for all theoretical
and some experimental data is cr = 214.3 GHz/second, where the radar system is
assumed to operate at S-band occupying the frequency range of fstop = 4.069 GHz
and fstart = 1.926 GHz.
9
2.2
The Range Migration Algorithm (RMA)
The data matrix s xn , ω(t) is processed by a SAR imaging algorithm to produce
an image matrix S X, Y , which is the resulting SAR image of the target scene of
interest. One of the more popular SAR imaging algorithms is known as the RMA.
The RMA for this dissertation was developed directly from [37]. The RMA follows
these four processing steps:
1. Cross range discrete Fourier transform (DFT).
2. Matched filter.
3. Stolt interpolation.
4. Inverse discrete Fourier transform (IDFT) into resulting image.
The details of each of these processing steps will be discussed in this section by
example. A single point scatterer is defined and a SAR data matrix is calculated
in Section 2.2.1. The cross range DFT step will be described in Section 2.2.2. The
matched filter process will be described in Section 2.2.3. Section 2.2.4 will describe
how the Stolt interpolation is used for proper frequency domain data formatting.
Section 2.2.5 will describe how to produce an image from the Stolt interpolated data
using the IDFT.
2.2.1
Simulation of a single point scatterer
Using the data collection geometry in Section 2.1 the approximation for a single point
scatterer located located at (xt , yt ) with a reflection magnitude of at creates the SAR
data matrix
ω(t)
−j2 c
s xn , ω(t) = at e
q
(xn −xt )2 +yt2
.
(2.3)
Where c is the speed of light approximately 3 · 108 m/second. Without loss of generality let at = 1. Locating the point scatter at xt = 0 feet cross range from rail
10
center (where x = 0 is the center of the rail) and yt = −10 feet downrange produces a
simulated SAR data matrix. Using these values, this was calculated using the MATLAB program in Appendix A, where the real value of the SAR data matrix points
are shown in Figure 2.2, and the phase of the SAR data matrix points is shown in
Figure 2.3. The downrange DFT of the SAR data matrix is shown in Figure 2.4.
This plot shows the wave-front curvature of the single point scatterer, which is in an
arc because the range to target varies over the aperture length L.
2.2.2
Cross range Fourier transform
The first step in the RMA is to calculate the DFT along the cross range axis of
the SAR data matrix s xn , ω(t) resulting in a wave number domain data matrix
s kx , ω(t) . In addition to the DFT, the simple substitution is made: kr = ω(t)/c.
This results in the SAR data matrix s(kx , kr ). Using the MATLAB programs in
Appendix B and Appendix C, the magnitude after the cross range DFT is shown in
Figure 2.5 and the phase after the cross range DFT is shown in Figure 2.6.
2.2.3
Matched filter
A 2D matched filter is then applied to the s(kx , kr ) matrix. The equation for the
matched filter used in the RMA is
smf (kx , kr ) = e
jRs
q
2
kr2 −kx
,
(2.4)
where Rs is the downrange distance to scene center. In this example Rs = 0, however
in practice using a rail SAR imaging system or a spatially diverse antenna array, Rs is
equal to the range to calibration target, shifting the scene center to a downrange line
located at distance Rs from the rail. Multiplying equation 2.4 by s kx , kr results in
smatched (kx , kr ) = smf (kx , kr ) · s kx , kr .
11
(2.5)
Using the MATLAB programs in Appendix B and Appendix C, the resulting phase
of smatched (kx , kr ) is plotted in Figure 2.7 and the resulting downrange DFT magnitude is plotted in Figure 2.8.
2.2.4
Stolt interpolation
The Stolt interpolation transforms the 2D SAR data after the matched filter
smatched (kx , kr ) from the radar transmitted chirp kr domain to the the spatial
wave number domain ky . The relationship between ky , kr , and kx is given by
q
2.
ky = kr2 − kx
(2.6)
A 1D interpolation must be conducted across all the downrange wave numbers kr to
map them onto ky thus resulting in the Stolt interpolated matrix sst (kx , ky ). Using
the MATLAB programs in Appendix B and Appendix C, the resulting phase after
the Stolt interpolation is shown in Figure 2.9.
2.2.5
Inverse Fourier transform in to image domain
In order to convert the Stolt interpolated matrix sst (kx , Ky ) into image domain
S(X, Y ), a subsection of the curved Stolt interpolated data shown in Figure 2.9
must be taken such that the resulting subsection is completely filled with data. The
resulting subsection of the data is shown in Figure 2.10.
The resulting SAR image data matrix S(X, Y ) is the 2D IDFT of the subsection of
Stolt interpolated data. Using the MATLAB programs in Appendix B and Appendix
C, the resulting SAR image of the point scatterer is shown in Figure 2.11.
2.3
Simulation of multiple point scatterers
Using equation 2.3 a SAR data matrix can be produced for a single point scatterer. It
is often useful to simulate multiple point scatterers to test a SAR imaging algorithm.
12
For this reason Equation 2.3 was modified to represent N scatterers
q
N
2
X
−j2ω(t) (xn −xti )2 +yti
.
s xn , ω(t) =
ati e
i=1
(2.7)
Using the MATLAB program in Appendix A, three point scatterers with reflection
amplitudes of at1 = at2 = at3 = 1 were simulated at the following locations cross
range and down range from the linear rail:
(xt1 , yt1 ) = (3, −10)
feet,
(xt2 , yt2 ) = (−3, −15) feet,
(xt3 , yt3 ) = (−2, −10) feet.
This simulated SAR data matrix was then fed into the data conditioning MATLAB
program in Appendix B and the RMA SAR imaging MATLAB program in Appendix
C resulting in the SAR image shown in Figure 2.12. From this, it is clear that the
RMA SAR algorithm written for this dissertation is capable of imaging a target scene.
13
Figure 2.1. Rail SAR data collection geometry.
14
Figure 2.2. Real values of the SAR data matrix for a single point scatterer.
15
Figure 2.3. Phase of the SAR data matrix for a single point scatterer.
16
Figure 2.4. Magnitude of simulated point scatterer after downrange DFT, showing
the wave-front curvature of the point scatterer.
17
Figure 2.5. Magnitude after the cross range DFT.
18
Figure 2.6. Phase after the cross range DFT.
19
Figure 2.7. Phase after the matched filter.
20
Figure 2.8. Magnitude after 2D DFT of matched filtered data.
21
Figure 2.9. Phase after Stolt interpolation.
22
Figure 2.10. A subsection of the phase after Stolt interpolation.
23
Figure 2.11. SAR image of a simulated point scatterer.
24
Figure 2.12. SAR image three simulated point scatterers.
25
CHAPTER 3
SCATTERING FROM A PERFECT ELECTRIC CONDUCTING
CYLINDER
Cylinders are often used for radar system calibration and for measuring system performance. It is for this reason that a two dimensional (2D) solution based on [61] to
the scattering from a perfect electric conducting (PEC) cylinder will be developed.
This solution will be verified from previous results. Using the solution developed in
this chapter a set of simulated SAR data will be created. The SAR imaging algorithm
from Chapter 2 will then be used to image this simulated cylinder.
3.1
T M z scattering solution to a 2D PEC cylinder
The radar system that will be used for acquiring experimental data will be using
vertically polarized transmit and receive antennas. The radar will be located many
wavelengths away from the target scene. It is for these reasons the plane wave approximation will be used for this solution to the scattering of a 2D PEC cylinder.
Since the radar polarization is vertical the plane wave polarization that will be used
for this solution is T M z .
The geometry for a T M z plane wave incident on a 2D PEC cylinder is shown
in Figure 3.1, where a is the radius of the 2D infinite cylinder centered along the
ẑ axis. The incident T M z plane wave is traveling towards the cylinder from the
y = −∞ direction towards x = 0, where the electric and magnetic field components
→
−
are transverse to the x̂ direction of propagation. The incident electric field E i is
→
−
polarized parallel to the ẑ axis and the incident magnetic field H i vector is parallel
to the −ŷ axis.
26
For a T M z incident plane wave, we can represent the incident electric field as
→
−i
E = ẑEo e−jko x ,
(3.1)
where Eo is the magnitude of the incident electric field. Applying the cartesian to
cylindrical coordinate transformation x = ρ cos φ, where ρ is the radius and φ is
the angle with respect to the x-axis in cylindrical coordinates, Equation 3.1 can be
re-written as
→
−i
E = ẑEo e−jko ρ cos φ .
(3.2)
Using the cylindrical wave transformation from [61] this can be re-written as
e−jkρ cos φ =
∞
X
j −n Jn (kρ)ejnφ .
(3.3)
n=−∞
Applying this result to Equation 3.2 results in
→
−i
E = ẑEo
∞
X
j −n Jn (ko ρ)ejnφ .
(3.4)
n=−∞
Applying Euler’s formula to the above results in
→
−i
E = ẑEo
∞
X
−n
j Jn (ko ρ) cos nφ + j · j −n Jn (ko ρ) sin nφ .
(3.5)
n=−∞
The second term in the above equation goes to zero because inside of the summation
term
sin (−n) + sin (n) = 0.
27
Applying this to Equation 3.5 results in
∞
X
→
−i
E = ẑEo
j −n Jn (ko ρ) cos nφ.
(3.6)
n=−∞
Equation 3.6 is symmetric about the n = 0 term of the summation. For this reason
symmetry can be used to reduce the summation from −∞ to ∞ to 0 to ∞. In
addition, the simplification j −n = (−j)n may be used. Applying both of these to
Equation 3.6 results in the simplified incident electric field equation
∞
X
→
−i
(−j)n εn Jn (ko ρ) cos nφ,
E = ẑEo
n=0
where
εn =


 1 for n = 0
(3.7)
.

 2 for n 6= 0
The total electric field around the cylinder shown in Figure 3.1 is defined as
→
− tot →
−
→
−
E
= E i + E s,
(3.8)
→
−
→
−
where E s is the scattered electric field from the cylinder. E s travels outward away
from the cylinder and is represented by the cylindrical traveling wave equations from
→
−
[61], thus the solution to the scattered field E s has the form
→
−s
E = ẑEo
∞
X
(2)
cn Hn (ko ρ),
(3.9)
n=−∞
where the unknown amplitude coefficients cn must be solved in order to find the
→
−
solution to the scattered field E s .
Applying boundary conditions to the surface of the cylinder at ρ = a and for
0 ≤ φ ≤ 2π as shown in Figure 3.2, the total electric field becomes zero at the
28
cylinder surface and Equation 3.8 can be re-written as
→
−
→
−
0 = E i + E s.
(3.10)
Substituting the incident field Equation 3.4 and the scattered field Equation 3.9 into
Equation 3.10, the above equation becomes
0 = ẑEo
∞
X
j −n Jn (ko a)ejnφ + ẑEo
n=−∞
∞
X
(2)
cn Hn (ko a).
(3.11)
n=−∞
Combining the summation terms and suppressing both ẑ and Eo , the expression
becomes
∞
X
0=
−n
(2)
j Jn (ko a)ejnφ + cn Hn (ko a) .
(3.12)
n=−∞
The expression within the summation is zero for all n to enforce the boundary conditions on the surface of the cylinder, therefore the above equation can be simplified
further by considering 3.12 on a mode-by-mode basis
(2)
0 = j −n Jn (ko a)ejnφ + cn Hn (ko a).
(3.13)
Finally, the solution for cn is
cn =
−(j)−n Jn (ko a)ejnφ
.
(2)
Hn (ko a)
(3.14)
Substituting Equation 3.14 into Equation 3.9 results in the scattered field solution
to the 2D PEC cylinder
→
−s
E = ẑEo
∞
X
−(j)−n Jn (ko a) jnφ
(2)
Hn (ko ρ)
e
.
(2)
Hn (ko a)
n=−∞
29
(3.15)
Applying Euler’s equation, the scattered field becomes
→
−s
E = ẑEo
∞
X
−(j)−n Jn (ko a) (2)
Hn (ko ρ)
cos nφ + j sin nφ .
(2)
Hn (ko a)
n=−∞
(3.16)
The sine term of the above equation vanishes because inside of the summation
sin (−n) + sin (n) = 0,
and thus Equation 3.16 can be written
→
−s
E = ẑEo
∞
X
−(j)−n Jn (ko a)
(2)
Hn (ko ρ)
cos nφ.
(2)
(k
a)
H
n=−∞
o
n
(3.17)
Equation 3.17 is symmetric about the n = 0 term of the summation, thus symmetry
can be used to reduce the summation from −∞ to ∞ to 0 to ∞. Furthermore, the
substitution j −n = (−j)n can be made. Applying both of these to Equation 3.17
leaves the simplified scattered electric field equation
∞
X
→
−s
Jn (ko a) (2)
E = −ẑEo
(−j)n εn
Hn (ko ρ) cos nφ,
(2)
H
(k
a)
n=0
o
n
where
εn =


 1 for n = 0
(3.18)
.

 2 for n 6= 0
3.2
Echo width of a 2D PEC cylinder
In this section the 2D radar cross section, known as echo width, of the cylinder will be
calculated in order to verify the results from Section 3.1 with the results in [61]. This
verification will show that the 2D cylinder model is valid to be used in the theoretical
development shown later in this dissertation.
30
The echo width equation from [61] is defined as
→
#
−
E s 2
σ2D = lim 2πρ →
.
ρ→∞
−
E i 2
"
(3.19)
In the eventual theoretical imaging geometry and experimental setup the PEC
cylinder target will be located approximately 20 feet or greater from the rail SAR
imaging system. It was for this reason that rather than taking the limit as ρ → ∞,
it was decided to let ρ = 20 feet and modify Equation 3.19 to
→
#
−
E s 2
σ2D ≈ 2πρ →
−
E i 2
"
.
(3.20)
ρ=20 ft
Substituting the incident field Equation 3.7 and the scattered field solution Equation 3.18 into Equation 3.20 letting the frequency be 2 GHz (the low end of the
approximately 2 GHz to 4 GHz radar transmit chirp) and the radius of the PEC
cylinder a = 0.6λ a MATLAB program was written (see Appendix D) to calculate
and plot the echo width of a 2D PEC cylinder with T M z plane wave incidence. The
resulting echo width shown in Figure 3.3 matches the results shown in [61], therefore
this 2D PEC cylinder model for plane wave T M z incidence is valid for use in the
theoretical developement shown throughout this dissertation.
3.3
Range profiles of a 2D PEC cylinder
Using Equation 3.18, a MATLAB program was written (see Appendix D) to simulate
a 2 GHz to 4 GHz chirped radar system measuring the range profile of a cylinder
located 30 feet downrange for various cylinder radii values a. The IDFT was taken
of the time harmonic scattered field solution over 256 frequency data points between
2 GHz and 4 GHz. Theses range profile results are summarized in Table 3.1.
From these results it is clear that 2D PEC cylinders of various radii can be sim-
31
ulated downrange from a chirped radar system. Due to this result it is now possible
to simulate a SAR image of a 2D PEC cylinder.
3.4
SAR image of a the 2D PEC cylinder model
Using the 2D cylinder model developed in this chapter a SAR data set was simulated
using the geometry shown in Figure 2.1 and using the MATLAB program in Appendix
E for a cylinder of radius a = 3 inches located 30 ft downrange from the center of
the rail. The simulated data set was then fed into the MATLAB data conditioning
program in Appendix B and processed by the MATLAB coded RMA SAR imaging
algorithm in Appendix C. The resulting SAR image is shown in Figure 3.8. A second
SAR image of a cylinder with a radius of a = 6 inches located 30 ft downrange from
center rail is shown in Figure 3.9. From these results it is clear that a 2D PEC
cylinder can be modeled effectively and a simulated SAR image produced from the
model.
32
Table 3.1. Summary of range profile results.
cylinder radius
a = 23.62 inch
a = 1.96 inch
range profile in dB real data range profile
Figure Figure 3.4
Figure Figure 3.5
Figure Figure 3.6
Figure Figure 3.7
Figure 3.1. T M z incident plane wave on the PEC cylinder geometry.
33
Figure 3.2. 2D T M z incident plane wave on the PEC cylinder geometry.
34
Figure 3.3. Bistatic echo width of a 2D PEC cylinder with T M z plane wave incident
electric field observed at a distance of 20 ft from the cylinder.
35
Figure 3.4. Range profile of a 23.62 inch radius cylinder with T M z plane wave
incidence, showing the downrange time delay of the front face of the cylinder.
36
Figure 3.5. Real value of the range profile of a 23.62 inch radius cylinder with T M z
plane wave incidence.
37
Figure 3.6. Range profile of a 1.96 inch radius cylinder with T M z plane wave incidence, showing the downrange time delay of the front face of the cylinder.
38
Figure 3.7. Real value of the range profile of a 1.96 inch radius cylinder with T M z
plane wave incidence.
39
Figure 3.8. SAR image of the 2D cylinder model with a radius of a = 3 inches, located
30 ft downrange.
40
Figure 3.9. SAR image of the 2D cylinder model with a radius of a = 6 inches, located
30 ft downrange.
41
CHAPTER 4
A DIELECTRIC SHEET MODEL USING WAVE MATRICES
A dielectric sheet model of a lossy-dielectric slab will be developed in this chapter
using wave matrix theory from [63]. In order to develop this model, reflection and
transmission of plane waves at a dielectric boundary will first be discussed. From this
the general wave matrix method will be presented. Using the wave matrix method
a solution to the reflection and transmission of an air-dielectric-air interface will be
developed. With this model, simulated 2 GHz to 4 GHz range profiles will be acquired
for both normal and oblique incidences. Finally, a simulated SAR data set of this
dielectric model will be acquired. A SAR image of the simulated dielectric sheet
will be created using the SAR imaging algorithm developed in Chapter 2. This SAR
image will provide reference data for a lossy-dielectric slab of equal thickness imaged
by a rail SAR imaging system.
4.1
Reflection and transmission at a dielectric boundary
In order to properly examine reflection and transmission coefficients an example of a
single layer air-dielectric problem will be explored. In this problem there is an airdielecric boundary, where there exists an incident plane wave, reflected plane wave,
and transmitted plane wave as shown in Figure 4.1. It is important to note that
the radar system that will be used later in this dissertation for measured results uses
vertically polarized antennas. For this reason it was decided that all solutions in this
chapter will be for T M z plane waves. For convenience the model will be developed
for normal incidence and then modified for oblique incidence.
42
4.1.1
Incident, reflected and transmitted electric fields
For a T M z incident plane wave traveling along the x̂ direction the incident electric
field is
→
−i
E = ẑEo e−jko x ,
(4.1)
where Eo is the magnitude of the incident electric field and the free space wave
number is
√
ko = ω µo o .
(4.2)
In this, ω = 2πf , f is the frequency of interest, µo is the free-space permeability and
o is the free-space permittivity. For a T M z plane wave reflected from the interface,
the electric field is
→
−r
E = ẑEr ejko x ,
(4.3)
where Er is the magnitude of the reflected electric field. For a T M z plane wave
transmitted into the dielectric, the electric field is
→
−T
E = ẑET e−jkx ,
(4.4)
where ET is the magnitude of the transmitted electric field. The wave number inside
of the dielectric is
r σ
k = ω µo o r +
,
jω
(4.5)
where r is the relative complex permittivity and σ is the dielectric conductivity in
units of mho/m which accounts for the loss of the dielectric material.
43
4.1.2
Incident, reflected and transmitted magnetic fields
The time harmonic version of Faraday’s law from Maxwell’s equations relating the
electric field to the magnetic field is written as
→
−
→
−
∇ × E = −jωµo H .
(4.6)
Applying Equation 4.6 to the incident electric field Equation 4.1 results in the incident
magnetic field
→
−i
ko
Eo e−jko x .
H = −ŷ
ωµo
(4.7)
√
Substituting ko = ω µo o into the above equation results in
√
ω µo o
→
−i
H = −ŷ
Eo e−jko x .
ωµo
(4.8)
Letting ω cancel out and simplifying further the incident magnetic field is
→
−i
H = −ŷ
r
o
Eo e−jko x .
µo
(4.9)
At this point is is important to note that the impedance of free-space is defined as
r
Zo =
µo
o
(4.10)
and the admittance of free-space is the reciprocal of the impedance
1
Yo =
=
Zo
r
o
.
µo
(4.11)
Thus, applying Equation 4.11 to Equation 4.9 results in the equation for the incident
magnetic field
→
−i
H = −ŷYo Eo e−jko x .
44
(4.12)
Applying Faraday’s law Equation 4.6 to the reflected electric field Equation 4.3
results in the reflected magnetic field
→
−r
ko
H = ŷ
Er ejko x .
ωµo
(4.13)
√
Substituting the free-space wave number equation ko = ω µo o into the above equation results in
√
ω µo o
→
−r
H = ŷ
Er ejko x .
ωµo
(4.14)
Letting ω cancel out and simplifying further
→
−r
H = ŷ
r
o
Er ejko x .
µo
(4.15)
Applying the free-space admittance Equation 4.11 to the above results in the equation
for the reflected magnetic field
→
−r
H = ŷYo Er ejko x .
(4.16)
Applying Faraday’s law Equation 4.6 to the transmitted electric field Equation
4.4 results in the transmitted magnetic field into the dielectric medium
→
−T
k
H = −ŷ
E e−jkx .
ωµo T
(4.17)
Substituting the dielectric wave number k Equation 4.5 into the above results in the
transmitted field within the medium
→
−T
H = −ŷ
r σ
ω µo o r + jω
ωµo
45
ET e−jkx .
(4.18)
Letting ω cancel out and simplifying further
v
u
u o r + σ
t
→
−T
jω
H = −ŷ
ET e−jkx .
µo
(4.19)
At this point is is important to note that the TEM wave impedance of the dielectric
is defined as
Z=
µo
s
σ
o r + jω
(4.20)
and the admittance of the dielectric is the reciprocal of the impedance
v
u
u o r + σ
t
1
jω
Y = =
.
Z
µo
(4.21)
Applying the dielectric admittance Equation 4.21 to the Equation 4.19 results in the
equation for the transmitted magnetic field
→
−T
H = −ŷY ET e−jkx .
4.1.3
(4.22)
Boundary conditions at an air-dielectric interface
Applying the boundary condition at the discontinuity interface in Figure 4.1 at the
plane x = 0 results in the total electric field equation
h→
h→
−i →
− i
− i
E + Er
= ET
x=0
x=0
(4.23)
and the total magnetic field equation
h→
h→
−i →
− i
− i
H + Hr
= HT
.
x=0
x=0
(4.24)
Substituting the incident, reflected and transmitted electric field Equations 4.1,
46
4.3 and 4.4 into the above Equation 4.23 results in
h
i
h
i
−jk
x
jk
x
−jkx
o
o
ẑEo e
+ ẑEr e
= ẑET e
.
x=0
x=0
(4.25)
Substituting x = 0 and letting ẑ drop out results in the boundary equation
Eo + Er = ET .
(4.26)
Considering the total magnetic field Equation 4.24 and substituting in the incident, reflected and transmitted magnetic field Equations 4.12, 4.16 and 4.22 results
in the following
h
− ŷYo Eo e−jko x + ŷYo Er ejko x
i
x=0
h
i
= − ŷY ET e−jkx
x=0
.
(4.27)
Substituting x = 0, letting ŷ and −1 drop out results in
Yo Eo − Yo Er = Y ET .
(4.28)
The dielectric admittance Y in Equation 4.21 gives
v
u
r r
u o r + σ
t
o
σ
jω
Y =
=
r +
.
µo
µo
jωo
(4.29)
Substituting the free-space admittance Yo Equation 4.11 into the above results in a
slightly different form for the free-space admittance equation
r
Y = Yo r +
47
σ
.
jωo
(4.30)
Applying the above equation to Equation 4.28 results in
r
Yo Eo − Yo Er = Yo r +
σ
E .
jωo T
(4.31)
Canceling out the Yo terms and collecting terms on the left results in the boundary
equation
r
Eo − Er − r +
4.1.4
σ
E = 0.
jωo T
(4.32)
Transmission and reflection coefficients
By definition (as was indicated in [63]) the reflection coefficient Ri is the ratio of
reflected to incident electric field amplitudes at the interface plane x = 0, shown in
Figure 4.1,
Er
Ri =
.
Eo
(4.33)
In the case of an air-dielectric boundary, R1 is the reflection coefficient for a wave
incident from region 1 onto region 2 (as shown in Figure 4.1) which provides the
following relation
Er = R1 Eo .
(4.34)
By definition (as was indicated in [63]), the transmission coefficient Tij is the
ratio of transmitted to incident electric field amplitudes at the interface plane x = 0,
shown in Figure 4.1,
E
Tij = T .
Eo
(4.35)
In the case of an air-dielecric boundary, T12 is the transmission from region 1 to
region 2 (as shown in Figure 4.1) which provides the following relation
T12 Eo = ET .
48
(4.36)
Substituting the reflection coefficient Equation 4.34 and the transmission coefficient Equation 4.36 into the boundary Equation 4.26 results in
Eo + R1 Eo = T12 Eo .
Dividing both sides by Eo results in the equation relating T12 and R1
1 + R1 = T12 .
(4.37)
Substituting the reflection coefficient Equation 4.34 and the transmission coefficient Equation 4.36 into the boundary Equation 4.32 results in
r
Eo − R1 Eo − r +
σ
T Eo = 0.
jωo 12
Again dividing both sides by Eo results in a second equation relating T12 and R1 :
r
σ
1 − R1 = r +
T .
jωo 12
(4.38)
Equations 4.37 and 4.38 are a set of two equations and two unknowns. Solving
these equations for the reflection coefficient R1 results in
σ
r + jω
o
q
R1 =
σ
1 + r + jω
o
1−
q
(4.39)
and solving Equations 4.37 and 4.38 for the transmission coefficient T12 results in
T12 =
2
q
.
σ
1 + r + jω
o
(4.40)
The reflection coefficient Equation 4.39 can be written in terms of dielectric ad-
49
mittances in each medium
q
q
q
σ
σ
σ
1 − r + jω
Y
−
Y
r + jω
r + jω
o
o
o
o Yo
o
q
q
q
=
·
=
.
R1 =
σ
σ
σ
Y
o
1 + r + jω
1 + r + jω
Yo + Yo r + jω
o
o
o
1−
Substituting the dielectric admittance Equation 4.30 into the above results in the
reflection coefficient R1 in terms of dielectric admittances
Yo − Y
R1 =
.
Yo + Y
(4.41)
The reflection coefficient Equation 4.41 can also be written in terms of dielectric
impedances in each medium
Zo − Zo
1
1
1−
Yo − Y
Zo − Z Zo
Z
Z
=
·
= o
=
R1 =
1
1 Zo
Zo + Zo
Yo + Y
1+
Zo + Z
Zo
Z
Zo
Z · Z,
Zo Z
Z
which results in a simplified reflection coefficient R1 in terms of dielectric impedances
Z − Zo
R1 =
.
Z + Zo
(4.42)
The transmission coefficient Equation 4.40 can be written in terms of dielectric
admittances in each medium
T12 =
2
2
Yo
2Y
q
q
qo
=
·
=
.
σ
σ
σ
Yo
1 + r + jω
1 + r + jω
Yo + Yo r + jω
o
o
o
Substituting the dielectric admittance Equation 4.30 into the above results in
2Yo
T12 =
.
Yo + Y
(4.43)
The transmission coefficient Equation 4.43 can also be written in terms of dielectric
50
impedances in each medium
2
2
Zo
Z
2Yo
Zo
=
=
·
· ,
T12 =
1
1
Z
Yo + Y
Zo
1 + Zo Z
Zo + Z
which results in a simplified transmission coefficient T12 in terms of dielectric
impedances
2Z
T12 =
.
Z + Zo
(4.44)
If the geometry shown in Figure 4.1 were reversed and the incident electric field
wave was traveling from region 2 to region 1 then the reflection coefficient would be
found by simply swapping the admittance terms Y and Yo in Equation 4.41 thus
giving us
Y − Yo
.
R2 =
Y + Yo
(4.45)
In a similar fashion, the admittance terms Y and Yo in Equation 4.43 can be swapped
to find the transmission coefficient for a wave traveling from region 2 to region 1
2Y
T21 =
.
Y + Yo
4.2
(4.46)
Wave matrices
Looking again at the air-dielectric boundary problem solved in Section 4.1 and shown
in Figure 4.1, let a T M z plane wave of amplitude c1 be incident from region 1 to
region 2. At the same time, another T M z plane wave is incident from region 2 to
region 1 with an amplitude of b2 . This is a different approach for the same problem
solved in Section 4.1. The geometry for this approach is shown in Figure 4.2.
The total reflected field in region 1 has the value b1 and is due to the transmitted
field from region 2 to region 1 represented by T21 b2 . Coefficient b1 is also due to
the incident plane wave in region 1 of amplitude c1 reflecting off of the boundary at
51
x = 0 which is represented by R1 c1 . The resulting reflected field amplitude equation
becomes
b1 = R1 c1 + T21 b2 .
(4.47)
The total reflected field in region 2 has the value c2 and is due to the transmitted
field from region 1 to region 2 represented by T12 c1 . Coefficient c2 is also due to
the incident plane wave in region 2 of amplitude b1 reflecting off of the boundary at
x = 0 which is represented by R2 b2 . The resulting reflected field amplitude equation
becomes
c2 = R2 b2 + T21 b2 .
(4.48)
Rearranging the variables in Equations 4.48 and 4.47 results in
c1 =
R b
c2
− 2 2
T12
T12
(4.49)
and
R
R1 R2 b2 + 1 c 2 .
b1 = T21 −
T12
T12
(4.50)
Equations 4.49 and 4.50 can be re-written in matrix form








−R2
1  1
  c2   A11 A12   c2 
 c1 
,


 =
=


T
12
A21 A22
b2
b2
b1
R1 T12 T21 − R1 R2
(4.51)
where the values of matrix A are simply
1
,
T12
(4.52)
−R2
A12 =
,
T12
(4.53)
R
A21 = 1
T12
(4.54)
A11 =
52
and
T T − R1 R2
.
A22 = 12 21
T12
(4.55)
The reflection coefficient R1 from region 1 to region 2 is the negative of the
reflection R2 coefficient from region 2 to 1 as shown by Equations 4.41 and 4.45, thus
Yo − Y
Y − Yo
Yo − Y
R1 =
= −1 · R2 = −1 ·
=
.
Yo + Y
Y + Yo
Yo + Y
From this R1 is related to R2 where
R1 = −R2 .
(4.56)
Looking at the A12 Equation 4.53 and applying the above relation results in
R
−R2
= 1.
A12 =
T12
T12
(4.57)
Similar to the T12 Equation 4.37 for transmission from region 1 to region 2, the
opposite is true when transmission from region 2 to 1 occurs, where the equation for
T21 becomes
T21 = 1 + R2 .
(4.58)
Applying the above result Equation 4.37 and Equation 4.56 to the A22 Equation 4.55
results in the simplification
(1 + R1 )(1 − R1 ) + R12
T T − R1 R2
(1 + R1 )(1 + R2 ) − R1 R2
A22 = 12 21
=
=
,
T12
T12
T12
that is,
A22 =
53
1
.
T12
(4.59)
Placing Aij Equations 4.52, 4.57, 4.54, and 4.59 back into the matrix A results in




1  1 R1  
 c1 



=
T12 R
1
b1
1

c2 
.
b2
(4.60)
Looking at the geometry in Figure 4.2, an incident plane wave traveling from
region 1 to region 2 has an amplitude of c1 at the interface x = 0. Similarly, a plane
wave being reflected from region 2 into region 1, or due to the transmission from
region 2 to region 1, has an amplitude of b1 at the interface x = 0. At some location
inside of region 2 at x = x1 the plane wave amplitudes of the incident and reflected
waves are phase shifted, where c2 and b2 represent the phase shifted amplitudes of
the incident and reflected waves shifted to a new terminal plane at x = x1
c2 = c1 e−jkx1
and
b2 = b1 ejkx1 .
Solving the above for c1 and b1 yields the terminal shifted wave amplitudes
c1 = c2 ejkx1
(4.61)
b1 = b2 e−jkx1 .
(4.62)
and
By definition [63] the electrical length is the phase difference between a wave traveling from one location to another in a dielectric medium (free space or otherwise).
Electrical length θ is linear in this case (however there are other cases where it is
not, and these will not be covered here) and is defined as the wave number times the
54
linear distance between the two locations in the dielectric medium. For the case of
normal incidence according to the geometry in Figure 4.2, the electrical length is
θ = kx.
(4.63)
Substituting the electrical length θ1 = kx1 into the terminal shifted wave amplitude
Equations 4.61 and 4.62 results in
c1 = c2 ejθ1
(4.64)
b1 = b2 e−jθ1 .
(4.65)
and
Placing the above equations in to matrix form results in

 c1

b1


jθ
0
  c2 
  e 1
.

=
0
e−jθ1
b2


(4.66)
The above matrix provides the complex amplitude of the traveling waves incident
and reflected at a terminal shifted plane. Such a terminal shifted pair of forward and
reflected wave amplitudes could be used to be incident on another boundary located
at x = x1 . This phase shifted terminal plane concept is shown in Figure 4.3, and
can be used to add a second dielectric layer (or many more layers) to the two region
dielectric boundary problem shown in Figure 4.2 by multiplying the matrix Equation
4.60 and the matrix Equation 4.66 resulting in the following matrix




1  1 R1  
 c1 

=


T1 R1 1
b1
55
ejθ1
0

0

  c2 

.
e−jθ1
b2
(4.67)
Notice the change in notation, where the transmission coefficient from region 1 to
region 2 has become T12 = T1 . From now on this notation will be used for the
transmission coefficients to more conveniently show the general case for the wave
matrix solution of multiple dielectric layers.
Multiplying out the above matrix Equation 4.67 results in


−jθ
1 
R1 e
1 



=
T1 R ejθ1
−jθ1
e
1

 c1

b1

ejθ1

c2 
,
b2
(4.68)
which is the solution to the incident and reflected wave amplitudes at some depth
in region 2. Using this concept of phase shifted terminal planes it is now possible
to provide a general wave matrix solution for an n-layered material composed of
n number of dielectric layers of a finite thickness. Where the finite thickness is
represented by the electrical length θi . The general wave matrix solution is





jθ
−jθ
n
i   cn+1 
 c1  Y 1  e i Ri e

=


.
T
jθ
−jθ
i
i
i
b1
Ri e
e
bn+1
i=1
4.3
(4.69)
A dielectric sheet model based on wave matrices
In this section a model of a lossy-dielectric slab will be developed. Using the wave
matrix theory developed in Section 4.2 an air-dielectric-air layered planar dielectric
solution is developed. This solution is fed a set of frequency dependent dielectric
property data from a set of lossy-dielectric model data from [64] so that the dielectric
model has practical lossy-dielectric slab properties. This lossy-dielectric slab model
is essential for analysis of the through-dielectric radar imaging problem. In the next
chapter this model will be combined with the cylinder model from Chapter 3 to
produce simulated SAR imagery of a cylinder behind a lossy-dielectric slab. The
eventual simulation of a cylinder behind a lossy-dielectric slab will provide essential
56
information for developing radar system specifications and understanding imaging
limitations.
In order to use the wave matrix theory developed in Section 4.2, the incident
field and transmitted field impedances must be found for oblique incidence on a three
layered problem, where the dielectric layers are air-dielectric-air as shown in Figure
4.4. Using these impedances, it is a matter of substitution to apply the wave matrix
method to find the solution to the air-dielecric-air problem.
Looking at the geometry for an oblique incidence in Figure 4.4, the intrinsic
impedance of the dielectric layer is η, and the intrinsic impedance of free space is
ηo , where
r
µo
o
(4.70)
ηo
µ
=q
.
σ
σ
o r + jω
r + jω
o
(4.71)
ηo =
and
s
η=
It is important to note that at normal incidence the dielectric impedance as defined
in Equation 4.20 equals the intrinsic impedance of the dielectric layer η. Also at
normal incidence the impedance of free space as defined in Equation 4.10 equals the
intrinsic impedance of free space ηo . However, in this section the air-dielecric-air wave
matrix problem will be solved for an oblique incidence where this is not the case. An
oblique incidence causes changes in the effective dielectric impedance with respect to
the plane wave direction.
Looking at Figure 4.4, a T M z plane wave is incident on the dielectric layer at
an incident angle of φi and represented by the incident electric field Equation 4.1
(rewritten here for convenience)
→
−i
E = ẑEo e−jko x .
57
Since the incident plane wave is at an angle φi with respect to the x̂ direction
of propagation, the dielectric can be thought of as rotated by angle φi with respect
to the xy plane. This is a rotated coordinate system, where the incident plane wave
in the xy axis is rotated onto the dielectric axis x0 y 0 using the standard coordinate
rotation matrix [65], where the xyz coordinates as a function of x0 y 0 z 0 are
z = z0,
x = x0 cos φi − y 0 sin φi ,
(4.72)
y = x0 sin φi + y 0 cos φi .
Substituting Equation 4.72 into the incident electric field Equation 4.1 results in
0
0
→
−i
E = zˆ0 Eo e−jko (x cos φi −y sin φi ) .
(4.73)
Applying the time harmonic version of Faraday’s law from Maxwell’s equations, Equation 4.6, with respect to the rotated coordinate axis x0 y 0 z 0 to the above Equation 4.73
results in the incident magnetic field
r
−
→
0
0
o
ˆ
i
H = − x0
sin φi Eo e−jko (x cos φi −y sin φi )
µo
r
0
0
o
ˆ
0
cos φi Eo e−jko (x cos φi −y sin φi ) . (4.74)
−y
µo
And from Equation 4.70
1
=
ηo
r
o
.
µo
Substituting this into the above Equation 4.74 results in the incident magnetic field
58
equation in coordinate rotated space
−
→
0
0
1
H i = −xˆ0 sin φi Eo e−jko (x cos φi −y sin φi )
ηo
0
0
1
− yˆ0 cos φi Eo e−jko (x cos φi −y sin φi ) . (4.75)
ηo
Dividing the incident electric field in the zˆ0 direction (Equation 4.73) by the incident
magnetic field (Equation 4.75) in the yˆ0 direction results in the impedance of the
incident wave traveling in the rotated xˆ0 direction (normal to the surface of the
dielectric)
−E i 0
z = η sec φ .
Z1 =
o
i
i
H 0
y
(4.76)
Due to Snell’s law of refraction, the direction of propagation for the transmitted
electromagnetic fields is refracted towards the normal vector n̂ in Figure 4.4 by an
angle φr inside of the dielectric region [61], [63]. For this reason the transmitted
electric field must be represented in terms of φr . Applying the coordinate rotation
matrix Equation 4.72 with respect to φr to the transmitted field Equation 4.4 results
in
0
0
→
−T
E = zˆ0 ET e−jk(x cos φr −y sin φr ) .
(4.77)
Applying the time harmonic version of Faraday’s law from Maxwell’s equations, Equation 4.6, with respect to the rotated coordinate axis x0 y 0 z 0 to the above Equation 4.77
results in the transmitted magnetic field
−−→
H T = −xˆ0
s
σ
o r + jω
0
0
sin φr ET e−jk(x cos φr −y sin φr )
µo
s
σ
o r + jω
0
0
ˆ
0
−y
cos φr ET e−jk(x cos φr −y sin φr ) . (4.78)
µo
59
From Equation 4.71
s
1
=
η
σ
o r + jω
.
µo
Substituting this into the above Equation 4.78 results in the incident magnetic field
equation in coordinate rotated space
−−→
0
0
1
H T = −xˆ0 sin φr ET e−jk(x cos φr −y sin φr )
η
0
0
1
− yˆ0 cos φr ET e−jk(x cos φr −y sin φr ) . (4.79)
η
Dividing the transmitted electric field in the zˆ0 direction (Equation 4.77) by the
transmitted magnetic field (Equation 4.79) in the yˆ0 direction results in the impedance
of the transmitted wave traveling in the rotated xˆ0 direction (normal to the surface
of the dielectric) inside of the dielectric
−E T0
z = η sec φ .
Z2 =
r
T
H 0
y
(4.80)
It is useful to normalize the impedances of the incident and transmitted waves
with respect to the impedance of the incident wave in free space. For the incident
field in free space the normalized impedance is simply 1, where
1=
Z1
.
Z1
(4.81)
For the transmitted field in the dielectric the normalized impedance becomes
Z=
Z2
η sec φr
=
=
Z1
ηo sec φi
q ηo
σ sec φr
r + jω
o
ηo sec φi
.
Simplifying the above results in the normalized dielectric impedance with respect to
60
incidence angle φi and angle of refraction φr
cos φi
.
Z=q
σ cosφ
r + jω
r
o
(4.82)
The above equation for normalized impedance is difficult to use in practice because it
is a function of both incidence angle φi and angle of refraction φr . Snell’s law must
be applied in this case in order to simplify the equation and make it a function of φi
and the dielectric properties only. The equation for Snell’s law [61] relates angle φi
to φr as follows
ko sin φi = k sin φr .
(4.83)
Solving for the angle of refraction φr yields
φr = sin−1
!
sin φi
q
σ
r + jω
o
.
(4.84)
Apply this result by evaluating the cos φr term in the denominator of Equation 4.82
"
cos φr = cos sin−1
q
σ
r + jω
o
Applying the trigonometric identity cos (sin−1 x) =
v
u
u
cos φr = t1 −
sin2 φi
σ =
r + jω
o
!#
sin φi
q
p
.
1 − x2 results in
σ − sin2 φ
r + jω
i
o
q
.
σ
r + jω
o
Substituting the above result for cos φr in to Equation 4.82 results in the normalized
impedance of the dielectric with a plane wave incident at an angle of φi
Z=q
cos φi
.
σ − sin2 φ
r + jω
i
o
61
(4.85)
The normalized impedance reflection coefficient for an incident plane wave traveling from free-space in to the dielectric is
Z −1
.
R1 =
Z +1
(4.86)
The normalized impedance reflection coefficient for an incident plane wave traveling
from the dielectric to free-space is
1−Z
R2 =
.
Z +1
(4.87)
The transmission coefficient for an incident plane wave traveling from free-space in
to the dielectric is
T1 = 1 + R1 .
(4.88)
The transmission coefficient for a plane wave traveling out from inside of the dielectric
to free-space is
T2 = 1 + R2 .
(4.89)
Applying the above reflection and transmission coefficients in Equations 4.86
through 4.89 for the air-dielectric-air problem to the general form of the wave matrix
solution Equation 4.69 in Section 4.2 results in the following wave matrix solution:

 c1

b1



−jθ
i   1 R2   c3 
R1 e
1 



,
=

T1 T2 R ejθi
−jθi
R
1
b
e
2
3
1


ejθi
(4.90)
where the electrical length for a dielectric with a thickness d for an oblique incidence
angle φi is
r
θ = ko d
r +
σ − sin2 φi .
jωo
(4.91)
Using the wave matrix Equation 4.90, a theoretical radar target can be located on
62
the opposite side of the dielectric slab by simply solving for c3 and b1 . In Equation
4.90, b3 is a function of radar target function Γ where
b3 = Γc3 .
(4.92)
The use of this technique will be shown in the next chapter, however it is important
to mention it here in order to explain why Equation 4.90 is being solved for both c3
and b1 .
Solving Equation 4.90 for c3 and b1 results in the solution to an air-dielecric-air
interface for an oblique incidence angle:
c1 T1 T2
ejθ + R1 R2 e−jθ + Γ R2 ejθ + R1 e−jθ
(4.93)
i
c3 h
R1 ejθ + R2 e−jθ + Γ R1 R2 ejθ + e−jθ .
T1 T2
(4.94)
c3 =
b1 =
4.4
Simulated range profiles of the dielectric sheet model
In this section practical lossy-dielectric properties will be fed into the dielectric model
developed in Section 4.3. This will result in a theoretical model of a lossy-dielectric
slab of finite thickness. Range profiles will then be taken of this dielectric slab model.
These simulated range profiles will provide insight into what radar returns are expected from a lossy-dielectric slab standing by itself. Two range profiles will be taken,
one at normal incidence and the other at an oblique incidence. The range profile geometry is shown in Figure 4.6. In both cases (normal and oblique incidences), the
radar is 20 ft from the surface of the slab.
The simulated range profile data set is calculated by solving Equations 4.92
through 4.94 for b1 . The parameters and specifications used to simulate these range
profiles of the dielectric slab are shown in Table 4.1.
63
The incident plane wave is T M z and for this reason the wave amplitude coefficient
at the dielectric boundary c1 is
c1 = Eo e−jko r1 .
(4.95)
The received scattered field is simply a plane wave traveling back from the dielectric surface to the radar receiver (the theoretical radar system in this case is assumed
to be mono-static), and thus the received scattered field equation is
Es = b1 e−jko r1 .
(4.96)
The IDFT is taken of Es , and the resulting time domain range profile is shown. All
of these calculations were done using the MATLAB program shown in Appendix F.
The resulting range profile for a normally incident φi = 0 plane wave is shown
in Figure 4.7. Looking at this result it is clear that the strongest reflection was due
to the surface of the dielectric, which is 20 ft downrange from the radar system. It
is also interesting to note the second, but barely noticeable, reflection from the back
side of the dielectric is shown in Figure 4.7 to be the smaller reflection just to the
right of the large surface reflection.
The resulting range profile for an oblique incident φi = π
6 plane wave is shown in
Figure 4.8. The results for this range profile are nearly identical to that of Figure 4.7.
Looking at Figure 4.8 it is clear that the strongest reflection was due to the surface
of the dielectric, and the second much smaller reflection from the back side of the
dielectric.
From the these theoretical range profile results presented, it is clear that the
dielectric model is functioning properly for both normal and oblique incident waves.
Based on these results it is now possible to simulate a SAR image of the dielectric to
64
see what a concrete slab might look like when imaged by an 8 foot linear rail SAR
imaging system.
4.5
Simulated SAR image of the dielectric slab model
In this section simulated SAR imagery is created of the dielectric sheet model developed in this chapter. This is done by acquiring 48 evenly spaced range profiles across
an 8 foot long simulated linear rail placed in front of a lossy-dielectric slab at a range
of 20 feet.
The imaging geometry here is somewhat complicated. Looking at Figure 4.9 it
is assumed that some unknown radar target is located at a distance d3 downrange
from the rail and exactly at the rail center L/2. In the next chapter a cylinder will be
placed at this location, for now however, there will be nothing located at this position
making the radar target function Γ = 0. It is important to image the target scene as
if there were a target there behind the slab. This is done for comparison purposes.
In the next chapter the target scene with a target behind the slab can be compared
to the results in this chapter for a target scene without a target behind the slab using
the exact same imaging geometry which facilitates the use of coherent background
subtraction in the simulated data.
In Figure 4.9 the radar travels down the linear rail of length L, acquiring range
profiles at evenly spaced increments located at rail positions x(n) from 0 to L. The
incident angle φi is dependent upon the radar position on the rail x(n) and the location of the unknown radar target relative to the rail. From this, the x(n) dependent
incident angle function φi (n) is
"
d3
#
,
φi (n) = cos−1 q
−L + x(n)2 + d2
2
3
65
(4.97)
where r1 (n) is the distance from the radar itself to the dielectric slab
r1 (n) =
d1
,
cos φi (n)
(4.98)
and r3 (n) is the distance from the opposite side of the slab to the unknown target
location
d − d1 − d
r3 (n) = 3
.
cos φi (n)
(4.99)
Where d is the thickness of the dielectric slab and d1 is the distance from the linear
rail to the surface of the dielectric slab. From these equations an x(n) dependent
incident field amplitude can be derived
c1 (n) = Eo e−jko r1 (n) .
(4.100)
The x(n) dependent scattered field equation is
Es (n) = b1 (n)e−jko r1 (n) ,
(4.101)
where b1 (n) is dependent upon the x(n) dependent impedance equation
Z(n) = q
cos φi (n)
.
σ − sin2 φ (n)
r + jω
i
o
(4.102)
The x(n) dependent reflection and transmission coefficients are
Z(n) − 1
,
R1 (n) =
Z(n) + 1
(4.103)
1 − Z(n)
R2 (n) =
,
Z(n) + 1
(4.104)
T1 (n) = 1 + R1 (n)
(4.105)
66
and
T2 (n) = 1 + R2 (n).
(4.106)
Using the above equations, the x(n) dependent amplitude coefficients become
c3 (n) =
c1 (n)T1 (n)T2 (n)
ejθ(n) + R1 (n)R2 (n)e−jθ(n) + Γ R2 (n)ejθ(n) + R1 (n)e−jθ(n)
(4.107)
and
b1 (n) =
c3 (n) h
R (n)ejθ(n) + R2 (n)e−jθ(n)
T1 (n)T2 (n) 1
i
+ Γ R1 (n)R2 (n)ejθ(n) + e−jθ(n) . (4.108)
Where the x(n) dependent electrical length θ(n) is
r
θ(n) = ko d
r +
σ − sin2 φi (n).
jωo
(4.109)
In addition, the radar target function Γ relates b3 (n) to c3 (n) by the equation
b3 (n) = Γc3 (n).
(4.110)
In this case Γ = 0.
Using the model derived shown in Equations 4.97 through 4.110, a simulated SAR
image data set was calculated using the parameters shown in Table 4.2.
The MATLAB program used to calculate the SAR data set is shown in Appendix
G. Where, first the MATLAB program in Appendix G was run, after which the data
conditioning MATLAB program in Appendix B was run followed by the RMA SAR
imaging algorithm MATLAB program in Appendix C.
Figure 4.10 shows the theoretical image of a 4 inch thick lossy-dielectric slab,
67
Table 4.1. Parameters and specifications used for simulating range profiles of the
dielectric slab.
The radar target function Γ = 0
Permittivity of the slab r = 5 based on [64]
Conductivity σ is a function of frequency based on [64] (see Figure 4.5)
Incident angle φi = 0 for normal, φi = π
6 for oblique
Thickness of dielectric d = 3.94 inches
Distance from radar to slab face r1 = 20 ft
Chirp frequency: 2 GHz to 4 GHz in 256 steps
Incident wave amplitude Eo = 1
Table 4.2. A simulated SAR image of the dielectric slab was calculated using these
parameters and specifications.
Permittivity of the slab r = 5 based on [64]
Conductivity σ is a function of frequency based on [64] (see Figure 4.5)
Distance from rail to front of dielectric d1 = 20 ft
Distance from rail to unknown target center d3 = 30 ft
The unknown radar target function Γ = 0
Linear rail length L = 8 ft
Number of evenly space range profiles across rail length: 44
Chirp frequency: 2 GHz to 4 GHz in 256 steps
Incident wave amplitude Eo = 1
68
where d = 4 inches. It is clear from this image that both the front and back sides of
the slab show up in the image. However, the back of the slab is greatly attenuated
due to the conductivity loss of the lossy-dielectric slab model.
These results show that it is possible to simulate a SAR image of a dielectric
slab. The next logical step is to image both a dielectric slab and a radar target on
the opposite side of the slab. This will be done in the next chapter with interesting
results to follow.
69
Figure 4.1. Incident, reflected and transmitted fields from an air-dielectric interface.
70
Figure 4.2. Incident, reflected and transmitted fields from an air-dielectric interface
represented by wave amplitude coefficients.
71
Figure 4.3. The wave matrix geometry for multiple dielectric layers.
72
Figure 4.4. The air-dielectric-air geometry for oblique plane wave incidence.
73
Figure 4.5. Conductivity of the lossy-dielectric slab model, where r = 5.
74
Figure 4.6. Geometry for simulated range profile data of a lossy-dielectric slab.
75
Figure 4.7. Range profile of a 3.94 inch thick simulated slab at incidence angle φi = 0.
76
Figure 4.8. Range profile of a 3.94 inch thick simulated slab at incidence angle φi = π
6.
77
Figure 4.9. Simulated SAR imaging geometry of slab only.
78
Figure 4.10. SAR image of a 4 inch thick lossy-dielectric slab model.
79
CHAPTER 5
SIMULATION OF A THROUGH-DIELECTRIC SLAB RADAR
IMAGE
Combining the dielectric model developed in Chapter 4 with the 2D PEC cylinder
model developed in Chapter 3 a complete model of a 2D PEC cylinder behind a
dielectric slab will be developed in this chapter. The dielectric slab in this case will
have the electromagnetic properties found in [64]. Range profiles of a cylinder behind
a dielectric slab will first be discussed in Section 5.1. In Section 5.2 a complete
model with a 2D PEC cylinder located behind a dielectric slab will be developed and
simulated SAR imagery will be presented. These results will be used to determine
radar system specifications and design architecture in later chapters.
5.1
Simulated range profiles of a 2D PEC cylinder behind a dielectric
slab
In this section the lossy-dielectric slab model developed in Chapter 4 will be combined
with the 2D PEC cylinder model developed in Chapter 3 and tested by acquiring
simulated range profiles of the cylinder behind the slab at normal incidence and
oblique incidence. These simulated range profiles will provide insight into what radar
returns are expected from a cylinder behind a lossy-dielectric slab.
The geometry is shown in Figure 5.1 where a dielectric slab is placed between a
radar system and a PEC cylinder. The radar system is located 20 ft from the front of
the slab and the cylinder is located 10 ft from the back side of the slab. The thickness
of the slab is d. Simulated range profiles using this geometry are acquired for both
normal and oblique incidences.
The simulated range profile data is calculated by solving Equations 4.92 through
4.94 for b1 , where the radar target function Γ is that of a 2D PEC cylinder which was
80
solved in Chapter 3. Phase shifting the terminal plane of the cylinder some distance
r3 from the terminal plane of the dielectric slab
Eshif ted = e−j2ko r3
and multiplying this phase shifted plane by the solution to the 2D PEC cylinder
Equation 3.18 (ignoring the vector direction since all solutions here are T M z ) results
in the radar target function for a 2D PEC cylinder
Γ = −e−j2ko r3
∞
X
Jn (ko a) (2)
(−j)n εn
Hn (ko ρ) cos nφ,
(2)
Hn (ko a)
n=0
where
εn =
(5.1)


 1 for n = 0

 2 for n 6= 0
and the bi-static observation angle φ = −π.
The incident plane wave is T M z and thus the wave amplitude coefficient at the
dielectric boundary c1 is
c1 = Eo e−jko r1
(5.2)
where r1 is the distance from the radar system to the surface of the dielectric wall.
The received scattered field is a plane wave traveling back from the dielectric
surface to the radar receiver (the theoretical radar system in this case is assumed to
be mono-static), and thus the received scattered field equation is
Es = b1 e−jko r1 .
(5.3)
The IDFT is taken of Es for a number of test frequencies which emulate a FMCW
radar transmit chirp. Parameters used to simulate range profiles for normal and
81
oblique incidence is shown in Table 5.1.
A MATLAB program was written to simulate the range profiles of a cylinder
behind a dielectric slab and is shown for reference in Appendix H. All results presented
here were calculated using this program.
The resulting range profile for a normally incident φi = 0 plane wave is shown in
Figure 5.2. The location of the front of the slab is clearly indicated at approximately
40 nS. The location of the front of the cylinder is also clearly shown at approximately
60 nS. The cylinder is approximately 32 dB below the initial reflection off of the front
of the slab. This result is of particular interest in determining radar dynamic range,
transmit power and sensitivity requirements.
The resulting range profile for an oblique incident φi = 7.6◦ plane wave is shown
in Figure 5.3. The results for this range profile are nearly identical to that of Figure
5.2. Looking at Figure 5.3 the front of the dielectric slab shows up where expected at
approximately 40 nS. The cylinder reflection is also clearly indicated at approximately
60 nS. The cylinder reflection is 32 dB below the initial reflection from the surface of
the slab.
Based on these theoretical results it is clear that a minimum of 32 dB of dynamic
range will be required to see such a cylinder behind a 3.94 inch thick lossy dielectric
slab. These simulated results are interesting because it would not be difficult to
replicate in a laboratory setting using a tall metal pipe in place of the 2D cylinder.
5.2
Simulation of a through-dielectric slab radar image
In this section simulated SAR imagery of a 2D PEC cylinder placed behind a lossydielectric slab will be developed. Using the same procedure as was shown in Section
4.5, a 2 GHZ to 4 GHz chirped radar system is simulated. This radar system acquires
48 evenly spaced range profiles across an 8 foot long simulated linear rail placed in
front of the dielectric slab. A 2D PEC cylinder is placed behind the lossy-dielectric
82
slab at some distance.
The imaging geometry here is similar to what was used in Section 4.5. Looking at
Figure 5.4 a 2D PEC cylinder of radius a is located at a distance d3 downrange from
the rail and exactly at the rail center L
2 . The 2D PEC cylinder has a radar target
function Γ. The radar travels down the linear rail of length L acquiring range profiles
at evenly spaced increments located at rail positions x(n) from 0 to L. The incident
angle φi is dependent upon the radar position on the rail x(n) and the location of the
unknown radar target relative to the rail. And from this the x(n) dependent incident
angle function φi (n) is re-written here from Section 4.5 for convenience
"
d3
#
,
φi (n) = cos−1 q
−L + x(n)2 + d2
2
3
(5.4)
where r1 (n) is the distance from the radar itself to the dielectric slab
r1 (n) =
d1
cos φi (n)
(5.5)
and r3 (n) is the distance from the opposite side of the slab to the unknown target
location
d − d1 − d
r3 (n) = 3
.
cos φi (n)
(5.6)
Where d is the thickness of the dielectric slab and d1 is the distance from the linear
rail to the surface of the dielectric slab. From these equations the x(n) dependent
incident field amplitude coefficient can be derived
c1 (n) = Eo e−jko r1 (n) .
83
(5.7)
The x(n) dependent scattered field equation is
Es (n) = b1 (n)e−jko r1 (n) ,
(5.8)
where b1 (n) is dependent upon the x(n) dependent impedance equation
Z(n) = q
cos φi (n)
.
σ − sin2 φ (n)
r + jω
i
o
(5.9)
The x(n) dependent reflection and transmission coefficients are given by
Z(n) − 1
R1 (n) =
,
Z(n) + 1
(5.10)
1 − Z(n)
R2 (n) =
,
Z(n) + 1
(5.11)
T1 (n) = 1 + R1 (n)
(5.12)
T2 (n) = 1 + R2 (n).
(5.13)
and
Using the above equations the x(n) dependent amplitude coefficients become
c3 (n) =
c1 (n)T1 (n)T2 (n)
(5.14)
ejθ(n) + R1 (n)R2 (n)e−jθ(n) + Γ R2 (n)ejθ(n) + R1 (n)e−jθ(n)
and
c3 (n) h
b1 (n) =
R1 (n)ejθ(n) + R2 (n)e−jθ(n)
T1 (n)T2 (n)
i
jθ(n)
−jθ(n)
+ Γ R1 (n)R2 (n)e
+e
. (5.15)
84
Where the x(n) dependent electrical length θ(n) is given by
r
r +
θ(n) = ko d
σ − sin2 φi (n).
jωo
(5.16)
In addition, the radar target function Γ relates b3 (n) to c3 (n) by the equation
b3 (n) = Γc3 (n).
(5.17)
In this case a 2D PEC cylinder is located behind the slab and has the x(n) dependent
radar target function (derived from Equation 5.1)
Γ(n) = −e−j2ko r3 (n)
∞
X
J (ko a)
(2)
(−j)i εi i
Hi (ko ρ) cos nφ,
(2)
Hi (ko a)
i=0
where
εn =


 1 for n = 0
(5.18)
.

 2 for n 6= 0
Using the model derived in Equations 5.4 through 5.17 a simulated SAR image
data set was calculated using the parameters shown in Table 5.2.
The MATLAB program used to calculate the SAR data set is shown in Appendix
I. In this, first the MATLAB program in Appendix I was run after which the data
conditioning MATLAB program in Appendix B was run followed by the RMA SAR
imaging algorithm MATLAB program in Appendix C.
Two simulated SAR images were calculated using a slab thickness of d = 4 inches
and for cylinders of radius a = 3 inches (see Figure 5.5) and a = 6 inches (see Figure
5.6).
The amplitude of the cylinder in both images is significantly less than the lossydielectric slab. The dynamic range required to display these image results is great
(notice the amplitude scale when comparing imagery) requiring 50 to 60 dB of image
85
Table 5.1. The substitutions shown here were used to simulate range profiles of a
cylinder behind a dielectric slab.
Range from the radar to the front of the slab r1 = 20 ft
Range from back side of slab to cylinder center r3 = 10 ft
Permittivity of the wall r = 5 based on [64]
Cylinder radius a = 3 inches
Conductivity σ is a function of frequency based on [64] (see Figure 4.5)
Incident angle φi = 0 for normal, φi = 7.6◦ for oblique
Thickness of dielectric d = 3.94 inches
Chirp frequency: 2 GHz to 4 GHz in 256 steps
Incident wave amplitude Eo = 1
Table 5.2. The substitutions shown here were used to simulate SAR image data of a
cylinder behind a dielectric slab.
Permittivity of the slab r = 5 based on [64]
Conductivity σ is a function of frequency based on [64] (see Figure 4.5)
Distance from rail to front of dielectric d1 = 20 ft
Distance from rail to unknown target center d3 = 30 ft
The unknown radar target function Γ = 0
Linear rail length L = 8 ft
Number of evenly space range profiles across rail length: 48
Chirp frequency: 2 GHz to 4 GHz in 256 steps
Incident wave amplitude Eo = 1
86
dynamic range. This observation is consistent with the simulated range profiles in
Section 5.1.
The simulated SAR imagery of cylinders in this section are not much different
than the simulated cylinder without a slab in Section 3.4. This result is promising
showing that the lossy-dielectric slab does not significantly distort the SAR image
of an object behind the slab using this geometry where the rail SAR is located 20
feet away from the slab. The cylinder image is, however, greatly attenuated. These
results indicate that radar sensitivity and dynamic range appear to be the greatest
challenge to imaging behind a lossy-dielectric slab with the same dielectric properties
from [64].
5.3
Simulation of a through-dielectric slab radar image using coherent
background subtraction
Coherent background subtraction is typically used in small rail SAR systems such
as the systems presented in [59] [60]. In these systems the background clutter is
significant compared to the target scene. Reducing or eliminating background clutter from radar imagery is achieved by the use of background subtraction. Background subtraction works by first measuring the target scene producing data ma
trix sback x(n), ω(t) . After this, by placing targets in the target scene and mea
suring again producing data matrix sscene x(n), ω(t) . The resulting backgroundsubtracted data set is the difference between the target scene with targets placed and
the target scene before the targets were placed:
stargets x(n), ω(t) = sscene x(n), ω(t) − sback x(n), ω(t) .
(5.19)
It is difficult to notice the location of the cylinders in the images shown in Figure
5.5 and Figure 5.6. For this reason a simulated data set using background subtraction
87
will be developed in this section by applying Equation 5.19, where:
sback x(n), ω(t) = dielectric slab model from Section 4.5,
sscene x(n), ω(t) = dielectric slab and cylinder model from Section 5.2.
Two simulated images were created using background subtraction with a slab of
thickness d = 4 inches using a cylinder with a radius of a = 3 inches (see Figure 5.7)
and a = 6 inches (see Figure 5.8).
In comparing the simulated imagery of cylinders in free space in Figure 3.8 and
Figure 3.9 to the background subtracted cylinders behind a lossy dielectric slab in
Figure 5.7 and Figure 5.8, it is interesting to note that the cylinder images look
similar but the return amplitudes are 25 to 30 dB less behind a lossy dielectric slab
than in free space. This is due to the attenuation inside of the slab and the boundary
conditions of the slab reflecting most of the incident field. It is also interesting to
note that the imagery of cylinders behind a slab are shifted slightly downrange by
a few inches compared to the imagery of cylinders in free space. This is due to the
round trip delay through the dielectric slab, where inside the slab the waves travel
at a slower velocity causing the target location to be delayed slightly in downrange.
These results will be discussed further and used to develop a set of radar system
design specifications and architecture requirements in the next chapter.
88
Figure 5.1. Geometry of simulated range profile data.
89
Figure 5.2. Range profile of a lossy-dielectric slab in front a 3 inch radius cylinder at
normal incidence.
90
Figure 5.3. Range profile of a lossy-dielectric slab in front a 3 inch radius cylinder at
an oblique incidence.
91
Figure 5.4. Simulated SAR imaging geometry.
92
Figure 5.5. SAR image of a simulated target scene made up of a 3 inch radius cylinder
target behind a 4 inch thick lossy-dielectric slab.
93
Figure 5.6. SAR image of a simulated target scene made up of a 6 inch radius cylinder
target behind a 4 inch thick lossy-dielectric slab.
94
Figure 5.7. SAR image of a simulated target scene made up of a 3 inch radius cylinder
target behind a 4 inch thick lossy-dielectric slab using background subtraction.
95
Figure 5.8. SAR image of a simulated target scene made up of a 6 inch radius cylinder
target behind a 4 inch thick lossy-dielectric slab using background subtraction.
96
CHAPTER 6
RADAR SYSTEM DESIGN REQUIREMENTS AND THEORETICAL
ANALYSIS
A theoretical model provides insight into the numerous practical design challenges
present when developing a measurement system. This chapter was written as a general
discussion of the basic design requirements which will drive the system designs shown
in later chapters. In this chapter theory meets the reality of system design where
simulated radar data is produced using the models developed in Chapters 3, 4, and
5. Using this simulated data some general design guidlines and an overall design
philosophy will be developed.
6.1
Flash and through-lossy-dielectric slab attenuation
When measuring a target behind a dielectric slab the strongest scattered return signal
is the slab itself. The strong scattered signal from the slab will be referred to as the
’flash.’
The flash is the single greatest technical challenge when measuring targets on the
other side of a dielectric slab. The flash saturates radar receivers, causing desensitization. The flash sets the dynamic range of a radar system by peaking the digitizer,
causing the least significant bits (LSB’s) of the digitizer to be above the level of the
scattered returns from the targets behind the slab.
In order to examine the flash more closely the model in Section 5.1 was utilized
where the distance from the radar to the slab is 20 feet, slab thickness is d = 4
inches and the distance from the radar to the 2D PEC cylinder is 30 feet. Using
this theoretical setup the effects of flash are shown in Figure 6.1 (b) for a cylinder
of a = 3 inches in radius and Figure 6.2 (b) for a cylinder of a = 6 inches in radius.
For comparative purposes the simulated range profile model of a 2D cylinder in free
97
space developed in Section 3.3 is shown in Figure 6.1 (a) for a cylinder of radius a = 3
and Figure 6.2 (a) for a cylinder of radius a = 6 inches. Looking at these figures, the
flash from the lossy dielectric slab is significantly greater than the cylinder behind
the slab. The slab is the target located at approximately 40 nS down range and the
cylinder is the target located at approximately 60 nS downrange.
Targets behind a lossy-dielectric slab scatter significantly less energy back to the
radar system compared to the same targets in free space. This is because a large
amount of attenuation occurs when a pulse emitted from a radar transmitter travels
through a slab. This loss is also due to the radar transmitter signal reflecting off of
the air-slab boundary. This attenuation is upwards of 10 dB for a 4 inch thick lossydielectric slab and much greater attenuation occurs for thicker lossy-dielectric slabs.
What is left of the transmitted signal on the other side of the slab travels towards
the target then scatters off the target. The scattered signal then has to pass back
through the slab experiencing further attenuation (another 10 dB or so) due to the
loss tangent of the lossy slab material and reflection off of the air-slab boundary. This
results in a reduction of the scattered return magnitude of the target compared to the
same target in free space, typically 15 dB to 30 dB lower. Through-lossy-dielectric
slab attenuation is evident in the case of a 4 inch thick slab shown in Figure 6.1 and
Figure 6.2, where in both cases the return amplitude of the cylinder behind the slab
is approximately 15 dB lower than the same cylinder in free space. For a 12 inch thick
lossy-dielectric slab the scattered signal from the cylinder is attenuated significantly
to the point of being undetectable using the theoretical model developed in Section
5.1, this is shown in Figure 6.4.
The flash and the through slab attenuation severely impede system performance.
The flash limits dynamic range, the through slab attenuation places targets below
the receiver noise floor and limits receiver sensitivity.
98
6.2
The necessity of a range gate
There is one sure way to get rid of the flash, that is with the use of a time domain
rang-gate. A time domain range-gate will switch on the radar receiver at just the
right time so that it is receiving the scattered signals from the targets behind the
slab and not the flash from the slab. If a time domain range-gate were used on the
theoretical range profiles shown in Figure 6.1 (b), Figure 6.2 (b) and Figure 6.4 (b)
then there would be no flash at 40 nS down range.
With the flash eliminated a radar system could use its full dynamic range to acquire data on the targets located behind the slab increasing sensitivity of the resulting
measurement. A time domain range gate is a necessity for through-lossy-dielectric
slab radar imaging.
6.3
In-line attenuator approximation
Through-slab attenuation limits receiver sensitivity. One way to look at targets on the
other side of a lossy-dielectric slab is to ignore the flash using a time domain rangegate, and focus on the through-slab attenuation. It will be shown in this section
that the distortion effects on a target placed on the other side of a lossy-dielectric
slab are minimal. This is due to the lossy properties of the dielectric. If the flash
can be ignored then the through slab imaging problem simply becomes an issue of
receiver sensitivity, where what amounts to an in-line attenuator is placed between
the transmit and receive paths of the scattered radar signals.
6.3.1
Comparison of through-lossy-dielectric slab and free-space range
profile results
When looking at the simulated results for a lossy-dielectric slab in Figure 6.1 (b) and
Figure 6.2 (b) it is observed that the scattered magnitude of the cylinder behind a slab
is simply attenuated, shifted downrange by a few nano-seconds, but not distorted.
99
This is due to the fact that the dielectric slab is lossy with a frequency dependent
conductivity plotted in Figure 4.5. There is little to no distortion in the range profile
of a cylinder located behind this dielectric slab.
6.3.2
Lack of multi-bounce in simulated range profile results
If the conductivity were set to σ = 0, then using the model developed in Section 5.1
the range profile of a lossless dielectric slab with thickness d = 4 inches, distance from
radar to wall of 20 feet, distance from radar to cylinder 30 feet, is shown in Figure
6.3. Scattering from the front and back of the slab can be seen in these results at
approximately 40 nS down range. The scattering from the lossless slab shows up as
a double-bounce in time domain. Similarly, scattering from the cylinder located at
approximately 60 nS downrange also shows up as a double bounce.
Increasing the slab thickness to d = 12 inches, making the conductivity σ = 0
and using the model developed in Section 5.1, the range profile of a a = 3 inch radius
cylinder behind a lossless 12 inch thick dielectric slab is shown in Figure 6.5. It is
clear from these results that the double bounce starting at 40 nS down range is the
front and back of the slab and the double bounce starting at approximately 65 nS is
the cylinder causing multiple echos due to the scattered fields traveling in and out of
the dielectric.
6.3.3
Comparison of lossly and lossless dielectric range profiles
In summary; if a dielectric slab were lossless then the range profile results would be
extremely distorted with a double-bounce for the slab and each target in the scene.
This was shown in Figure 6.3 and Figure 6.5. As it turns out the dielectric of interest
from [64] is a very lossy material and so the range profile of this dielectric slab is much
less full of distortion as shown in Figure 6.1 and Figure 6.2. This is because the lossy
properties of the dielectric cause attenuation to signals that bounce more than once
through the slab. Based on these results it appears that signal attenuation due to
100
the radar signals traveling through a lossy slab, not distortion due to the lossy slab,
is the limiting factor to detecting a cylinder located behind a lossy-dielectric slab.
6.3.4
In-line attenuator approximation applied to theoretical range profile results
Based on the results presented in this section there is very little distortion affecting
the range profile of a cylinder behind a lossy-dielectric slab. For this reason the
dielectric slab can be approximated as simply an attenuator in-line with the target
scene. This in-line attenuator approximation is accurate only if a time domain rangegate is utilized (which eliminates the flash from the slab) in the radar system design.
The in-line attenuator approximation is shown in Figure 6.16, where for a d = 4 inch
thick slab there is 7.5 dB of attenuation from the transmitted signal to the target
scene and 7.5 dB of additional attenuation due to the scattered target scene back
through the slab to the radar receiver.
6.4
Using the RMA for through-dielectric slab imaging
It will be shown in this section that free-space SAR imaging algorithms or other freespace radar algorithms are accurate when radar imaging through a finite thickness
lossy-dielectric slab at stand-off ranges of 20 feet to 30 feet.
6.4.1
Comparison of simulated free-space and through-dielectric slab
SAR imagery
Using the model developed in Section 5.3 and SAR processed using the RMA developed in Chapter 2 a comparison of the theoretical background subtracted SAR
imagery of a cylinder behind a d = 4 inch thick lossy-dielectric slab to that of a
cylinder in free-space based on the imaging geometry shown in Figure 5.4 is shown
in Figure 6.6 for a cylinder of radius a = 3 inches. The same is shown in Figure 6.7
for a cylinder of radius a = 6 inches. In both cases the cylinder image appears to
101
be the same as it is in free-space except that the return amplitude of the cylinder
is significantly lower by approximately 15 dB. There is little noticeable distortion in
the simulated SAR image of a cylinder behind a lossy-dielectric slab, except that the
cylinder image is slightly wider and shifted downrange by a small amount.
A similar result can be found when increasing the slab thickness to d = 12 inches
and is shown in Figure 6.12. Again, there is little noticeable distortion in the theoretical SAR image of a cylinder behind a lossy-dielectric slab except that the cylinder
image is noticeably wider and shifted downrange by a small amount. However, the
increase width of the cylinder is not so wide that the image is severely degraded.
These results show that according to the model when imaging through a lossydielectric slab the free space RMA SAR imaging algorithm is effective in imaging a
target scene. This is due to the fact that dielectric is lossy and the radar system is
located at a stand-off range of 20 feet to 30 feet.
6.4.2
Comparison of simulated free-space and through-lossless-dielectric
slab SAR imagery
The phenomena of multi-bounce shown in Section 6.3.2 occurs in simulated SAR
imagery of a cylinder behind a finite thickness lossless-dielectric slab. This causes
image blurring and distortion making imaging through a lossless-dielectric at a standoff range difficult using a free space imaging algorithm such as the RMA.
Using the theoretical model developed in Section 5.3, SAR processed using the
RMA developed in Chapter 2 and letting the dielectric conductivity σ = 0 for the
imaging geometry shown in Figure 5.4, a simulated image of a cylinder with radius
a = 3 through a dielectric slab of thickness d = 4 inches was created. The results for
this are shown in Figure 6.8. It is clear from this image that multi-bounce effects of
a lossless-dielectric affect the SAR image by blurring the position of the cylinder.
A similar result can be found by increasing the slab thickness to d = 12 inches and
is shown in Figure 6.13. This image is interesting because it shows the cylinder in two
102
different locations down range. This result is due to the increased thickness of the
dielectric and the lossless properties of the dielectric. The lossless properties of the
dielectric are causing an increase in the multi-bounce effects inside of the dielectric
by not attenuating the multi-bounce effects. These multi-bounce effects are showing
up in the resulting simulated SAR image.
6.4.3
Simulated offset through-lossy-dielectric slab imagery
Theoretical SAR imagery of an offset PEC cylinder was created using the model
developed in Section 5.3, and SAR processed using the RMA developed in Chapter
2. This model uses the offset imaging geometry shown in Figure 6.9. A simulated
image of a cylinder offset in cross range (not centered with rail) with radius a = 3
through a lossy-dielectric slab with thickness d = 4 inches was created, results are
shown in Figure 6.10. It is clear from this image that little noticeable image distortion
is present when imaging a cylinder that is slightly offset in cross range.
Using the same imaging geometry, but changing the slab thickness to d = 12
inches, similar results were found and shown in Figure 6.14. Some minor differences
are noticeable including some downrange target shifting and the cylinder image is
slightly wider than free space. With those exceptions there is little to no noticeable
image distortion is present.
6.4.4
Offset through-lossless-dielectric slab imagery
The resulting simulated SAR image becomes blurred and distorted when there is no
loss in the dielectric slab imaging geometry shown in Figure 6.9. This result is shown
in Figure 6.11, where a simulation of a cylinder with radius a = 3 is imaged behind a
lossless-dielectric slab of thickness d = 4 inches. Multiple bounce effects occur which
are similar to those found in Section 6.4.2.
A second case was tested with more dramatic effect where the lossless-dielectric
was increased to d = 12 inches in thickness. This resulting imagery is shown in Figure
103
6.15.
6.4.5
Summary of using the RMA for through-dielectric slab imaging
From the results in Section 6.4.1 it is clear that the in-line attenuator approximation
is effective when ranging or imaging cylinders through lossy-dielectrics at stand-off
ranges. It also holds true for SAR imaging geometries where the target is slightly
shifted in cross range, as was shown in Section 6.4.3.
It was also shown in Sections 6.4.2 and 6.4.4, that if dielectric slabs were lossless
then the RMA would not be useful.
Based on these simulated results, imagery from a lossy dielectric slab is better
than imagery from a lossless slab. These results show that free space SAR algorithms
such as the RMA developed in Chapter 2 are very effective at imaging a target scene
behind a finite thickness lossy-dielectric slab when the radar system is placed at a
stand-off range from the slab.
6.5
Summary of the general design requirements
By analyzing results generated from the models developed in Chapters 3, 4 and 5,
two major design specifications were realized in this chapter:
• Time domain range gate
• High sensitivity receiver
In addition to these specifications it has been assumed throughout this dissertation
that the following system specifications will be adhered to:
• Chirped radar system, from 2 GHz to 4 GHz
• Linear rail SAR imaging geometry using an 8 foot linear rail where the range
profile data is acquired once every 2 inches across the rail
104
In addition to these specifications it was found that the RMA free-space SAR
imaging algorithm works extremely well for imaging radar scenes through a lossydielectric slab with a finite thickness when the radar is placed at a stand-off range.
A unique design approach will be shown in the next chapter that meets these
design requirements fully. It will be shown that this system is capable of imaging
through lossy-dielectric slabs and other material.
105
(a)
(b)
Figure 6.1. Simulated range profiles of a cylinder with radius a = 3 inches in free-space
(a) and behind a 4 inch thick lossy-dielectric slab (b).
106
(a)
(b)
Figure 6.2. Simulated range profiles of a cylinder with radius a = 6 inches in free-space
(a) and behind a 4 inch thick lossy-dielectric slab (b).
107
(a)
(b)
Figure 6.3. Simulated range profiles of a cylinder with radius a = 3 inches in free-space
(a) and behind a 4 inch thick lossless-dielectric slab (b).
108
(a)
(b)
Figure 6.4. Simulated range profiles of a cylinder with radius a = 3 inches in free-space
(a) and behind a 12 inch thick lossy-dielectric slab (b).
109
(a)
(b)
Figure 6.5. Simulated range profiles of a cylinder with radius a = 3 inches in free-space
(a) and behind a 12 inch thick lossless-dielectric slab (b).
110
(a)
(b)
Figure 6.6. Simulated SAR imagery of a 2D cylinder with radius a = 3 inches
in free-space (a) and behind a 4 inch thick lossy-dielectric slab using background
subtraction (b).
111
(a)
(b)
Figure 6.7. Simulated SAR imagery of a 2D cylinder with radius a = 6 inches
in free-space (a) and behind a 4 inch thick lossy-dielectric slab using background
subtraction (b).
112
(a)
(b)
Figure 6.8. Simulated SAR imagery of a 2D cylinder with radius a = 3 inches in
free-space (a) and behind a 4 inch thick lossless-dielectric slab using background
subtraction (b).
113
Figure 6.9. The 2D PEC cylinder is offset in cross range from the center of the rail
to show the theoretical effects of an offset target behind a dielectric slab.
114
(a)
(b)
Figure 6.10. Simulated SAR imagery of a 2D cylinder offset by approximately 2 feet
with radius a = 3 inches in free-space (a), behind a 4 inch thick lossy-dielectric slab
using background subtraction (b).
115
(a)
(b)
Figure 6.11. Simulated SAR imagery of a 2D cylinder offset by approximately 2 feet
with radius a = 3 inches in free-space (a), behind a 4 inch thick lossless-dielectric slab
using background subtraction (b).
116
(a)
(b)
Figure 6.12. Simulated SAR imagery of a 2D cylinder with radius a = 3 inches
in free-space (a) and behind a 12 inch thick lossy-dielectric slab using background
subtraction (b).
117
(a)
(b)
Figure 6.13. Simulated SAR imagery of a 2D cylinder with radius a = 3 inches in
free-space (a) and behind a 12 inch thick lossless-dielectric slab using background
subtraction (b).
118
(a)
(b)
Figure 6.14. Simulated SAR imagery of a 2D cylinder offset by approximately 2 feet
with radius a = 3 inches in free-space (a), behind a 12 inch thick lossy-dielectric slab
using background subtraction (b).
119
(a)
(b)
Figure 6.15. Simulated SAR imagery of a 2D cylinder offset by approximately 2 feet
with radius a = 3 inches in free-space (a), behind a 12 inch thick lossless-dielectric
slab using background subtraction (b).
120
Figure 6.16. A simple attenuation model for the lossy-dielectric slab for use in determining system specifications.
121
CHAPTER 7
HIGH SENSITIVITY RANGE-GATED FMCW RADAR
ARCHITECTURE
A radar architecture was developed which includes a range-gate to eliminate the flash
off of the slab and a high sensitivity receiver to overcome through-slab attenuation.
The system presented in this chapter fulfills the design specifications outlined in
Chapter 6. It will be shown in this chapter that a traditional time domain range-gate
reduces receiver sensitivity when it is short in time duration (as what is uniquely
required for small radar imaging applications). The radar architecture developed in
this chapter is unusual in that it allows for a small duration time domain range-gate
while at the same time increasing receiver sensitivity.
7.1
Noise bandwidth and receiver sensitivity
According to standard receiver design theory [67] [68] the sensitivity is directly related
to the IF bandwidth. The narrower the IF bandwidth the better the sensitivity.
According to [67] for an ideal receiver (disregarding the effects of all the stages) the
sensitivity of a SSB or CW receiver can be determined by calculating the minimum
detectable signal (MDS) which is related directly to the IF bandwidth:
M DSdBm = −174 + 10 log10 Bn + N F,
(7.1)
where:
Bn = noise bandwidth of the receiver (Hz), which will be assumed to be the
IF bandwidth in this chapter for an ideal receiver:
−174 dBm is the available thermal noise power per Hz at room temperature of
122
290◦ K :
N F = 3.3dB front end noise figure for a typical Mini-Circuits broad band
amplifier that might be used at the 2 GHz to 4 GHz band for
through-lossy-dielectric slab imaging.
From this equation the relation is obvious; the narrower the IF bandwidth Bn the
more sensitive the receiver.
7.2
Time domain range-gate and receiver sensitivity
A typical coherent time domain range-gated radar system [34] might utilize a pulsed
IF radar architecture such as the one shown in Figure 7.1. Where a pulse is produced
with the wave-form generator, fed into the transmit mixer MXR1. MXR1 is fed by
OSC3 which is a microwave frequency oscillator operating close to the frequency of
interest to make a measurement. OSC2 and OSC3 are phase locked to master clock
oscillator OSC1. The output of MXR1 is fed into the transmitter which contains
power amplifiers, filters, and RF switches. The output of the transmitter is fed into
a duplexer which is connected to the antenna. The antenna is switched between the
transmitter and receiver circuits through the duplexer. Inside the duplexer there are
circulators and transmit and receive switches. The receive port of the duplexer feeds
the front end receiver low noise amplifier LNA1. The output of LNA1 is fed into
MXR2. MXR2 is driven by OSC3 so it is perfectly coherent with MXR1. The output
of MXR2 is fed into the IF amplifier AMP1. AMP1 adds gain to the system and
pushes the IF into the IF filter FL1. FL1 is a band limiting filter, the bandwidth of
this filter depends on the minimum range gate that the system is designed to handle.
The bandwidth of FL1 sets the equivalent noise bandwidth of the receiver following
Equation 7.1. The bandwidth depends on the minimum pulse time that the radar is
expected to detect. A typical specification would be Bn = 1/T where T = range gate
123
time duration in seconds. The output of FL1 is fed into a splitter which then feeds
into the IQ demodulator made up of MXR3, MXR4, and fed by OSC2. The output
of the IQ demodulator is baseband, DC to whatever the bandwidth of FL1 is. This
signal is amplified by Video Amp 1 and Video Amp 2 then digitized. The digitizers
sample at or above the minimum sampling rate of Bn for an IQ demodulator, where
Bn is the IF bandwidth of the radar system. The range-gate is pulsed at a high rate.
During each pulse OSC3 is tuned to a different frequency making up a range profile of
frequency domain data. The radar could also span a number of frequencies during a
single pulse, which would require a higher receiver equivalent noise bandwidth Bn . If
10 frequency steps were made during one pulse then the radar receiver would require
a bandwidth of Bn = 10/T .
7.2.1
Example of a 40 nS time domain range-gate in a pulsed IF radar
system ranging a target through a dielectric slab
As was shown in Chapter 6, a range-gate is necessary to image a target behind a
dielectric slab. Examining the likely imaging scenario shown in Figure 7.2, where
radar range to slab d1 = 20 feet and the cylinder target is located d3 = 30 feet
from the radar, the maximum length range-gate for this geometry is less than 40 nS.
Looking at this range profile example in Figure 7.3 the location of the slab, target and
multipath are clearly shown. The multipath is due to the transmitted pulse bouncing
off the slab, back to the radar system itself, then off the radar again and back to the
slab. This multipath return would be detecting the radar equipment and anything or
anyone near the radar equipment rather than targets behind the slab. Multipath is
a serious problem when imaging through dielectric slabs, causing the slab to act like
a mirror producing imagery of targets that appear to be behind the slab but which
are actually in front of the slab inside of the desired range-gate. It is for this reason
that the range-gate shown in Figure 7.3 be no greater than 40 nS long. The shorter
the range-gate the less likely multi-path returned signals will be detected, the better
124
it is for through-slab imaging.
Since the range-gate must be T = 40 nS or less then the minimum IF bandwidth
of a typical pulsed IF radar design for this scenario would be Bn = 1/T = 25 MHz.
According to Equation 7.1 the ideal receiver sensitivity would be -96.7 dBm.
7.2.2
Example of a 20 nS time domain range-gate in a pulsed IF radar
system ranging a target through a dielectric slab
It is likely that a through-dielectric slab imaging system would be placed 10 feet in
front of a slab, or closer. Such a geometry is explored here where looking at Figure
7.2 the dielectric slab is located d1 = 10 feet from the radar system and the target is
located at d3 = 15 feet from the radar system. A range profile result of this geometry
is shown in Figure 7.4 where the reflection off of the slab is located at 20 nS, the
target is located at 30 nS and the multi-path begins at 40 nS. The multi-path in this
case would be the radar system itself and anything or anyone standing behind the
radar system as measurements are acquired. The geometry here is tighter in spacing
than the previous section. For this reason the range-gate must be shorter in duration,
approximately 20 nS or less, in order to avoid detecting the slab or multi-path.
For a range-gate of T = 20 nS or less the minimum IF bandwidth would be
Bn = 1/T = 50 MHz. According to Equation 7.1 the ideal receiver sensitivity would
be -93.7 dBm.
7.2.3
Time domain range-gates and their limitations
Based on results in Sections 7.2.1 and 7.2.2 it was shown that short duration rangegates are required to eliminate the flash from the dielectric slab and to remove unwanted multi-path reflections. The tighter the down range image geometry the smaller
the range-gate. In most through-dielectric slab imaging scenarios the down range geometry will be very close requiring the use of small duration range-gates. The smaller
the range-gate the less sensitive the radar receiver. This sets a limit on radar system
125
performance if conventional design architecture, such as pulsed IF, or UWB impulse
(where the receiver bandwidth would exceed 500 MHz) were used. However, a much
less conventional method will be shown in the next section that implements a short
duration range-gate while at the same time increasing, rather than decreasing, receiver sensitivity.
7.3
High sensitivity range-gated FMCW radar architecture
It will be shown in this section that a chirped radar (rather than a time domain
pulsed IF system) used with some creative design could implement a short duration
time domain range-gate using readily available low cost high frequency parts while
at the same time increasing receiver sensitivity.
7.3.1
FMCW radar
Rather than using a time domain pulse, frequency modulated continuous wave
(FMCW) radar uses a linearly modulated voltage controlled oscillator (VCO) amplified and transmitted out towards a target scene to range targets. Figure 7.5 shows
a block diagram of a typical FMCW radar, such as that found in [60]. Range to target
information from an FMCW radar is in the form of low frequency beat tones at or
near the audio frequency range. This is accomplished by a linear ramp modulated
VCO OSC1. The output of OSC1 is fed directly to the transmit antenna ANT1 and
transmitted out toward a target. The chirped signal transmitted out to the target
scene is represented by the equation:
T X(t) = cos 2π(fosc + cr t)t ,
where:
fosc = start frequency of linear ramp modulated VCO and
126
(7.2)
cr = radar chirp rate.
The transmitted signal is radiated out towards the target scene and bounces off of a
target. The round trip time from the transmitter antenna ANT1 to the target, back
to the receiving antenna ANT2, is tdelay . This time shifted delayed transmit signal
T X(t − tdelay ) is amplified by LNA1 and fed into the RF port of MXR1.
Some of the power from OSC1 is coupled off using CLPR1 and fed into the LO
port of MXR1 making the receiver coherent with the VCO OSC1 so that T X(t) is
multiplied by T X(t − tdelay ). This is represented by the video amplifier output
equation (disregarding amplitude coefficients)
V ideo(t) = T X(t) · T X(t − tdelay ),
V ideo(t) = cos 2π(fosc + cr t)t · cos 2π(fosc + cr t)(t − tdelay ) .
The higher frequency term is ignored because the IF port of a practical mixer could
not produce the resulting microwave frequencies. The resulting video signal is the
audio frequency beat tone which is directly proportional to the chirp rate cr and the
round trip radar to target and back time delay tdelay plus a DC phase term:
V ideo(t) = cos 2πfosc tdelay + cr ttdelay .
(7.3)
If there are a variety of targets down range then the signal V ideo(t) will be a superposition of beat tones at various frequencies and amplitudes. All of this is digitized
and fed into signal processing and data conditioning hardware.
7.3.2
High sensitivity range-gated FMCW radar system design theory
The range to target information from an FMCW radar system is in the form of low
frequency beat tones. For this reason it is possible to implement a short duration
range gate in an FMCW radar system by simply placing a band pass filter (BPF) on
127
the output of Video Amp1 in Figure 7.5. However, this is difficult to implement in
practice because it is extremely difficult to design effective high Q bandpass filters at
base-band. Much higher performance BPF’s are available in the form of widely used
IF communications filters which operate at high frequencies. These filters are found
in two way radios, TV sets and radio receivers. Examples of these IF filters include:
• crystal filters
• ceramic filters
• SAW filters
• mechanical filters
These communications IF filters typically operate at standard IF frequencies of 10.7
MHz, 21.4 MHz, 455 KHz, 49 MHz and etc. These communications filters are high
Q, where Q is defined as [70]:
Q=
fc
,
B
(7.4)
where:
fc = center frequency of the BPF and
B = −3 dB bandwidth of the filter.
A typical operating frequency of a crystal filter would be fc = 10.7 MHz with a
bandwidth of B = 7.5 KHz. The resulting Q of this filter would be Q = 1426.7. High
Q’s such as this are extremely difficult to achieve with BPF designs at base-band
audio frequencies. The design shown in this section uses high Q IF filters to create a
short duration range gate, while at the same time, reduces receiver noise bandwidth
Bn causing a dramatic increase in receiver sensitivity. With this design; the shorter
duration the range gate, the more sensitive the radar receiver.
128
A simplified block diagram of the high sensitivity range-gated FMCW radar system is shown in Figure 7.6. In the following explanation amplitude coefficients will be
ignored. OSC1 is a high frequency tunable oscillator which could be anything from
a PLL synthesizer to an old vacuum tube signal generator. The frequency output of
OSC1 is fBF O which can be represented by the equation:
BF O(t) = cos 2πfBF O t .
(7.5)
The output of OSC1 is fed into the IF port of MXR1. The LO port of MXR1 is
driven by OSC2. OSC2 is a 2 GHz to 4 GHz voltage tuned YIG oscillator (YIG
oscillators are VCO’s which are capable of producing highly linear voltage tuning
slopes). OSC2 is FM modulated by a linear ramp input, where the output of OSC2
can be represented by the equation:
LO(t) = cos 2π(2 · 109 + cr t)t .
(7.6)
OSC1 and OSC2 are mixed together in MXR1 to produce the transmit signal which
is then amplified by power amplifier PA1. The output of PA1 is fed into the transmit
antenna ANT1 and propagated out towards the target scene. The transmitted signal
out of ANT1 is T X(t), where:
T X(t) = LO(t) · BF O(t),
T X(t) = cos 2π(2 · 109 + cr t)t · cos 2πfBF O t .
129
After some simplification this becomes
T X(t) = cos 2π(2 · 109 + cr t)t + 2πfBF O t + cos 2π(2 · 109 + cr t)t − 2πfBF O t .
(7.7)
The transmitted waveform T X(t) is radiated out to the target scene, reflected off of
a target, delayed by some round trip time tdelay and propagated back to the receiver
antenna ANT2. The received signal at ANT2 is represented by the equation:
RX(t) = cos 2π(2 · 109 + cr t)(t − tdelay ) + 2πfBF O (t − tdelay )
+ cos 2π(2 · 109 + cr t)(t − tdelay ) − 2πfBF O (t − tdelay ) . (7.8)
The output of ANT2 is amplified by LNA1 and fed into MXR2. The LO port of
MXR2 is fed by OSC2. The IF output of MXR2 is the product
IF (t) = LO(t) · RX(t).
Evaluating this product results in
IF (t) =
cos 2π(2 · 109 + cr t)t · cos 2π(2 · 109 + cr t)(t − tdelay ) + 2πfBF O (t − tdelay )
+cos 2π(2 · 109 + cr t)t ·cos 2π(2 · 109 + cr t)(t − tdelay ) − 2πfBF O (t − tdelay ) .
(7.9)
130
Multiplying out the terms in the above equation results in
IF (t) =
cos 2π(2 · 109 + cr t)(t − tdelay ) + 2πfBF O (t − tdelay ) + 2π(2 · 109 + cr t)t
+ cos 2π(2 · 109 + cr t)(t − tdelay ) + 2πfBF O (t − tdelay ) − 2π(2 · 109 + cr t)t
+ cos 2π(2 · 109 + cr t)(t − tdelay ) − 2πfBF O (t − tdelay ) + 2π(2 · 109 + cr t)t
+ cos 2π(2 · 109 + cr t)(t − tdelay ) − 2πfBF O (t − tdelay ) − 2π(2 · 109 + cr t)t .
(7.10)
As a practical consideration the IF port of MXR2 can not output microwave frequencies so the high frequency terms can be dropped resulting in:
IF (t) =
cos 2π(2 · 109 + cr t)(t − tdelay ) + 2πfBF O (t − tdelay ) − 2π(2 · 109 + cr t)t
+ cos 2π(2 · 109 + cr t)(t − tdelay ) − 2πfBF O (t − tdelay ) − 2π(2 · 109 + cr t)t .
(7.11)
Expanding out the terms inside of the cosine argument results in:
IF (t) = cos 2π(2 · 109 + cr t)t − 2π(2 · 109 + cr t)tdelay
9
+ 2πfBF O (t − tdelay ) − 2π(2 · 10 + cr t)t
+ cos (2π(2 · 109 + cr t)t − 2π(2 · 109 + cr t)tdelay
9
− 2πfBF O (t − tdelay ) − 2π(2 · 10 + cr t)t . (7.12)
131
Letting the high frequency terms cancel out:
9
IF (t) = cos − 2π(2 · 10 + cr t)tdelay + 2πfBF O (t − tdelay )
9
+ cos − 2π(2 · 10 + cr t)tdelay − 2πfBF O (t − tdelay ) . (7.13)
As another practical consideration the DC blocking capacitors in the IF amplifier
AMP1 will reject the DC phase terms, resulting in:
IF (t) = cos − 2πcr ttdelay + 2πfBF O t + cos − 2πcr ttdelay − 2πfBF O t .
Simplifying the arguments in the cosine terms:
IF (t) = cos 2π(fBF O − cr tdelay )t + cos 2π(fBF O + cr tdelay )t .
(7.14)
IF (t) is fed into the high Q IF filter FL1. FL1 is a high Q communications
bandpass filter. FL1 has a center frequency of fc and a bandwidth of BW . OSC1 is
set to a frequency such that fBF O ≥ BW
2 + fc causing FL1 to pass only the lower
sideband of IF (t), thus causing the output of FL1 to be:
F IL(t) =


 cos 2π(f

 0
−BW
BW
BF O − cr tdelay )t if 2 + fc < fBF O − cr tdelay < 2 + fc
.
for all other values
(7.15)
BW
Only beat frequencies in the range of −BW
2 + fc < fBF O − cr tdelay < 2 + fc
are passed through IF filter FL1. Since it was shown in Section 7.3.1 that in an
FMCW radar system the range to target is directly proportional to the beat frequency
132
cr tdelay , then the band limited IF signal (which is is proportional to downrange target
location) is effectively a hardware range-gate.
Increasing the bandwidth of FL1 increases the range-gate duration. Decreasing
the bandwidth of FL1 decreases the range-gate duration. It is for this reason that the
range-gate is adjustable if a number of different bandwidth filters were used switched
in and out, of the IF signal chain.
If fBF O were increased then the filter FL1 passes only signals that fit the equality
in Equation 7.15. Since the cr t term is subtracted from fBF O then the cr t term
would have to be greater in size to compensate for a higher fBF O frequency in order
to let the IF signals pass through FL1. Thus, the filter FL1 would only pass beat
tones further down range but at the same range duration in length if the frequency
fBF O were increased. So the range-gate is adjustable in physical downrange location
(physical down range time delay).
In addition to these desirable properties the narrow bandwidth of FL1 greatly
increases the receiver sensitivity according to Equation 7.1. For the pulsed IF radar
discussed in Section 7.2 the shorter the range-gate the worse the receiver sensitivity
became. The opposite is true for the range-gated FMCW radar architecture shown in
this section because according to Equation 7.15 the narrower the IF bandwidth the
shorter the range-gate. According to Equation 7.1, the narrower the IF bandwidth
the more sensitive the receiver. The radar architecture presented in this section has
accomplished both high sensitivity and short duration range-gating without the loss of
receiver sensitivity performance. For the design presented in this section the receiver
sensitivity increases the shorter the range gate becomes.
FL1 might be a filter with fc = 10.7 MHz and BW = 7.5 KHz. According to
Equation 7.1 the receiver sensitivity would be -131.9 dB, which is significantly higher
than a pulsed IF radar system. At the same time, the bandwidth would allow for a
range-gate duration of only 9.375 nS for a chirp rate of cr = 800 GHz/second. This
133
sensitivity performance is significantly greater than a pulsed IF radar system with a
9.375 nS range-gate, which according to 7.1 would be approximately -90.4 dBm ideal.
One last step occurs in the signal chain shown in Figure 7.6 where the output of
FL1 is downconverted to base band through MXR3. The LO port of MXR3 is driven
by OSC1 so the output of MXR3 is fed through Video Amp1 and can be represented
by the equation:
V ideo(t) = BF O(t) · F IL(t).
Video Amp1 is an active low pass filter, rejecting the higher frequency component of
the cosine multiplication, resulting in video output signal:
V ideo(t) =


BW
 cos 2πcr t
t
if −BW
delay
2 + fc − fBF O < cr tdelay < 2 + fc − fBF O .

 0
for all other values
(7.16)
The result is a range gated base-band FMCW video signal similar to (except without
the DC phase term) Equation 7.3. This is identical to a traditional FMCW system
discussed in Section 7.3.1 except that this signal is band limited by a high Q bandpass
filter with an adjustable center frequency which effectively range-gates the video signal
that is fed into the digitizer.
7.4
High sensitivity range-gated FMCW radar architecture conclusions
and advantages
The high sensitivity range-gated FMCW radar architecture presented in this chapter
was shown to be capable of producing a short duration range-gate while at the same
time dramatically increasing receiver sensitivity compared to a pulsed IF radar system
of similar range-gate duration. The range-gate in this case requires the use of low134
cost high frequency parts, many of which can easily be fabricated by hand directly
from references such as [70]. Such a range-gated high sensitivity system would be
inexpensive to implement and require a minimum quantity of expensive microwave
components.
This architecture will be used to implement complete imaging radar systems in the
next chapters. These include an S-band through-dielectric slab imaging system, an
X-band free space imaging system, and a near real-time high speed spatially diverse
antenna array imaging system.
135
Figure 7.1. Block diagram of a typical coherent pulsed IF radar system where the IF
bandwidth must be wide enough to capture the returned pulse from the target scene.
136
Figure 7.2. Side view of a typical through-dielectric slab imaging geometry.
137
Figure 7.3. Example of a 40nS range-gate where the slab is located at d1 = 20 feet
down range, target is located d3 = 60 nS down range and mulipath (from slab to
radar and back again) is shown 80 nS down range.
138
Figure 7.4. Example of a 20nS range-gate where the slab is located at d1 = 10 feet
down range, target is located d3 = 30 nS down range and mulipath (from slab to
radar and back again) is shown 40 nS down range.
139
Figure 7.5. A direct conversion FMCW radar system.
140
Figure 7.6. Simplified block diagram of the high sensitivity range-gated FMCW radar
system.
141
CHAPTER 8
S-BAND THROUGH-DIELECTRIC SLAB RAIL SAR IMAGING
SYSTEM
Using the high sensitivity range-gated FMCW radar architecture developed in Chapter 7, an S-band (2 GHz to 4 GHz) through-dielectric slab radar imaging system will
be shown in this chapter. This radar system is a linear rail SAR much like [60] where
a radar sensor is mounted on a linear rail and moved automatically down the rail
acquiring range profiles of the target scene at evenly spaced increments across the
rail. The imaging geometry of this type of rail SAR utilized in a through dielectric
slab imaging scene is shown in Figure 5.4. All topics related to this radar system will
be discussed in this chapter including theory of operation, parts lists and schematics.
It is important to note that power distribution, pin-outs and specifics of various
toggle switches and indicator lamps will be omitted. Only linear power supplies were
used throughout this system. The various modules use a wide variety of voltages and
the details of how power was routed to the various modules will not be covered in
this dissertation.
A picture of the S-band through dielectric slab rail SAR imaging system is shown
in Figure 8.11. The entire system block diagram is shown in Figure 8.1, Figure 8.2,
Figure 8.3, Figure 8.4, Figure 8.5 and Figure 8.6. A list of parts in these Figures is
shown in Table 8.1 and Table 8.2.
8.1
Radar control and data acquisition
Looking at Figure 8.6 all radar system control is done using a PC running Labview.
Software written in Labview moves the radar system down the rail one increment at
a time by communicating with the Motor Controller through the RS232 port. The
motor controller moves the rail. Once the move is complete the computer uses the
142
PCI6014 NIDAQ Card to simultaneously and coherently trigger a linear ramp using
the CTR0 pin while at the same time digitizing the Video Output from the radar IF
using the AI0 pin at a rate of 200 KSPS with 16 bits of resolution.
The ramp generator (schematic shown in Figure 8.10) receives an inverted pulse
input from CTR0 and outputs a linear ramp which FM modulates the YIG Oscillator.
The ramp generator is an adjustable Wilson current mirror [78] which drives a 0.1 uF
capacitor. The current mirror is set using a 10 turn precision potentiometer which
allows for the chirp rate cr to be adjusted. An N-channel MOSFET is wired in
parallel with the capacitor on the output of the current mirror. The MOSFET shorts
the capacitor when CTR0 is high and presents an open-circuit to the capacitor when
CTR0 is low. Thus when CTR0 is low the radar begins to chirp. A picture of the
motion control, data conditioning, ramp generator and power supplies is shown in
Figure 8.12.
The output of the Ramp Generator feeds the VCO control voltage input of the
Yig Oscillator. The YIG Oscillator is a Weinschel Engineering 430A Sweep Oscillator
with a 432A RF Unit capable of 2 GHz to 4 GHz operation. The output of the YIG
Oscillator is fed into SPLTR1 in the power splitter chassis.
The Beat Frequency Oscillator (BFO) is a Hewlett Packard HP3325A Synthesizer/Function Generator set to approximately 10.7 MHz CW (frequency is changed
by 10’s of KHz depending on range gate location, see Section 7.3.2 for details). The
BFO output is fed into the radar IF. The radar system is chirped at a chirp rate of
approximately cr = 200 · 109 Hz/second with a chirp time of T = 10 mS allowing for
a maximum detection range of approximately 150 feet for an ADC sample rate of 200
KSPS. Chirp time is variable and can be increased at the cost of reducing maximum
detectable range. A picture of the BFO, Yig oscillator and the power splitter chassis
is shown in Figure 8.13.
143
8.2
S-band transmitter signal chain
The YIG Oscillator output in Figure 8.6 is fed out to the power splitter and delay
line chassis shown in Figure 8.3. The YIG Oscillator output is then fed into a -3
dB splitter SPLTR1. SPLTR1 is a Mini-Circuits ZN2PD2-50-S, 500 MHz to 5 GHz,
2-way -3 dB splitter. Half of the output is fed into the transmitter front end shown
in Figure 8.4. The other half is fed through a coaxial delay line DELAY1 and out to
the receiver front end Figure 8.5. The LO output to the transmitter front end from
SPLTR1 is fed directly into the LO port of MXR1 on the transmitter front end. A
picture of the transmitter front end is shown in Figure 8.14.
The BFO output from Figure 8.6 is fed into the BFO input in Figure 8.1 where a
picture of the radar IF is shown in Figure 8.16. The BFO signal is fed through CLPR1
and into the LO port of MXR3. Some of the power from the BFO is coupled out
of CLPR1, which is a Mini-Circuits ZX30-12-4, 5 MHz to 1 GHz, -12 dB directional
coupler. -12 dB of the BFO is coupled off and fed out of the IF to the adjustable
attenuator ATTN3 shown in Figure 8.2. ATTN3 allows for the adjustment of drive
power to the transmitter front end. ATTN1, ATTN2 and ATTN3 are located on the
top of the radar system so as to make easy adjustments while measuring. A picture
of this setup is shown in Figure 8.17
The output of ATTN3 is fed into the IF port of MXR1 inside of the transmitter
front end in Figure 8.4. MXR1 is a Mini-Circuits ZEM-4300MH, 300 MHz to 4300
MHz, Level 13 (+13 dBm LO). The LO from the YIG Oscillator is mixed with the
BFO in MXR3. The output product from MXR3 is amplified by AMP1 which is
a Mini-Circuits ZJL-4G, Gain=11 dB, IP1=12 dBm, NF=5.5 dB. The output of
AMP1 is fed through FL2 which is a Mini-Circuits VLP-41, 4.1 GHz LPF. FL2 filters
transmitter harmonics. The transmitter signal is fed out through CLPR2 then to the
transmitter antenna ANT1. CLPR2 is a Midwest Microwave, 2 GHz to 4 GHz, -20
144
dB directional coupler. Some power is coupled off of the transmit signal for diagnostic
purposes. Transmitter power is approximately 10 dBm and varies upwards of 4 dB
over the bandwidth of the chirp due to the transmitter signal chain and YIG oscillator
amplitude responses.
The transmit signal radiates out of ANT1 which is a vertically polarized linearly
tapered slot antenna (LTSA), developed from work in [71], [72], [73], [74], [75] and
[76]. Transmit signals are radiated out of ANT1 and propagate into the target scene.
8.3
S-band receiver signal chain
The transmitted signal is reflected off of the target scene and received by ANT2.
ANT2 is a vertically polarized LTSA identical to the transmitter antenna ANT1.
The output of ANT2 is fed into the receiver front end shown in Figure 8.5. A picture
of the receiver front end is shown in Figure 8.15. The output of ANT2 is fed into
FL3 which is a Mini-Circuits VHF-1200, 1200 MHz HPF. This signal is amplified by
LNA1 which is a Mini-Circuits ZX60-6013E amplifier, 20 MHz to 6 GHz, Gain=14
dB, NF=3.3 dB. The output of LNA1 is filtered by FL4 which is a Mini-Circuits
VHF-1200, 1200 MHz HPF. The output of FL4 is filtered by FL5 which is a MiniCircuits VLP-41, 4.1 GHz LPF. The output of FL5 is fed into the RF port of MXR2.
MXR2 is a Mini-Circuits ZEM-4300MH, 300 MHz to 4300 MHz, Level 13 (+13 dBm
LO). The LO port of MXR2 is driven by the LO output of the power splitter shown in
Figure 8.3. The resulting IF output from MXR2 is the de-chirped frequency domain
data centered at the BFO frequency and filtered by FL1. FL1 is a Mini-Circuits SLP10.7, 11 MHz LPF. The output of FL1 is amplified by AMP2 which is a Mini-Circuits
ZFL-1000VH, Gain=20 dB, IP1=25 dBm, NF=4.5 dB.
The IF output from the receiver front end is fed into the IF shown in Figure 8.1.
The receiver IF signal is filtered through the selectable crystal filter switch matrix
XTAL Filter Mux2. The schematic of XTAL Filter Mux2 is shown in Figure 8.7.
145
This is a PiN diode selectable filter matrix with three different IF filters of center
frequency fc = 10.7 MHz and selectable bandwidths of B = 7.5 KHz, B = 15 KHz
and B = 30 KHz. There is also a ’through’ line which bypasses the filters. Isolation
between filters is high, measured to be > 90 dB. The filter mux is capable of handling
upwards of 23 dBm of power before the PiN diodes begin to distort (switch on and
off causing undesirable mixer products in the IF signal chain).
The output of the XTAL Filter Mux2 is fed into FL6 which is a Mini-Circuits
PLP-10.7, 11 MHz LPF. The output of XTAL Mux2 is amplified by AMP3 which
is a Mini-Circuits ZKL-2R5, Gain=30 dB, IP1=15 dBm, NF=5 dB. The output of
AMP3 is fed out of the IF and through the Kay Model 20 adjustable attenuator
ATTN2. The output of ATTN2 is fed back into the IF and amplified by AMP5.
AMP5 is a Mini-Circuits ZHL-6A, Gain=25 dB, IP1=22 dBm, NF=9.5 dB.
The output of AMP5 is fed into XTAL Filter Mux1 which is identical to XTAL
Filter Mux2. The output of XTAL Mux1 is fed into the RF port of MXR3. MXR3 is
a Mini-Circuits RAY-6U, Level 23 (+23 dBm LO). The schematic for MXR3 is shown
in Figure 8.8. The LO port of MXR3 is fed by the BFO so as to shift the IF down
to base band. The output of MXR3 is amplified by the video amplifier VideoAmp1.
The schematic for VideoAmp1 is shown in Figure 8.9. VideoAmp1 is made up of an
amplification stage and a 4th order active LPF optimized for uniform delay [79].
The output of the video amplifier is fed directly into the AI0 digitizer input pin on
the PCI6014 data acquisition card. This digitized video data is the frequency domain
range profile data which is acquired at every rail position.
ATTN1 and ATTN2 are adjusted to maximize the voltage swing on the digitizer
input so as to take advantage of as much of the digitizer dynamic range as possible
without saturating the receiver signal chain. Due to the high LO drive level of the
front end mixer MXR2 and the IF mixer MXR3 the receiver does not saturate unless
there is -5 dBm of power at the antenna terminals. The receiver sensitivity with
146
ATTN1 and ATTN2 set to 0 dB is approximately -131.9 dBm theoretical on the lowest
bandwidth setting of B = 7.5 KHz according to Equation 7.1. Receiver sensitivity
was measured to be < −125 dBm. The system dynamic range was measured to be
> 120 dB.
8.4
Calibration and background subtraction
The radar system is calibrated before measurements are made. The calibration is
done to sharpen imagery and not an exact sphere target calibration (this was due
to fact that a large calibration sphere sizable enough to achieve the desired signal to
noise ratio in a high clutter environment was not available in the laboratory at the
time). The calibration procedure is simple. A 5 foot tall 3/4 inch diameter copper
pole is used as a cal target. This cal target is used because it is easily detectable
at S-band using a vertically polarized antenna system. The cal target is placed at a
known location down range from the radar sensor. A range profile is taken of the pole
represented by spole ω(t) . The pole is then removed and a background range profile
is taken, the result is represented by scalback ω(t) . The background is subtracted
from the pole range profile resulting in a clean range profile of the pole only, where
scal ω(t) = spole ω(t) − scalback ω(t) .
(8.1)
The cal data is referenced to a theoretical point scatterer which is represented by the
equation
−j2kr Rpole
scaltheory ω(t) = e
,
(8.2)
where Rpole is the range to the cal pole and kr = ω(t)/c. The cal factor is calculated
by the equation
scaltheory ω(t)
.
scalf actor ω(t) =
scal ω(t)
147
(8.3)
This cal factor is multiplied by each range profile acquired across the rail.
Coherent background subtraction using Equation 5.19 is utilized on all measured
data.
8.5
S-band SAR data processing
Data form the PC running control software in Labview is saved in the MATLAB
.MAT matrix format. From this data both calibration and coherent background
subtraction are implemented using the MATLAB program written and shown in
Appendix J. Once the data is calibrated and backgrond subtracted the RMA SAR
algorithm developed in Chapter 2 and written in MATLAB in Apprendix C is then
run to produce the resulting image.
148
149
4
3
2
1
A
A
SEL1
Xtal Filter Mux1
SEL3
Xtal Filter Mux2
1
AMP5
IF Input
A
B
Video Output
FL9
FL8
FL7
SEL2
G
4
SEL4
C
C
F
3
FL6
MXR3
Video Output
VideoAmp1
FL10
AMP3
Figure 8.1. Block diagram of the IF.
FL13
FL12
FL11
B
D
D
C
1
E
D
E
BFO Input
1
CLPR1
2
Out to ATTN3
Input from ATTN2
B
1
Out to ATTN2
E
4
3
2
1
150
4
3
2
1
A
A
1
1
2
3
I
B
3
C
ATTN3
ATTN2
ATTN1
C
Figure 8.2. Attenuator chassis.
BFO output to transmitter
front end
D
Input from BFO CLPR1
Output to radar IF
2
C
B
Input from radar IF
A
Output to radar IF
M
IF input from RX front end
B
D
D
E
E
4
3
2
1
151
A
B
C
1
1
J
D
LO Output to TX Front End
E
E
2
1
Figure 8.3. Power splitter and delay line.
4
-3dB
SPLTR1
Y
DELA
LO Output to RX Front End
D
4
1
2
L
C
3
N
RF Input from Yig Oscillator
B
3
2
1
A
152
4
3
2
1
A
BFO in
A
I
3
J
2
LO in
MXR1
FL2
C
C
CLPR2
-20 dB Out
2
1
TX Out
D
H
D
Figure 8.4. S-band transmitter front end.
B
AMP1
B
H
To TX Out
1
E
ANT1
E
4
3
2
1
153
4
3
2
1
A
A
L
2
MXR2
FL5
LO in
FL4
AMP2
LNA1
C
M
2
C
IF Out
FL3
1
D
D
RX in
Figure 8.5. S-band receiver front end.
B
FL1
B
1
E
Receive Ant Connection
to Rx in
K K
ANT2
E
4
3
2
1
154
A
CTR0 pin
PCI6014
NIDAQ Card
Ramp
Generator
AI0 pin
Motor Controller
RS232
PCI Bus
PC
B
B
Yig
Oscillator
Beat
Frequency
Oscillator
C
C
E
F
1
N
D
LO Output to Power Splitter
1
Video Input to Digitizer
2
BFO Output
1D Linear Rail
D
E
E
4
3
2
1
Figure 8.6. High level block diagram of the PC connected to motion control and data acquisition/triggering.
4
3
2
1
A
155
Component
AMP1
AMP2
AMP3
AMP5
ANT1
ANT2
ATTN1
ATTN2
ATTN3
Beat Frequency Oscillator
CLPR1
CLPR2
Delay Line
FL1
FL2
FL3
FL4
FL5
FL6
FL7
FL8
FL9
Table 8.1. S-band modular component list.
Description
Figure (if applicable)
Mini-Circuits ZJL-4G, Gain=11 dB, IP1=12 dBm, NF=5.5 dB
NA
Mini-Circuits ZFL-1000VH, Gain=20 dB, IP1=25 dBm, NF=4.5 dB
NA
Mini-Circuits ZKL-2R5, Gain=30 dB, IP1=15 dBm, NF=5 dB
NA
Mini-Circuits ZHL-6A, Gain=25 dB, IP1=22 dBm, NF=9.5 dB
NA
Linear Tapered Slot Antenna, vertically polarized [71]-[77]
NA
Linear Tapered Slot Antenna, vertically polarized [71]-[77]
NA
Surplus 3 position attenuator: thru/-30 dB/load
NA
Kay Model 20 adjustable attenuator
NA
Surplus rotary attenuator
NA
Hewlett Packard HP3325A Synthesizer/Function Generator
NA
MIni-Circuits ZX30-12-4, 5 MHz to 1 GHz, -12 dB directional coupler
NA
Midwest Microwave, 2 GHz to 4 GHz, -20 dB directional coupler
NA
Misc. microwave coax cables
NA
Mini-Circuits SLP-10.7, 11 MHz LPF
NA
Mini-Circuits VLP-41, 4.1 GHz LPF
NA
Mini-Circuits VHF-1200, 1200 MHz HPF
NA
Mini-Circuits VHF-1200, 1200 MHz HPF
NA
Mini-Circuits VLP-41, 4.1 GHz LPF
NA
Mini-Circuits PLP-10.7, 11 MHz LPF
Figure 8.7
ECS-10.7-7.5B, 4 pole crystal filter, fc = 10.7 MHz, B = 7.5 KHz
Figure 8.7
ECS-10.7-15B, 4 pole crystal filter, fc = 10.7 MHz, B = 15 KHz
Figure 8.7
ECS-10.7-30B, 4 pole crystal filter, fc = 10.7 MHz, B = 30 KHz
Figure 8.7
156
Component
FL10
FL11
FL12
FL13
LNA1
Motor Controller
MXR1
MXR2
MXR3
PC
PCI-6014
Ramp Generator
SPLTR1
VideoAmp1
XTAL Filter Mux1
XTAL Filter Mux2
Yig Oscillator
Description
Mini-Circuits PLP-10.7, 11 MHz LPF
ECS-10.7-7.5B, 4 pole crystal filter, fc = 10.7 MHz, B = 7.5 KHz
ECS-10.7-15B, 4 pole crystal filter, fc = 10.7 MHz, B = 15 KHz
ECS-10.7-30B, 4 pole crystal filter, fc = 10.7 MHz, B = 30 KHz
Mini-Circuits ZX60-6013E, 20 MHz to 6 GHz, Gain=14 dB, NF=3.3 dB
RMV SPRT232-ST stepper motor controller
Mini-Circuits ZEM-4300MH, 300 MHz to 4300 MHz, Level 13 (+13 dBm LO)
Mini-Circuits ZEM-4300MH, 300 MHz to 4300 MHz, Level 13 (+13 dBm LO)
Mini-Circuits RAY-6U, Level 23 (+23 dBm LO)
PC running Labview, all system control software was written in Labview
National Instruments PCI-6014 data acquisition and IO card
Wilson current mirror [78] based linear ramp generator
Mini-Circuits ZN2PD2-50-S, 500 MHz to 5 GHz, 2-way -3 dB splitter
Active LPF, based using the MAX414 op-amp
Pin diode selectable crystal filter mux
Pin diode selectable crystal filter mux
Weinschel Engineering 430A Sweep Oscillator and 432A RF Unit
Table 8.2. S-band modular component list (continued).
Figure (if applicable)
Figure 8.7
Figure 8.7
Figure 8.7
Figure 8.7
NA
NA
NA
NA
Figure 8.8
NA
NA
Figure 8.10
NA
Figure 8.9
Figure 8.7
Figure 8.7
NA
157
Figure 8.7. Schematic of IF filter switch matrix XTAL Filter Mux1 and XTAL Filter Mux2.
158
4
3
2
1
A
A
1
BNC Female
J3
B
2
1
U1
5
4
6
2
J2
1
8
7
C
D
1
J1
2
D
IF
BNC Female
BNC Female
LO, +23 dBm
Mini-Circuits
RAY-6U
(bot view)
3
C
Figure 8.8. Schematic of double balanced mixer module MXR3.
RF
2
B
E
E
4
3
2
1
159
4
3
2
1
1
2
2
2
A
F
3
2
5
6
+
-
In
Out
U2a
1
2
2
R8
2
1
nF
9
+
-
nF
2.67
2 10
0.25w.
680Ω
R6
0.25w.
3.9K
2.67
2
0.25w.
680Ω
B
1
R7
1
uF 4.7uF
100
+5 V Regulator Out
B
1
2
C
MAX414
U2c
8
F
4.7u
-12V
0.25w.
330Ω
R9
C
F
0.1u
R11
2
R12
2
R10
nF
D
F
+
-
nF
2.67
2 12
13
0.1u
0.25w.
620Ω
1
0.25w.
3.9K
1
5.7V
2.67
2
0.25w.
620Ω
1
R1
0.25w.
390Ω
1
Q1
2N3906, PNP
FeedThru2
D
E
R13
2
2
1
E
BNC
Female
1
J1
2
4
3
Video Out
to Digitizer
1Kw .
0.25
U2d
14
2
1
MAX414
0.25w.
3K
1
F R2
4.7u
-5 V Regulator Out
Figure 8.9. Schematic of the video amplifier VideoAmp1.
MAX414
U2b
7
CCW 10K CW
NC
3
GND
L4705
U1
MAX414
11
4
0.25w.
8.2K
R4
5 1wΩ.
0.25
1
+
-
1
FeedThru1
0.1u
R3
Video
Input from 1
MXR3
J2
R5
NC
0.25w.
2.7K
BNC
Female
1
A
NC
F
4.7u
+12V
160
4
3
2
1
A
*
J2
1
2
B
MTP3055E
4.7uF
0.1uF
+15V
1Kw .
0.25
B
C
F
0.1u
2N3906
*
2N3906
C
-15V
3
2
+
-
4
7
+15V
0.1uF
AD817
U1
6
5
D
4.7uF
1
0.1uF
4.7uF
D
Figure 8.10. Schematic of the Ramp Generator.
BNC Female
K
1
R1
2
250
2N3906
*These should
be a matched
pair
10 turn pot,
high precision
Pulse Input from
CTR0 Pin on
PCI6014 Data
Acquisition Card
A
1
J1
2
BNC Femal
E
Ramp Output
to Yig
Oscillator
E
4
3
2
1
Figure 8.11. The S-band rail SAR imaging system.
Figure 8.12. Data conditioning hardware, motion control and power supplies.
161
Figure 8.13. From the top down: Hewlett Packard HP3325A Synthesizer/Function
Generator BFO, the power splitter chassis and the Weinschel Engineering 430A Sweep
Oscillator with the 432A RF Unit.
162
Figure 8.14. The S-band transmitter front end.
Figure 8.15. The S-band receiver front end.
163
(a)
(b)
Figure 8.16. Radar IF (a), inside of the radar IF (b).
Figure 8.17. ATTN1, ATTN2 and ATTN3 mounted on the attenuator chassis.
164
CHAPTER 9
THROUGH-DIELECTRIC AND FREE-SPACE S-BAND RAIL SAR
IMAGING RESULTS
Using the S-band rail SAR imaging system developed in Chapter 8 radar imaging
through two types of lossy dielectrics and free-space will be shown in this chapter.
Comparisons between simulations and measured data will be shown. Many interesting
results are presented beginning with free-space imagery and low transmit power freespace imagery. Imaging through a 4 inch thick lossy-dielectric slab will be shown
where image scenes made up of objects as small as 12 oz aluminum cans and 6 inch
tall bolts are imaged. It will be shown that imaging similar target scenes through an
unknown lossy-dielectric slab is possible.
9.1
Free-space imaging results
The rail SAR was first setup in a free-space imaging scenario where targets were
placed down range from the rail and imaged in free-space without a dielectric slab in
the target scene. A transmit power of approximately 10 milliwatts was used for the
imagery shown in this section. Experiments were conducted to compare measured
cylinder imagery to simulated. Other targets were also imaged in free space.
9.1.1
Comparison of measured and simulated free-space cylinder imagery
Simulated imagery of cylinders using the model developed in Chapter 3 are compared
to measured SAR imagery of cylinders. This comparison shows that the imaging
system functions properly in free-space. The geometry of this experiment is shown
in Figure 9.1 where d3 is the range from the rail SAR to the cylinder and d4 is the
cross range location of the cylinder with respect to the center of the rail.
Experimental SAR imagery of an a = 3 inch radius cylinder located at approxi-
165
mately d3 = 12 feet and d4 = −0.5 feet is shown in Figure 9.2b. A simulated SAR
image of a 2D cylinder with the same dimensions is shown in Figure 9.2a. Measured
data closely matches simulated in this case.
Experimental SAR imagery of an a = 6 inch radius cylinder located at approximately d3 = 12 feet and d4 = −0.5 feet is shown in Figure 9.3b. A simulated SAR
image of a 2D cylinder with the same dimensions is shown in Figure 9.3a. Measured
data closely matches simulated data in this case.
9.1.2
Free-space SAR imagery of various targets
It was shown in the previous section that measured and simulated free-space cylinder
imagery is in agreement. For this reason free-space SAR imagery was acquired on
a number of other other targets. A SAR image of an a = 4.3 inch radius sphere is
shown in Figure 9.4a. Shown in Figure 9.4b is a SAR image of a group of 6 inch tall
3/8 inch diameter carriage bolts in a block ‘S’ pattern. A picture of carriage bolts is
shown in Figure 9.5a and a picture of the block ‘S’ target scene is shown in Figure
9.5b.
Both target groups are relatively small at S-band and demonstrate the S-band rail
SAR free-space imaging abilities.
166
Figure 9.1. Free-space SAR imaging geometry.
167
(a)
(b)
Figure 9.2. SAR imagery of an a = 3 inch radius cylinder in free-space; simulated
(a), measured (b).
168
(a)
(b)
Figure 9.3. SAR imagery of an a = 6 inch radius cylinder in free-space; simulated
(a), measured (b).
169
(a)
(b)
Figure 9.4. Measured free-space SAR imagery of an a = 4.3 inch radius sphere (a), a
group of 6 inch tall 3/8 inch diameter carriage bolts in a block ‘S’ pattern (b).
170
(a)
(b)
Figure 9.5. Picture of 6 inch tall 3/8 inch diameter carriage bolts (a), target scene of
carriage bolts in a block ‘S’ pattern (b).
171
9.2
Low power free-space imaging results
In order to test the high sensitivity radar architecture developed in Chapter 7 it was
decided to rail SAR image a number of target scenes using extremely low transmit
power. In the previous section rail SAR imagery was acquired at a transmit power
level of approximately 10 milliwatts. In this section rail SAR imagery is acquired
using 5 picowatts to 10 nanowatts of transmit power at the antenna terminals.
9.2.1
Comparison of measured and theoretical low power free-space cylinder imagery
Using 100 picowatts of transmit power the S-band rail SAR acquired imagery of two
different cylinders. This imagery is compared to simulated imagery from the cylinder
model in Chapter 3 in this subsection. The imaging geometry for this experiment is
shown in Figure 9.1 where both cylinders are located at approximately d3 = 12 feet
and d4 = 0.5 feet.
Figure 9.6a shows a simulated SAR image of an a = 3 inch radius cylinder. Figure
9.6b shows a measured SAR image of an a = 3 inch radius cylinder using 100 picowatts
of transmit power. Simulated and measured data are in close agreement and therefore
reducing transmit power has little effect on the resulting SAR image in this case.
Figure 9.7a shows a theoretical SAR image of an a = 6 inch radius cylinder.
Figure 9.6b shows a measured SAR image of an a = 6 inch radius cylinder using 100
picowatts of transmit power. Simulated and measured data are in close agreement
and therefore reducing transmit power has little effect on the resulting SAR image in
this case.
From these results it was shown that reducing the radar transmit power has little
effect on the resulting SAR image for cylinders of radius a = 3 inch and a = 6
inch. These results also show that the free-space model is in agreement with the
measurements acquired using low transmit power.
172
9.2.2
Low power free-space SAR imagery of various targets
In the previous section the radar system has been shown to be capable of imaging
cylinders in free space with extremely low transmit power. For this reason it was
decided to try imaging a variety of other target scenes using varying levels of transmit
power.
Figure 9.8 shows the SAR image of an a = 4.3 inch radius sphere using 100
picowatts of transmit power.
A group of 6 inch tall 3/8 inch diameter carriage bolts setup in a block ‘S’ pattern
were imaged at various low transmit powers. A picture of this target scene is shown in
Figure 9.5. The bolts imaged using the full transmit power of 10 milliwatts is shown
in Figure 9.4b. The same target scene of bolts imaged using a transmit power of only
10 nanowatts is shown in Figure 9.9. Little difference is noticeable between this and
the full power image. The target scene of bolts imaged using a transmit power of 100
pico-watts is shown in Figure 9.10. Very little difference is noticeable between this
image and the full power image. The target scene of bolts imaged using a transmit
power of only 5 pico-watts is shown in Figure 9.11. The top row of bolts in this image
begins to fade into the noise.
From these results it was shown that the S-band rail SAR imaging system is very
sensitive when operating in free-space without a dielectric slab in the target scene.
173
(a)
(b)
Figure 9.6. Simulated SAR image of an a = 3 inch radius cylinder in free-space (a),
measured SAR image of an a = 3 inch radius cylinder using a transmit power of 100
picowatts (b).
174
(a)
(b)
Figure 9.7. Theoretical SAR image of an a = 6 inch radius cylinder in free-space (a),
measured SAR image of an a = 6 inch radius cylinder using a transmit power of 100
picowatts (b).
175
Figure 9.8. SAR image an a = 4.3 inch radius sphere in free-space using 100 picowatts
of transmit power.
Figure 9.9. SAR image of a group of carriage bolts in free-space using 10 nanowatts
of transmit power.
176
Figure 9.10. SAR image of a group of carriage bolts in free-space using 100 picowatts
of transmit power.
Figure 9.11. SAR image of a group of carriage bolts in free-space using 5 picowatts
of transmit power.
177
9.3
Through-lossy-dielectric slab imaging results
Results are presented in this section of various targets which are imaged through a
lossy-dielectric slab. A picture of the slab used for these measurements with a target
scene of 12 oz aluminum cans behind it is shown in Figure 9.12. A picture of the
imaging geometry with the S-band rail SAR in the foreground is shown in Figure
9.13. For these measurements the radar system was transmitting approximately 10
milliwatts of power at the antenna terminals and it was located approximately 29.5
feet from the slab.
9.3.1
Comparison of measured and simulated through-lossy-dielectric
slab imagery of cylinders
In this subsection measured and simulated through-lossy-dielectric slab imagery of
cylinders is compared. The simulated imagery was calculated from the model developed in Chapter 5. The geometry of this target scene is shown in Figure 9.14 where
d1 = 29.5 feet, d3 = 37.5 feet and d4 = 1.1 feet.
A measured through-lossy-dielectric slab image of a cylinder with a radius of a = 3
inches is shown in Figure 9.15b. A simulated through-lossy-dielectric slab image of a
cylinder using the same dimensions is shown in Figure 9.15a. Through-slab measured
imagery and theoretical imagery are in agreement. There is greater image clutter
in the measured image compared to the theoretical. This is due to the high clutter
environment from which the measured data was acquired where the cylinder was likely
interacting with the background subtracted clutter.
A measured through-lossy-dielectric slab image of an a = 6 inch radius cylinder
is shown in Figure 9.16b. A simulated through-lossy-dielectric slab image of a cylinder using the same dimensions is shown in Figure 9.16a. Through-slab measured and
simulated imagery are in agreement. Increased image clutter is noticeable in the measured image. This is likely due to the highly cluttered laboratory environment from
178
which the measured data was acquired where the cylinder was probably interacting
with the background subtracted clutter.
In both cases the measured results are nearly identical to the simulations. This
shows that the rail SAR is effective at imaging targets on the opposite side of a lossydielectric slab. This also shows that the plane wave approximation used to develop
the through-lossy-dielectric slab model works effectively for modeling this geometry.
9.3.2
Through-lossy-dielectric slab SAR imagery of various targets
It was decided to image a number of targets behind the lossy-dielectric slab. In
these experiments the slab was located at d1 = 29.5 feet from the rail SAR imaging
system. For these measurements the transmitter was set to an output power level
of approximately 10 milliwatts. Targets were placed at various locations behind the
lossy-dielectric slab.
A sphere with a radius of a = 4.3 inches was imaged through the slab, results are
shown in Figure 9.17. The location of the sphere is clearly shown in this image.
An image scene made up of 12 oz aluminum soft drink cans configured in a block
‘S’ is shown in Figure 9.12. A through slab radar image was taken of this target scene
and is shown in Figure 9.18. Most of the cans are visible in this image with some
fading into the background clutter.
A target scene made up of three 6 inch tall 3/8 inch diameter carriage bolts
(picture of a carriage bolt is shown in Figure 9.5a) placed in a diagonal was imaged
behind the slab. A radar image of this target scene is shown in Figure 9.19. The
location of each of the three bolts is clearly shown in this image.
From these results it was shown that targets as small as 12 oz aluminum soft
drink cans and 6 inch tall bolts can be imaged behind a 4 inch thick lossy-dielectric
slab using 10 milliwatts of radar transmit power with the radar located at a stand-off
range of 29.5 feet from the slab.
179
Figure 9.12. The 4 inch thick lossy-dielectric slab.
180
Figure 9.13. Through-lossy-dielectric slab image scene where the S-band rail SAR is
located 29.5 feet from the slab.
181
Figure 9.14. Through-slab rail SAR imaging geometry.
182
(a)
(b)
Figure 9.15. Theoretical SAR image of an a = 3 inch radius cylinder behind a 4 inch
thick lossy-dielectric slab (a), measured SAR image of an a = 3 inch radius cylinder
behind a 4 inch thick lossy-dielectric slab (b).
183
(a)
(b)
Figure 9.16. Theoretical SAR image of an a = 6 inch radius cylinder behind a 4 inch
thick lossy-dielectric slab (a), measured SAR image of an a = 6 inch radius cylinder
behind a 4 inch thick lossy-dielectric slab (b).
184
Figure 9.17. Sphere with a radius of a = 4.3 inches imaged behind a 4 inch thick
lossy-dielectric slab.
Figure 9.18. 12 oz aluminum beverage cans in a block ‘S’ configuration imaged behind
a 4 inch thick lossy-dielectric slab.
185
Figure 9.19. Diagonal row of three 6 inch tall 3/8 inch diameter carriage bolts imaged
behind a 4 inch thick lossy-dielectric slab.
186
9.4
Through an unknown lossy-dielectric slab imaging results
It was shown in the previous section that the through-dielectric slab model and measurements were in agreement. It was for this reason that it was decided to test the
performance of the S-band rail SAR imaging system on an unknown dielectric slab.
This dielectric is not a homogeneous one and it has less loss than the lossy-dielectric
slab measured in the previous section. Nothing is known about the electromagnetic
properties or the physical composition of this dielectric. A picture of the unknown
lossy dielectric slab used in this experiment is shown in Figure 9.20. The transmit
power level in these experiments was approximately 10 milliwatts. The unknown slab
was located approximately d1 = 10 feet downrange from the rail SAR.
A measured radar image of a cylinder with a radius of a = 3 inches located behind
the slab is shown in Figure 9.21. A measured radar image of a cylinder with a radius of
a = 6 inches behind the slab is shown in Figure 9.22. For both images the position of
the cylinders is clearly shown and some image distortion due to the slab is noticeable
compared to the free space imagery in Figure 9.2b and Figure 9.3b.
Figure 9.23 shows the through-slab measured radar image of a sphere with a radius
a = 4.3 inches. Noticeable distortion is present around the sphere compared to its
free space image in Figure 9.4a.
A through-slab radar image of a group of 6 inch tall 3/8 inch diameter bolts in a
block ‘S’ configuration is shown in Figure 9.24. The position of each bolt is shown
however there is more clutter present compared to the free space image in Figure
9.4b.
A variety of target scenes were shown to be imaged behind the unknown lossy
dielectric slab in this section. Targets as small as 6 inch tall bolts were detectable.
These results show the flexibility of this radar system to image target scenes through
unknown dielectric slabs.
187
(a)
(b)
Figure 9.20. Experimental setup for imaging through an unknown lossy-dielectric
slab (a), a close up view of the unknown lossy-dielectric slab (b).
188
Figure 9.21. A radar image of a cylinder with a radius of a = 3 inches behind the
unknown dielectric slab.
Figure 9.22. A radar image of a cylinder with a radius of a = 6 inches behind the
unknown dielectric slab.
189
Figure 9.23. A radar image of a sphere with a radius of a = 4.3 inches behind the
unknown dielectric slab.
Figure 9.24. A target scene of carriage bolts in a block ‘S’ configuration imaged
behind the unknown dielectric slab.
190
9.5
Low-power through an unknown lossy-dielectric slab imaging results
The radar imaging through an unknown lossy dielectric slab results from the previous
section are replicated in this section using only 10 nanowatts of transmit power at
the antenna terminals.
A radar image of a cylinder with a radius of a = 3 inches behind the unknown
slab is shown in Figure 9.25. A radar image of a cylinder with a radius of a = 6
inches behind the uknown slab is shown in Figure 9.26. The image distortion appears
to be the same except for slightly less cross range clutter compared to the full power
images in Figure 9.21 and Figure 9.22. The position of the cylinders is clearly shown
and some image distortion due to the slab is noticeable compared to the free-space
cylinder imagery in Figure 9.2b and Figure 9.3b.
Figure 9.27 shows the radar image of a sphere with a radius of a = 4.3 inches
behind the unknown slab. This image is similar to the full power sphere image in
Figure 9.23. Noticeable distortion is present around the sphere compared to its freespace image in Figure 9.4a.
A radar image of a group of carriage bolts in a block ‘S’ configuration behind the
unknown slab is shown in Figure 9.28. This image appears to have less clutter than
the full power image in Figure 9.24. This is probably due to target interactions with
the unknown slab. The location of the bolts are clearly detectable, however there is
more clutter present compared to the free-space image in Figure 9.4b.
These low transmit power results were nearly identical to the full power results.
This is an interesting observation and shows how effectively the S-band rail SAR can
image through an unknown dielectric slab.
191
Figure 9.25. A cylinder with a radius of a = 3 inches radar imaged behind an unknown
slab using 10 nanowatts of transmit power.
Figure 9.26. A cylinder with a radius of a = 6 inches radar imaged behind an unknown
slab using 10 nanowatts of transmit power.
192
Figure 9.27. A sphere with a radius of a = 4.3 inches radar imaged behind an uknown
slab using 10 nano-watts of transmit power.
Figure 9.28. A target scene of carriage bolts in a block ‘S’ configuration radar imaged
behind an unknown slab using 10 nanowatts of transmit power.
193
9.6
Discussion of S-band rail SAR imaging results
In this chapter it was shown that the high sensitivity range-gated FMCW radar
architecture developed in Chapter 7 is effective operating as an S-band rail SAR in
free space, through a lossy-dielectric slab and through an unknown-dielectric slab.
The theoretical model of a cylinder in free-space was shown to be accurate compared to measured data using full transmit power and low transmit power. The
S-band rail SAR has proven itself to be a highly sensitive machine. It was shown in
this chapter that imaging objects in free-space as small as carriage bolts using only
5 picowatts of transmit power is possible.
Imaging through a lossy-dielectric slab was achieved. The theoretical model was
in agreement with measured results. The S-band rail SAR could image target scenes
made up of objects as small as soda cans and carriage bolts behind a 4 inch thick
lossy-dielectric slab.
Imaging through an unknown dielectric slab was shown to be possible using the
S-band rail SAR. Image scenes made up of objects as small as aluminum beverage
cans and 6 inch tall bolts were imaged through the unknown slab. These results were
replicated using only 10 nanowatts of transmit power.
In this chapter the model was shown to agree with the measured results. The high
sensitivity range-gated FMCW radar architecture was shown to be very effective in
imaging through lossy-dielectric slabs and in free space. These results are encouraging
and it is for this reason that the next step in through-slab radar imaging development
will be to build a high speed antenna array so that near real-time radar imagery of
what is behind an unknown slab can be displayed to the radar operator. This near
real-time system development will be shown later in this dissertation.
194
CHAPTER 10
X-BAND RAIL SAR IMAGING SYSTEM
An X-band front end was developed which plugs directly into the IF and control
systems of the S-band rail SAR developed in Chapter 8 changing it into an X-band
rail SAR imaging system. This X-band rail SAR imaging system design was based on
the high sensitivity range-gated FMCW radar architecture developed in Chapter 7. It
will be shown in this chapter that the X-band rail SAR is capable of imaging target
scenes made up of pushpins using only 10 nanowatts of transmit power. Imaging
results from the X-band rail SAR will be compared to a direct conversion FMCW
radar system from [60]. In this chapter it will be shown that the high sensitivity
range-gated FMCW radar architecture is highly effective at X-band for applications
such as RCS measurement.
10.1
High sensitivity range-gated FMCW X-band radar front end
A block diagram of the X-band front end is shown in Figure 10.1 and a list of all
components in Table 10.1. A picture of the X-band front end is shown in Figure 10.3.
The X-band front end was developed to plug into the radar IF and control assemblies
from Chapter 8. This allows the X-band front end to operate according to the high
sensitivity range-gated FMCW radar architecture developed in Chapter 7.
In order to operate at X-band the YIG Oscillator is modified by removing the
432A RF Unit from the Weinschel Engineering 430A Sweep Oscillator and replacing
it with the 434A RF Unit which is capable of linear tuning from 7.835 GHz to 12.817
GHz. The chirp rate cr is increased significantly when using the 434A RF Unit
because the 434A RF unit covers more frequency than the S-band 432A RF Unit but
the chirp time remains the same.
The power splitter chassis shown in the block diagram in Figure 8.3 is no longer
195
used when using the 434A RF Unit. The output of the 434A RF Unit is fed directly
into the front end YIG Oscillator LO Input port shown in the X-band front end block
diagram in Figure 10.1. From this the LO is fed through CLPR4 which is a Omni
Spectra X-band -10 dB directional coupler. The output of CLPR4 is fed through
ATTN4 and ATTN6 which are 3 dB and 6 dB attenuators. The output of ATTN6
feeds CIRC1 which is a UTE Microwave X-band isolator. The output of CIRC1 is
fed into the LO port of MXR4 which is a Watkins Johnson M31A mixer. Some of the
signal is coupled off of CLPR4 and fed through CIRC2 which is a UTE Microwave
X-band isolator. The output of CIRC2 is fed through a coaxial delay line DELAY2.
The output of DELAY2 feeds the LO port of the mixer MXR5.
The coupled BFO output from ATTN3 in Figure 8.2 feeds into the BFO In port on
the X-band front end block diagram in Figure 10.1. The BFO mixes with the X-band
LO and the output of MXR4 is fed into AMP6 which is a Microwave Components
Corporation MH858231, Gain=25 dB, IP1=23 dBm. The output of AMP6 is fed
through CIRC3 which is an X-band isolator. The output of CIRC3 is fed out of the
front end assembly to either a step attenuator ATTN5 or a through line. ATTN5
is a Narda Microline Step Attenuator Model 705-69. ATTN5 is used to reduce the
transmit power when acquiring low power rail SAR measurements. The output of
either the through line or ATTN5 is fed through CLPR3 then to the transmitter
antenna ANT3. CLPR3 is a -20 dB directional coupler. ANT3 is a Microtech 205297
X-band horn with a WR90 waveguide flange. The X-band microwave signals fed
into ANT3 are radiated out towards the target scene. Some of the transmit signal is
coupled off CLPR3 for diagnostic purposes.
Scattered X-band signals from the target scene are received by ANT4 which is a
Microtech 205297 X-band horn with a WR90 waveguide flange. The output of ANT4
is amplified by LNA2 which is an Amplica Inc. XM553403, Gain=20dB, IP1=25 dBm.
The output of LNA2 is fed into the RF port of MXR5 which is a TRW Microwave
196
MX18533. The IF output of MXR5 is amplified by AMP7 which is a Mini-Circuits
MAR-4, Gain = 8 dB, NF=6 dB, IP1=12.5 dBm, schematic shown in Figure 10.2 .
The output of AMP7 is fed into filter FL14 which is a Mini-Circuits PBP-10.7, 10.7
MHz bandpass filter, schematic also shown in Figure 10.2. The IF output of FL14 is
fed out to the IF input of ATTN1 in the attenuator chassis shown in Figure 8.2.
X-band rail SAR imaging is achieved by plugging the X-band front end in to the
existing S-band radar IF and mounting the X-band front end on to the linear rail.
The same imaging algorithm and MATLAB codes are used for SAR imaging except
that a few numbers are changed. When using the X-band front end the radar chirp
rate is cr = 500 · 109 Hz/S. The transmit power is approximately 10 milliwatts and
adjustable down to picowatts. The aperture spacing across the linear rail is different:
the radar traverses only 90 inches (rather than 96 inches at S-band) acquiring range
profile data every 1 inch (rather than every 2 inches at S-band).
197
198
4
3
2
1
N
1
CLPR4
ATTN4
2
A
Yig Oscillator
LO Input
I
BFO In
A
CIRC2
B
CIRC1
MXR4
M
2
CIRC3
FL14
C
AMP7
4
P
O
O
3
IF Out
DELAY2
AMP6
ATTN5
1
Attenuator
C
Or
D
LNA2
CLPR3
3
ANT3
ANT4
-20 dB Out
P
O
MXR5
5
P
5
Thru
4
D
Figure 10.1. Block diagram of the X-band front end.
ATTN6
B
E
E
4
3
2
1
199
Component
AMP6
AMP7
ANT3
ANT4
ATTN4
ATTN5
ATTN6
CIRC1
CIRC2
CIRC3
CLPR3
CLPR4
DELAY2
FL14
LNA1
MXR4
MXR5
Table 10.1. X-band front end modular components list.
Description
Microwave Components Corporation MH858231, Gain=25 dB, IP1=23 dBm
Mini-Circuits MAR-4, Gain = 8, NF=6 dB, IP1=12.5 dBm
Microtech 205297, X-band horn with WR90 waveguide flange
Microtech 205297, X-band horn with WR90 waveguide flange
3 dB in line attenuator
Narda Microline Step Attenuator, Model 705-69
Midwest Microwave 6 dB in line attenuator
UTE Microwave X-band isolator
UTE Microwave X-band isolator
Unknown surplus X-band isolator
Unknown surplus X-band -20 dB directional coupler
Omni Spectra X-band -10 dB directional coupler
Coaxial delay line
Mini-Circuits PBP-10.7, 10.7 MHz bandpass filter
Amplica, Inc. XM553403, Gain=20dB, IP1=25 dBm
Watkins Johnson M31A
TRW Microwave MX18533
Figure (if applicable)
NA
Figure 10.2
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
Figure 10.2
NA
NA
NA
200
4
3
2
1
A
RF in
A
2
0.01uF
B
AMP7
MAR4
U2
0.01uF
C
2
1
0.01uF
RFC
0w0.Ω
0 . 215
FeedThru1
FL14
1
R1
2
+12V
C
U1
5
4
6
Mini-Circuits
PBP-10.7
(bot view)
3
D
D
8
7
2
J1
1
BNC
Male
RF out
Figure 10.2. Schematic of the module that contains AMP7 and FL14.
J2
1
BNC
Female
B
E
E
4
3
2
1
(a)
(b)
Figure 10.3. X-band front end (a), X-band front end in operation with adjustable
transmit attenuator ATTN5 in line (b).
201
10.2
Free-space X-band imaging
A 1:32 scale model F14 and a group of pushpins were imaged using the X-band rail
SAR. Approximately 10 milliwatts of transmit power was used for the measurements
shown in this section. All measurements were acquired outdoors. A picture of the
outdoor experimental setup is shown in Figure 10.4. The targets were placed on the
styrofoam table down range in front of the rail.
A rail SAR image of a 1:32 scale model F14 aircraft coated in aluminum foil was
measured. The resulting image is shown in Figure 10.5. This image clearly shows an
aircraft slightly off-axis. Many details are noticeable including the nose and wings.
Much of the metal surface was illuminated in this radar image.
A rail SAR image of a group of pushpins was acquired. Pushpins are the small
plastic and metal thumbtacks that are used to hold up papers and posters. A picture
of a pushpin is shown in Figure 10.6a. A picture of a target group made up of pushpins
is shown in Figure 10.6b. A radar image of a group of pushpins was acquired and is
shown in Figure 10.7. The location of all pushpins is clearly shown. Looking at this
radar image it is possible to count each of the individual pushpins. Some cross range
blurring on the pushpins making up the ‘G’ is noticeable. This was likely due to wind
gusts on the day this image was acquired. Fading out is noticeable at the bottom of
the ‘92.’ This is due to the -3 dB cutoff of the range-gate which happened to be set
too close to the bottom of the target scene.
A zoomed out radar image of the group of pushpins is shown in Figure 10.8. No
additional downrange clutter and very little crossrange clutter is present in this image.
This image demonstrates the effectiveness of the range-gate in an X-band imaging
scenario.
The results shown in this section demonstrate the effectiveness of the high sensitivity range-gated FMCW architecture used for X-band high resolution SAR imaging
202
applications. A system such as this could be used for radar cross section measurements
(RCS) or other applications requiring a sensitive radar system with a range-gate.
203
Figure 10.4. The X-band rail SAR and target scene.
Figure 10.5. X-band rail SAR image of a 1:32 scale F14 model.
204
(a)
(b)
Figure 10.6. One pushpin (a), image scene of ‘GO STATE’ in pushpins (b).
205
Figure 10.7. X-band rail SAR image of a group of pushpins.
Figure 10.8. Zoomed-out X-band rail SAR image of a group of pushpins.
206
10.3
Low power free-space X-band imaging
In Section 9.2 it was shown that extremely low transmit power could be used to image
a target scene at S-band. For this reason it was decided to place an attenuator in the
transmitter signal chain of the X-band front end and image groups of pushpins using
low transmit power.
An X-band rail SAR image was acquired of a group of pushpins using only 100
nanowatts of transmit power. This result is shown in Figure 10.9. This image is
nearly identical to the full power image of a group of pushpins shown in Figure 10.7
(except for the text in the image). Clutter and signal-to-noise are nearly identical in
both images.
An X-band rail SAR image was acquired of a group of pushpins using only 10
nanowatts of transmit power. This result is shown in Figure 10.10. The pushpins
further down range in this image have faded into the noise compared to the full power
image shown in Figure 10.7. All pushpins in this image are clearly visible.
An X-band rail SAR image was acquired of a group of pushpins using only 1
nanowatt of transmit power. This result is shown in Figure 10.11. Only a few
pushpins on the bottom row of letters are visible. This transmit power level is too
low for use in imaging small targets such as pushpins.
These low power results demonstrate the effectiveness of the high sensitivity rangegated radar architecture applied to X-band. A low power radar system such as this one
could be used for RCS measurements and detection applications such as automotive
radar. Based on the low-power results presented in this section a radar system using
this architecture could be developed to easily meet various federal regulations for
transmit power allowing for the widespread use of high performance low power radar
sensors.
207
Figure 10.9. X-band rail SAR image of a group of pushpins using 100 nanowatts of
transmit power.
Figure 10.10. X-band rail SAR image of a group of pushpins using 10 nanowatts of
transmit power.
208
Figure 10.11. X-band rail SAR image of a group of pushpins fading into the noise
using 1 nanowatt of transmit power.
209
10.4
Comparison of high sensitivity range-gated FMCW to a typical
FMCW radar imaging system
A low cost X-band rail SAR imaging system was developed in [60] capable of imaging
small objects such as scale model aircraft and pushpins. The radar system developed
for [60] was a simple direct conversion FMCW system. FMCW radar architecture
was discussed in Section 7.3.1. In this section imagery from the radar system developed in [60] will be compared to radar imagery produced by the X-band rail SAR
developed in this chapter which is based on the high sensitivity range-gated FMCW
radar architecture.
Figure 10.12 shows a radar image of a 1:32 scale model F14 aircraft from [60].
Figure 10.5 shows a radar image of the same model acquired using the X-band rail
SAR. Both images are in agreement.
Figure 10.13 shows a radar image of a pushpin target scene from [60]. Figure 10.7
shows a radar image of a pushpin target scene measured using the X-band rail SAR.
The image from [60] in Figure 10.13 shows more clutter and the amplitude return of
the pushpins in the last few rows is shown fading into the noise.
A zoomed-out pushpin image from [60] is shown in Figure 10.14. Much clutter is
present down range and some cross range. By contrast, the zoomed out image acquired by the X-band rail SAR shown in Figure 10.8 contains no noticeable downrange
clutter due to the range gate and much less cross range clutter.
In this section it was shown that the high resolution range-gated FMCW radar
architecture is more effective compared to direct conversion FMCW for small rail
SAR applications where down range clutter must be eliminated and high sensitivity
is required.
210
Figure 10.12. SAR image of a 1:32 scale model F14 using a direct conversion FMCW
radar system.
Figure 10.13. SAR image of a group of pushpins using a direct conversion FMCW
radar system.
211
Figure 10.14. Zoomed out SAR image of a group of pushpins using a direct conversion
FMCW radar system showing the presence of significant clutter outside the target
scene of interest.
212
10.5
Discussion of X-band rail SAR imaging results
In this chapter it was shown that the high sensitivity range-gated FMCW radar
architecture developed in Chapter 7 can be applied to the X-band frequency range.
This application was shown to be an effective use of the high resolution range-gated
FMCW radar architecture. High resolution imagery of model aircraft and groups of
pushpins were acquired. Extremely low transmit power imagery of pushpin target
scenes were acquired. It was shown that this X-band rail SAR imaging system is
more effective than previous direct conversion FMCW radar systems [60] in reducing
downrange clutter. With its simplicity of design this radar architecture could be used
for RCS measurements, automotive radar, or other radar sensor applications where
low transmit power, high sensitivity and range gating are attractive features on a
low-budget.
213
CHAPTER 11
HIGH SPEED SAR IMAGING ARRAY
It takes 7 to 10 minutes to acquire data using the S-band rail SAR imaging system
developed in Chapter 8. This is too slow for practical applications that might result
from this research. In this chapter a near real-time proof-of-concept array based SAR
imaging system is developed that meets these requirements. This system is capable
of acquiring data and processing a SAR image of a target scene continuously on a
computer screen with an update rate of up to 1 image every 1.9 seconds. This allows
for the ability to quickly locate and track targets behind dielectric slabs.
It will also be shown in this chapter that the high sensitivity range-gated FMCW
radar architecture developed in Chapter 7 is capable of operating at high speeds.
The S-band radar system developed in Chapter 8 will be used as the radar sensor
connected to the high speed imaging array developed in this chapter. This radar will
be shown to operate at high speeds while imaging targets in free-space and through
a lossy-dielectric slab in near real-time.
11.1
SAR on an array
When imaging a target scene using a rail SAR the radar sensor physically moves
down a linear track acquiring range profile data at known locations (see Figure 2.1).
For the case of the S-band rail SAR imaging system developed in Chapter 8 the radar
sensor is mounted on an 8 foot long linear rail and it acquires a range profile once
ever 2 inches. Rather than physically moving the radar down a rail it is possible to
electronically switch the radar antenna (assuming a mono-static radar in this case)
between 48 antenna elements which are evenly spaced every 2 inches apart. This
array would be equivalent to a rail SAR acquiring data every 2 inches 48 times down
an 8 foot long rail, see Figure 11.1.
214
The problem with this simple array is that it requires a large number of elements;
48 elements, one for every rail position. The S-band radar system developed in
Chapter 8 is a bi-static radar which uses a separate transmit and receive antennas.
When the transmit and receive elements are close to each other relative to the range
to target scene they are equivalent to one single element at the mid-point. This is
known as the phase center. The phase center is equivalent to a mono-static radar
antenna at the mid-point between a closely spaced transmit and receive element as
shown in Figure 11.2. In order to make the equivalent array in Figure 11.1 using a bistatic radar it would require 48 transmit antennas and 48 receive antennas for a total
of 96 antennas as shown in Figure 11.3. This is a large number of antenna elements,
expensive and complicated to implement. However, when using a bi-static radar,
thoughtful design can utilize careful positioning of bi-static elements to maximize the
number of phase centers with the fewest antenna elements possible.
A better use of transmit and receive elements is shown in Figure 11.4 where using
the spacing shown and combinations of two receive elements and six transmit elements
yields 12 phase centers. This is done by switching the receiver on to only one receive
element at a time and by switching the transmitter on to only one transmit element
at a time. By switching combinations of transmit and receive elements the number of
phase centers created is maximized. The combinations are shown in Figure 11.4 by
straight lines running through their corresponding receive element, transmit element,
and resulting phase center. This is a much lower cost option. A large array is built
for a relatively low cost in the next section by cascading a number of these together
to create a 44 phase center array using only eight receive elements and 13 transmit
elements.
215
Figure 11.1. A radar device electronically switched across an array of evenly spaced
antenna elements is equivalent to a rail SAR.
216
Figure 11.2. When bi-static transmit and receive elements are close together (relative
to target range) spaced apart by x it is equivalent to a single mono-static element at
the mid point x/2.
217
Figure 11.3. Side view of a simple bi-static antenna array using 48 transmit and 48
receive elements for a total of 96 elements.
218
219
Figure 11.4. A more advanced bi-static antenna array producing 12 mono-static phase centers at the expense of only 2 receive
elements and 6 transmit elements (units in inches).
11.2
Array implementation
All design details of the high speed SAR array are presented in this section. These
include the array physical layout, antenna switch block diagram and the connections
between the array and the existing S-band radar system developed in Chapter 8.
11.2.1
Array antenna spacing and physical layout
Using the concept developed in Section 11.1 a large array was developed by cascading
several of the array sections from Figure 11.4 in series. The layout of the resulting
large array is shown in Figure 11.5 and Figure 11.6. This array is effectively 88 inches
long with a cross range sample spacing of 2 inches producing 44 equivalent phase
centers across the aperture. All antenna combinations used to create the 44 phase
centers are shown in Figure 11.5 and Figure 11.6. The physical location of each phase
center is also shown in this figure.
11.2.2
Array implementation and interface
This array was designed to plug into the existing S-band radar system developed
in Chapter 8. A modification was made to the system block diagram in Figure 8.6
by adding a National Instruments PCI-6509 digital I/O (DIO) card to control the
antenna array. This modification is shown in Figure 11.7. A picture of this array
built out of vertically polarized LTSA antenna elements is shown in Figure 11.12.
The antenna element configuration is noticeable in Figure 11.12a.
All antennas in this array are identical and based on the LTSA design from [71].
These elements were produced in volume by a printed circuit board manufacturer to
be identical to each other. The LTSAs are built out low cost readily available 1/16
inch thick FR-4 circuit board material with the resulting LTSA pattern etched on
the surface. The LTSA layout is shown in Figure 11.10 and a picture of a completed
receive LTSA with antenna feed is shown in Figure 11.11. The antenna is fed by
soldering a low loss CNT100a coaxial cable to the area shown in the layout in Figure
220
11.10. In this the center conductor of the coaxial feed is soldered to the top copper
trace and the shield is soldered to the bottom copper trace. An optimum feed point
is found by sliding the coax cable back and forth down the 4 inch long slot until
optimum SWR is achieved for the desired bandwidth. In the case of this LTSA
design the antenna covers 1 GHz to 4 GHz with an SWR ≤ 2 : 1 across the entire
band. Each of the 21 LTSA antennas in the array were individually tuned using this
method.
The feed lines running to the transmit LTSAs (ANT1 through ANT13) are fed
directly into the switch matrix. The feed lines running from the receive LTSAs
(ANT14 through ANT21) feed directly into a small pre-amplifier mounted on each of
the receive LTSAs. This is shown in Figure 11.11b.
The LTSA feed lines run directly into the switch matrix box which is located
on the back of the array shown in Figure 11.13. The block diagram of the switch
matrix is shown in Figure 11.8 and Figure 11.9 with a parts list in Table 11.1. The
switch matrix switches the S-band transmitter and receiver front ends to the correct
antenna combination to create the desired phase center. The switch matrix that
controls these antenna elements is made up of two different sub-switch matrices: a
transmitter switch matrix with block diagram shown in Figure 11.8 and a receive
switch matrix with block diagram shown in Figure 11.9.
The transmitter front end shown in Figure 8.4 feeds directly into the transmit
switch matrix RF input port shown in Figure 11.8. This is fed through a -10 dB
Narda directional coupler CLPR1 into SW1 which is a Mini-Circuits ZSWA-4-30DR,
DC-3 GHz 4-way GaAs Switch. SW1 branches out into three other identical switches
SW2, SW3 and SW4 all of which are controlled by DIO from the PCI-6509 card.
Some power is coupled off CLPR1 and fed out for diagnostic purposes. The output
of these switches is fed directly to all of the transmit antenna elements which make
up the transmit sub-array consisting of the bottom row of LTSA’s shown in Figure
221
11.12. All solid state switches in both switch matrices are designed to operate only
up to 3 GHz however it was found through laboratory tests that these switches could
operate up to 4 GHz with a 2 dB increase in insertion loss and a 5 dB reduction in
port to port isolation.
The receiver front end shown in Figure 8.5 is connected directly to the receive
switch matrix shown in Figure 11.9 through the RF output port. Received signals
detected by the antennas and amplified by LNA1 through LNA8 are fed into SW6
and SW7 which are Mini-Circuits ZSWA-4-30DR, DC-3 GHz 4-way GaAs Switches
controlled by DIO from the PCI-6509 card. The output of these switches is fed into
SW5 which is a Mini-Circuits ZSDR-230, DC-3 GHz PiN Diode Switch controlled by
DIO from the PCI-6509 card.
The PCI-6509 card selects the proper switch combination to turn on the correct
antenna pair to make the required phase center. This is done by sending a 32 bit
hexadecimal word out of the DIO ports on the PCI-6509. The pin-out of this hex
word and its connection to the switches is shown in Table 11.2 and Table 11.3. Table
11.4 and Table 11.5 are look up tables which show the hex code required to turn
on a given phase center number. In these tables the transmit and receive element
combination for a given phase center and the hex code which is sent to the switch
matrix box used to switch on this phase center is shown.
When the S-band front ends are connected to the high speed SAR array there is
no reduction in receiver sensitivity due to the low-noise amplifiers mounted on the
receive elements. There is a significant reduction in transmit power due to the loss
through the transmitter switch matrix. This loss is around 8 to 10 dB across the 2
GHz to 4 GHz band. The result is a transmit power of about 1 to 2 milliwatts. This
is an estimation since it is difficult to probe the power at the transmitter antenna
feeds which are soldered directly to the coaxial feed lines. All data presented in this
chapter was acquired using this reduced transmit power of about 1 or 2 milliwatts.
222
11.2.3
Processing and control software
Labview software was written to rapidly acquire data across the array then calculate a radar image. This software also calibrates the array and performs coherent
background subtraction. The RMA SAR algorithm written in MATLAB code from
Appendix C was re-written in Labview.
Background subtraction is identical to that presented in Section 5.3. The calibration procedure is much different than for the rail SAR. All phase centers must be
calibrated because of the small 1 or 2 dB inconsistencies compounding throughout the
signal path due to hand-soldered antenna feeds, LNA’s and the switch matrix losses.
The calibration is a 2-D calibration where each of the 44 phase centers is calibrated.
This array calibration is done to sharpen imagery and is not an exact sphere
target calibration. This is due to the lack of an available large-enough calibration
sphere. A 5 foot tall 3/4 inch diameter copper pole is used as a calibration target.
The calibration target is placed at exactly 11 feet down range and centered to the
middle of the array. Range profile data is acquired across the array at each of the
44 phase centers. This data is represented by spole xn , ω(t) where xn is the cross
range phase center position. The pole is then removed and a background 2D range
profile data array is acquired. The result is represented by scalback xn , ω(t) . The
background is subtracted from the pole range profile data resulting in a 2D range
profile array of the pole only
scal xn , ω(t) = spole xn , ω(t) − scalback xn , ω(t) .
(11.1)
The calibration data is referenced to a theoretical point scatterer which is represented
by the equation
−j2kr Rpole
scaltheory xn , ω(t) = e
.
223
(11.2)
Where the Rpole is a 2D range to pole across the array represented by the equation
q
Rpole =
x2n + (11 · 0.3048)2 .
(11.3)
Where kr = ω(t)/c. The calibration factor is calculated by the equation
scaltheory xn , ω(t)
scalf actor xn , ω(t) =
.
scal xn , ω(t)
(11.4)
This 2D cal factor is multiplied by each 2D SAR data set acquired by the array.
The software written to run the array performs calibration then coherent background subtraction. Once this is complete the software continuously acquires data
from the array, performs the RMA, displays the image and starts over again. No
coherent integration is applied to the data. This system will produce one image every 1.9 seconds for a cr = 845.2 · 109 Hz/Sec or one image ever 2.8 seconds for a
cr = 422 · 109 Hz/Sec.
224
225
Figure 11.5. The high speed SAR imaging array physical layout (all units are in inches), antenna combinations and phase center
locations (1 of 2).
226
Figure 11.6. The high speed SAR imaging array physical layout (all units are in inches), antenna combinations and phase center
locations (2 of 2).
227
A
CTR0 pin
PCI6014
NIDAQ Card
PCI Bus
PC
AI0 pin
Ramp
Generator
PCI Bus
B
B
Yig
Oscillator
Beat
Frequency
Oscillator
PCI-6509
NIDAQ Card
C
2
BFO Output
3
E
32 Bit Hex Code Out to
Array Switch Matrix
D
F
1
N
D
LO Output to Power Splitter
1
Video Input to Digitizer
32 Lines of DIO
C
E
E
4
3
2
1
Figure 11.7. The overall radar system block diagram with modifications required to control the antenna array.
4
3
2
1
A
228
4
3
2
1
5
4
RFout1
RFout4
A
RFin
1
C6
Mini-Circuits
C3
ZX606013E-S+
C4
C5
RFout2
SW4
RFout3
ANT13 ANT12
A
4
B
P1,7
P1,5
P1,6
2
3
P1,4
1
ANT9
5
RFout1
RFout4
RFin
1
C6
Mini-Circuits
C3
ZX606013E-S+
C4
C5
RFout2
C6
7
6
5
4
8
7
6
5
C
2
3
2
RFout3
SW1
RFin
1
C5
RFout1
3
ANT7
Mini-Circuits
C3
ZX606013E-S+
C4
RFout4
RFout2
SW3
RFout3
4
5
4
ANT8
C
ANT6
P0,3
P0,0
P0,1
P0,2
P1,1
P1,2
P1,3
5
4
D
RFin
1
2
3
3
2
1
9
ANT3
ANT2
1
CLPR1
2
P0,4
P0,7
P0,5
P0,6
ANT1
E
H
E
Input from S-band
Front End TX Out
-10 dB Out
C6
C5
RFout1
Mini-Circuits
C3
ZX606013E-S+
C4
RFout4
RFout2
SW2
RFout3
ANT4
P1,0
ANT5
D
Figure 11.8. Block diagram of the transmitter switch matrix.
2
3
ANT11 ANT10
B
4
3
2
1
229
4
3
2
1
ANT21
A
A
1
LNA
ANT20
5
4
RFout1
RFout4
C6
B
2
3
4
Com
RF2
SW5
4
LNA
C
TTL
RF1
ANT17
Mini-Circuits
ZSDR-230
P2,7
P2,5
P2,6
2
3
P2,4
3
LNA
ANT18
1
ANT19
C
9
5
LNA
5
4
ANT16
RFout1
RFout4
P3,0
RFin
1
D
C6
C5
Mini-Circuits
C3
ZX606013E-S+
C4
RFout2
RFout3
SW6
6
LNA
D
8
7
6
5
P2,0
P2,1
P2,2
P2,3
7
LNA
ANT14
8
LNA
E
1
K
E
Out to S-band Receiver
Front End RX in Port
2
3
ANT15
Figure 11.9. Block diagram of the receiver switch matrix.
RFin
1
C5
Mini-Circuits
C3
ZX606013E-S+
C4
RFout2
SW7
RFout3
2
LNA
B
4
3
2
1
230
Table 11.1. High speed SAR imaging array modular component list.
Component
Description
ANT1 through ANT21
LTSA built on FR4
CLPR1
Narda -10 dB Directional Coupler
LNA1 through LNA8
Mini-Circuits ZX60-6013E-S+
SW1 through SW4, SW6, SW7 Mini-Circuits ZSWA-4-30DR, DC-3 GHz 4-way GaAs Switch
SW5
Mini-Circuits ZSDR-230, DC-3 GHz PiN Diode Switch
Figure (if applicable)
Figure 11.10
NA
NA
NA
NA
231
Figure 11.10. Layout of the LTSA used for both the transmit and receive elements in the high speed SAR array (units in
inches).
Table 11.2. 32 bit hex word for communicating with the switch matrix (1 of 2).
Transmitter Switch Matrix
Port 1
6
5
4
3
SW4
C5 C3 C4 C6 C5 C3
7
2
1
0
7
SW3
C4 C6 C5 C3
Port 0
6
5
4
3
SW2
C4 C6 C5 C3
2
1 0
SW1
C4 C6
Table 11.3. 32 bit hex word for communicating with the switch matrix (2 of 2).
Port 3
0
SW5
TTL
Receiver Switch Matrix
Port 2
7
6
5
4
3
2
1
SW7
SW6
C5 C3 C4 C6 C5 C3 C4
0
C6
Table 11.4. High speed SAR array hex look-up table (1 of 2).
Phase Center
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Receive Element Transmit Element
14
1
15
1
14
2
15
2
14
3
15
3
14
4
15
4
14
5
15
5
16
3
17
3
16
4
17
4
16
5
17
5
16
6
17
6
16
7
17
7
16
8
17
8
232
Hex Code
1010001
1020001
1010012
1020012
1010022
1020022
1010042
1020042
1010082
1020082
1040022
1080022
1040042
1080042
1040082
1080082
1040104
1080104
1040204
1080204
1040404
1080404
Table 11.5. High speed SAR array hex look-up table continued (2 of 2).
Phase Center
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
Receive Element Transmit Element
18
6
19
6
18
7
19
7
18
8
19
8
18
9
19
9
18
10
19
10
18
11
19
11
20
9
21
9
20
10
21
10
20
11
21
11
20
12
21
12
20
13
21
13
233
Hex Code
0100104
0200104
0100204
0200204
0100404
0200404
0100804
0200804
0101008
0201008
0102008
0202008
0400804
0800804
0401008
0801008
0402008
0802008
0404008
0804008
0408008
0808008
(a)
(b)
Figure 11.11. A receiver LTSA antenna element with LNA mounted on the antenna
(a), close up view of LNA mounted at the end of the LTSA (b).
234
(a)
(b)
Figure 11.12. The front of the array showing the vertically polarized LTSA antennas
(a), the array at an angle showing the ’V’ of the LTSA elements pointing out towards
the target scene (b).
235
Figure 11.13. The back side of the array showing the LTSA antenna feeds running to
the switch matrix box mounted on the back of the array.
236
Figure 11.14. The near real-time S-band radar imaging system.
237
11.3
Array measured data
A number of images were acquired using the high speed SAR imaging array. The
Labview software written for this system can save MATLAB .MAT matrix files of
the RMA processed SAR image. The resulting imagery presented here were acquired
using the high speed array and are displayed using MATLAB for convenience. The
chirp rate for all measurements presented in this section is cr = 422 · 109 Hz/Sec
which provided a radar image of the target scene once every 2.8 seconds. Coherent
integration was not used while acquiring the data presented in this section.
11.3.1
Comparison of free-space simulations and measured data
Imagery of cylinders was acquired using the high speed SAR array. This imagery is
compared to simulated imagery of cylinders using the model developed in Chapter 3.
Simulations were conducted by programming the theoretical model to the same
cross range and down range sample spacing used in the high speed array imaging system. SAR imagery of cylinders was simulated by running three MATLAB programs.
Simulated free-space cylinder data was calculated using the MATLAB program in
Appendix E. The data conditioning program was run in Appendix B followed by the
RMA imaging algorithm in Appendix C which produced the resulting simulated SAR
image.
Figure 11.15a is a simulated radar image of a cylinder with radius a = 3 inches in
free-space. Figure 11.15b is a measured radar image of a cylinder in free-space with
radius a = 3. Both images are in agreement and the cylinder cross range and down
range resolution is similar in both cases.
Figure 11.16a is a simulated radar image of a cylinder with a radius of a = 6 inches
in free-space. Figure 11.16b is a measured radar image of a cylinder with the same
radius in free-space. Both theoretical and measured images are in close agreement.
Both simulated cylinders were close to the measured cylinders. This shows that
238
the high speed SAR imaging array is an effective SAR imaging system.
239
(a)
(b)
Figure 11.15. SAR imagery of a a = 3 inch radius cylinder in free space: simulated
(a), acquired from the high speed array (b).
240
(a)
(b)
Figure 11.16. SAR imagery of a a = 6 inch radius cylinder in free space: simulated
(a), acquired from the high speed array (b).
241
11.3.2
Free-space imagery
System performance was found to be close to simulation in the previous section. For
this reason it was decided to image a number of other various target scenes.
Figure 11.17 is a SAR image of a sphere in free space. This is in agreement with
the radar image of the same sphere acquired by the S-band rail SAR in Figure 9.4a.
Figure 11.18 is a SAR image of a carriage bolt target scene configured in a block
‘S.’ This is in agreement with the same target scene acquired by the S-band rail SAR
in Figure 9.4b.
Figure 11.19 is a SAR image of a 12 oz aluminum beverage can. This target will
be imaged behind a lossy-dielectric slab later in this chapter.
Groups of three different sized nails were imaged to determine the smallest target
detectable. Figure 11.20 is a SAR image of a group of 3 inch tall nails in a block ‘S’
configuration. All nails are clearly visible in this image where the top row further
down range is fading out. Figure 11.21 is a SAR image of a group of 2 inch tall nails
in the same configuration. The location of each nail is clearly shown. Figure 11.22 is a
SAR image of a group of 1.25 inch tall nails in a block ‘S’ configuration. The bottom
two rows are clearly shown and a few of the nails at the top row are noticeable. This
image shows that the array is a very sensitive high speed SAR imaging device capable
of detecting and providing the location of targets as small as 1.25 inch tall nails.
242
Figure 11.17. SAR image of an a = 4.3 inch radius sphere in free-space.
Figure 11.18. SAR image of a group of 6 inch long 3/8 inch diameter carriage bolts
in free-space.
243
Figure 11.19. Image of a 12 oz aluminum beverage can in free-space.
Figure 11.20. Image of a group of 3 inch tall nails in free-space.
244
Figure 11.21. Image of a group of 2 inch tall nails in free-space.
Figure 11.22. Image of a group of 1.25 inch tall nails in free-space.
245
11.3.3
Comparison of through-dielectric slab simulations and measured
data
A comparison of measured and theoretical through-lossy-dielectric slab imagery will
be shown in this section.
Measured SAR imagery were acquired by placing the high speed SAR array at a
range of 20 feet from a d = 4 inch thick lossy-dielectric slab. Targets were placed at
various ranges on the opposite side of the slab.
Simulated data of cylinders behind a lossy-dielectric slab was calculated by using
background subtraction and the MATLAB programs in Appendix I and G. This
simulated data was fed into the data conditioning MATLAB program in Appendix B
and processed by the RMA in Appendix C resulting in the simulated SAR imagery.
Figure 11.23a shows a simulated SAR image of a cylinder with a radius of a = 3
inches behind a d = 4 inch thick lossy-dielectric slab. Figure 11.23b shows a measured
SAR image of the cylinder behind a d = 4 inch thick lossy-dielectric slab. The
measured is in agreement with the theoretical except for an increase in clutter. This
may be due to the background subtraction interacting with the target.
Figure 11.24a shows a theoretical SAR image of a cylinder with a radius of a = 6
inches behind a d = 4 inch thick lossy dielectric slab. Figure 11.24b shows a measured
image of the cylinder behind a d = 4 inch thick lossy-dielectric slab. The measured
is in agreement with the theoretical with some noise and clutter present.
In both cases the measured data from the high speed SAR array was is agreement
with the simulated. Increased noise is present in these images compared to the rail
SAR imagery of the same targets shown in Figure 9.15b and Figure 9.16b. This may
be due to the transmit loss caused by the switch matrix. The rail SAR feeds more
power to the transmit antenna than the high speed SAR imaging array and this might
be the reason for the differences.
246
(a)
(b)
Figure 11.23. Imagery of an a = 3 inch radius cylinder behind a d = 4 inch thick
lossy-dielectric slab: simulated (a), acquired from the high speed array (b).
247
(a)
(b)
Figure 11.24. Imagery of an a = 6 inch radius cylinder behind a d = 4 inch thick
lossy-dielectric slab: simulated (a), acquired from the high speed array (b).
248
11.3.4
Through-dielectric slab imagery
In this section through lossy-dielectric slab imagery was acquired on a number of
other miscellaneous targets using the high speed array.
Figure 11.25 is a through lossy dielectric slab image of a 12 oz aluminum beverage
can. The location of this can is clearly shown behind the lossy-dielectric slab.
Figure 11.26 is a through lossy-dielectric slab image of a single carriage bolt.
This shows that the high speed SAR array is capable of detecting and indicating the
location of small objects behind lossy-dielectric slabs.
Figure 11.27 is a through-lossy-dielectric slab image of a sphere with radius a = 4.3
inches. This image is close to the through slab rail SAR image of the same sphere
shown in Figure 9.17.
This data shows that the high speed SAR array is capable of accurately imaging
target scenes made up of small objects through a lossy-dielectric slab.
249
Figure 11.25. SAR image of a 12 oz aluminum beverage can through a d = 4 inch
thick lossy-dielectric slab.
Figure 11.26. SAR image of a 6 inch tall 3/8 inch diameter carriage bolt through a
d = 4 inch thick lossy-dielectric slab.
250
Figure 11.27. SAR image of an a = 4.3 inch radius sphere through a d = 4 inch thick
lossy-dielectric slab.
251
11.4
Discussion of the high speed SAR imaging array
The results in this chapter prove the concept of attaching the high sensitivity range
gated FMCW radar system to a high speed antenna array to create near real-time
imagery of targets behind a dielectric slab. This system operates at an image re-fresh
rate of one image every 1.9 seconds to 2.8 seconds (depending on chirp rate settings).
This is very fast for a SAR imaging system. The concept was proven here, however, it
would not be difficult to increase the imaging re-fresh rate to more than 5 or 6 images
per second by upgrading to more sophisticated data acquisition hardware. A high
speed real-time SAR imaging system that is capable of imaging through dielectric
slabs at such a rate of speed has numerous applications.
252
CHAPTER 12
CONCLUSIONS AND FUTURE WORK
A near real-time through-lossy-dielectric slab radar imaging system was developed in
this dissertation which is different from most previous work. It uses a modified FMCW
radar design which is very sensitive, capable of range-gating and using extremely low
levels of transmit power. One of the largest switched antenna arrays built for throughslab imaging was developed for this system. It operates at a stand-off range of 20 to
30 feet from the slab. This system is capable of near real-time through-slab imaging
using an airborne SAR algorithm. It was shown that this radar design is flexible and
can be adapted to other frequency bands such as X-band.
The entire development process was shown in this dissertation. Starting with a
through-slab imaging model and ending with three functional proof-of-concept prototypes.
The model was effective at generating simulated rail SAR data of a 2D PEC cylinder placed behind a dielectric slab. Simulated and measured data were in agreement
for both the rail SAR and the antenna array. Future modeling work could be to
develop a periodic metal structure made up of PEC rods inside of a lossy dielectric
slab. This might also lead to the modeling of multiple layers of lossy-dielectric slabs.
One unexpected result from this research was the ability for both the S-band and
X-band radar systems to image small targets using extremely low transmit power.
Future work with low power radar imaging has many applications including automotive radar, marine and aviation navigational radar and unmanned ground vehicle
radar. This equipment could use nano-watts or pico-watts of transmit power to avoid
obstacles or image unknown terrain at a short distance. These systems would not be
difficult to implement. Some possible future work might be to simply plug the X-band
253
front end directly into a consumer X-band marine radar to see how far the radar could
range targets using a PPI scope with only 10 mili-watts of transmit power.
The near real-time imaging system has numerous applications including earthquake victim location through building rubble, ground penetrating radar or other
through-lossy-dielectric slab applications. The data resulting from this system was
shown to be in agreement with the model for both free-space and through-lossydielectric slab imaging. This system was shown to be capable of imaging various
small targets behind a slab. Possible future work for the near real-time through-slab
imaging system includes:
1. Increasing the rate at which data can be read from the digitizer. This can be
done by upgrading to a National Instruments PXI bus digitizer or by
developing a proprietary high speed data acquisition system. By doing this the
image refresh rate should be increased from one image every 1.9 seconds to at
lest 1 or 2 per second or better. A rate of 5 or 6 images per second is desirable.
2. Increase digitizer sampling speed to greater than 200 KSPS which would
increase the maximum alias-free down range of the radar system.
3. Developing a better antenna design for the array such as a Vivaldi.
4. Increased transmit power to at least 1 or 2 watts which would allow the
system to be capable of imaging through a number of dielectric slabs.
5. Increasing the chirp bandwidth to cover 1-4 GHz.
6. Using multiple BFO carriers and IF filters to allow for multiple simultaneous
range gates.
The research area of through-dielectric slab imaging is diverse. It was decided
to focus on S-band through-dielectric slab radar imaging in this dissertation. An
254
entire proof of concept program was developed to completion starting with modeling
and ending with three operational prototypes. The model drove all design aspects
and the results were in agreement with the model. The resulting near real-time
imaging system was very effective at imaging objects as small as metal rods and 12
oz aluminum soda cans placed behind a lossy-dielectric slab. More work could be
done in this area and the future shows great potential for the eventual fielding of
reliable through-dielectric slab imaging systems.
255
APPENDICES
256
APPENDIX A
MATLAB CODE FOR SIMULATING MULTIPLE POINT
SCATTERERS
The following MATLAB program was written to simulate SAR data of three different
point scatterers at various locations in a target scene.
%Range Migration Algorithm from ch 10 of Spotlight Synthetic
%Aperture Radar
%Signal Processing Algorithms, Carrara, Goodman, and Majewski
clear all;
c = 3E8; %(m/s) speed of light
%**********************************************************************
%radar parameters
fc = 3E9; %(Hz) center radar frequency
B = 2E9; %(hz) bandwidth
Rs = 0; %(m) y coordinate to scene center (down range)
Xa = 0; %(m) beginning of new aperture length
L = 8*.3048; %(m) aperture length
Xa = linspace(-L/2, L/2, 48); %(m) cross range position of
%radar on aperture L
Za = 0;
Ya = Rs; %THIS IS VERY IMPORTANT, SEE GEOMETRY FIGURE 10.6
fsteps = 500;
257
%**********************************************************************
%create SAR if data according to eq 10.4 and 10.5 (mocomp
%to a line) ignoring RVP term
%target parameters, 3 targets
at1 = 1;
xt1 = 2*.3048;
yt1 = -10*.3048;
zt1 = 0;
at2 = 1;
xt2 = 0*.3048;
yt2 = -5*.3048;
zt2 = 0;
at3 = 1;
xt3 = -2*.3048;
yt3 = -10*.3048;
zt3 = 0;
%Rt and Rb for 3 targets according to equation 10.26
Rb1 = sqrt((Ya - yt1)^2 + (Za - zt1)^2);
Rt1 = sqrt((Xa - xt1).^2 + Rb1^2);
%Rt1 = sqrt(yt1^2 + (Xac-xt1).^2);
Rb2 = sqrt((Ya - yt2)^2 + (Za - zt2)^2);
Rt2 = sqrt((Xa - xt2).^2 + Rb2^2);
%Rt2 = sqrt(yt2^2 + (Xac-xt2).^2);
258
Rb3 = sqrt((Ya - yt3)^2 + (Za - zt3)^2);
Rt3 = sqrt((Xa - xt3).^2 + Rb3^2);
%Rt3 = sqrt(yt3^2 + (Xac-xt3).^2);
Kr = linspace(((4*pi/c)*(fc - B/2)), ((4*pi/c)*(fc + B/2)), fsteps);
%according to range defined on bottom of page 410
for ii = 1:fsteps %step thru each time step to find phi_if
for jj = 1:size(Xa,2) %step thru each azimuth step
phi_if1(jj,ii) = Kr(ii)*(Rt1(jj) - Rs);
phi_if2(jj,ii) = Kr(ii)*(Rt2(jj) - Rs);
phi_if3(jj,ii) = Kr(ii)*(Rt3(jj) - Rs);
end
end
sif1 = at1*exp(-j*phi_if1);
sif2 = at2*exp(-j*phi_if2);
sif3 = at3*exp(-j*phi_if3);
sif = sif1+sif2+sif3; %superimpose all three targets
clear sif1;
clear sif2;
clear sif3;
clear phi_if1;
clear phi_if2;
clear phi_if3;
s = sif;
save thruwall s;
259
%view simulated range history after range compression (figure 10.7) OK
%for ii = 1:size(sif,1);
%
sview(ii,:) = fftshift(fft(sif(ii,:)));
%end
%***********************************************************************
%a note on formatting, our convention is sif(Xa,t)
%**************************************************************
%plot the real value data for the dissertation
set(0,’defaultaxesfontsize’,13);
imagesc(Kr*c/(4*pi*1E9), Xa/.3048, real(sif));
colormap(gray);
ylabel(’x position (ft)’);
xlabel(’recieved chirp frequency (GHz)’);
title(’real values of the single point scatter SAR data matrix’);
colorbar;
cbar = colorbar;
set(get(cbar, ’Title’), ’String’, ’V/m’,’fontsize’,13);
print(gcf, ’-djpeg100’, ’real_value_of_single_pt_scatterer_raw_data.jpg’);
260
APPENDIX B
MATLAB CODE FOR OPENING SAR DATA TO BE PROCESSED
BY THE RMA
The following MATLAB program was used throughout the dissertation to open simulated SAR data and condition that data to be processed by the RMA MATLAB
program shown in appendix C.
%Range Migration Algorithm from ch 10 of Spotlight Synthetic
%Aperture Radar
%Signal Processing Algorithms, Carrara, Goodman, and Majewski
clear all;
c = 3E8; %(m/s) speed of light
%*********************************************************************
load thruwall s; %load variable sif %for image data
sif = s; %image without background subtraction
clear s;
%clear sif_sub;
%***********************************************************************
%radar parameters
fc = 3E9; %(Hz) center radar frequency
B = 4E9 - 2E9; %(hz) bandwidth
cr = 2E9/10E-3; %(Hz/sec) chirp rate
Tp = 10E-3; %(sec) pulse width
%VERY IMPORTANT, change Rs to distance to cal target
Xa = 0; %(m) beginning of new aperture length
delta_x = (2*1/12)*0.3048; %(m) 2 inch antenna spacing
261
L = delta_x*(size(sif,1)); %(m) aperture length
Xa = linspace(-L/2, L/2, (L/delta_x)); %(m) cross range
%position of radar on aperture L
Za = 0;
t = linspace(0, Tp, size(sif,2)); %(s) fast time, CHECK SAMPLE RATE
Kr = linspace(((4*pi/c)*(fc - B/2)), ((4*pi/c)*(fc + B/2)), (size(t,2)));
Rs = 0; %(m) y coordinate to scene center (down range),
%make this value equal to distance to cal target
Ya = Rs; %THIS IS VERY IMPORTANT, SEE GEOMETRY FIGURE 10.6
%*************************************************e***********************
%Save background subtracted and callibrated data
save sif sif delta_x Rs Kr Xa;
%clear all;
%run IFP
SBAND_RMA_IFP;
262
APPENDIX C
RMA MATLAB CODE
This is the RMA SAR imaging algorithm coded in MATLAB. In order to image theoretical data you must first run the data conditioning program in Appendix B. In order
to image measured data using calibration you must first run the data conditioning
program in appendix J.
%Range Migration Algorithm from ch 10 of Spotlight
%Synthetic Aperture Radar
%Signal Processing Algorithms, Carrara, Goodman, and Majewski
%***********************************************************************
%a note on formatting, our convention is sif(Xa,t)
% YOU MUST RUN THIS FIRST TO CAL AND BACKGROUND SUBTRACT DATA:
%RMA_FINAL_opendata
%load data
clear all;
load sif;
figcount = 1;
close_as_you_go = 0;
do_all_plots = 0;
set(0,’defaultaxesfontsize’,13); %set font size on plots
263
%so we can see it in the dissertation
% NOTE: the function ’dbv.m’ is just dataout = 20*log10(abs(datain));
%***********************************************************************
if do_all_plots == 1,
figure(figcount);
S_image = angle(sif);
imagesc(Kr, Xa, S_image);
colormap(gray);
title(’Phase Before Along Track FFT’);
xlabel(’K_r (rad/m)’);
ylabel(’Synthetic Aperture Position, Xa (m)’);
cbar = colorbar;
set(get(cbar, ’Title’), ’String’, ’radians’,’fontsize’,13);
print(gcf, ’-djpeg100’, ’phase_before_along_track_fft.jpg’);
if close_as_you_go == 1,
close(figcount);
end
figcount = figcount + 1;
end
%along track FFT (in the slow time domain)
%first, symetrically cross range zero pad so that the radar can squint
zpad = 256; %cross range symetrical zero pad
szeros = zeros(zpad, size(sif,2));
for ii = 1:size(sif,2)
index = (zpad - size(sif,1))/2;
264
szeros(index+1:(index + size(sif,1)),ii) = sif(:,ii); %symetrical
%zero pad
end
sif = szeros;
clear ii index szeros;
S = fftshift(fft(sif, [], 1), 1);
%S = fftshift(fft(sif, [], 1));
clear sif;
Kx = linspace((-pi/delta_x), (pi/delta_x), (size(S,1)));
if do_all_plots == 1,
figure(figcount);
S_image = dbv(S);
imagesc(Kr, Kx, S_image, [max(max(S_image))-40,
max(max(S_image))]);
colormap(gray);
title(’Magnitude After Along Track FFT’);
xlabel(’K_r (rad/m)’);
ylabel(’K_x (rad/m)’);
cbar = colorbar;
set(get(cbar, ’Title’), ’String’, ’dB’,’fontsize’,13);
print(gcf, ’-djpeg100’, ’mag_after_along_track_fft.jpg’);
if close_as_you_go == 1,
close(figcount);
end
figcount = figcount + 1;
265
end
if do_all_plots == 1,
figure(figcount);
S_image = angle(S);
imagesc(Kr, Kx, S_image);
colormap(gray);
title(’Phase After Along Track FFT’);
xlabel(’K_r (rad/m)’);
ylabel(’K_x (rad/m)’);
cbar = colorbar;
set(get(cbar, ’Title’), ’String’, ’radians’,’fontsize’,13);
print(gcf, ’-djpeg100’, ’phase_after_along_track_fft.jpg’);
if close_as_you_go == 1,
close(figcount);
end
figcount = figcount + 1;
end
if do_all_plots == 1,
figure(figcount);
S_image = dbv(fftshift(fft(S, [], 2), 2));
imagesc(linspace(-0.5, 0.5, size(S, 2)), Kx, S_image,
[max(max(S_image))-40, max(max(S_image))]);
colormap(gray);
title(’Magnitude of 2-D FFT of Input Data’);
xlabel(’R_{relative} (dimensionless)’);
266
ylabel(’K_x (rad/m)’);
cbar = colorbar;
set(get(cbar, ’Title’), ’String’, ’dB’,’fontsize’,13);
print(gcf, ’-djpeg100’, ’mag_after_2D_fft.jpg’);
if close_as_you_go == 1,
close(figcount);
end
figcount = figcount + 1;
end
%**********************************************************************
%matched filter
%create the matched filter eq 10.8
for ii = 1:size(S,2) %step thru each time step row to find phi_if
for jj = 1:size(S,1) %step through each cross range in the
%current time step row
%phi_mf(jj,ii) = -Rs*Kr(ii) + Rs*sqrt((Kr(ii))^2 - (Kx(jj))^2);
phi_mf(jj,ii) = Rs*sqrt((Kr(ii))^2 - (Kx(jj))^2);
Krr(jj,ii) = Kr(ii); %generate 2d Kr for plotting purposes
Kxx(jj,ii) = Kx(jj); %generate 2d Kx for plotting purposes
end
end
smf = exp(j*phi_mf); %%%%%%%%%%%%
%smf = exp(-j*phi_mf); %%%%%%%%%%% THIS IS THE KEY ISSUE !!!!!
%note, we are in the Kx and Kr domain, thus our convention is S_mf(Kx,Kr)
267
%appsly matched filter to S
S_mf = S.*smf;
%clear smf phi_mf;
if do_all_plots == 1,
figure(figcount);
S_image = angle(S);
imagesc(Kr, Kx, S_image);
colormap(gray);
title(’Phase After Matched Filter’);
xlabel(’K_r (rad/m)’);
ylabel(’K_x (rad/m)’);
cbar = colorbar;
set(get(cbar, ’Title’), ’String’, ’radians’,’fontsize’,13);
print(gcf, ’-djpeg100’, ’phase_after_matched_filter.jpg’);
if close_as_you_go == 1,
close(figcount);
end
figcount = figcount + 1;
end
clear S;
if do_all_plots == 1,
figure(figcount);
268
S_image = dbv(fftshift(fft(S_mf, [], 2), 2));
imagesc(linspace(-0.5, 0.5, size(S_mf, 2)), Kx, S_image,
[max(max(S_image))-40, max(max(S_image))]);
colormap(gray);
title(’Magnitude of 2-D FFT of Matched Filtered Data’);
xlabel(’R_{relative} (dimensionless)’);
ylabel(’K_x (rad/m)’);
cbar = colorbar;
set(get(cbar, ’Title’), ’String’, ’dB’,’fontsize’,13);
print(gcf, ’-djpeg100’,
’mag_after_downrange_fft_of_matched_filtered_data.jpg’);
if close_as_you_go == 1,
close(figcount);
end
figcount = figcount + 1;
end
%**********************************************************************
%perform the Stolt interpolation
%NOTICE:
Must change these parameters!!!!
%Ky_even = linspace(6, 13, 1028); %create evenly spaced Ky for
%interp for book example
%Ky_even = linspace(334.5, 448, 512); %create evenly spaced Ky for
%interp for real data
%Ky_even = linspace(200, 515, 512); %create evenly spaced Ky for
%interp for real data
269
%FOR DATA ANALYSIS
%kstart = 42.5; %for 1 to 3 ghz
%kstop = 118.5; %for 1 to 3 ghz
kstart =85.4; %for 2 to 4 ghz
kstop = 153.8; %for 2 to 4 ghz
%FOR DISSERTATION TO SHOW STOLT WORKING
%kstart = 50;
%kstop = 200;
Ky_even = linspace(kstart, kstop, 512); %create evenly spaced Ky
%for interp for real data
Ky_eeven = linspace(kstart, kstop, zpad); %make this same size as
%kx so we can find downrange
clear Ky S_St;
for ii = 1:size(Kx,2)
Ky(ii,:) = sqrt(Kr.^2 - Kx(ii)^2);
%S_st(ii,:) = (interp1(Ky(ii,:), S_mf(ii,:), Ky_even)).*H;
S_st(ii,:) = (interp1(Ky(ii,:), S_mf(ii,:), Ky_even));
end
S_st(find(isnan(S_st))) = 1E-30; %set all Nan values to 0
clear S_mf ii Ky;
if do_all_plots == 1,
270
figure(figcount);
S_image = angle(S_st);
imagesc(Ky_even, Kx, S_image);
colormap(gray);
title(’Phase After Stolt Interpolation’);
xlabel(’K_y (rad/m)’);
ylabel(’K_x (rad/m)’);
cbar = colorbar;
set(get(cbar, ’Title’), ’String’, ’radians’,’fontsize’,13);
print(gcf, ’-djpeg100’, ’phase_after_stolt_interpolation.jpg’);
if close_as_you_go == 1,
close(figcount);
end
figcount = figcount + 1;
end
%*********************************************************************
%perform the inverse FFT’s
%new notation:
v(x,y), where x is crossrange
%first in the range dimmension
clear v Kr Krr Kxx Ky_even;
%v = fftshift(ifft2(S_st,(size(S_st,1)*1),(size(S_st,2)*1)),1);
v = ifft2(S_st,(size(S_st,1)*4),(size(S_st,2)*4));
xx = sqrt(Kx.^2 + Ky_eeven.^2);
%bw = (3E8/(4*pi))*(max(xx)-min(xx));
bw = 3E8*(kstop-kstart)/(4*pi);
max_range = (3E8*size(S_st,2)/(2*bw) - Rs)/.3048;
figure(figcount);
271
S_image = dbv(v);
imagesc(linspace(-1*(zpad*delta_x/2)/.3048, 1*(zpad*delta_x/2)
/.3048, size(v, 1)), linspace(0,-1*max_range, size(v,2)),
flipud(rot90(S_image)), [max(max(S_image))-15,
max(max(S_image))-0]);
colormap(gray); %MUST DO THIS FOR DISSERTATION FIGS
%colormap(’default’);
title(’Final Image’);
ylabel(’Downrange (ft)’);
xlabel(’Crossrange (ft)’);
cbar = colorbar;
set(get(cbar, ’Title’), ’String’, ’dB’,’fontsize’,13);
print(gcf, ’-djpeg100’, ’final_image.jpg’);
if close_as_you_go == 1,
close(figcount);
end
figcount = figcount + 1;
clear cbar close_as_you_go figcount jj do_all_plots;
v = ifft2(S_st); %creat an un-zero padded version of the image
%clear S_st;
save lastimage v max_range zpad delta_x;
%save the set of un-zero padded image data
272
APPENDIX D
MATLAB CODE FOR CALCULATING PEC CYLINDER ECHO
WIDTH AND RANGE PROFILES
The following MATLAB program was written to calculate and plot the echo width
(σ2D /λ) and the range profile of a 2D PEC cylinder for an T M z plane wave incidence.
%Scattering of a PEC cyldiner from sec 11.5.1 in Balanis
%this program measures echo width and a range profile
clear all;
close all;
set(0,’defaultaxesfontsize’,18);
%constants
mu0 = 4*pi*10^-7; %(H/m) free space permeability
eps0 = 8.85418E-12; %(F/m) free space permittivity
E0 = 1; %(V/m) incident field magnitude
c = 3E8; %(m/sec) speed of light
%frequency chirp
fstart = 2E9; %(Hz) start freq
fstop = 4E9; %(Hz) stop freq
f = linspace(fstart, fstop, 256); %(Hz) frequency chirp
w = 2*pi*f;
k00 = w*sqrt(mu0*eps0); %free space wave number
273
%cylinder distance from radar
r3n = 20*.3048; %(m)
%cylinder parameters
a = 0.6; %(m) radius of the cylinder
N = 200; %must be high enough for convergance on larger cylinders
%***********************************************
%calculate the range profile here
%***********************************************
phi_rp = -pi; %observation angle mono-static
for ii = 1:size(k00,2) %collect the chirp data
k0 = k00(ii);
%calculated the scattered field
sum = 0;
for jj = 1:N
n = jj - 1;
if n == 0
epsn = 1;
else
epsn = 2;
end
sum = sum + (-j)^n*epsn*besselj(n,k0*a)*
besselh(n,2,k0*r3n)/besselh(n,2,k0*a)*cos(n*phi_rp);
end
Es(ii) = E0*-1*sum;
end
274
%accont for the plane wave delay incident on the cylinder
Es = Es.*exp(-j*k00*r3n);
%plot the range profile
figure
bw = fstop-fstart;
max_time = size(Es,2)/(bw);
plot(linspace(0, max_time/1E-9, size(Es,2)),real(ifft(Es)),’k’);
grid on;
title([’radius of cylinder ’,num2str(a),’m and distance ’,
num2str(r3n/.3048),’ft’]);
xlabel(’time (ns)’);
ylabel(’scattered electric field E^s (V/m)’);
print(gcf, ’-djpeg100’, ’cylinder_realrp_pt6m.jpg’);
figure
bw = fstop-fstart;
max_time = size(Es,2)/(bw);
plot(linspace(0, max_time/1E-9, size(Es,2)),dbv(ifft(Es)),’k’);
grid on;
title([’radius of cylinder ’,num2str(a),’m and distance ’,
num2str(r3n/.3048),’ft’]);
xlabel(’time (ns)’);
ylabel(’scattered electric field E^s (dB)’);
print(gcf, ’-djpeg100’, ’cylinder_dbrp_pt6m.jpg’);
%*********************************************
%calculate the echo width here
275
%*********************************************
clear Ei Es w f;
N = 400; %need more N terms to converge the incident field
phii = linspace(0, pi, 100); %(rad) rip through a bunch of obsv. angles
lambda = 3E8/2E9; %(m) wavelength at 2 GHz (low freq of radar)
a = 0.6*lambda; %(m) diameter of cylinder
r3n = 20*.3048; %(m) distance to cylinder
w = 2*pi*c/lambda; %(rad/sec) radial frequency in terms of wavelength
k0 = w*sqrt(mu0*eps0); %free space wave number in terms of wavelength
for ii = 1:size(phii,2)
phi = phii(ii);
%calculate the scattered field
sum = 0;
for jj = 1:N
n = jj - 1;
if n == 0
epsn = 1;
else
epsn = 2;
end
sum = sum + (-j)^n*epsn*besselj(n,k0*a)*
besselh(n,2,k0*r3n)/besselh(n,2,k0*a)*cos(n*phi);
end
Es(ii) = E0*-1*sum;
%calculate the incident field
sum = 0;
for jj = 1:N
276
n = jj - 1;
if n == 0
epsn = 1;
else
epsn = 2;
end
sum = sum + (-j)^n*epsn*besselj(n,k0*r3n)*cos(n*phi);
end
Ei(ii) = E0*sum;
end
sigma = 2*pi*r3n*(abs(Es).^2)./(abs(Ei).^2);
figure
semilogy(phii*180/pi,sigma/lambda,’k’)
axis([0 180 1 25]);
title(’echo width (units in \sigma_{2D}/\lambda) of a
0.6\lambda radius cylinder’);
xlabel(’observation angle (degrees)’);
ylabel(’echo width’);
grid on;
print(gcf, ’-djpeg100’, ’cylinder_echowidth.jpg’);
277
APPENDIX E
MATLAB CODE FOR SIMULATING SAR DATA OF THE 2D PEC
CYLINDER MODEL
The following MATLAB code was written to simulate SAR data for the 2D PEC
cylinder model developed in Chapter 3. The resulting data from this program can be
fed into the data conditioning program in Appendix B and the RMA SAR imaging
algorithm written in Appendix C resulting in a simulated SAR image of a 2D PEC
cylinder.
%theoretical RAIL SAR data for imaging the
%Balanis cylinder model
clear all;
close all;
%constants
mu0 = 4*pi*10^-7; %(H/m) free space permeability
eps0 = 8.85418E-12; %(F/m) free space permittivity
sigma = 0.13; %(S/m) conductivity
E0 = 1; %(V/m) incident field magnitude
%frequency chirp
fstart = 2E9; %(Hz) start of chirp
fstop = 4E9; %(Hz) stop of chirp
f = linspace(fstart, fstop, 256); %(Hz) frequency chirp
w = 2*pi*f;
278
k00 = w*sqrt(mu0*eps0); %free space wave number
%geometry constants
d3 = 20*.3048; %(m) distance from rail to target center
L = 8*.3048; %(m) length of the rail
inc = linspace(-L/2, L/2, 48); %(m) location of radar
%on the rail, accross all rail positions
%first, build the phi_n matrix, the incident angle
phi_n = acos(d3./(sqrt(inc.^2 + d3^2)));
%next, build the r matricies
r3n = d3./(cos(phi_n));
%cylinder parameters
a = (3/12)*.3048; %(m) radius of the cylinder
N = 400; %must be high enough for proper
%convergance on larger cylinders
phi_rp = -pi; %observation angle mono-static
for ii = 1:size(k00,2)
k0 = k00(ii);
%calculated the scattered field of the cylinder
sum = 0;
for jj = 1:N
n = jj - 1;
if n == 0
279
epsn = 1;
else
epsn = 2;
end
sum = sum + (-j)^n*epsn*besselj(n,k0*a)*
besselh(n,2,k0*r3n)/besselh(n,2,k0*a)*cos(n*phi_rp);
end
%Es plus some plane wave range between that and the boundar
s(:,ii) = -1*sum.*exp(-j*k0*r3n);
end
figure
bw = fstop-fstart;
max_time = size(s,2)/(bw);
plot(linspace(0, max_time/1E-9, size(s,2)),dbv(ifft(s(16,:))));
%index of 16 was choosen to make nearly normal incidence
grid on;
title([’radius of cylinder ’,num2str(a),’m and distance ’
,num2str(r3n(16)/.3048),’ft’]);
xlabel(’time (ns)’);
ylabel(’scattered electric field E^s (dB)’);
save thruwall s;
280
APPENDIX F
MATLAB CODE FOR CALCULATED A SIMULATED RANGE
PROFILE OF A LOSSY-DIELECTRIC SLAB
The following MATLAB program was written simulate a range profile of the lossydielectric slab model developed in Chapter 4.
%Wave Matrix method for a lossless dielectric sheet at some
%oblique angle of incidence (from Collin)
clear all;
close all;
set(0,’defaultaxesfontsize’,13);
%*****************************
%constants
%*****************************
mu0 = 4*pi*10^-7; %(H/m) free space permeability
eps0 = 8.85418E-12; %(F/m) free space permittivity
E0 = 1; %(V/m) incident field magnitude
%dielectric properties
%epsr = 1.00005;%relative dielectric constant of layer 1
epsr = 5;
d = 0.1; %(m) thickness of the dielectric region 2
phi_i = 0*pi/180; %(rad) incidence angle
281
%frequency chirp and wave number calculations
fstart = 2E9; %(Hz) start freq
fstop = 4E9; %(Hz) stop freq
f = linspace(fstart, fstop, 256); %(HZ) chirp frequency
w = 2*pi*f; %(rad/m) radial freq
%conductivity approximation (from a model)
%sigma = 0.219 + (f - 1E9)*.036/1E9; %porosity factor of 0.15
sigma = 0.1194 + (f - 1E9)*.0222/1E9; %porosity factor of 0.10
%sigma = 0;
%wave number calculations
k0 = w*sqrt(mu0*eps0); %(rad/m) wave number in free space
k = w.*sqrt(mu0*(epsr*eps0+sigma./(j*w)));
%*****************************
%calculations
%*****************************
%calculate the normalized dielectric impedance wrt incidence angle
%perpendicular polarization (TMz)
Z = cos(phi_i)./sqrt((epsr + sigma./(j*w*eps0)) - (sin(phi_i))^2);
%calculate the relection coefficients wrt incidence angle
R1 = (Z-1)./(Z+1);
R2 = (1-Z)./(1+Z);
282
%calculate the transmission coefficients wrt incidence angle
T1 = 1 + R1;
T2 = 1 + R2;
%calculate the elecrical length of layer 1, phase length of layer 1
theta = d*sqrt(epsr + sigma./(w.*j*eps0) - (sin(phi_i)).^2).*k0;%;
%calcualte c1, the incident fileld at the first boundary face
r = 10*.3048; %(m) distance from interface
c1 = E0*exp(-j*k0*r);
%calculate the radar target function
r_target = 10*.3048; %(m) range to target away from the wall
rcs = 0; %0 for the case of dielectric only imaging and range profiles
rcs = rcs.*exp(-j*2*k0*r_target);
%calculate c3 the tranmitted into region 3 coeficient wrt phase legnth of
%layer 1
c3 = T1.*T2.*c1./(exp(j*theta)+R1.*R2.*exp(-j*theta)+
rcs.*(R2.*exp(j*theta)+R1.*exp(-j*theta)));
%calculate b1 the reflected coefficient, wrt phase length of layer 1
b1 = (c3./(T1.*T2)).*(R1.*exp(j*theta)+R2.*exp(-j*theta)
+rcs.*(R1.*R2.*exp(j*theta)+exp(-j*theta)));
%plane wave takes some distance to get back to receiver,
%thus the received field is
283
Es = b1.*exp(-j*k0*r);
%plot the conductivity
figure
plot(f/1E9, sigma,’k’);
grid on;
title([’conductivity \sigma
over freqeuncy range,
with \epsilon_r = ’,num2str(epsr)]);
xlabel(’frequency (GHz)’);
ylabel(’conductivity’);
print(gcf, ’-djpeg100’, ’dielectric_conductivity.jpg’);
%plot the time domain figure
figure
bw = fstop-fstart;
max_time = size(Es,2)/(bw);
plot(linspace(0, max_time/1E-9, size(Es,2)),dbv(ifft(Es)),’k’);
grid on;
title([’incident angle \phi_{ i} = ’,num2str(phi_i*180/pi),’ (deg)’]);
xlabel(’time (ns)’);
ylabel(’scattered electric field E^s (dB)’);
print(gcf, ’-djpeg100’, ’dielectric_only_deg_incidence.jpg’);
284
APPENDIX G
MATLAB CODE FOR SIMULATING RAIL SAR DATA OF THE
DIELECTRIC SLAB MODEL
The following MATLAB program was written to simulate rail SAR data of the dielectric slab model developed in Chapter 4.
%rail SAR data is theoretically calculated using this program for a
%dielectric wall ONLY
%this is the final simulation program
clear all;
%close all;
set(0,’defaultaxesfontsize’,18);
%*****************************
%constants
%*****************************
mu0 = 4*pi*10^-7; %(H/m) free space permeability
eps0 = 8.85418E-12; %(F/m) free space permittivity
E0 = 1; %(V/m) incident field magnitude
%dielectric properties
epsr = 5;%relative dielectric constant of layer 1
%frequency chirp and wave number calculations
285
fstart = 2E9; %(Hz) start freq
fstop = 4E9; %(Hz) stop freq
f = linspace(fstart, fstop, 256); %(HZ) chirp frequency
w = 2*pi*f; %(rad/m) radial freq
%conductivity approximation (from a model)
%sigma = 0.219 + (f - 1E9)*.036/1E9; %porosity factor of 0.15
sigma = 0.1194 + (f - 1E9)*.0222/1E9; %porosity factor of 0.10
%sigma = 0;
%wave number calculations
k0 = w*sqrt(mu0*eps0); %(rad/m) wave number in free space
k = w.*sqrt(mu0*(epsr*eps0+sigma./(j*w)));
%****************************************
%unique geometry and incidence angles
%****************************************
%geometry constants
d1 = 10*.3048; %(m) distance from rail to wall
d = 4/12*.3048; %(m) thickness of wall
d3 = 20*.3048; %(m) distance from rail to target center
L = 8*.3048; %(m) length of the rail
inc = linspace(-L/2, L/2, 48); %(m) location of radar on the rail, accross
%all rail positions
%inc = 4*.3048
%first, build the phi_n matrix, the incident angle
286
phi_n = acos(d3./(sqrt(inc.^2 + d3^2)));
%next, build the r matricies
r1n = d1./(cos(phi_n));
r3n = (d3-d1-d)./(cos(phi_n));
%cylinder parameters
a = (3/12)*.3048; %(m) radius of the cylinder
N = 200; %must be high enough for proper convergance on larger cylinders
%************************************************
%calculatons
%************************************************
for n = 1:size(inc,2)
phi_i = phi_n(n);
n
%calculate the normalized dielectric impedance wrt incidence angle
%perpendicular polarization (TMz)
Z = cos(phi_i)./sqrt((epsr + sigma./(j*w*eps0)) - (sin(phi_i))^2);
%calculate the relection coefficients wrt incidence angle
R1 = (Z-1)./(Z+1);
R2 = (1-Z)./(1+Z);
287
%calculate the transmission coefficients wrt incidence angle
T1 = 1 + R1;
T2 = 1 + R2;
%calculate the elecrical length of layer 1, phase length of layer 1
theta = d*sqrt(epsr + sigma./(w.*j*eps0) - (sin(phi_i)).^2).*k0;%;
%calcualte c1, the incident fileld at the first boundary face
r = r1n(n); %(m) distance to wall
c1 = E0*exp(-j*k0*r);
rcs = 0; %0 for the case of dielectric only imaging and range profiles
%rcs = rcs.*exp(-j*2*k0*r3n(n));
%calculate c3 the tranmitted into region 3 coeficient wrt
% phase legnth of layer 1
c3 = T1.*T2.*c1./(exp(j*theta)+R1.*R2.*exp(-j*theta)+rcs.*
(R2.*exp(j*theta)+R1.*exp(-j*theta)));
%calculate b1 the reflected coefficient, wrt phase length of layer 1
b1 = (c3./(T1.*T2)).*(R1.*exp(j*theta)+R2.*exp(-j*theta)+
rcs.*(R1.*R2.*exp(j*theta)+exp(-j*theta)));
%plane wave takes some distance to get back to receiver,
%thus the received field is
Es(n,:) = b1.*exp(-j*k0*r);
288
end
s = Es;
save thruwall s;
SBAND_RMA_opendata
289
APPENDIX H
MATLAB CODE FOR ACQUIRING A SIMULATED RANGE
PROFILE OF A PEC CYLINDER ON THE OPPOSITE SIDE OF A
DIELECTRIC SLAB
The following MATLAB program was written to simulate a range profile of a PEC
cylinder on the opposite side of a dielectric slab. This code was based on the model
developed in Chapter 5.
%Range profile data is theoretically calculated using this program for a
%dielectric wall and cylinder
%this is the final simulation program
clear all;
%close all;
set(0,’defaultaxesfontsize’,18);
%*****************************
%constants
%*****************************
mu0 = 4*pi*10^-7; %(H/m) free space permeability
eps0 = 8.85418E-12; %(F/m) free space permittivity
E0 = 1; %(V/m) incident field magnitude
%dielectric properties
epsr = 5;%relative dielectric constant of layer 1
290
%frequency chirp and wave number calculations
fstart = 2E9; %(Hz) start freq
fstop = 4E9; %(Hz) stop freq
f = linspace(fstart, fstop, 256); %(HZ) chirp frequency
w = 2*pi*f; %(rad/m) radial freq
%conductivity approximation (from a model)
%sigma = 0.219 + (f - 1E9)*.036/1E9; %porosity factor of 0.15
sigma = 0.1194 + (f - 1E9)*.0222/1E9; %porosity factor of 0.10
%sigma = 0;
%wave number calculations
k0 = w*sqrt(mu0*eps0); %(rad/m) wave number in free space
k = w.*sqrt(mu0*(epsr*eps0+sigma./(j*w)));
%****************************************
%unique geometry and incidence angles
%****************************************
%geometry constants
d1 = 20*.3048; %(m) distance from rail to wall
d = 4/12*.3048; %(m) thickness of wall
d3 = 30*.3048; %(m) distance from rail to target center
L = 8*.3048; %(m) length of the rail
%inc = linspace(-L/2, L/2, 48); %(m) location of radar on the rail, accross
%all rail positions
%inc = 4*.3048
291
inc = [0 -L/2] %take range profiles at only these locations
%first, build the phi_n matrix, the incident angle
phi_n = acos(d3./(sqrt(inc.^2 + d3^2)));
%next, build the r matricies
r1n = d1./(cos(phi_n));
r3n = (d3-d1-d)./(cos(phi_n));
%cylinder parameters
a = (3/12)*.3048; %(m) radius of the cylinder
N = 400; %must be high enough for proper convergance on larger cylinders
phi_rp = -pi; %IMPORTANT: this is the monostastic observation angle
%************************************************
%calculatons
%************************************************
for n = 1:size(inc,2)
phi_i = phi_n(n);
n
%calculate the normalized dielectric impedance wrt incidence angle
%perpendicular polarization (TMz)
Z = cos(phi_i)./sqrt((epsr + sigma./(j*w*eps0)) - (sin(phi_i))^2);
%calculate the relection coefficients wrt incidence angle
292
R1 = (Z-1)./(Z+1);
R2 = (1-Z)./(1+Z);
%calculate the transmission coefficients wrt incidence angle
T1 = 1 + R1;
T2 = 1 + R2;
%calculate the elecrical length of layer 1, phase length of layer 1
theta = d*sqrt(epsr + sigma./(w.*j*eps0) - (sin(phi_i)).^2).*k0;%;
%calcualte c1, the incident fileld at the first boundary face
r = r1n(n); %(m) distance to wall
c1 = E0*exp(-j*k0*r);
%calculate the radar target function
%calculated the scattered field of the cylinder
for ii = 1:size(k0,2)
sum = 0;
for jj = 1:N
nn = jj - 1;
if nn == 0
epsn = 1;
else
epsn = 2;
end
sum = sum + (-j)^nn*epsn*besselj(nn,k0(ii)*a).*besselh(nn,2,k0(ii)*
r3n(n))./besselh(nn,2,k0(ii)*a).*cos(nn*phi_rp);
293
end
rcs(ii) = sum;
end
rcs = -1*exp(-j*k0*r3n(n)).*rcs;
%calculate c3 the tranmitted into region 3 coeficient wrt
%phase legnth of layer 1
c3 = T1.*T2.*c1./(exp(j*theta)+R1.*R2.*exp(-j*theta)
+rcs.*(R2.*exp(j*theta)+R1.*exp(-j*theta)));
%calculate b1 the reflected coefficient, wrt phase length of layer 1
b1 = (c3./(T1.*T2)).*(R1.*exp(j*theta)+R2.*exp(-j*theta)
+rcs.*(R1.*R2.*exp(j*theta)+exp(-j*theta)));
%plane wave takes some distance to get back to receiver,
%thus the received field is
Es(n,:) = b1.*exp(-j*k0*r);
end
phi_i = phi_n(1);
%plot the time domain figure for normal incidence
figure
bw = fstop-fstart;
max_time = size(Es,2)/(bw);
plot(linspace(0, max_time/1E-9, size(Es,2)),dbv(ifft(Es(n,:))),’k’);
grid on;
294
title([’incident angle \phi_{ i} = ’,num2str(phi_i*180/pi),’ (deg)’]);
xlabel(’time (ns)’);
ylabel(’scattered electric field E^s (dB)’);
print(gcf, ’-djpeg100’, ’rp_cylinder_wall_normal_inc.jpg’);
phi_i = phi_n(2);
%plot the time domain figure for oblique incidence
figure
bw = fstop-fstart;
max_time = size(Es,2)/(bw);
plot(linspace(0, max_time/1E-9, size(Es,2)),dbv(ifft(Es(n,:))),’k’);
grid on;
title([’incident angle \phi_{ i} = ’,num2str(phi_i*180/pi),’ (deg)’]);
xlabel(’time (ns)’);
ylabel(’scattered electric field E^s (dB)’);
print(gcf, ’-djpeg100’, ’rp_cylinder_wall_oblique_inc.jpg’);
295
APPENDIX I
MATLAB CODE FOR SIMULATING SAR DATA OF A PEC
CYLINDER ON THE OPPOSITE SIDE OF A DIELECTRIC SLAB
The following MATLAB program was written to simulate rail SAR data of a PEC
cylinder on the opposite side of a dielectric slab. This code was based on the model
developed in Chapter 5.
%rail SAR data is theoretically calculated using this program for a
%dielectric wall and cylinder
%this is the final simulation program
clear all;
%close all;
set(0,’defaultaxesfontsize’,18);
%*****************************
%constants
%*****************************
mu0 = 4*pi*10^-7; %(H/m) free space permeability
eps0 = 8.85418E-12; %(F/m) free space permittivity
E0 = 1; %(V/m) incident field magnitude
%dielectric properties
epsr = 5;%relative dielectric constant of layer 1
296
%frequency chirp and wave number calculations
fstart = 2E9; %(Hz) start freq
fstop = 4E9; %(Hz) stop freq
f = linspace(fstart, fstop, 256); %(HZ) chirp frequency
w = 2*pi*f; %(rad/m) radial freq
%conductivity approximation (from a model)
%sigma = 0.219 + (f - 1E9)*.036/1E9; %porosity factor of 0.15
sigma = 0.1194 + (f - 1E9)*.0222/1E9; %porosity factor of 0.10
%sigma = 0;
%wave number calculations
k0 = w*sqrt(mu0*eps0); %(rad/m) wave number in free space
k = w.*sqrt(mu0*(epsr*eps0+sigma./(j*w)));
%****************************************
%unique geometry and incidence angles
%****************************************
%geometry constants
d1 = 20*.3048; %(m) distance from rail to wall
d = 4/12*.3048; %(m) thickness of wall
d3 = 30*.3048; %(m) distance from rail to target center
L = 8*.3048; %(m) length of the rail
inc = linspace(-L/2, L/2, 48); %(m) location of radar on the rail, accross
%all rail positions
%inc = 4*.3048
297
%first, build the phi_n matrix, the incident angle
phi_n = acos(d3./(sqrt(inc.^2 + d3^2)));
%next, build the r matricies
r1n = d1./(cos(phi_n));
r3n = (d3-d1-d)./(cos(phi_n));
%cylinder parameters
a = (3/12)*.3048; %(m) radius of the cylinder
N = 400; %must be high enough for proper convergance on larger cylinders
phi_rp = -pi; %IMPORTANT: this is the monostastic observation angle
%************************************************
%calculatons
%************************************************
for n = 1:size(inc,2)
phi_i = phi_n(n);
n
%calculate the normalized dielectric impedance wrt incidence angle
%perpendicular polarization (TMz)
Z = cos(phi_i)./sqrt((epsr + sigma./(j*w*eps0)) - (sin(phi_i))^2);
%calculate the relection coefficients wrt incidence angle
R1 = (Z-1)./(Z+1);
R2 = (1-Z)./(1+Z);
298
%calculate the transmission coefficients wrt incidence angle
T1 = 1 + R1;
T2 = 1 + R2;
%calculate the elecrical length of layer 1, phase length of layer 1
theta = d*sqrt(epsr + sigma./(w.*j*eps0) - (sin(phi_i)).^2).*k0;%;
%calcualte c1, the incident fileld at the first boundary face
r = r1n(n); %(m) distance to wall
c1 = E0*exp(-j*k0*r);
%calculate the radar target function
%calculated the scattered field of the cylinder
for ii = 1:size(k0,2)
sum = 0;
for jj = 1:N
nn = jj - 1;
if nn == 0
epsn = 1;
else
epsn = 2;
end
sum = sum + (-j)^nn*epsn*besselj(nn,k0(ii)*a).
*besselh(nn,2,k0(ii)*r3n(n))
./besselh(nn,2,k0(ii)*a).*cos(nn*phi_rp);
end
299
rcs(ii) = sum;
end
rcs = -1*exp(-j*k0*r3n(n)).*rcs;
%rcs = 0; %0 for the case of dielectric only imaging
%and range profiles
%rcs = rcs.*exp(-j*2*k0*r3n(n));
%calculate c3 the tranmitted into region 3 coeficient wrt phase
%legnth of layer 1
c3 = T1.*T2.*c1./(exp(j*theta)+R1.*R2.*exp(-j*theta)
+rcs.*(R2.*exp(j*theta)+R1.*exp(-j*theta)));
%calculate b1 the reflected coefficient, wrt phase length of layer 1
b1 = (c3./(T1.*T2)).*(R1.*exp(j*theta)+R2.*exp(-j*theta)
+rcs.*(R1.*R2.*exp(j*theta)+exp(-j*theta)));
%plane wave takes some distance to get back to receiver,
%thus the received field is
Es(n,:) = b1.*exp(-j*k0*r);
end
s = Es;
save thruwall s;
SBAND_RMA_opendata
300
APPENDIX J
MATLAB CODE FOR OPENING MEASURED CALIBRATION AND
SAR DATA
The following MATLAB program was written to open measured SAR data and calibration data. This program calibrates the SAR data and conditions the data to be
fed into the MATLAB RMA program in appendix C.
%Range Migration Algorithm from ch 10 of Spotlight Synthetic Aperture Radar
%Signal Processing Algorithms, Carrara, Goodman, and Majewski
clear all;
c = 3E8; %(m/s) speed of light
%*********************************************************************
%load IQ converted data here
load rback2 s; %load variable sif %for background subtraction cal data
%*********************************************************************
%perform background subtraction
sif_sub = s;
load rsphere s; %load variable sif %for image data
sif = s-sif_sub; %perform coherent background subtraction
%sif = sif_sub; %image just the background
%sif = s; %image without background subtraction
clear s;
clear sif_sub;
301
%***********************************************************************
%radar parameters
fc = (4.069E9 - 1.926E9)/2 + 1.926E9; %(Hz) center radar frequency
B = (4.069E9 - 1.926E9); %(hz) bandwidth
cr = 2E9/10E-3; %(Hz/sec) chirp rate
Tp = 10E-3; %(sec) pulse width
%VERY IMPORTANT, change Rs to distance to cal target
Rs = (37.1)*.3048; %(m) y coordinate to scene center (down range),
%make this value equal to distance to cal target
Xa = 0; %(m) beginning of new aperture length
delta_x = 2*(1/12)*0.3048; %(m) 2 inch antenna spacing
L = delta_x*(size(sif,1)); %(m) aperture length
Xa = linspace(-L/2, L/2, (L/delta_x)); %(m) cross range
%position of radar on aperture L
Za = 0;
Ya = Rs; %THIS IS VERY IMPORTANT, SEE GEOMETRY FIGURE 10.6
t = linspace(0, Tp, size(sif,2)); %(s) fast time, CHECK SAMPLE RATE
Kr = linspace(((4*pi/c)*(fc - B/2)), ((4*pi/c)*(fc + B/2)), (size(t,2)));
%************************************************************************
%callibration
load rcal37pt1_2 s; %load callibration file to standard target
s_cal = s;
load rcalback_2 s; %load background data for cal to standard target
s_cal = s_cal - s; %perform background subtraction
cal = s_cal;
302
%calculate ideal cal target parameters
%target parameters, 3 targets
at1 = 1; %amplitude of cal target
xt1 = 0;
yt1 = (37.1)*.3048; %(m) distance to cal target
zt1 = 0;
%Rt and Rb for 1 cal target according to equation 10.26
Rb1 = sqrt((Ya - yt1)^2 + (Za - zt1)^2);
xa = 0;
Rt1 = sqrt((xa - xt1).^2 + Rb1^2);
Kr = linspace(((4*pi/c)*(fc - B/2)), ((4*pi/c)*(fc + B/2)), (size(t,2)));
%according to range defined on bottom of page 410
for ii = 1:size(t,2) %step thru each time step to find phi_if
phi_if1(ii) = Kr(ii)*(Rt1 - Rs);
end
cal_theory = at1*exp(-j*phi_if1);
clear phi_if1;
%calculate the calibration factor
cf = cal_theory./(cal);
%apply the cal data
for ii = 1:size(sif,1)
sif(ii,:) = sif(ii,:).*cf; %turn off cal
end
303
%Save background subtracted and callibrated data
save sif sif delta_x Rs Kr Xa;
%clear all;
%run IFP
SBAND_RMA_IFP;
304
BIBLIOGRAPHY
305
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