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The generalized spatial correlation algorithm for self-calibration of microwave antenna arrays

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The generalized spatial correlation algorithm for self-calibration
o f microwave antenna arrays
Borsari, Geordi K enneth, Ph.D.
University of Pennsylvania, 1993
Copyright ©1993 by Borsari, Geordi Kenneth. All rights reserved.
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
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THE GENERALIZED SPATIAL CORRELATION ALGORITHM FOR
SELF-CALIBRATION OF MICROWAVE ANTENNA ARRAYS
Geordi Kenneth Borsari
A DISSERTATION
IN
ELECTRICAL ENGINEERING
Presented to the Faculties of the University of Pennsylvania in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy
1993
r
y
' Um
/ \
Bq^nardD. Steinberg.
Supervisor of Dissertation
Sohrab Rabii
Graduate Group Chairperson
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COPYRIGHT
© Geordi Kenneth Borsari
1993
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TO MICHELE, MADELINE, AND MY PARENTS, KEN AND DIANA
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A ck n o w le d g m e n ts
My utmost thanks, appreciation, and regards go to my advisor Dr. Bernard D.
Steinberg. His confidence and belief in me over the past four years has truly helped to
make it possible to see this work to completion.
I also wish to thank my fellow members of the Valley Forge Research Center,
namely Dr. Rich Pauls, Randy Perlow, and Dr. Qing Zhou, all of whom freely gave their
time to discuss various aspects of this research. Special thanks go to Randy Perlow for the
multitude of discussions we had regarding all aspects of this work. I also want to thank
Shirley Levy for all of her administrative help and Donald Carlson and Walter Borders for
all of their help with the experimental data sets.
More thanks than I could ever say go to my wonderful wife, Michele, who lived
through this with me and always stood by and supported me through both good and bad
times. I know it's been a long road.
Lastly, I want to thank my parents, Ken and Diana, who always supported me
through all of my studies. They sacrificed a lot to allow me to attain this goal.
To all of you, Thank You.
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ABSTRACT
THE SPATIAL CORRELATION CLASS OF SELF-CALIBRATION,
PHASE CORRECTING ALGORITHMS
Geordi K. Borsari
Bernard D. Steinberg
High resolution microwave imaging systems generally require the use of large
aperture array antennas. Such large aperture systems inevitably experience phase errors in
the measured data. The Spatial Correlation class of algorithms is one class of algorithms
that attempts to remove these phase errors from the recorded data set. This work develops
a general algorithm (GSCA) that characterizes this class and allows its characteristics and
properties to be studied. The generalized theory is applied to successfully self-calibrate
experimental data sets that could not be calibrated successfully with existing theory. The
GSCA reveals that the Spatial Correlation class is divided into two sub-classes of
algorithms. Extensive simulations are used to compare performances of algorithms from
each sub-class in the presence of receiver noise and element position errors.
The
performance curves presented as a result of the performance study represent the first such
set of performance curves available and can be used as design curves for system designers.
In the last chapter the GSCA is extended for self-calibration with near-field data.
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Table of Contents
C hapter 1:
Introduction — 1
1.0
Objective...................................................................................................1
1.1
Contributions............................................................................................2
1.2
History of the Spatial Correlation Class of Algorithms....................... 5
1.3
The M icrowave Imaging System................................................... 8
1.4
The Experimental D ata............................................................................ 12
1.5
Dissertation Outline..................................................................................12
1.6
Dissertation Summary............................................................................. 14
C hapter 2:
Existing Spatial Correlation-Based Algorithm s
— 15
2.0
In tro d u c tio n ...........................................................................................15
2.1
The Spatial Correlation Class Defined.......................................... 17
2.2
The Spatial (Auto)Correlation Function of an Incoherent Source
Distribution............................................................................................. 17
2.3
The Muller-Buffington Algorithm..........................................................19
2.4
2.5
2.6
2.3.1
The Image Sharpness Function..............................................19
2.3.2
Maximizing the Image Sharpness Function in the
A perture D om ain.............................................................. 22
The Unit-Lag Spatial Correlation Algorithm..................................24
2.4.1
Practical Implementation of the Unit-Lag S C A ....................26
2.4.2
An Iterative Unit-Lag SCA..................................................... 27
The Multiple-Lag Spatial Correlation Algorithms................................. 28
2.5.1
The Least-Squares Multiple-Lag S C A .................................. 29
2.5.2
The Iterative Multiple-Lag SCA..................................... 33
2.5.3
The Full and Partial Multiple-Lag SCA's..............................35
Other Spatial Correlation-Based Algorithms..................................36
2.6.1
The Shear Averaging Algorithm............................................ 36
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2.7
Chapter 3:
2.6.2
The Phase Gradient Algorithm................................................37
2.6.3
The Flax and O'Donnell Algorithm................................. 38
2.6.4
Phase Closure From Radio Astronomy..................................39
S u m m ary .................................................................................................40
The G eneralized Spatial Correlation A lgorithm
— 42
3.0
In tro d u c tio n ........................................................................................... 42
3.1
The Energy Conservation Algorithm...................................................... 43
3.2
3.3
3.1.1
Development of the E C A ......................................................... 43
3.1.2
Performance of the ECA with Simulated D ata.......................46
3.1.3
Performance of the ECA with Experimental Data.............. 47
The Generalized Spatial Correlation Algorithm..................................... 57
3.2.1
Re-Interpreting the ECA Objective Function.......................... 57
3.2.2
Concepts of the Generalized Spatial Correlation
Algorithm..................................................................................58
3.2.3
Development of the Generalized Spatial Correlation
Algorithm................................................................................. 61
3.2.4
The Sub-Classes of the Spatial Correlation C lass................. 65
The Relationship of the GSCA to Existing Spatial Correlation
Algorithms................................................................................................68
3.3.1
Revisiting Tsao's Energy Conservation Algorithm.............68
3.3.2
Use of A Narrow Interval of Integration
with Experimental D ata........................................................... 71
3.3.3
Deriving the Modified Muller-Buffington
Algorithm From The GSCA ............................................85
3.3.4
The Unit-Lag SCA and the G SC A .........................................86
3.4
Commonality of the Maximization Procedures........................................87
3.5
S u m m ary ...................................................................................................88
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C h ap ter 4:
Noise and Element Position Error Performance
C om parision — 90
4.0
In tro d u c tio n ..........................................................................................90
4.1
Element Position Errors..........................................................................91
4.1.1
Introduction.............................................................................. 91
4.2.1
The Element Postion Error Simulation................................... 93
4.1.3
Main Beam Loss Due to Element Position Errors................. 98
4.1.4
An Approximation to the SCA Performance Curves............. 107
4.2
Receiver Noise Analysis........................................................................110
4.3
Analysis of the Solutionsof the Parametric Sub-Class and
theM M B.................................................................................................122
4.4
C h ap ter 5:
4.3.1
Linearization of the MMB and the
Parametric Sub-Class..............................................................122
4.3.2
The GSCA and the MLSCA.............................................131
S u m m ary ................................................................................................. 133
Self-C alibrating with the GSCA Using
N ear-F ield Data — 135
5.0
In tro d u c tio n ........................................................................................... 135
5.1
The Near Field of a Linear A rray............................................................136
5.2
Elimination of the Effects of Near-Field Phase Curvature
on the GSCA............................................................................................138
5.2.1
Mathematical Characterization of the Near-Field Effects
on the GSCA Objective Function........................................... 138
5.2.2
Robustness of the Unit-Lag Spatial Correlation
Algorithm................................................................................. 143
5.2.3
The Near-Field Weighting Function.................................. 144
5.2.4
The Effect of a Dominant Scatterer on the
Near-Field Weighting Function.............................................. 146
5.2.5
Use of the Near-Field Weighting Function with
Practical High-Resolution Imaging Systems.........................153
5.3
Performance of the GSCA with Near-Field D ata...................................157
5.4
S u m m ary ..................................................................................................163
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C h ap ter 6:
C o n clu sio n s — 165
6.0
S u m m ary ................................................................................................. 165
6.1
Conclusions.............................................................................................. 165
6.2
Suggestions for Future W o rk .................................................................. 170
A p p en d ices —
171
A:
Derivation of the Variance of the Correlation Phase Noise
Resulting From Additive Receiver Noise.............................................. 173
B:
Main Beam Relative Gain Curves For Array Element Position
E rro rs .......................................................................................................180
R e fe re n c e s
.................................................................................................................. 202
In d e x
206
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LIST OF TABLES
Table 3.1
Summary of Results with Simulated D ata............................. 76
Table 4.1
Parameters of Main Beam Loss and the Values
Used in the Simulations.......................................................... 96
Table 4.2
Largest Correlation Lag Necessary to Achieve Peak
Performance.............................................................................97
Table 4.3
Summary of the Utility of the Spatial Correlation
Algorithms.............................................................................104
Table 4.4
Performance of the ECA and M M B .....................................112
Table 4.5
Algorithm Preferences for Various Situations..................... 113
Table 5.1
Order of Magnitude Approximation of System
Parameters.............................................................................154
Table 5.2
System Parameters for VFRC ISAR Imaging System.... 154
Table 5.3
System Parameters for the VFRC 83-Meter
Imaging System.................................................................... 154
Table 5.4
Results of Calibration with the GSCA and Unit-Lag
SCA with no Dominant Scatterer and a Data Set
Originating in the Near F ield.............................................. 157
Table 5.5
Results of Calibration with the GSCA and Unit-Lag
SCA with a Small Dominant Scatterer and a Data Set
Originating in the Near F ield...............................................160
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LIST OF ILLUSTRATIONS
Figure 1.1
Diagram of Imaging System ......................................................................... 9
Figure 1.2
Diagram of a typical experimental data set in matrix form. The
vertical axis represents elements in the phased array. The horizontal
axis represents range to the target.
Consequently, any row
represents a sampled range profile received at a particular element
and any column represents the data received across the array from a
particular range.............................................................................................. 10
Figure 1.3
The self-calibration scheme: p's represent channel phase errors, jV s
represent estimates of the phase errors, w's represent standard
beamsteering weights....................................................................................11
Figure 2.1
Illustration of the geometry used in deriving
equation (2.3).
(Adapted from Reference 2 )............................................................... 18
Figure 2.2
Illustration of the Muller-Buffington optical phase correcting
scheme. (Reprinted from Reference 24).................................................... 20
Figure 2.3
Illustration of the performance of the Muller-Buffington phase
correcting scheme using simulated data.
(Reprinted from Reference 2 4 )....................................................................21
Figure 2.4
Illustration of the performance of the Muller-Buffington phase
correcting scheme using experimental data.
(Reprinted from
Reference 24)................................................................................................ 21
Figure 2.5
Illustration of the correlation measurement between elements n and
(n+1).............................................................................................................. 27
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Figure 2.6
Illustration of correlation measurements of lags Z j, Z2>^3
Figure 2.7
The form of the matrix A and vector b from equation
Figure 3.1
Illustration of image formation with no phase errors present in the
29
(2.9).................33
imaging system...............................................................................................45
Figure 3.2
Illustration o f image formation with phase errors
present in the
imaging system...............................................................................................45
Figure 3.3
Images of rangebin #1 of simulated far-field data set. (a) Image
using undistorted data, (b) Image using distorted data, (c) Image
using
Figure 3.4
calibrated d ata.............................................................................48
Images of rangebin #4 of simulated far-field data set. (a) Image
using undistorted data, (b) Image using distorted data, (c) Image
using
Figure 3.5
calibrated data............................................................................. 49
Images of rangebin #6 of simulated far-field data set. (a) Image
using undistorted data, (b) Image using distorted data, (c) Image
using
Figure 3.6
calibrated...d ata............................................................................. 50
2-D images from an experimental data set. (a) Image obtained using
the
Unit-Lag SCA.
Au = 30.5 mrads.
(b) Image obtained using the ECA with
(c)
Image obtained using the ECA with
Au = 0.66 m rads..........................................................................................52
Figure 3.7
Cross-range images of the dominantscatterer rangebin of the ISAR
data set using the (a) Unit-Lag SCA. (b) ECA with Au = 30.5
mrads. (c) ECA with Au = 0.66 mrads.......................................................53
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Figure 3.8
Cross-range image of the dominant scatterer rangebin of a SAR data
set using the (a) Unit-Lag SCA. (b) ECA with Au = 30.5 mrads.
(c) ECA with Au = 0.66 mrads.................................................................... 55
Figure 3.9
Average image intensity of the undistorted simulated data......................... 58
Figure 3.10
Illustration of the result of the convolution of a rectangle function
with a sinc-squared function.........................................................................67
Figure 3.11
Image of rangebin #1 of the simulated data after calibration with
Au = 17.24 mrads.........................................................................................70
Figure 3.12
Average image intensity of the undistorted simulated data when a
dominant scatterer exists in the data set....................................................... 73
Figure 3.13
Image of rangebin #1 with a dominant scatterer included in the
averaging process.......................................................................................... 75
Figure 3.14
Image of rangebin #1 of the simulated data set with Au=16.7 mrads
and a dominant scatterer present...................................................................75
Figure 3.15
Average image intensity after calibration.......................................................76
Figure 3.16
Images of three rangebins of simulated data with integration interval
equal to [13,180] mrads................................................................................ 77
Figure 3.17 Images of three rangebins of simulated data with integration interval
equal to [-167,0] m rads....................................................................... 78
Figure 3.18 Average image intensity of the ISAR data when the dominant
scatterer rangebin is included in the averaging process.............................. 80
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Figure 3.19
Average image intensity of the ISAR data when the dominant
scatterer rangebin is not included inthe averaging process.................... 80
Figure 3.20
Cross-range image of the dominant scatterer rangebin of ISAR data
with Au=0.66 mrads and the dominant scatterer rangebin absent
from the averaging process................................................................. 82
Figure 4.1
Theoretical relative gain curve for the DSA with the results o f four
experiments showing agreement with the theory. (Reprinted from
Reference [45]).............................................................................................. 94
Figure 4.2
Simulated relative gain curve produced by the simulation of Section
4.1.2, showing very good agreement with the theoretical DSA curve
and the experimental results of Figure 4.1..................................................94
Figure 4.3
Relative gain curves for various numbers of correlation lags and a
dominant scatterer present in the data set. (a) Relative gain curves
for non-parametric sub-class,
(b) Same as (a) with L=1 curve
removed, (c) Relative gain curves for the parametric sub-class for
the same case as (a).......................................................................................97
Figure 4.4
Relative gain curves for various numbers of correlation lags and no
dominant scatterer present in the data set. (a) Relative gain curves
for non-parametric sub-class,
(b) Relative gain curves for the
parametric sub-class for the same case as (a)............................................98
Figure 4.5
Main beam gain curves for ufov = 0.017 rads., o Ax = X. (a )L = l
and (b) L=29................................................................................................. 100
Figure 4.6
Main beam gain curves for ufov = 0.167 rads., a Ax = X. (a) L=1
and (b) L=29................................................................................................. 104
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Figure 4.7
Difference between the DSA and SCA relative gain curves for fieldsX
X
of-view simulated and for (a) g rms position error and (b) - rms
X
position error and for (c) ^ nns position error and (d)
X rms
p o sition e rro r............................................................................................... 106
Figure 4.8
The phase error profile across the array used in the noise analysis
simulations...................................................................................................... 108
Figure 4.9
RMS residual phase error after calibration for (a) 23 dB SNR (b) 13
dB SNR (c) 3 dB SNR with no dominant scatterer present in the
data set.............................................................................................................113
Figure 4.10
Main beam relative gain after calibration for (a) 23 dB SNR (b) 13
dB SNR (c) 3 dB SNR with no dominant scatterer present in the
data set.............................................................................................................114
Figure 4.11
Average sidelobe level after calibration for (a) 23 dB SNR (b) 13 dB
SNR (c) 3 dB SNR with no dominant scatterer present in the data
set.....................................................................................................................115
Figure 4.12
Peak sidelobe level after calibration for (a) 23 dB SNR (b) 13 dB
SNR (c) 3 dB SNR with no dominant scatterer present in the data
set.....................................................................................................................116
Figure 4.13
RMS residual phase error after calibration for (a) 23 dB SNR (b) 13
dB SNR with a dominant scatterer present in the data set...........................117
Figure 4.14
Main beam relative gain after calibration for (a) 23 dB SNR (b) 13
dB SNR with a dominant scatterer present in the data set...........................118
Figure 4.15
Average sidelobe level after calibration for (a) 23 dB SNR (b) 13 dB
SNR with a dominant scatterer present in the data set.................................119
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Figure 4.16
Relative difference between the GSCA and the MLSCA solutions
131
Figure 5.1
Illustration of the effect of sever near-field phase curvature...................140
Figure 5.2
Error-corrected phases of the correlation lags 1 through 40. These
lags were computed from the dominant scatterer rangebin of the
experimental data after calibration had been applied........................... 147
Figure 5.3
Error-corrected phases of the correlation lags 41 through 70 and 100
through 110.
These lags were computed from the dominant
scatterer rangebin of the experimental data after calibration had been
applied..............................................................................................................148
Figure 5.4
Phase of averaged correlation coefficient of lag 1. Averaged over
rangebins 30-60. Indeed, dominant scatterer bin dominates the
averaging process........................................................................................... 149
Figure 6.5
Unwrapped undistorted phases of correlation lags 1 - 10 of the
simulated data with a dominant scatterer present in the data set............. 150
Figure 6.6
Unwrapped error-corrected phases of the correlation lags 1 - 10 of
the simulated data with a dominant scatterer present in the data set........... 150
Figure 6.7
Unwrapped error-corrected phases of lags 1 - 10 of the simulated
data without a dominant scatterer present in the data set.............................151
Figure 5.8
Correlation function and weighting function taken from experimental
data with a dominant scatterer and FOV=0.66 mrads. Solid line
represents correlation function from experimental data set. Dashed
line represents weighting function with FOV=0.66 mrads. Dasheddotted line represents weighting function with full FOV=30.5
mrads............................................................................................................... 154
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Figure 5.9
GSCA image with rectangular weighting function o f extent
Au=0.167. There is no dominant scatterer present and the data set
originates in the near field..............................................................................157
Figure 5.10
Unit-Lag SCA image with no dominant scatterer and a near-field
data set............................................................................................................. 157
Figure 5.11
Estimated correlation function and the sine weighting function when
no dominant scatterer is present in the data set................................... 158
Figure 5.12
(a) GSCA image with ECA approximation. Dominant Scatterer
am p(l)=1.9 Field-of-view = 0.01724 (b) Unit-Lag SCA Image
with Dominant Scatterer amp(l)=1.9 FOV = 0.01724......................... 160
Figure 5.13
Estimated correlation function and the sine weighting function when
a dominant scatterer is present in the data set...............................................160
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Preface
A Brief Description of the Chapter Contents.
The first section of Chapter 1 contains an outline of the significant contributions of
this work. The remainder of the chapter provides a background of the development of the
Spatial Correlation Class along with details concerning the experimental data sets used in
this research.
Chapter 2 describes the algorithms known to belong or be related to the Spatial
Correlation class prior to this research. This chapter explains each algorithm for reference
in future chapters and establishes notation to be used throughout the remaining chapters.
Chapter 3 is the main body of this work. In this chapter the Generalized Spatial
Correlation Algorithm (GSCA) is introduced, derived, and shown to characterize the entire
Spatial Correlation class. The GSCA is used to show that Tsao's Energy Conservation
Algorithm (ECA) is a member of the Spatial Correlation class. The GSCA is shown to
successfully self-calibrate experimental data when the ECA failed. The reasons for the
ECA failure is found in the GSCA theory.
Chapter 4 compares the performances of algorithms from each sub-class in the
presence of receiver noise and random element position errors through extensive
simulations. The plots of the expected relative gain curves in Appendix B represent the
first compilation of such curves that can be used for system design purposes.
Chapter 5 addresses the problem of performing self-calibration using near-field data
sets with multiple-lag correlation algorithms. It is shown in this chapter that near-field data
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sets are not as restrictive as originally thought and in most practical systems, modification
of the algorithms for near-field data sets is not necessary. It is shown that the presence of a
dominant scatterer is sufficient to eliminate any adverse effects on the calibration process
caused by near-field data.
Chapter 7 is a summary of the conclusion drawn from this work along with
suggestions for future work.
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Chapter 1
Introduction
1.0 Objective
The first working procedure for se lf calibration of a distorted radar phased antenna
array was the Dominant Scatterer Algorithm (DSA), which required the existence of a
scatterer with small physical size and large radar cross section (RCS) somewhere in the
field-of-view [45]. This limitation is eliminated in algorithms that are based on the
autocorrelation function of the measured wavefront of the backscatterered radiation field.
This work develops a new self-calibration algorithm for a distorted radar phased antenna
array that:
1) subsumes all other published correlation-based self-calibration
procedures;
2) includes an algorithm, based upon Parseval's Theorem, hitherto
believed to be the basis of another type of self-calibration procedure;
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3) shows that the performance of these algorithms parallels that of the DSA
and is as good as the DSA when a dominant scatterer is present in the
data set.
1.1 Contributions
The primary contributions of this research are as follows:
1)
Development o f the Generalized Spatial Correlation Algorithm (GSCA)
(from Ch. 3)
M aximization of this algorithm's objective function provides a means for
determining the estimates of the phase and element position errors present in the imaging
system. The GSCA is shown to include all published algorithms that operate upon the
spatial correlation measurements of the wavefront. These procedures are called spatial
correlation algorithms. Examples are the Modified Muller-Buffington Algorithm [49], the
Unit-Lag Spatial Correlation Algorithm [1,3], the Shear Averaging Algorithm [20], and the
Least-Squares Multiple Lag Algorithm [48]. Each of these algorithms is described in
Chapter 2. It also includes Tsao's Energy Conservation Algorithm (ECA) [55,56] which is
based upon Parseval's Theorem and therefore had hitherto been thought to operate upon
different properties of the wavefront statistics.
The analysis herein discloses that spatial correlation algorithms can be divided into
model-based and non-model-based groups called parametric and non-parametric sub­
classes. The ECA belongs to the former and the Modified Muller-Buffington belongs to
the latter. Analysis in Chapter 4 shows that the parametric sub-class provides slightly
superior performance, provided that the model is correctly chosen. Lastly, because the
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GSCA objective function embodies all of the known spatial correlation algorithms, new
inter-relationships between these algorithms are revealed.
2)
Study o f Element Position Error and Receiver Noise Tolerances
(from Ch.. 4, and Appendix B)
The GSCA objective function is a highly nonlinear function of the phase error
estimates and the phase noise of the measured correlation values. Analysis shows that
spatial correlation algorithms subsumed within the GSCA yield least-squares solutions for
the phase error estimates when the phase noise variance is small. This makes algorithms
from the parametric sub-class particularly attractive since they yield the same solution as the
MLSCA under these conditions without the phase unwrapping process required by the
MLSCA. Computer simulations lead to design data showing effects of noise and element
position errors on mainbeam gain and sidelobe level as functions of field-of-view size, scan
angle, rms element position error, and the number of correlation lags used in the calibration
process. Performance curves are provided in Appendix B for design engineers.
It is found that the main beam reconstruction performance of the GSCA is
equivalent to the DSA when a dominant scatterer is in the field-of-view, and is ^ to 1 dB
poorer without it. As a consequence the mainbeam gain performance can be estimated at
any scan angle based on DSA theory plus a single measurement made at boresight.
Additional contributions are:
3)
Modification o f the GSCA fo r Self-Calibration with Near-Field Data (from Ch. 4)
Since the GSCA is a multiple-lag algorithm, the stationarity of the random complex
radiation field measured in the array aperture must be addressed. The algorithm must be
modified to account for the non-stationarity of the random complex field measured in the
array aperture that occurs when the source distribution is located in the near-field of the
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array. The modification is adapted from Subbaram's work [49] with the Modified MullerBuffington multiple-lag algorithm. It is shown that under certain conditions it may be
necessary to include a weighting function (termed the "near-field weighting function") in
the GSCA objective function to remove the effects of the near-field phase curvature present
in the recorded data set. Additionally, this chapter establishes that near-field data sets are
not as restrictive as originally thought and in most practical systems modification of the
algorithms (i.e., the use of the near-field weighting function) for near-field data sets is not
necessary.
4)
The Commonality o f the Maximization Procedures (from Ch. 3)
Each of the spatial correlation algorithms referred to above was independently
invented or derived for differing circumstances and different conditions. Understandably,
the methods of weight vector adaptation are also different. Since all of the published spatial
correlation based algorithms are subsumed within the GSCA, only one efficient method is
needed to maximize the objective function. A generic non-linear optimization package can
be used. However, the methods developed specifically for individual algorithms are now
known to be generally applicable to all algorithms belonging to the GSCA class. These
methods exploit Hermitian symmetry properties that exist in the objective functions,
thereby producing efficient optimization methods. Whereas previously a different efficient
optimization method had to be implemented for each algorithm, now with the GSCA theory
only one method need be implemented and various spatial correlation algorithms can be
used by changing the weighting function in software.
5)
Analysis o f the Spatial Correlation Sub-Classes (from Ch. 4)
The GSCA objective function is a highly nonlinear function of the phase error
estimates and the phase noise of the measured correlation values. Because of this no
significant mathematical theory regarding level of additive receiver noise or the level of
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element position error that can be tolerated by spatial correlation algorithms can be derived.
Analysis of algorithms from the two sub-classes shows that both algorithms yield
approximate least-squares solutions. It is shown that the least-squares problem solved by
the parametric algorithm is the same as the one solved by the MLSCA. However, using the
parametric algorithm instead of the MLSCA eliminates the phase unwrapping requirement
of the MLSCA. The analysis also shows that Attia's ULSCA is the unit-lag algorithm of
the parametric class and is not a degenerate version of the MMB as previously claimed in
[29]. Lastly, the analysis predicts the performance of the parametric algorithm to be
superior to the performance of the non-parametric algorithm. All of these results are
observed in the computer simulations presented in the same chapter.
1.2 History of the Spatial Correlation Class of
Algorithms
In high-resolution microwave imaging, each element of a phased array antenna
samples the received complex field generated by the source or scatterer distribution. Phase
errors are inherently present in the signals in each receiver channel because of element
position errors, propagation medium distortion, multipath, electrical mistunings, etc.
These errors degrade the mainlobe gain of the array, increase its sidelobes and sometimes
cause multiple lobing. The result is severe distortion of the image. A crucial step in the
imaging process is self-calibration of the system for the purpose of removing the phase
errors.
Removal of the phase errors can be accomplished by the signal processor of the
imaging system. If information regarding the system phase errors is known, the signal
processor can account for the phase errors when processing the received data. If no such
information about the sytem phase errors is known (as is generally the case), then
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measurements made of the wavefronts received from one or more beacons or sources
positioned at a known locations can disclose the phase errors present in the system. Once
the phase errors are known the signal processor can remove them when processing the
data. This is a survey type of calibration; it requires the positions of the sources to be
known .
In the general radar situation a priori information of this nature is not available.
Instead, self-calibration must be employed. Self-calibration, also known as adaptive
beamforming, is an adaptive procedure in which calibration is accomplished by extracting
information concerning the channel phase errors directly from the measured data set, i.e.,
the received complex field measured at every element in the array. Any algorithm designed
to self-calibrate an array system must operate on the error-corrupted data set received by the
distorted array.
The idea of self-calibrating optical telescopes to compensate for atmospheric
turbulence was introduced in 1953 by Babcock [4], In 1954 Green [23] proposed an
analogous method in the field of X-ray crystallography. Rogstad [38], in 1968, presented
another technique to correct for atmospheric distortion in an optical interferometer. The
spatial correlation type of algorithm was introduced in 1974 by Muller et al. [31] who
proposed a real-time self-calibration method to correct atmospherically degraded telescope
images; they reported the first observatory results using their method in 1977 [12], The
Muller-Buffington technique involved operations upon the image intensity distribution; it
was shown to be equivalent to spatial correlation operations in the aperture plane in 1977
[25]. Since the early 1980's self-calibration techniques have been used in radio astronomy
to compensate for wavefront distortion induced by spatial variations in the refractive index
of the propagation medium [10], [11], [17],[37].
The Valley Forge Research Center (VFRC) independently developed methods to
perform the task of self-calibration for high-resolution microwave imaging systems
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[ 1 , 3 , 2 8 ,2 9 , 4 0 ,4 1 , 4 2 ,4 5 , 4 8 ,4 9 , 5 4 ,5 6 ] .
The m easurem ent instrum ent is
known as the "radio camera" and was first described by Steinberg in 1973 [41]. The 1973
Steinberg paper described a technique to self-calibrate a large phased array used to obtain
high resolution images of microwave scatterers/sources. The first radio camera experiment
demonstrating the feasibility of microwave self-calibration was performed one year later
[41]. The algorithm upon which most subsequent radio camera experiments are based was
published in 1981 [42] with further details and examples in [28, 29,44, 45, 48, 52]. It
is known as the Dominant Scatterer Algorithm, (DSA).
Based on the analysis by Hamaker et al. [25] of Muller's theorem [31], Attia, in
1984, developed the (Unit-Lag) Spatial Correlation Algorithm (ULSCA) [1 ,3 ,4 8 ].
Unlike the DSA, which operated upon the wavefront from a rangebin having a dominant
scatterer, it exploits the properties of the spatial correlation function of the received complex
field generated by the source distribution being imaged.
Other algorithms that do so are
the Muller-Buffington method [29], the Modified Muller-Buffington Algorithm (MMB)
[29],[49], the Multiple-Lag Spatial Correlation Algorithm [48],[49], and the Energy
Conservation Algorithm (ECA) [55],[56]. Until this work the ECA was not known to
belong to the Spatial Correlation Class.
Muller et al. showed that an incoherent source distribution can be imaged without
errors by a distorted telescope objective compensated by information derived from the
intensity distribution in the image plane. However, the source distribution, or scene, to be
imaged in radar is usually coherent, meaning that the phase relationships between scatterers
is unchanged from pulse to pulse, provided that the transmitter and receiver locations
remain fixed. Therefore, a method of emulating an incoherent source distribution had to be
devised to apply Muller's concept to radar imaging situations.
Two methods to "decohere" the source distribution have been developed at VFRC.
The first procedure is range bin diversity, developed by Tsao [54] and Attia [3]. It is based
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on the assumption that some portion of the radar rangebin data is statistically
homogeneous. The radiation field received from each such rangebin can be considered as
snapshots of a radiation field originating from an equivalent incoherent source distribution.
In this manner multiple rangebin radar data can emulate data received from an incoherent
source distribution.
The second procedure is transmitter location diversity, which was developed by
Subbaram [51] in 1986. In this technique successive radar pulses are transmitted from
different antenna elements in the array. Consequently, a scatterer is illuminated by a
different phase from pulse to pulse. As a result, the re-radiation phase relationships vary
and the received waveforms will appear as if they originated from an incoherent source
distribution.
1.3 The Microwave Imaging System
The goal of the radio camera is to obtain high angular resolution images at
microwave wavelengths. Since angular resolution is proportional to the wavelength
divided by the length of the array, large aperture arrays are required for microwave systems
to possess high angular resolution. For example, to obtain an image with the same
resolution of optical cameras, a microwave "camera" aperture must be 3 to 6 orders of
magnitude larger than that of an optical camera. Practicalities prevent building a monolithic
antenna such as a dish antenna with the necessary size; consequently distributed, sparse
phased arrays or synthetic apertures are required.
Figure 1.1 illustrates the relevant phased array geometry. The array is assumed to
be linear (one-dimensional) and lie within an x-y plane with origin located at some
reference element within the array. The x-axis contains the array and the y-axis is in the
transverse direction. The angle of arrival 0 of energy from a source is measured from the
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u fo v
Illuminated
Target
Field
Transmitter
Signal Processor
Display
Figure 1.1
Diagram of Imaging System
array normal. The echoes from a transmitted pulse are collected over several rangebins in
all the receive channels. The transmitter is assumed to be separate from the phased array
(i.e., bistatic radar) and it illuminates an angular sector called the field-of-view (FOV)
significantly larger than the beamwidth of the array. One can envision a set of received data
in the form of a matrix as shown in Figure 1.2. The signal processor is responsible for
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self-calibration and forming the phase-corrected image. The system is assumed to be
narrowband and consequently complex exponential notation is appropriate.
Echo
Range
N Elements
Figure 1.2
Diagram of a typical data set in matrix form. The vertical axis
represents elements in the phased array. The horizontal axis
represents range to the target. Consequently, the nth row
represents the sampled range profile received by the nth
element and the klh column represents the data received across
the array from the k* rangebin.
The combined effects of element position errors, electrical mistunings, etc., can be
grouped into an undesired complex gain present in each channel, provided that the field-ofview is not excessively large. The complex gains are spatially random but temporally
constant (i.e., the gain in any given channel is constant while the received complex
radiation field is being measured). Steinberg [47] showed that amplitude variations have a
significantly smaller effect on image quality than phase variations. Consequently, the
complex gain errors reduce to channel phase errors (3n, n = 1,2,...,N.
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10
The signal
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processor estimates the (3n's and subtracts the estimates (3n's from the recorded data.
Figure 1.3 illustrates this model.
i,y
V V VV V V V
w
Figure 1.3
The self-calibration scheme:
P's represent channel phase
A
errors, P's represent estimates of the phase errors, w's represent
standard beamsteering weights.
The corrected data set approximates samples o f the radiation field measured by an
undistorted phased array. Therefore standard electronic-scanning beamsteering weights wn
may be applied to scan the beam over the region of interest and produce a high-resolution
image without distortion. If the phase errors are not estimated exactly, residual phase
errors exist in each channel and some image distortion will remain. However, the level of
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distortion will be significantly decreased by self-calibration . One objective of this work is
to study the accuracy of the phase error estimation for spatial correlation algorithms.
1.4 The Experimental Data
All of the experimental radar data sets were obtained at the field site of the Valley
Forge Research Center. The inverse synthetic aperture radar (ISAR) [15] data sets were
obtained at 9.6 GHz (A. =3.123 cm) with a 4-ft parabolic transmit/receive antenna. The
data sets were echoes from airplanes flying into the Philadelphia International Airport. The
experiment is reported in [14],[53] and more details can be found in [13],[15]. The radar
was a low power (250 watts peak), short pulse (7 ns or 1 m) fully coherent radar. Each
echo trace was sampled at a 200 MHz rate following coherent quadrature demodulation,
and quantized to 8 bits in each receiving channel. The short pulse length produced a range
cell of 1.05 meters and the system recorded 120 range bins, spaced by 0.75 meters, from
512 radar echo waveforms. Other details concerning specific data sets are presented where
necessary.
1.5 Dissertation Outline
This dissertation is divided into 6 chapters, three of which present new work. This
first chapter is an introduction to the self-calibration problem. A brief history of self­
calibration techniques is presented. Additionally, array geometry, notations, and the
structure of a general imaging system is also presented in this chapter.
The second chapter provides a detailed background of self-calibration based upon
the spatial autocorrelation function and describes all the known spatial correlation based
algorithms found in the literature. Several algorithms described were developed at the
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Valley Forge Research Center (VFRC). They include Atria's Unit-Lag Spatial Correlation
Algorithm, Subbaram's extensions to Attia's unit-lag algorithm, the Multiple-Lag Spatial
Correlation Algorithm, and the Modified Muller-Buffington Algorithm. Other spatial
correlation algorithms described are the basic Muller-Buffington Algorithm (optics and
radio astronomy), the Shear Averaging Algorithm (SAR imaging), the Phase Gradient
Algorithm (SAR imaging), the Flax-O'Donnell algorithm (medical ultrasound imaging) and
phase closure (radio astronomy). This establishes a set o f spatial correlation algorithms
and this background is used in the development of the generalized algorithm presented in
Chapter 3.
In Chapter 3 Tsao's Energy Conservation Algorithm (ECA) is discussed. An
important limitation of the ECA discovered experimentally during this research provides the
insight that leads to the development of the Generalized Spatial Correlation Algorithm
(GSCA) derived in this chapter. The GSCA includes the ECA along with the algorithms
described in Chapter 2. Hitherto, Tsao's algorithm was not believed to be a spatial
correlation algorithm.
Chapter 4 provides performance results from computer simulations of the effects of
random element position errors and additive element receiver noise. Analysis of the
objective functions of algorithms of the two sub-classes explains the superior performance
of the parametric algorithm over the non-parametric algorithm and explains the performance
difference between the unit-lag algorithms of the two sub-classes. The analysis also
indicates that the parametric algorithm is favorable over the MLSCA in the presence of
small variance phase noise in the measurement of the spatial correlation values. Appendix
B supplements this chapter with 40 plots of expected main beam relative gain versus scan
angle for the two algorithms.
The GSCA assumes a set of incoherent sources in the far field of the phased array.
However, in most practical high-resolution microwave imaging situations, the sources are
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in the near field of the array. All self-calibration algorithms focus the array as well as
correct errors. The backscatterered radiation field is a spatially non-stationary random
process in the near field, however, and therefore, care must be taken in how the GSCA is
applied. In Chapter 5 the effects o f having the source distribution in the near field are
presented. It is found that the low-order correlation lags may still be used while the higherorder lags are unsuited for use in the self-calibration process. The suitability of a particular
correlation lag depends on a variety of parameters, the most important of which is the size
of the field-of-view being illuminated by the transmitting antenna.
Chapter 6 contains a summary and conclusions drawn from this research along with
suggestions for future work in this area.
1.6 Dissertation Summary
The objective of this work is to understand the use of measurements of the spatial
autocorrelation function of a spatially random complex radiation field received in the
aperture of a phased array antenna to self-calibrate the array antenna of a high resolution
microwave imaging system. This work investigates properties relating to all known spatial
correlation algorithms.
The algorithms in this group are all relatively scene independent and are designed to
work in situations where there is no dominant scatterer present in the data set. Because of
the strong similarities between all these algorithms a unifying theory exists which firmly
classifies them into one class and characterizes the class. This theory is developed and
presented throughout this document.
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Chapter 2
Existing Spatial Correlation-Based
Algorithms
In 1974 Muller et al. [31] described in the optical literature a class of image
sharpening functions that, upon maximization, yield a set of phase corrections that cancel
the channel phase errors. In optics the phase errors are caused mainly by atmospheric
turbulence. One particular objective function which Muller had significant success with
([16],[17]) is the integral over the image plane of the squared image intensity, i.e.,
Jj* I2(x,y)dxdy
(2.1)
Muller proved that this function reaches its absolute maximum when all phase errors have
been eliminated [31]. A much simpler and more insightful proof is presented by Hamaker
et al. in [25]. Hamaker showed that the maximum of (2.1) will be obtained when the
random radiation field measured in the array aperture is a spatially stationary random
process.
The beauty of Muller and Buffington's objective function (2.1) is that it does not
depend on the characteristics of the source distribution being imaged. There are no
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constraints or assumptions made on the characteristics of the source distribution in either
proof ([25] or [31]). However, the Muller-Buffington algorithm is restricted to use with
incoherent radiation fields and in radar imaging the received radiation field is rarely
incoherent. The rangebin diversity and transmitter location diversity techniques, developed
by Attia [3] and by Subbaram [49] respectively, can be used to successfully emulate an
incoherent radiation field. These techniques, along with Hamaker's proof, open the way
toward a scene independent self-calibration algorithm for use in high-resolution microwave
imaging.
This chapter describes all of the spatial correlation algorithms known in the
literature. These algorithms are subsumed within the GSCA in Chapter 3. The first section
defines the Spatial Correlation class of algorithms for use throughout the remainder of the
document. The second section derives the spatial correlation function of the received
random radiation field and its Fourier transform relationship with the source intensity
distribution.
The third section presents Hamaker's proof that maximizing (2.1) is
equivalent to requiring the received random radiation field to be a stationary random
process. The ideas of these two sections are then combined in the fourth, fifth, and sixth
sections where the ULSCA, the MLSCA, the Shear Averaging, the Phase Gradient, the
Flax-O'Donnell, and the Phase Closure algorithms are discussed.
The latter four
algorithms stem from the fields of SAR imaging, ultrasound imaging, and radio
astronomy.
2.1 The Spatial Correlation Class Defined
In this document a "class of algorithms" is defined as:
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any set of algorithms that make use of a common property or
principal to perform a specific task.
The Spatial Correlation class of algorithms is defined as:
the set of algorithms that exploits information embedded in
the measured spatial correlation values to estimate phase
errors present in a phased array system.
2.2 The Spatial (Auto)Correlation Function of an
Incoherent Source Distribution
Figure 2.1 shows a source distribution, J(0), at range pD from the coordinate
system origin. The source distribution is assumed to cover an angular extent A0 shown by
the dashed lines in the figure. The radiation field suitably measured along the x-axis is
given by the scalar diffraction integral as
g-jkp(x)^
/•
e(x) =
d0
J(0)
PM
(2.2)
J
0
provided all of the sources produce the same polarization. The distance p(x) is given by
the law of cosines as p2(x) = p2 + x2 - 2xp0c o s(| -0) and p(x) = ‘s j p2 + x2 -2x p o sin(0 )
since cos(^ -0) = sin(0). If p0 > x, then the source distribution is said to be located in the
far-field of the antenna and the distance p(x) can be well approximated as p(x) = p0 - xu
L 2
where u = sin0. Any distance satisfying p0 >
meters ( L ^ denotes the length of the
array in meters) is generally considered to be in the far field [46]. Additionally, if A0 <3 1
then u = 0 and (2.2) can be written as
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e(x) =
J(u) eikxu du
(2.3)
Au
after dropping a complex constant term. If J(u) is a deterministic quantity, the radiation
field is also deterministic and (2.3) states that a Fourier transform relationship exists
between the measured radiation field and the far-field source distribution. However, if J(u)
is a random quantity, then the measured radiation field is also a random quantity and the
field measured along the x-axis is a spatially random process. The autocorrelation function
of this random process is given by
R(x1,x2) = E{e(x1)e*(x2)}
f E{J(u)J*(v)} e>k[xl u ' X2V1 du dv
(2.4)
Incoherent
Source
Distribution
P(x)
Figure 2.1
Illustration of the geometry used in deriving equation 2.3. (Adapted from [3])
18
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If the source distribution is composed of incoherent sources E{J(u)J (v)}
I(u)5(u-v)
where I(u) represents the intensity distribution of the sources. Then
I(u)S(u-v)
R( x j , x 2) =
" X2V^ du dv
= f I(u) eJk[xl ' x2]u du
= R(X l- x 2) = R(X)
(2.5)
where X = xj - x2 . Since (2.5) shows only a dependence on the difference between
positions xj and x2, the measured random radiation field is a wide sense stationary random
process when the source distribution is located in the far field. Equation (2.5) also shows
that the spatial autocorrelation function is proportional to the Fourier transform of the
source intensity distribution, I(u). This Fourier relationship between the angular intensity
distribution and the spatial correlation function is commonly known as the Van-Cittert
Zemike Theorem [6] and is analogous to the Weiner-Khinchine theorem.
2.3 The Muller-Buffington Algorithm
2.3.1 The Image Sharpness Function
The concepts involved in the Muller-Buffington Algorithm are illustrated in
Figure2.2. Shown in the figure is an undistorted radiation field that suffers unknown
random phase perturbations from atmospheric turbulence, spatial variations in the index of
refraction, etc. The imaging lens corrects for geometrical phase differences but does not
correct for the random phase perturbations that consequently propagate through the system
to the adjustable phase shifters. Without the bank of adjustable phase shifters the image
19
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would be completely distorted. If the phase shifters can adjust the phase of each ray
appropriately, then the unperturbed phasing will be restored and an undistorted image will
be formed.
Phase restored
Image
Plane
Phase disturbed
Adjustable
Phase
Shifter
Controller
J j = JJ I2(x,y) dxdy
Figure 2.2
Illustration of the Muller-Buffington optical phase correcting scheme. (Reprinted from [31])
Muller et al. measured the image sharpness defined by (2.1) and adjusted the bank
of adjustable phase shifters until a maximum image sharpness was obtained. They proved
that when the phase shifters were adjusted so that a maximum image sharpness was
obtained, all the random phase perturbations had been removed from the radiation field and
the resulting image was undistorted.
Figures 2.3 and 2.4 show examples of the results that Muller et al. obtained with
this algorithm. Figure 2.3 shows results from simulated data. The distorted image is
shown in the top part of the graph and the phase corrected image is underneath it. In the
simulation three point source were simulated and phase perturbations were added. The
distorted image does not show the presence of the three point sources. However, after
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maximizing (2.1), the resulting image clearly shows the three point sources. Figure 2.4
shows a similar graph of experimental data obtained from Leuschner Observatory in
California. Both the upper and lower images are of the star Sirius. As in Figure 2.3, the
upper image contains phase perturbations and is formed with no phase corrections applied.
The lower image is what was obtained after (2.1) had been maximized and the phase
corrections had been applied. The image clearly shows the presence of a point source
indicating the presence of the star which can not be seen in the upper image.
l o ) F t i d b e c k OFF
1 2 3 3 0 PST)
(o) Spackla
potlirn
lb) Fiidbock ON 12300 PST)
tb) Corr*et»d
Imoga
-OB
-0.6
-0.4
-0 2
0
0.2
0.4
0.6
0B
P osition In im ag o p lan* ( a r c i o c )
Taken from JOSA Septem ber 1974. pp. 1200-1210
P e t i t i o n in t h t
im o g * p l o n * , s e c o n d * of ore
Figure 2.4
Figure 2.3
2.3.2 M axim izing the Image Sharpness Function in the Aperture
D om ain
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In [31] Muller and Buffington proved that (2.1) is maximum when all of the
random phase errors are removed from the incident radiation field. In 1977 Hamaker et al.
[25] made the connection between maximizing (2.1) in the image domain and enforcing a
stationarity constraint in the aperture domain.
Consider the Muller-Buffington objective function of (2.1) in one dimension. It is
well known from Fourier optics [6] that the intensity, I(u), is a Fourier transform pair with
the product of the visibility function, V(X) in the aperture, and the optical transfer function,
T(X). The optical transfer function is related to the pupil function, P(X), through
T(X) =
P(x)P*(x+X) dx
( 2 .6 )
Since this Fourier relationship exists, Parseval's Theorem states
I2(u) du = f IV(x)l2 IT(x)l2 dx
(2.7)
V(X) is determined solely by the source intensity distribution and consequently maximizing
(2.1) is equivalent to maximizing T(X) for all X. Let the undistorted pupil function, P0, be
real and assume that the atmospheric distortions affect only the phases of the measured
radiation field as assumed in [31]. Therefore, P(x) can be expressed as
P(x) = P0(x)eiP(x)
( 2 .8 )
and the optical transfer function, T (X ), can be expressed as
T(X) = f P0(x)P0(x+X )ej [ PW ' P(X+X)] dx
(2.9)
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Equation (2.9) shows that T(X) is a summation of phasors each with a phase given by the
argument of the complex exponential. The magnitude (and therefore the square magnitude)
of T(X) will be maximum when all the phasors are cophased. This requires the complex
exponential to be independent of the variable x. The consequences of requiring the
argument of the complex exponential to be independent of x can be seen by expanding P(x)
in a Taylor series about some arbitrary point, "a", as
p(x) = a + p'(x)|a (x-a) + p”( x )|a
+ .. .
(2.10)
The quantity P(x) - (3(x+X) will be independent of x if P(x) is composed only of a constant
and linear term. A constant phase error does not affect the imaging system and is therefore
of no consequence. The presence of a linear phase error across the imaging system
introduces a pointing error (a shift in image location) but does not distort the image.
Therefore, maximization of (2.1) yields a diffraction limited image with a possible shift in
image location.
Since (2.9) is a summation of complex exponentials and requiring that the quantity
P(x) - P(x+X ) be independent of x demonstrates that measurements of the same visibility
value between different pairs of elements should be equal. An unperturbed radiation field
originating from an incoherent source distribution should yield the same correlation values
independent of the location in the array at which they are measured. This means the
unperturbed radiation field measured in the aperture should be a stationary random process.
In the optics field, Muller's algorithm provided a method of enforcing such a constraint on
the radiation field without directly measuring the correlation values. In the microwave
imaging field it provided the basis for the Spatial Correlation class of self-calibration
algorithms.
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2.4 The Unit-Lag Spatial Correlation Algorithm
The simplest algorithm in the Spatial Correlation class is the Unit-Lag SCA [3].
The basis o f the algorithm lies in the realization that the phases of the spatial correlation
values measured in the aperture contain information regarding the channel phase errors. As
(2.4) and (2.5) show, the spatial correlation values measured in the aperture depend only
on the difference between the positions used in making the measurement when the source
distribution is incoherent, in the far field of the array, and the imaging system is free of
errors.
Assume that s(u) is a source distribution satisfying both the incoherence and farfield requirements. The intensity distribution of s(u) is I(u) and the corresponding spatial
correlation function in the antenna aperture is R (X ) where X is a continuous variable.
However, the phased array is a sampled aperture and the corresponding correlation
function is also sampled. Denote the sampled spatial correlation function as R(jt) = R(jtd)
where t (an integer) represents the lag index and d represents the interelement spacing
(i.e., the sampling interval). Such notation implies that the array is periodic. This is a
sufficient, but not necessary, condition for the unit-lag SCA to work. In general, the
following unit-lag algorithm can also calibrate an aperiodic array.
If the source
distribution, s(u), is symmetrical about the optical axis, then the correlation function will be
real except for a linear phase component associated with the pointing direction of the beam.
The correlation function develops an imaginary component when channel phase errors exist
in the imaging system. Therefore, the phases of the correlation measurements contain all
the necessary information about the channel phase errors to calibrate the system.
Consider an N-element array antenna. The N signals present in each channel are in
error because of any or all of the problems mentioned in Chapter 1. These corrupted
signals can be expressed as e(n) = e (n )e'^ n where e(n) represents the signal that would
24
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be obtained in the nlh channel if no phase errors were present. Therefore, any weight
vector of the form q(n) = e*Pn will correct the signals and any procedure that yields the
values of Pn solves the calibration problem.
The unit-lag spatial correlation coefficient of the error-free sample between the nth
and the (n+ l)111elements is
R (l) =E{e(n)e*(n+l)}
(2.11)
which is assumed to be real in [3]. With phase errors present in the imaging system the
measured correlation value is
R’(n ,l) = E{e(n)e!i!(n+l)ej[Pn+1' Pn] } = E{e(n)e*(n+l)}ei[Pn+rPn]
and
= R ( l) e P n+rPn]
(2.12)
arg{ R (n,l) j = (3n+1 ~(3n
(2.13)
In an N-element array there are N -l lag-1 correlation values available but there are N
unknown phase errors. This is enough information to calibrate the array because one can
freely choose the phase of any one element in the array.
Denote the reference element as element #1 and let the phase error (ij be zero.
From (2.13)
(32 = arg ( r '( 1 ,1 ) ) , P3 = arg ( r '( 2 , 1)) + arg ( r ’(1,1)),
etc.
(2.14)
The phase error in the n1*1 channel is
n
a r g ( R '( p - U ) )
Pn = £
(2.15)
p=2
25
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2.4 .1 P r a c tic a l Im plem en tation o f the U nit-Lag SCA
Equations (2.12) through (2.15) demonstrate that the system can self-calibrate
provided an ensemble exists over which an expected value can be taken and second-order
statistics can be defined. This requires the measurement of a radiation field that is a random
process and a sequence of snapshots will represent an ensemble of sample functions of that
process. However, in radar applications the backscattered field is rarely a random process
because the echoes from successive transmissions are substantially identical.
A method to obtain such an ensemble is to consider the radiation fields from
successive rangebins as different realizations of a random process. Attia showed that this
can be successfully done provided the source distributions in the rangebins being
considered are statistically similar [3]. If the scene being imaged satisfies this criterion,
then the imaging radar should collect as many rangebins as it can and the spatial average in
range
K
(2.16)
can be formed to approximate the expected value in (2.12). In (2.16) the subscript "Ic” is
the rangebin index and "K" is the total number of rangebins averaged. This process is
shown in Figure 2.5. As long as each correlation measurement is independent, in range,
of the other correlation measurements, then (2.16) is an unbiased and consistent estimator
t
»
of R (n ,l) and will converge to R (n ,l) as K—> °°.
In essence, the more rangebins
collected from a statistically homogeneous scene, the better the Unit-Lag SCA should
perform.
2 .4 .2 A n I te r a tiv e U nit-Lag SCA
26
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When the array is periodic an iterative algorithm can be used to calibrate the system.
In a periodic array the distance between neighboring elements is constant and the received
radiation field is spatially sampled at constant intervals. In this case, when the unit-lags are
computed, the correlation distances corresponding to each computed value are equal. Since
the radiation field is wide sense stationary (as shown by (2.5)) the correlation values must
PnO
O P „ +l
t
Figure 2.5
Illustration of the correlation measurement between elements n and (n+1).
depend only on the correlation distance and not on the position in the array at which the
value is computed. Therefore, in an undistorted (P; = 0 , for all i) periodic array , all unitlag correlation values must be equal, i.e.,
r
'(1,1) = R (2 ,l) = . . . = r '(N-1,1)
(2.17)
As seen from (2.12) the phase errors alter the individual measurements. Any
method that forces convergence to (2.17) will calibrate the imaging system. One such
method is to maximize the objective function [1]
27
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N-l
JA =
X RW)e ‘K P n + l
‘ Pn)
(2.18)
n=l
It is easily verified from (2.12) and (2.18) (and proved in [1]) that J A will be maximum
A
A
when Pn = Pn modulo 2rc. In practice the maximization is done by substituting R (n,l)
a
of (2.16) for R (n ,l) in (2.18) and adjusting the phase error estimates, Pn, until the global
maximum is attained. This algorithm is termed "iterative" because maximization of (2.18)
is usually not a one-step process, but instead requires changing the values of the parameters
(the pn's) several times until no change in the value of the objective function (2.18) is
observed.
2.5 The M ultiple-Lag
Algorithms
Spatial
Correlation
In addition to the unit-lag correlation values there are N-2 other correlation lags
measurable in an N element array and all provide information regarding the phase errors.
Figure 2.6 illustrates the measurement of the higher-order lags. In an ideal environment
(zero noise and K = °o) these lags provide no more information than is already contained
in the unit-lag measurements and are therefore of no use in the self-calibration process.
However a limited sample size (K < °°) produces a statistical variability to the correlation
estimates and additive noise further increases the output variance.
28
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Figure 2.6
Illustration of correlation measurements of lags Lj,
L^,
Element receiver noise can corrupt the correlation measurements in two ways.
First, if the signal-to-noise ratio (SNR) is low the received signals in each channel will be
phase rotated by the channel noise and the resulting correlation measurements will be in
error. Additionally if the true correlation value given by (2.11) is close, or equal, to zero
then channel noise will certainly corrupt the measurement of the phase of this value.
Redundancy offered by the higher-order correlation lags reduces the effects of element
receiver noise and finite rangebin averaging. The current multiple-lag algorithm of
Subbaram [48],[49] exists in two forms. One form establishes a least-squares problem and
the other form maximizes an objective function. These two forms are discussed in the two
following sections.
2 .5 .1 The L e a s t Squ ares M u ltip le-L a g SCA
The discussion regarding (2.12) and (2.13) can be generalized to show
R W ) = lR0t)lej[V*+Pn+* 'Pn]
(2.19)
29
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where the quantity V/, denotes the true phase of the lag Z correlation value given by
R (Z ) = lR ( Z ) le ^ = E {e(n)e*(n+Z )}. Let <|)(n+Z,n) represent the phase of the lag Z
correlation value R (n,Z) such that
<Kn+Z,n) = arg (R (n ,/.)) =
+ p n+* - p n
(2.20)
When noise is present and the finite spatial average in (2.16) is formed to approximate the
ensemble average, the measured correlation lag is corrupted and equations (2.19) and
(2.20) become
R (n,Z ) = |R(Z)lej[V*+Pn+rPn+5(n+*,n)]
and
(2.21)
$(n+Z ,n) = arg (R (n,Z )) = V/, + Pn+Ji - Pn + S (n + Z ,n )
(2.22)
where 8(n+Z,n) represents the phase noise induced by the element receiver noise and the
rangebin averaging.
The unit-lag correlation phase,
can be estimated as
A
V! = kdu0 « V i where uQ is the direction towards which the transmitter is pointed, but
the remaining correlation phases, Yjt, L > 1, are unknown quantities and must be removed
[48, pg. 243]. This can be accomplished by forming the quantity
y(n,jt) = $ (n + Z + l,n + l) - $(n+Z,n)
(2.23)
= Pn+jt + l ■ Pn+1 - Pn+jt + Pn + 8(n+£ + l,n + l) - 8 (n + /.,n )
In an N-element array there are N-jt measurable lag-/, correlation values. Therefore, N-Z -1
equations of the form of (2.23) can be formed for any particular lag. The equations
resulting from (2.23) can be combined with the N-l unit-lag equations to produce an
overdetermined system of equations of the form
30
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$( 2 , 1) - ^
<J>(3,2) -
= 0 2 - 1$!
= $3 - $2
Y(l>2) = $4 ' $2 ' $3 +
$1
(2-24)
y(l»N-2) = (3n - $2 - $n- 1 + Pi
When phase noise is not present (i.e., S(n+j£,n) = 0) the equations generated by (2.23)
are linear combinations of the unit-lag phases [48, pg. 248] and they provide no new
information. However, in the presence of noise these equations are no longer consistent
with the unit-lag equations. Consequently, the higher-order lags do provide additional
information and a least-squares solution can be found. This solution effectively averages
over the additionalobservations provided by the measurements of the high-order lags and
smoothes the effects of the noise.
Care must be taken when forming the overdetermined system of equations. Since
the channel phase errors, Pn, can take on any value within the range -it to 7t, significant
problems can occur because of modulo 271 ambiguities when the quantity y(n, t ) is
calculated. Additionally the quantity $(n+jt,n) itself may wrapped by 2% when measured.
Any wrapping of the measured phases creates an inconsistency in the set of equations that
exists even when there is no noise present. This inconsistency can introduce more noise
into the system and therefore the wrapping must be eliminated before finding the leastsquares solution. Steinberg and Subbaram present a method of unwrapping the 2n
ambiguities in Ref. [48] that is summarized in the following paragraph.
Phase unwrapping can be accomplished by noting that the higher-order correlation
phases are linear combinations of the unit-lag correlation phases. The equality between
y(r\JL) and the unit-lag phases can be shown to be
31
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y(n,jt) = $ (n + jt+ l,n + jt) - <j)(n+l,n) + 2nm
- 8 ( n + jt+ l,n + /) + S (n+l,n)
+ 8(n + jt+ l,n + l) - 8(n+jt,n)
(2.25)
When the noise term is smaller than jt, the integer m in (2.25) can be determined as
a
y(n,jt) - [$(n+jt + l,n+jt) - $(n+l,n)]
m = ---------------2k
round
(2.26)
where m is the estimate of m and "round" implies rounding to the nearest integer. Once m
is determined from (2.26), y(n ,t) can be unwrapped as
Y ( n / ) l u = y (n ,jt) - 2 j t m
(2.27)
Once a consistency between all of the equations is established by unwrapping the 2n
ambiguities, the least-squares solution can be found from the equation
A
Ap = b
Let L
(2.28)
represent thehighest-order lag used in (2.28). Let every lag for I < L be included
in (2.28). Then A is an ^ £ nl a
x N - l j matrix containing the coefficients of the
a
P's in equation (2.24), P is an (N-l x 1) vector containing the estimates of the phase
A
rJ'
errors (i.e., P = [ P 2,P3,P4 , . . . , P n ]
)> and b is an ^ j^NL -
x
vector
containing the unwrapped values of the phases of the measured spatial correlation lags. A
is a sparse matrix containing only l's, -l's and 0's and possessing a lot of structure. The
general form of A and b are shown in Figure 2.7 for lags one and two. The system of
equations in (2.28) contains only N-l unknowns (p2 through pN) because the phase error
in the reference channelisassumed to be zero. The least-squares solutionto (2.28) is
P = A# b = (A ^ 'V b
(2.29)
32
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where A
$
represents the pseudoinverse of A . The negative of (3 should be applied to the
0
0
...
0
0
0
0
1 0
0
...
0
0
0
0
o
o
o
o
-1
0
o
1
o
phased array to calibrate the imaging system.
•
•
• • •
• • •
•••
•
•
• •
• •
•
♦
0
0
0 0 . ..
0
0 -1
1
-1
1
1 0
. ..
0
0
0 0
1 . ..
0
0
0 0
1 -1 -1
•
•
_ 0
• • 1
• • •
0
0 0
Figure 2.7
1 -1 -1
Vi
A
0(3,2) - Vj
A
0(4,3) - Vi
•
b =
$(N,N-1) - V,
Y(2,2)Iu
•
•
•
• • • •
. ..
$(2,1) -
1_
y(N-3,2))a
The matrix A and vector b from equation 2.9
This algorithm is known as the Multiple-Lag SCA (MLSCA) and was developed by
Subbaram in [48].
2 . 5 . 2 The M o d if ie d M u ller-B u ffin gton A lg o rith m (M M B)
The least-squares algorithm described in the previous section is non-iterative and
the estimates of the phase errors are computed in one step. The pseudoinverse of A can be
computed once and stored since it is independent of the measured data. However, for
arrays containing a large number of elements the calculation of the pseudoinverse can be
extremely difficult and time consuming. For this reason Subbaram's MMB algorithm may
be more attractive.
33
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The iterative algorithm follows Muller's work very closely. However, instead of
maximizing an objective function in the image domain, an equivalent objective function is
maximized in the lag (correlation) domain. The derivation of this algorithm is similar to the
work of Hamaker et al. in [25] since they originally established the connection between
maximization of (2.1) and the equivalent effects in the lag domain.
Consider Muller's objective function and the derivation of the spatial correlation
function of a far-field incoherent source distribution presented in Section 2.1. It was
shown that the spatial (auto)correlation function and the intensity distribution are Fourier
transform pairs. Therefore Muller's objective function can be written as
I2(u) du = 7{I(uM (u)}|x=o
= R(X) *R(X)lx=0
R2(X ) dX
(2.30)
where J denotes the Fourier transform, * represents convolution, and the equality of the
second line in (2.30) is a result of the convolution property of the Fourier transform.
When the aperture is discrete and not continuous, as with a phased array, (2.30) becomes
w i >t=\< « i 2
<2-3»
where L represents the maximum lag included in the calibration process (L < N-l).
Because of the phase errors the measured values of R(Z) are not equal and depend
on both the element index, n, and the lag index, i , as shown in (2.21). An estimate of
R(Z) in (2.30) can be formed by averaging the N-Z different values measured in the array
for a particular lag, Z, as
34
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A
ROO =
X
A
(2.32)
* W ) e'J'(Pn+* ’ Pn)
n=l
Substituting (2.32) into (2.31) yields a lag domain objective function to be maximized that
is the analogous to Muller's image domain objective function
L
MMB
.«
N' z
£
|
1=1
X
a ,
8 (n jl)e 'j(P” +* ' Pn> |
n=l
Using R ( n /) from (2.21) and temporarily neglecting the noise term, 8(n+Z,n), it is easily
A
verified that (2.33) reaches its maximum when (3n exactly equals the phase error Pn.
A
Therefore applying a complex weight with phase -pn to the nth channel successfully
calibrates the array.
W hen the noise term, 8 ( n + / , n ) ,
A
is not neglected
T
(3 « (3 = [ P2>P3»P4>*• *’P n I
at the maximum of (2.33) and the effects of the noise will
be smoothed by the inclusion of the higher-order correlation lags in (2.33). The procedure
A
is iterative because the phase errors are obtained by adjusting and re-adjusting the p's until
the maximum of (2.23) is reached.
The iterative procedure used to maximize (2.33) requires an initial estimate o f the
A
values of the P's. The estimates that the unit-lag algorithm yields are generally used for
this initial guess. The function JMMB is maximized by any desirable method until the
change in the value of the (2.33) is less than some prescribed tolerance.
2 .5 .3
The F u ll a n d P a r tia l M u ltiple-L ag SCA's
Measurement of the N-l unit-lag correlation phases is sufficient to perform self­
calibration. Measuring any high-order lags in addition to the unit-lags provides redundant
measurements that can be used to smooth noise effects. It is not necessary that all the
available correlation lags be incorporated into the multiple-lag self-calibration process. As
many lags as are needed to satisfactorily self-calibrate the system may be used in the
35
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multiple-lag algorithms. The number of lags to incorporate into the algorithms generally
depends on several factors, such as the strength o f the undistorted correlation values and
the SNR per array element. The integration of these (and other) factors into determining
the number of lags to use in the multiple-lag algorithms is discussed in more detail in the
following chapters.
Throughout the remainder of this report the term "full multiple-lag algorithm" will
denote the use of all available correlation lags (L = N -l) with any of the multiple-lag
algorithms. The particular algorithm under discussion will be apparent from the context.
Likewise the term "partial multiple-lag algorithm" will denote the use of only a selection of
the available correlation lags (L < N -l) and the algorithm and lags being discussed will be
apparent from the context.
2.6 Other Spatial Correlation-Based Algorithms
2 .6 .1 The S h ea r A v e ra g in g A lg o rith m
In 1989, J. R. Fienup published another spatial correlation-based algorithm called
the Shear Averaging Algorithm [20] for use with SAR imaging systems. This algorithm
closely resembles Attia's unit-lag algorithm and degenerates to exactly the Unit-Lag SCA
when the rangebin data are range compressed prior to using the algorithm. Assuming the
data have been range-compressed the Shear Averaging Algorithm forms the quantity
K
R ( n , 0 = X £k(n + ^ k ( n)
(2-34)
k=l
which yields Attia's algorithm when i = 1 which is the case generally used. If the data in
the azimuth direction are highly oversampled, then using I > 1 may be beneficial. Like all
the spatial correlation algorithms, this algorithm assumes that a one-dimensional phase
36
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error exists which is independent of the range dimension. The estimate of the phase error
at the nlh dem ent is found from
,n<N-l
(2.35)
p=i
if the first array element is indexed as the zeroth element.
2 .6 .2 The P h a se G ra d ie n t A lg o rith m
The Phase Gradient Algorithm (PGA) was developed by Eichel et al. [19] in 1989
at Sandia National Laboratories primarily for the self-calibration of SAR images corrupted
by unknown phase errors. The PGA solves for the derivative of the phase error profile
across the array and integrates along the array to determine the phase error at the nlh
element. The algorithm iterates between the image domain and the aperture domain until
convergence is reached.
One iteration of the algorithm includes the following steps. The image of each
corrupted rangebin is formed and the most prominent scatterer in each rangebin is shifted to
the origin. A processing window symmetric about the origin is used to window the image
of each rangebin and the inverse Fourier transform is taken to obtain aperture domain data
(2.36)
where ^ ( n ) is the target related phase of the k th rangebin. For a general complex
function
y(x) = |y (x )|eJ')W
(2.37)
the derivative of the phase is given by the identity
37
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d-o(x) _ Im {y*(x)y"(x)}
5------dx
|y(x)f
(2.38)
Equation (2.38) is used to form a least-squares estimate o f the derivative of the phase error
profile
£ l m { g;(x)g;(x)}
P'(x) = -*
i
a
xk(x
k
(2.39)
In the case of an array (sampled aperture) (2.39) becomes
£ l m { g ‘(n)g (n + 1)}
ji(n +1) - jl(n) =
5--------
(2.40)
I g 4(“ )
k
The right-hand side of (2.40) can be shown [2] to reduce to the result of the Unit-Lag SCA
when the phase errors are small enough that sin((3n+1 - (Jn ) » (3n+1 - (3n.
2 .6 .3
The F le a a n d O 'D o n n ell A lgorith m
In 1988 Flax and O'Donnell [21] introduced a self-calibration algorithm into the
ultrasound literature that makes use of the unit-lag spatial correlation measurements to form
narrow transmit and receive beams on a phased array. In the ultrasound imaging field,
phase errors arise because of the different velocities of sound in the different tissue layers
of the human body. The Flax and O'Donnell algorithm calculates the channel phase errors
using the unit-lag SCA, applies the conjugates of these phase errors to the phased array and
re-transmits. On receive, the unit-lag algorithm is again used to calculate the phase errors
and the process is repeated. Flax and O'Donnell claim that with each transmission and
application of the Unit-Lag SCA the shapes of the transmit and receive beams improve
towards a narrow beam.
38
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Using the notation of [21], the authors form the quantity
C(x) = (A(x)A*(x - Ax))
(2.41)
where A(x) represents the signal received at position x in the array, < > denotes an
ensemble average and Ax is taken to be the inter-element spacing. The phase of (2.41)
contains information regarding the difference in time of arrival of the signals at two adjacent
sensors. To calibrate the ith channel, the time-of-arrival differences are summed from the
reference element to the i1*1 element and the total time difference is converted to radians,
conjugated and applied to the ith channel. Although it is never specifically stated that the
unit-lag spatial correlation measurements are being used to calibrate the array, equation
(2.41) is certainly nothing more than (2.11) re-expressed using different notation.
Consequently each stage of the Flax and O'Donnell algorithm utilizes the Unit-Lag SCA to
determine the phase errors.
2 .6 .4 P h a se C lo su re F rom R a d io A stron om y
Phase closure is a concept widely recognized in the radio astronomy field and was
originally coined by Jennison [26,27]. The term refers to a linear combination of observed
correlation phases around a closed loop of three or more antenna elements. The simplest
example of a closure phase is
* i j k = V ' * V + '*'ki
(2-42)
where 4 ^ = 4*^ + [3j - (3k is the phase of the correlation value measured between the j1*1
and the kltl elements; 4*^ is the phase of the true correlation value betweenthe j th and the
k ^ elements and(3j is the phase error in the channel of the j 111antenna element. It is easy to
verify that
^k^ijk^ij+ ^jk+ ^ki
(2.43)
39
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Many self-calibration algorithms in radio astronomy make use of the closure phases to
calibrate an imaging system. One such algorithm due to Schwab [39] implicitly makes use
of the closure phases by minimizing the mean square misfit between the measured (noisy)
correlation values and those predicted from a model, as modified by the phase estimates
R
s - nI Im
AXn,m
_
e K P»~Pu,)R
AXn,m
2 w„.m
(2.42)
The weights can be used to favor measurements that have good signal-to-noise ratios or
exclude those correlation values that are theoretically weak. Minimizing (2.42) is
equivalent to maximizing
S '
=2I
I
n
<2- « )
m
which can be re-expressed as
S '= 2
X
R n.m e"j(^
1-N <£<N -1
J
(2-44)
n
where the weights, wn m, and the model components, R^, have been assumed to depend
only on the difference n-m=£ and the Hermitian symmetry of the modeled and measured
correlation values has been exploited. Equation (2.44) describes a multiple-lag algorithm
of a form similar to Subbaram's MMB algorithm.
2.7 Summary
The goal of this chapter has been to establish the existence of a set of algorithms
that utilize information embedded in the spatial correlation values of a measured radiation
field to self-calibrate an array system. This is a prelude to developing the generalized
algorithm that includes all the algorithms as special cases. The use of correlation-based
algorithms is widespread and encompasses many fields of imaging including microwave
40
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(radar) imaging, ultrasound imaging, radio astronomy, and optics. All of the published
self-calibration algorithms known to the author that utilize information contained in the
measured spatial correlation values have been discussed in this chapter. The remainder of
this dissertation is devoted to generalizing this set into a general form and studying the
performance of this set of algorithms in the presence of additive noise and element position
errors.
When there are no errors in the measurements of the spatial correlation values, there
is no need to use a multiple lag algorithm. However, when the correlation measurements
are in error due either to the effects of receiver noise or insufficient rangebin averaging, the
redundant measurements of the higher-order correlation lags provide an additional
dimension of smoothing.
Consequently, the effect of the noise is lessened by
incorporating the higher-order lags into the calibration process. This is the benefit of the
multiple lag algorithms.
This chapter also provides an understanding of the existing spatial correlation
algorithms. The concept of the random radiation field received from a set of incoherent
sources being a stationary random process and Hamaker's proof connecting Muller's
objective function to the aperture domain are very important to the operation o f the entire
class of algorithms. Together they provide a basis for self-calibrating a microwave imaging
system. The notation developed in the discussions of the Unit-Lag SCA and the MultipleLag SCA will be followed throughout this report.
The forms of the both the Unit-Lag and Multiple-Lag algorithms presented in this
chapter are fundamental to the generalized algorithm developed in Chapter 3. Additionally,
the problems discussed in this chapter, such as the modulo
2k
ambiguities encountered
with the multiple-lag algorithm, are also inherent in the generalized algorithm and are
discussed further in the following chapters.
41
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Chapter 3
The Generalized Spatial Correlation
Algorithm
Each algorithm discussed in Chapter 2 is (1) spatial-correlation based and (2) has
slightly different desirable or undesirable characteristics for a particular imaging scenario.
This chapter develops a generalized spatial correlation algorithm (GSCA) that allows the
class to be characterized and studied as a whole instead of on a individual algorithm-toalgorithm basis. The impetus for the development of the GSCA was the failure of the
Energy Conservation Algorithm (ECA) to perform under certain conditions. Studies show
it to fall into the spatial correlation class of algorithms.
This chapter discusses the development of the GSCA and shows how the existing
spatial correlation algorithms discussed in Chapter 2 are characterized by the GSCA
objective function. The first section o f this chapter digresses slightly to present a
seemingly different self-calibration algorithm; the ECA. It proceeds in this order because
the study of this algorithm leads to the development of the GSCA objective function. The
second section develops a generalization of the ECA and shows it to be a generalized spatial
correlation-based algorithm which is now called the Generalized Spatial Correlation
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Algorithm. The third section demonstrates how the ECA and other already existing spatial
correlation algorithms are embodied by the GSCA and the fourth section discusses the need
for only one efficient maximization procedure for use with these algorithms. The fifth
section is a chapter summary.
3.1 The Energy Conservation Algorithm
3 .1 .1 D e v e lo p m e n t o f the E n erg y C o n serva tio n A lg o rith m
The ECA was developed by Tsao [55],[56] in the early 1980's. The algorithm is
based on Parseval's Theorem which relates the total energy in one domain to the total
energy in another domain provided the two domains are related by the Fourier transform.
In time-series analysis the familiar relationship is
+
oo
+
oo
J Ix(t)l2 dt = J |X(co)|2 dco
- oo
(3.1)
- oo
where X(co) is the Fourier transform of x(t). In imaging applications where the received
radiation field from the k th rangebin is the Fourier transform of the active source
distribution, s^ (u), the relationship is
4*
-f oo
00
J le^(x)l2 dx = J ls^(u)l2 du
- oo
(3.2)
- oo
When a phased array is used to measure the radiation field the discrete nature and finite
extent of the array are represented by a weighting function w(x) = ^ w ( n ) 8(x-xn) where
w(n) is the weight of the n ^ element located at position xn and 8(x-xn) represents the Dirac
delta function. These weights generally include beamsteering weights. Another weighting
function, q(x), is added for calibration purposes. When a passive source distribution is
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
illum inated by a transm itter with beamwidth A9 = Au = u2 - u l5 equation (3.2)
becomes
+
N
00
| I e^ (x)w(x)q(x)|2 dx = ^
le^ (n)w(n)q(n)|2
- 00
n=l
N
=X
n=1
w here
i(n)
rep resen ts
the
J
l ^ ( n) l 2 =
l ^ ( u ) l 2 du
(3.3)
Au
current
obtained
in
the
nlh
channel
and
A
q( x) =
q( n)8 (x-xn) = £ e
n8 (x-xn). The term s^ (u) is the image of source
distribution s^ (u) as seen through the combination of the transmitter and phased array
radiation patterns and can be expressed as
h
= X
®k (x n)e'-i(kuxn+^n) = ^
n=l
e^ (x n) e ^ n e'j(kuXn+^n)
(3.4)
n=l
following the notation of section 2.3.
Figures 3.1 and 3.2 illustrate theconcepts behind the ECA.When
there are no
phase errors in the imaging system, allthe energy inthearray aperture will transform to
within the Au angular sector in the image domain as shown by (3.3). However, when
phase errors are present some of the energy in the image domain will leak outside the Au
angular sector and the final equality in (3.3) will not exist. The ECA objective function
( 3 ,,
k
= l n=l
k=
1 Au
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Array
Image
Processor
With no phase errors
energy is located withn
the illuminated angular
sector in the image
domain
Image Plane
Figure 3.1
Illumination Beamwidth Au
Array
Image
Processor
Energy tails outside
Au because of
aperture Imperfections
Image Plane
Figure 3.2
-45-
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m easures
the
in eq u ality .
W hen
there
are
no
p h ase
erro rs
present
A
Jeca =
otherwise JECA > 0. The vector of phase error estimates, (3, is iteratively
A
tuned by the maximization procedure to the solution, P0, given by
Po = 1 P I J ECA ^ m inim ized }
(3.6)
A
The total energy in the aperture does not change with p because the amplitudes of
the currents are unchanged. Therefore, the summation in (3.5) over the square magnitudes
of the aperture currents is constant. Consequently, minimizing (3.5) is analogous to
maximizing
K
J EE CC AA,, =
,
S
1 h (")l d u
(3.7)
*=1 '
Au
and (3.6) is equivalent to
P0 = t P
j
1ECA2 *s maximized }
(3.8)
The procedure used to maximize (3.7) is given in [48]. The maximization procedure
A
requires an initial value of the vector P Q. The choice of the initial estimate determines
convergence time. Experimentation shows that convergence times can vary by a factor of 2
or 3. The Unit-Lag SCA provides a good initial estimate keeping convergence time to a
minimum.
3 .1 .2
P erfo rm a n ce o f the E C A w ith S im u la ted D a ta
Prior to this work, simulations were made by Tsao in 1983 [55] and in 1986 [56],
Subbaram in 1990 [50], and Patrick in 1990 [33], Figures 3.3 through 3.5 illustrate the
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performance of the ECA in the controlled environments of simulations. In each figure
part (a) shows the undistorted image of a simulated rangebin, part (b) shows the image
with distorted data, and part (c) shows the image after self-calibration. The rangebin of
Figure 3.3 contains a single point source located at uQ = 0. Two nearly equal and closely
spaced point sources are used in Figure 3.4; their locations are u0 i = 0 .0 6 1 4 and
u02 = 0.0372. The performance is excellent; both sources are fully resolved. Figure 3.5
shows the performance for two sources that differ by 20 dB in strength. The weaker
source is in the sidelobes of the stronger source (u0i = 0.0456, u02 = -0.0500). In this
case, too, the performance is highly satisfactory.
For consistency of comparison with the earlier work, the simulation used to
generate the rangebins is similar to the one used by Tsao (and later by Patrick). A 30element linear phased array with an inter-element spacing of two wavelengths was
simulated. A far-field data set consisting o f two point targets uniformly distributed
between Uj = -0.0835 and u2 = 0.0835 in each of ten rangebins was generated. Target
amplitudes were random variables uniformly distributed between 0 and 1 except in the
first rangebin where the second target was assigned an amplitude of zero. Random element
position errors were introduced to distort the array. The element position errors were
uniformly distributed, random variables independent in both the x- and y-directions and
also from element-to-element. The rms element position errors in Figure 3.3 - 3.5 are 7 ;
O
X
X
similar results were obtained for rms element position errors of ^ and
3 .1 .3 P erfo rm a n ce o f the ECA w ith E x p erim en ta l D a ta
Experiments disclosed that the ECA performs differently with high resolution
experimental data obtained from the VFRC laboratory. There are two primary reasons for
this performance. The first is the use of simulated far-field data in all the simulations. The
second is the assumption that only a restricted angular sector of angular extent Au is
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R angebin 01
(a)
u = sin (0)
Rangebin #1
-10
(b)
•20
.25
-30
-0.3
-01
0.3
- 0 .1
u = sin (0 )
Rangebin 01
(c)
-0.1
0
0.1
u = sin (0 )
Figure 3.3 Images of rangebin #1 of simulated data set. (a) Image using
undistorted data, (b) Image using distorted data, (c) Image
using calibrated data.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R angebin # 4
(a)
•25
-0.3
-0.2
LflJL
-0.1
0
0.1
02
0.1
02
u = sin (0 )
Rangebin #4
•10
(b)
•20
•25
-30
- 0.1
u
0.3
sin (0 )
Rangebin #4
-10
(C)
O
&
is
-15
•20
-25
•30
-0.3
- 0 .2
0.2
- 0.1
0.3
u = sin (0 )
Figure 3.4 Images of rangebin #4 of simulated data set. (a) Image using
undistorted data, (b) Image using distorted data, (c) Image
using calibrated data.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Rangebin #6
•10
(a)
I
&
•20
•30
-0.3
02
u = sin(0)
Rangebin #6
•10
(b)
-20
•25
•30
-0.3
0.3
• 0.1
u = sin(0 )
Rangebin #6
-10
(c)
-20
-25
-30
-0.3
0.2
-0 .1
0.3
u = sin(0 )
Figure 3.5 Images of rangebin #6 of simulated data set. (a) Image using
undistorted data, (b) Image using distorted data, (c) Image
using calibrated data.
50-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
illuminated and therefore re-radiates energy. In practice all transmitters have finite-extent
apertures and therefore possess sidelobes which illuminate targets at all angles of arrival.
Even when the illuminated target is in the clear and smaller in angular extent than the
transmitter beamwidth, the illuminated angular sector is the angular size of the target and
not the transmitter beamwidth.
These issues complicate the ECA. The far-field versus near-field issue is covered
in Chapter 5, where it is shown that the ECA does have the capability to calibrate the array
using near-field data provided a mild set of restrictions are satisfied. The effects of the
illuminated angular sector is the focus of this section. These effects did not manifest
themselves until the ECA was applied to experimental data. Recognizing these effects leads
to the development of the GSCA.
The ECA was applied to two types of experimental data. One type was ISAR data
obtained from a Boeing 727 aircraft [13] and the second is a set of high resolution radio
camera data from Phoenixville, Pennsylvania. To produce the ISAR image, a 128-element
array was synthesized with an inter-element spacing of 0.37 meters and 80 rangebins were
collected. The transmitting antenna beamwidth was 30.5 milliradians. The second type of
data set was obtained from a bistatic system with a 1.2 meter transmitting dish and a
synthetic-on-receive, 83-meter phased array. A single receiving antenna was time-shared
at 330 locations with an inter-element spacing of 0.2515 meters; 76 rangebins were
available.
The transmitting antenna beamwidth also was 30.5 milliradians and it
illuminated several streets containing houses in the town. Both systems operated at X-band
with a wavelength of 3.125 centimeters.
Figure 3.6 shows 2-D ISAR images. Part (a) used the SCA to calibrate, (b) the
ECA with Au, the integration interval, made equal to the transmitting beamwidth. The
Unit-Lag SCA phases were used as the initialization phases. Subtle differences exist
between these two images, differences that become more noticeable when a 1-dimensional
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.6 (a)
Figure 3.6 (b)
Figure 3.6 (c)
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slice is taken along one of the rangebins. Figures 3.7a and 3.7b show such slices from
rangebin #39 which contains the dominant scatterer of the data set located in an engine
slightly forward of the tail section. The dominant scatterer appears in the image as the large
lobe near u = 0, and the left wing of the plane appears to its left. One can see from the
images that there should be an interval in bin 39, between the dominant scatterer and the
wing, in which there is no airplane mass. This "null" interval can be seen in the SCA
image of Figure 3.7(a) in the close-in left-hand sidelobes that are between -20 and -30 dB
relative to the main lobe. This region appears about 10 dB higher in the ECA image. In
general, the ECA has filled-in the entire image within the integration interval. This is
exactly what the ECA was designed to do - maximize the image energy within the
specified angular sector. However, this algorithm has implicitly assumed that the radiation
pattern of the transmitting antenna is rectangular across the illuminated sector and that the
angular scatterer distribution is uniform. This is never the case in reality. Because o f this
implicit assumption, the ECA tries to push all of the image energy into the illuminated
sector, and because the objective function is unweighted in this interval, to distribute it
evenly throughout the sector. This is undesirable since all antennas radiate energy outside
of this sector through the sidelobes and targets that have structure, such as the airplane,
violate the assumption of uniform scatterer distribution. The target edges are lost or at least
blurred, and thus the target shape becomes difficult to distinguish because of the assumed
uniform angular distribution of transmitter energy.
Figure 3.7(c) shows an experiment in which Au is matched to the image width of
the dominant scatterer in a diffraction-limited image. In this case Au = 0.66 mrad, which is
on the order of one synthetic aperture beamwidth. Comparison with (a) shows the lobe
due to the dominant scatterer has been streamlined and the "null" region between the
dominant scatterer and the wing has been broadened. This results in a sharper, crisper,
image.
The distant sidelobe level is about -30 dB relative to the mainlobe in
both (a) and (c).
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Uml-Lag SCA Image
L ow i n te n sity re g io n
b e tw e e n e n g in e a n a wing
aE
(a)
•0.0s
-ao4
-ao3
-o.Q2
-aoi
aoi
ao2
ao3
aos
u = sin (0 )
ECA Image
AU
aE
(b)
-0.04
-0.03
-ao2
aoi
aoi
0.02
ao3
acw
u= sin(0)
ECA Image
E
1
•0.025
-0.02
-0.015
-0.01
-0.005
Figure 3.7 Cross-range images illustrating the importance of matching the
angular integration interval in the ECA to the target size.
Dominant scatterer rangebin of 1SAR data set. (a) Unit-Lag
SCA (b) ECA with Au = 30.5 mrads (c) ECA with
Au = 0.66 mrads
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The same phenomena occur with the Phoenixville, PA land data. When the
integration interval is the same size as the beamwidth of the transmitting antenna, there is
no improvement over the SCA image. Figure 3.8(a) shows the image of rangebin #76 (the
dominant scatterer bin) using the SCA. The ECA image obtained by using an integration
interval of 30.5 milliradians is shown in Figure 3.8(b). It can be seen in Figure 3.8b that
the energy within the integration interval has been increased over the SCA image.
However, the distribution of the energy is incorrect for a dominant scatterer. The nearly
uniform distribution of energy within Au, except in the mainlobe, is evident.
Figure 3.8(c) shows the image with an integration interval of 0.66 milliradians. In this
case the calibration has improved, as evidenced by a gain of between 2 and 3 dB in the
peak of the image.
The images obtained with a narrow integration interval are desirable when a strong
isolated scatterer in one rangebin dominates all the other echoes. In this case the simple
DSA is the algorithm of choice. However, there is no way to know a priori when such a
scatterer exists; hence the DSA is not generally applicable. In the next section the ECA is
shown to belong to the class of spatial correlation algorithms; a class that is suited to a
broader set of source distributions.
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Unil-Lag SCA Image
JS
£
(a)
S
- 0.06
-a o 2
0.02
u=si n( 0)
ECA Image
AU
(b)
•0.06
0.02
u=si n( 0)
ECA Image
AU
(c)
*
0.02
■0.06
u=si n( 9)
Figure 3.8 Cross-range images illustrating the importance of matching the
angular integration interval in the ECA to the target size.
Dominant scatterer rangebin of Limerick data set. (a) UnitLag SCA (b) ECA with Au = 30.5 mrads (c) ECA with
Au = 0.66 mrads
-56
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3.2 The Generalized Spatial Correlation Algorithm
3 .2 .1
R e-In terp retin g the EC A O b je c tiv e F unction
The anomalies between the simulation results and the experimental results can be
explained by rearranging the expression of the ECA objective function (3.7) and re­
interpreting the meaning of the objective function. Interchanging the order of integration
and summation leads to
K
(3.9)
Au
k =1
Equation (3.9) can be scaled without affecting the solution given by (3.7) Therefore, the
ECA can be characterized by maximizing the objective function
K
J
(3.10)
ECA
Au
k =1
The integrand in (3.10) is the average image intensity over all the measured rangebins.
Integration over the sector Au can be replaced with an integration over all u provided that a
rectangular windowing function is included in the integrand
K
rect(Au) ^ ^
Is^ (u)l
d
(3.11)
Jk = 1
where rect(Au) is unity within the range of Au and zero elsewhere. The integral (3.11) is
the image energy within the sector Au. It is maximized when the total image energy falls
within Au. In other words, JECA is maximized when the phase distortion in the aperture is
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corrected by the compensation weight vector such that the total image energy is confined to
the illuminated sector.
However, this alone does not guarantee that the image shape is a good copy of the
source distribution. The evidence from experiment indicates that such an objective function
molds the average image intensity to a compromise between the true intensity distribution
and the rectangle function , i.e.,
K
^ X
^*(u)l
Jfc= 1
~ rect(Au)
(3.12)
Examples are shown in Figures 3.7 and 3.8, the ECA is trying to "push" all of the received
energy into the specified angular sector in a manner that approximates a rectangular
function within the sector. Because in practice, sidelobes allow some amount of energy to
leak outside the mainlobe of any antenna, any algorithm that tries to restrict all the received
energy to be within a particular angular sector will not successfully calibrate the array
system. Recognition of this problem leads to the development that follows.
3 .2 .2 C o n cep t o f th e G en e ra lize d S p a tia l C o rrela tio n A lg o rith m
A more general calibration algorithm can be obtained by generalizing (3.11). The
ECA performed well in the simulations for several reasons. First, all of the simulated
radiation fields originated from within a very well defined angular sector. Second, 90% of
the simulated point sources possessed amplitudes of the same order of magnitude and
therefore the effects of any one source did not dominate the combined effects of all the
sources. Thus the pronounced effect of the dominant scatterer was absent from the
simulation problem. Third, the point sources were distributed throughout the entire fieldof-view (FOV). These three factors combined to yield an average intensity distribution that
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was approximately rectangular. Figure 3.9 shows the undistorted average intensity
distribution of the simulations. It is apparent that the energy is confined within the
illuminated angular sector and, except for the region near u = ±0.04, the squared
amplitude (intensity) is roughly of uniform level. Consequently, the undistorted average
intensity distribution roughly approximates a rectangle function and the presence of the
rectangle function in (3.11) is effective.
A verage Im age Intensity
120
100
*oo
3
•0.05
0
0.15
0.05
0.2
0.25
u = sin(0 )
Figure 3.9 Average image intensity of the undistorted simulated data
This suggests that the rectangle function in (3.11) should be generalized to
K
J
GSCA
lA
Is
P(u) 1 Y
k =1
,
p(u) g(u) du
(u )l
du
(3.13)
(3.14)
where the function g(u) represents the average intensity distribution obtained from K
rangebins and p(u) is a general weighting function particular to the characteristics of the
environment being imaged. The algorithm described by this objective function proves to be
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a desirable self-calibration algorithm when each rangebin of the complex source distribution
comprising the scene can be modeled as a realization of a random process.
Let the expected value of the random process, s(u), be denoted by E[s(u)]. Since K
realizations of s(u) are available an average intensity distribution,
K
■
2
(3.15)
g(u) = ^7 £ I Sfc(u)
K A=1
2
can be calculated to estimate the expected value of | s(u) | . The expected value represents a
hypothetical intensity distribution derived from a hypothetical incoherent source
distribution, sh(u).
In other words, the quantity Ejjs(u)|2 j represents an intensity
distribution that would be obtained from an equivalent single rangebin composed of
incoherent sources. If data were collected from such a source distribution, the radiation
field measured in the aperture would be a stationary random process and its spatial
correlation function would be independent of the position in the array at which it is
calculated.
Since g(u) is an unbiased, consistent estimator of E^'
74 '
j, g(u) will
approach E |js(u )|2 j as K —> «».
Assume K is large enough that g(u) = E[js(u)|2 j. Since E^|s(u)|2 j represents the
intensity distribution of an unknown distribution of incoherent sources, its inverse Fourier
transform,
1F_ 1 | e | | s ( u ) | 2 j | ,
measured in the aperture.
is the spatial autocorrelation function of the radiation field
Consequently, J - 1{g(u)} closely approximates the
autocorrelation function of the radiation field originating from sh(u) and is also
approximately independent of the position in the array at which it is calculated. When
phase errors are present g(u) * E |js(u )|2 j regardless of how large K is, in which case
J -1{ g( u) } will not be independent of the position in array. Consequently, the measured
radiation field is no longer a stationary random process.
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If the phase errors are somehow removed, then g(u) once again will equal
E[js(u)|2 ]. Likewise, if by any process, g(u) can be forced to approximately equal
E |js(u)[2
J, then the phase errors will have been removed by that process.
Consequently,
two separate but equivalent methods can be described to remove the phase errors. Equation
(3.13) can be used to mold g(u) to E^ |s(u)|2 j or the measured radiation field can be forced
to be stationary. The mathematical connection between these two concepts and the method
of forcing stationarity on the measured radiation field is described in the following section.
3 .2 .3
D e velo p m en t o f the G en era lized S p a tia l C o rre la tio n A lg o rith m
Let S denote the set of rangebins (realizations) being imaged. If S is illuminated by
a transmitting antenna having a radiation pattern fT(u), the angular reradiation distribution
from the k th rangebin is fT(u)s|t (u) and the received radiation field,
(x), in the array
aperture is ^7"-1 {fT(u)Syt ( u) }. Let fQ(u) denote the beam pattern of the undistorted
receiving array and let f(u) represent the beam pattern of the array with phase errors
present. The corresponding array weight vectors are the inverse Fourier transforms of the
radiation patterns, i.e., wQ(x) = j T ( f 0(u)) and w(x) = J ’ Hfiu)}. The distortion-free
current in the aperture is
iQfc(x) = w 0(x)e^ (x) = w0( x ) J ' 1{fT(u)sJt (u))
(3.16)
The image is related to these currents through the Fourier transform as
s k (u) = j { i fc(x)} = J { wQ(x) ^ _1 (fT(u)s^ (u))}
= f0( u ) * f T(u)s*(u)
where * means convolution.
(3.17)
The unaberrated intensity distribution is the squared
magnitude of (3.17)
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l ok <u ) = |f0(u) * fT(u)sjk(u)|2
Since
(3.18)
(u) is a realization of a random process, the intensity distribution is a random
quantity described by
I0(u) = | f0(u) * fT(u)s(u) | 2
(3.19)
Define p(u) as the expected image intensity distribution, i.e.,
P(u) = E { I0( u)} = E { I fQ(u) * fT(u)s(u) |2 }
(3.20)
Since (3.13) tends to mold g(u) to p(u), defining p(u) in this manner will cause the
algorithm to adjust the phase corrections until the average image intensity distribution is as
similar to the undistorted expected undistorted intensity distribution as possible. This
definition of p(u) eliminates the effects demonstrated by Figures 3.6(b), 3.7(b) and 3.8(b).
Now express (3.13) as an inverse Fourier transform
*
J =
p(u) g(u) du = J ' 1{ p(u)g(u)}1 ^
(3.21)
J
By the convolution property of the Fourier transform
J ' ^ p M g t u ) ) ! ^ = ( J _1{p(u)} * J ' M g(u) } )ljt=0
(3.22)
The first right-hand term is the autocorrelation function of the undisturbed received
radiation field, which is given by
J ' M p ( u ) } = 7 ' 1 { E { | f 0( u ) * f T(u)s(u)| 2 } }
(3.23)
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Evaluation of (3.23) requires knowledge of the second-order statistics of s(u). In general
these statistics are never known exactly and approximations need to be made. These
approximations are discussed in Sections 3.2.4 and 3.3.3.
The second right-hand term in (3.22), 7 ~ l { g (u )}, is
K
7 _1 { g( u)) = J _1 { ^
X
| s k (u)|2 }
k =1
X
= £
^ ( i s ^ u ) ! 2}
(3.24)
*=l
2
where |s^ (u)| is the measured intensity distribution of the image. In the absence of phase
errors, the inverse Fourier transform of |s^ (u)|
2
is the autocorrelation function of the
measured current distribution i^ (x) in the array aperture. As shown in Chapter 2, the
radiation field measured in the aperture is a stationary random process, when the source
distribution is located in the far field, and the corresponding correlation function does not
depend on the location of the measurement in the array. Consequently, the correlation
values of a particular lag are the same no matter where in the array they are measured.
When phase errors are present the measured radiation field is no longer a spatially
stationary random process and the spatial correlation values depend on both the lag number
and the element position number at which the they are measured. Therefore, (3.24) can be
expressed as
K
7 ' l i g( n) } = ^
X M
k =1
n ’Z) = ^ (n’/ )
(3-25)
A
w here R ( n , j t ) is the generalization o f (2.16) to multiple lags and
A
A^
Rk (n,jt) = e^ (n+Z.)e k (n). Since (3.25) should be independent of element position, n,
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when there are no phase errors, an estimate of the unaberrated value can be obtained by
averaging over n as in section 2.4.2 to obtain
R (it) = —
T
R ( n /) e i(Pn+£ ' W
(3.26)
The image domain objective function of (3.21) can now be converted to an aperture domain
objective
^GSCA =
* R^ l L=Q
(3.27)
by combining (3.21), (3.22), (3.23), (3.25) and (3.26), where wp(jt) = J 'M p( u ) } is
the weighting function expressed in equation (3.23). When the convolution operation is
explicitly written (3.27) simplifies to
=
Z
w p(jt) * <*) =
l =-L
Z
W p (o £
t=-L
R ( n / ) e j(P n+i ' ^ n)
(3.28)
n=l
A
where the Hermitian symmetry of R (-t') has been used to eliminate the negative argument.
Therefore maximizing (3.13) is the same as maximizing (3.28) which forces the random
radiation field to be stationary.
Equation (3.28) shows that J GSCA is a weighted sum of the spatial correlation
values measured in the array aperture. The function
is a general form of the
objective functions of all of the currently published spatial correlation-based algorithms.
-64-
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The weighting function Wp(jt) defines the specific algorithms and determines how well the
algorithms perform.
To implement any algorithm characterized by (3.28) Wp(Z) must be determined by
either (1) assuming a form for p(u) or (2) estimating p(u) from the measured data. These
two choices define two sub-classes within the Spatial Correlation class which are discussed
in the following section.
3 .2 .4
The S u b -C la sses o f the S p a tia l C o rrela tio n C lass
The weighting function, wp(jt), in (3.28) is the inverse Fourier transform of p(u).
Recall from (3.20) that p(u) = e {| fQ(u) * fT(u)s(u) I }.
The convolution can be
explicitly written to yield
p(u) = E {
Jf0 (x1)fTs(u-x1) dX] |
f*0(x2)f*s(u-x2) dx2 }
(3.29)
For compactness, the notation has been altered such that frs(u ) = fT(u)s(u) and
*
*
*
likewise, fTs(u) = fT(u)s (u). The product of the integrals in (3.29) can be combined into
the double integral
p (u ) =
jJ
f0 (Xi )f* (x2)fx(u-Xi)fj(u-x2)E { s(u-x1)s*(u-x2) } dX}dx2
(3.30)
Let tj = u-Xj and let ^ = u-x2 such that dtj = -dXj and dt^ = -dx2 . Therefore
p(u)=j J
V u -tl ) fo (u"^)fT (tl)fT<^)E t s( tl ) s* ('2 )} d ti d i2
(3 .3 i )
The simplest source model of radar clutter is a random distribution, independent in cross­
range, in which case E{s(tj)s*(<2)} ~ I(ti)8(t]-t2) and p(u) can be expressed as
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P (u) =
JJ
f0 (u -t1)f*(u-i2)fT( ti) 4 (t2)I(t1)5(t1-t2) d f 1 dt 2
= | I f0(u-*!) l2lfT( ti) l2 I(t1) d t 1
= lfT(u)l2 1(u) * lf0(u)I2
(3.32)
from which the weighting function, wp(Z), is obtained by taking the inverse Fourier
transform of (3.32)
w p(Z) =
-1 { p (u )} = f ~ l { lfT(u)l2 I(u ) * lfg(u)12 )
= [ y 1 {lfT(u)l2 } * J ' H I(u) 1 ] J ' l { lf0(u)l2 )
= [ J _1( lf T(u)l2 } * R (Z) ] 7 ~ x { lf0(u)l2 }
(3.33)
The weighting function depends on the spatial correlation function R(j£), which is
normally unknown. At this point two paths can be taken toward further evaluation of the
weighting function. One route is to assume a particular form of the intensity distribution
and therefore of R(jt). Another route is to substitute an estimate of R(Z) into (3.33). Both
of these methods yield viable algorithms each possessing advantages and disadvantages
over the other. These advantages and disadvantages along with the particular algorithms
obtained by the different methods are discussed in Section 3.3. Each method forms a sub­
class of algorithms as shown in Figure 3.10A. These two sub-classes have been termed
the parametric and non-parametric sub-classes respectively since one requires the choosing
o f a model (the form of the intensity distribution) and the other makes no such
assumptions. Briefly, the performance of the model-based form is somewhat superior
provided that the model is accurate. Future references in this document to these two sub­
classes refer to the manner in which (3.33) is evaluated.
-
66
-
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SCA Class
Parametric
Sub-Class
Non-Parametric
Sub-Class
A model is assumed for the
intensity distribution, I(u),
to determine Wp(jt).
No model is assumed.
Instead an estimate
ofR O t) is used.
Examples
MMB
ECA
ULSCA
Shear Averaging
Phase Closure
Flax-O'Donnell
Figure 3.10A The structure of the Spatial Correlation class as revealed by the GSCA and examples of
existing algorithms in the particular sub-classes.
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3.3 The Relationship of the GSCA to Existing
Algorithms
3 .3 .1
R evisitin g T sao's E n erg y C o n serva tio n A lg o rith m
The fact that (3.13) represents a general form of the objective function used by Tsao
and Patrick can be easily seen by setting p(u) equal to a rectangle function. The keen reader
will note that the resulting objective function, (3.10), does not necessarily imply a
rectangular transmit pattern. That is, the convolution of fQ(u) with the product of fT(u) and
s(u) is rectangular, but the transmitter pattern is not necessarily rectangular.
Consider the general form of p(u) from (3.32). If the intensity distribution is nearly
uniform within the angular sector illuminated by the transmitter and the transmitting antenna
possesses a rectangular radiation pattern then the product lfT(u)l2 I(u) will also be nearly
rectangular with an extent equal to the extent of the transmitting antenna radiation pattern.
In this case (3.32) becomes
p(u) = rect(AuT) * lf0(u)l2 = lfT(u)l2 * lfQ(u)I2
(3.34)
High-resolution imaging requires the receiving aperture to be large enough to produce a
very narrow receive beam. Generally, this receive beam is at least an order of magnitude
narrower than the angular sector illuminated by the transmitting antenna. Thus, in the
convolution in (3.34), the undistorted receiver array pattern fQ(u) will act essentially as an
impulse in u, in which case rect(AuT) * lfQ(u)I2 « rect(A uT) = lfT(u)l2. This effect is
demonstrated in Figure 3.10 where a rectangle function (dotted line) has been convolved
with a sine2 function (dashed line). The result is shown by the solid line in the figure. The
mainlobe of the sine2 function is approximately one-tenth the extent of the rectangle. The
convolution produces a function nearly rectangular in shape.
-
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O f course, even in simulation the function f0(u) is not exactly an impulse, and
consequently, the function p(u) is not precisely a rectangle function. In fact the complex
source distribution s(u) will modulate the rectangular transmitter pattern, and the
convolution with the finitely narrow, undistorted receive beam f0(u), will cause the
rectangular transmit pattern to spread in angle and deviate in shape from a rectangle.
However, because the spreading of the transmitter pattern is negligibly small due to the
narrowness of the undistorted receive pattern, and the magnitude of the distortion is small,
the function p(u) can be closely approximated by a rectangular function with an angular
extent equal to that of the transmitter pattern. This approximation yields a weighting
function p(u) = lfT(u)l2. Assuming that fT(u) is normalized, replacing p(u) with p(u) in
(3.13) yields the ECA objective function given by (3.10). If we view (3.13) as a GSCA
objective function, it is not quite the correct objective function for the particular simulations
of Tsao and Patrick. However, the sidelobes of the receive antenna power pattern produce
negligible leakage of energy from the mainlobe of the transmit power pattern and (3.10)
proves to be an excellent approximation to (3.13) in this case. Therefore, the results
observed by Tsao and Patrick do not contradict the GSCA and, in fact, should be expected
even though the objective function used is not the exact objective function predicted by the
GSCA theory.
This theory is supported by part of the discussion of the ECA in [48]. In the
process of deriving a method of maximizing (3.10) the authors manipulate the function
J eca int0
form
J ECa
=Z Z
P(m.D ei( ? " '
m=l n=l
^
(3.35)
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Convolution of N arrow Sine Squared with W ider Rectangle
1.4
1.2
0
1
0.8
0.6
£
0.4
0.2
0
20
40
60
80
100
120
140
S am ple Index
. 2
Figure 3.10. The result of the convolution of a rectangle function with a sine'
function. The solid line represents the final result.
where
K
A /,
\vA>*
* , ,
n(
\
\ j Ok (u2‘ ul) (xrTxm) • fk(u2"u l) (x n"xm)l V »
p(m ,n) = (u2-u ,) e 2
sin e ----------^-------------2*i e * (xm)e * (xn)
1
J
k=l
(3.36)
If a ^ scale factor is introduced into (3.35) then p(m,n) can be expressed as
j \ k (uj-uj) (xn-xm) . |'k(u2-u1)(xn-xrn)i a
sine
R(xm, xn)
p(m,n) = C e 2
(3.37)
by letting C = (u2-Uj) and
K
A
R( x m- x n) =
k
S
k=\
(3.38)
The function described by (3.35) can be re-mapped to include a lag index in the following
manner. Let m = n+1 such that I = m-n. Therefore, equation (3.35) becomes
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N
J'ECA
eca =
N-n
1 I
P(n+Z,n) ej(l^n ' ^ n+l)
(3.39)
n=l ji=l-n
The terms in (3.39) can be re-grouped and the objective function, JEq a , can be re­
expressed as
.
J g c A - l i ec <
ej
1
<“«><*»-»>
■ *»>
1=1 n=l
(3.40)
N-l
N -i
=X
w ( / ) X R (n, / )
1=1
n=i
N-l
=S
L=l
(3.41)
N-l
w O O R (/)
= ]L
wp(Z)R
(Z)
>
(3.42)
1=1
where the weighting function wp( / ) is a sine function which is the inverse Fourier
transform of the rectangle function rect(Au) in (3.10) as predicted by the GSCA theory.
Consequently, evidence that the ECA is a member of the Spatial Correlation class has
existed since the writing of [48] although it has not been recognized.
3 .3 .2
U se o f A N a rro w In terva l o f In teg ra tio n w ith E x p erim en ta l D a ta
Another anomaly that has been previously observed and discussed in section 3.1.2
is the high quality images obtained from the experimental data when the interval of
integration in (3.14) is reduced to the size of an undistorted receive beamwidth. The ECA
requires the integration interval to be the size of the transmitter beamwidth. However, the
images obtained by reducing the integration interval to the size of an undistorted receive
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beamwidth (nearly two orders of magnitude smaller) are superior to those obtained from
the ECA integration interval (Figures 3.6 (b) and (c)).
One of the most perplexing aspects of this phenomenon was that it was observed
with experimental high resolution microwave data, but not in the simulated data used by
Tsao and Patrick. In fact, when the integration interval was reduced in the simulations, the
images obtained were noticeably inferior to the ones obtained when an integration interval
the size of the transmitter beamwidth was used (see Figure 3.3 part (c), and Figure 3.11).
This effect can easily be explained by again considering the GSCA objective
function (3.13).
I m a g e o f R a n g e b i n #1 , F O V = 0 . 0 1 7 2 4
-10
-30
-35
' 4-0.25
-0.2
-0.15
-0.1
-0.05
0.05
0.15
0.2
0.25
u = sin(0)
Figure 3.11 Image of rangebin #1 of simulated data after calibration with
Au = 17.24 mrads. The transmitter beamwidth is 167 mrads.
Consider the final form of (3.32) and let I(u) = a + ^rect(Au') over the angular
interval Au > Au' where Au represents extent of I(u) and Au' represents the extent of the
rectangle function of amplitude !A.
p ( u ) = |f T(u)| [ a + j?rect(Au') ] * | f 0(u)|
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^rect(A u')
If A > a then (3.33) can be approximated by
p(u) = ^rect(Au') |fT(u)| * If0(u)I
(3.44)
Equations (3.43) and (3.44) model a situation where a Au' sector of the intensity
distribution is significandy stronger in square magnitude than the rest of the distribution. If
the width of the Au' sector is extremely small such that rect(Au') = 8(u-b) then the
function p(u) becomes
(3.45)
which shows that the appropriate weighting function is a scaled form of the product of the
power patterns of the undistorted receiving array and the transmitter array pattern. Ignoring
the scale factors the weighting function can be taken to be the power pattern of the
undistorted receiving array.
This situation models a data set containing a dominant scatterer. To calibrate such a
data set p(u) should take the form of the power pattern of the undistorted receiving array
steered to the location of the dominant scatterer, which is not known.
It turns out that the dominant scatterer location is not necessary to successfully selfcalibrate the array. Recall that (3.13) can be converted into an aperture domain objective
function through the Fourier transform and that the weighting function wp(jt) is the inverse
Fourier transform of p(u). When a dominant scatterer is present in the data set wp(jt)
evaluates to
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= ^ if T(b)|2 eJDfc f l { |f0(u )T
)
(3.46)
If the true location of the dominant scatterer is u=b0 and not u=b, then g(u) = lf0(u-b0)|2
and
{g(u)} becomes
= e jb°^ J l { If0(u )|2 }
(3.47)
and the aperture domain objective function develops a linear phase shift in t which does
not affect the calibration procedure.
Examples of the effects of a dominant scatterer on the function g(u) are shown in
Figure 3.9 and Figure 3.12. Figure 3.9 shows the error-free average image intensity, g(u),
of the simulated data of Tsao and Patrick in which all of the sources are of comparable
amplitudes. The extent of g(u) is approximately the field-of-view of the transmitting
antenna. If p(u) is approximated as a rectangle function the extent of p(u) should be made
equal to the transmitter beamwidth, yielding the ECA. The results of Tsao and Patrick
corroborate this theory. However, when one of the sources dominates the other sources,
the averaging process in g(u) is captured by the image of the dominant source and takes the
shape shown in Figure 3.12. Clearly, the rectangle that now best approximates the average
image intensity is one with a width equal to one beamwidth of the receive array. In this
case the GSCA theory yields an algorithm that is not the ECA.
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Average Image Intensity
120
100
o
T3
3
D.
E
<
•S
•0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
U = si n ( 0 )
Figure 3.9 Average image intensity of the undistorted simulated data with
the original source amplitudes. (Reprinted from page 58)
A v e r a g e I ma g e I nt ens i t y, g(u), wi t h Or i g i na l So u r c e A m p l i t u d e s
12000r
O.
6000
S3
4000
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
u = sin(9)
Figure 3.12 Average image intensity of the undistorted simulated data with
a dominant scatterer. Amplitude is 13 times the next largest
scatterer.
-7 5 -
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The amplitude of source #1 in the simulated data was originally 0.9 corresponding
to the average image intensity in Figure 3.9. After calibration using an narrow integration
interval (17.24 mrads = one array beamwidth) its image is shown in Figure 3.11. When
the amplitude of 0.9 was changed to 10.9 corresponding to the average image intensity in
Figure 3.12, the image improves as shown by Figure 3.13.
If the source amplitude is maintained at 10.9 and the integration interval is increased
to the size of the transmitter beamwidth (167 mrads) the resulting image (Figure 3.14) is
again inferior to the image obtained with a narrow integration interval (Figure 3.13). The
wide integration interval represents a p(u) with extent the size of the transmitter beamwidth
and the algorithm molds g(u) to best approximate this wide extent. Figure 3.15 shows the
final average image intensity. This image more closely approximates the wide rectangular
p(u) than does the function of Figure 3.12. Table 3.1 summarizes these results. Items in
boldface are results of combinations predicted from GSCA theory.
Table 3.1 Summary of Results with Simulated Data
Type of Intensity
Distribution
Narrow Integration Interval
Wide Integration Interval
Dominant Scatterer in FOV
Good Image
Poor Image
Poor Image
Good Image
Uniform Intensity
Distribution
Figures 3.16 and 3.17 demonstrate that the angular coordinate of the dominant
scatterer need not be known to successfully self-calibrate the array. Figure 3.19 shows
images of rangebin #'s 1,4, and 6 of the simulated data with the rectangular function p(u)
centered about u = 0.0965. In Figure 3.17 the interval is centered at u = -0.0835. The
array self-calibrates in both cases and the images are similar.
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I m a g e o f R a n g e b i n #1 w i t h D o m i n a n t S c a t t e r e r
•
10
‘5
CO
Q
-25
-30
-35
-40
•0,1
-0.05
0.05
0.1
0.15
0.2
0.25
u = sin(0)
Figure 3.13 Image of rangebin #1 with a dominant scatterer included in the
averaging process
I m a g e o f R a n g e b i n #1 wi t h D o m i n a n t S c a t t e r e r
-10
-15
•20
-25
-30
•35
.4 0 1 11.11 II II.. II 11 U II
-0.25
-0.2
-0.15
I
u J
-0.1
-0.05
.-------------tJ — L IL 1 L J_ 1 L .1 1
0
0.05
0.1
0.15
11, 11 II 11 II
0.2
0.25
u = sin(0)
Figure 3.14
Image of rangebin #1 of the simulated data set with
Au = 16.7 mrads and a dominant scatterer present in the
data set.
77-
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A v e r a g e I m a g e I n t e n s i t y A f t e r Ca l i b r a t i o n
7000
6000
*2
5000
4000
% 3000
2000
1000
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.15
0.2
0.25
u = sin(0)
Figure 3.15 Average image intensity after calibration. The function g(u)
has been molded to better resemble the rectangle function p(u).
-7 8 -
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Rangebin # 1
*10
•15
(a)
8
Q
-20
-25
•35
•0.05
U =
Q
0.05
sin(0)
0.15
02
025
0.15
02
025
0.15
02
025
R a n g e b in #4
•10
(b)
•25
•30
-35
.4oM h i ii ii u u i u y ii i i
-0.25
-02
-0.15
-0.1
-0.05
y
0.05
u ss s i n ( 0 )
R a n g e b i n #6
-10
-15
(C)
^
O
-20
*25
-30
-35
-40
•0.05
sin
Figure 3.16 Images of three rangebins of simulated data with integration
interval equal to [13,180] mrads.
79-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R a n g e b i n #1
-10
(a)
•25
•35
•0.05
Q
0.05
u = sin(0)
0.15
02
025
0.15
02
025
0.15
0.2
025
R a n g e b i n #4
•10
(b)
-25
•30
•35
.4 0 U 1-H II, II! I, I
-0.25
-0 2
-0.15
---------------■--------- I I I I. I
-0.1
-0.05
0
0.05
u = sin(0)
I II, I
0.1
R a n g e b i n #6
•10
(C)
’§
Q
•20
-25
-30
•35
-40
sin
Figure 3.17 Images of three rangebins of simulated data with integration
interval equal to [-167,0.0] mrads.
-
80
-
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The same results are found with the experimental data. The data set consists of
radar returns from a Boeing 727 aircraft flying into Philadelphia International Airport and
was obtained using inverse synthetic aperture techniques described in Sections 1.3 and 1.4
and in [40], at a range of approximately 2.7 km. A very strong specular echo comes from
an engine near the tail of the plane. Figure 3.18 shows the average error-corrected image
intensity of the experimental data set. This figure demonstrates that a dominant scatterer is
present in the data set. There is approximately an order of magnitude difference between
the intensity of the dominant scatterer and the next strongest source. When the dominant
scatterer rangebin is not included in the averaging process, the average error-corrected
image intensity is as shown in Figure 3.19.
The strongest source is now only
approximately 2.5 times as large as the next strongest source in intensity. In this case the
angular extent of g(u) has become more appropriately the width of the total field-of-view
since the intensity distribution is more closely modeled by the distribution of
Section 3.3.1.
Figure 3.20 shows the calibrated image of the dominant scatterer rangebin of the
ISAR data set. This image was obtained after calibrating the system using an integration
interval of Au = 0.66. The dominant scatterer rangebin was excluded from the calibration
process. The need to reduce the integration interval is evident when Figure 3.20 is
compared with Figure 3.7(c). Both figures used the same integration interval but Figure
3.7(c) includes the dominant scatterer rangebin in the calibration process while Figure 3.20
does not. Note that the lack of knowledge of the location of the dominant scatterer did not
hamper the self-calibration process.
-81 -
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xio3 A v e r a g e I m a g e I n t e n s i t y o f E x p e r i m e n t a l Data
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
u = sin(0)
Figure 3.18 Average image intensity of the ISAR data when the dominant
scatterer rangebin is included in the averaging process
xio5
A v e r a g e I m a g e I n t e n s i t y o f E x p e r i m e n t al Data
0.01
0.02
0.03
0.04
0.05
u = s in ( 0 )
Figure 3.19 Average image intensity of the ISAR data when the dominant
scatterer rangebin is not included in the averaging process
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82
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I ma ge o f D o m i n ^ ^ ^ ^ ^ e r e r Ra ngebi n
-0.025
-a ra
-0.01s
o.oi
-0.005
o
0.005
0.01
a o is
00 2
0025
u = sin(0)
Figure 3.7(c) Cross-range image of the dominant scatterer rangebin of ISAR
data set with Au = 0.66 mrads and the dominant scatterer
rangebin included in the averaging process.
(Reprinted from page 53)
I ma ge o f D om in an t Sc att ere r Ra ngebi n
1---------- .---------- .---------- ,---------- .---------- ,---------
7 0 1--------- 1
-----------
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
u = sin(0)
Figure 3.20 Cross-range image of the dominant scatterer rangebin of ISAR
data set with Au = 0.66 mrads and the dominant scatterer
rangebin absent from the averaging process
-83-
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Both the experimental and the simulated data agree with the theory of this section;
i.e., when a dominant scatterer is present in the field-of-view, the integration interval
should be reduced to the size of the receive array beamwidth and the image quality is
enhanced.
The GSCA explains the high quality images obtained from the experimental data
along with the low quality images obtained from the simulated data when a narrow
integration interval is used. It also suggests a method of determining when to reduce the
integration interval to one array beamwidth. The method is borrowed directly from the
DSA. Since the absolute location of the dominant scatterer does not need to be known, the
problem is detecting only the presence of a dominant scatterer. This can be done by
measuring the normalized amplitude variances of each rangebin as [29]
2
/
(3.48)
k
where N is the number of elements in the array. A rangebin with
< 0 . 1 2 contains a
k
dominant scatterer [24] and the integration interval must be reduced to the size of one array
beamwidth.
The DSA often is also applicable when (3.48) is less than 0.12. It is also much
simpler to implement. However, algorithms belonging to the Spatial Correlation class are
applicable under broader conditions than the DSA. The GSCA theory shows that any
algorithm within the Spatial Correlation class can be realized by altering the weighting
function Wp(jt). This is discussed in Section 3.4. Also, Chapter 4 shows that DSA
performance is attained by the spatial correlation algorithms even when a dominant scatterer
is present. Therefore, since the spatial correlation algorithms are applicable in more general
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circumstances than the DSA, and since there is no loss in performance level when a
dominant scatterer exists, use of the spatial correlation algorithms is preferable.
3 .3 .3
D e riv in g th e M o d ifie d M u ller-B u ffin gton A lg o rith m fr o m the G SC A
The objective function of (3.13) is a generalized form of the objective function of
the Modified Muller-Buffington Algorithm [49]
J MMB = t | R « ) f = i R « ) R ‘ (<)
f=l
(3.49)
f=l
which is the same as (3.28) if wp(Z) is set equal to R(Z). An important concern is what
assumptions are made in arriving at (3.49).
The function p(u) is a reference intensity distribution to which the average measured
intensity distribution, g(u), is to be matched. The reference intensity distribution is
obtained from the hypothetical incoherent source distribution, sh(u), as discussed in
Section 3.2.2. This intensity distribution contains no phase errors and is affected only by
the windowing of the array aperture. Consequently, p(u) is a diffraction-limited intensity
distribution of incoherent sources having an inverse Fourier transform that is the spatial
correlation function, Rh(Z), of the random radiation field. Therefore, Rh(Z) is the proper
weighting function, wp(jt), for the objective function of (3.28), i.e.,
R h( / ) = wpU ) = 7 ’ 1{ E { | f 0( u ) * f T(u) s(u)| 2 } }
(3.50)
and (3.26) can be expressed as
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L
L
£ w pOOROt) = X
t= L
R hOO
R ( l )
(3.51)
i=-L
In practice Rh(j£) is not likely to be known a p rio ri. In the MMB Rh(jt) is estimated from
the data as R(Z) and its substitution for wp(jt) in (3.28) yields the MMB objective function.
Therefore, the MMB can also be derived from the same generalized objective function and
is an example of an algorithm belonging to the non-parametric sub-class. No assumptions
are made on the form of p(u) or wp(jt). Instead wp(Z) is estimated from the data set
thereby requiring less information regarding s(u) than any algorithm from the parametric
sub-class. The trade-off is a performance that is slightly inferior to that of a parametric
algorithm, as shown in Chapter 4. However, the performance difference spans only 1 to 2
dB. Consequently the MMB is one of the more attractive algorithms in the spatial
correlation class since it is widely applicable and performs well.
3 .3 .4 The U n it-L ag S C A a n d the G SC A
The spatial correlation class as described by the GSCA objective function is
inherently a set of multiple-lag algorithms. Within the GSCA structure the ULSCA
materializes in one of two ways; (1) as a member of the parametric sub-class with
(3.52)
or (2) as an algorithm resulting from compensation for near-field conditions. The latter
situation is discussed in detail in Chapter 5 while the former is the subject of this section.
The function wp(jt) can be arbitrarily truncated to conform to (3.52). Since there is
no redundancy in this algorithm the noise performance is inferior to the multiple-lag
algorithms but is computationally simpler than any multiple-lag algorithm.
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It is important to note that the ULSCA belongs to the parametric sub-class. The
non-parametric sub-class also contains unit-lag algorithms. These are obtained by setting
L = 1 in (3.28). It is shown in Chapter 4 that these algorithms do not yield the ULSCA
solution.
3.4 Commonality of the Maximization Procedures
GSCA theory shows that a single objective function can be used for all the spatial
correlation algorithms. Yet each algorithm was developed with a customized method to
determine the estimates of the phase errors. The ULSCA forms a "running summation"
[3], the MLSCA makes use of a least squares solution [48], and the MMB maximizes an
objective function [29], as does the ECA [48].
There is no need, however, to customize a different maximization procedure to each
algorithm. Any of the existing procedures can be used to maximize the objective function
of any of the spatial correlation algorithms. A procedure that exploits the nearly sinusoidal
variation of the ECA objective function as a function of the phase error estimates Pn and
efficiently solves for the set of estimates is given in [48]. Since the ECA is now known to
belong to the spatial correlation class, all of the objective functions characterized by the
GSCA objective function can be maximized by this efficient procedure. The maximization
procedure designed for the MMB can also be used to efficiently maximize the GSCA
objective function. In conclusion, a single efficient maximization procedure can be
hardwired without restricting the set of spatial correlation algorithms available to the signal
processor.
The change from one spatial correlation algorithm to another can be
implemented simply by changing the general weighting function wp(jt).
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3.5 Summary
This chapter shows that, with the exception of those algorithms that use strong
point scatterers to cophase antennas, all published self-calibration algorithms belong to a
common (spatial correlation) class and that a single objective function leads to the estimates
of the phase errors.
Anomalies between simulation and experimental results with the Energy
Conservation Algorithm have indicated the theory of Tsao to be inadequate to self-calibrate
an array imaging system under general conditions. While simulation results verify that the
ECA is a viable self-calibration algorithm when the transmit radiation pattern is rectangular
and the field-of-view is uniformly filled with scatterers, the experimental results
demonstrate its weakness under more general and realistic conditions.
The ECA objective function (3.8) can be manipulated into the more general form
(3.10). The two components, p(u) and g(u), are interpreted as an expected image intensity
distribution and an average image intensity distribution respectively. The generalized,
image domain objective function can be Fourier transformed to an aperture domain
objective function composed of a general weighting function wp(jt) and an estimate of the
A
spatial correlation function of the measured radiation field, R (i). Therefore, the ECA and
its generalization are spatial correlation algorithms with a weighting function that depends
on the statistical properties of the source distribution. The objective function is a general
form of the objective functions of all of the existing spatial correlation algorithms.
Any particular algorithm within the Spatial Correlation class is determined by the
form of the general weighting function wp(jt). The ECA, the MMB algorithm, and the
ULSCA have all been shown to be specific cases of the GSCA.
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The GSCA theory has been used to explain the anomalies between the simulated
and experimental data. It has been shown that the ECA weighting function is incorrect
when a dominant scatterer is present as in the experimental data. The theory tells us that the
correct weighting function in u-space is one proportional to the receive array power pattern.
A simple method of determining when to change the ECA weighting function has been
presented. This method borrows the technique of detecting the presence of a dominant
scatterer from the DSA. Since it is only necessary to detect the presence of a dominant
scatterer and its location is not needed this is also a viable technique for determining when
to alter the ECA weighting function. Tests with both simulated and experimental data have
shown this theory to be correct.
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CHAPTER 4
Noise and Element Position-Error
Performance Analysis
In this chapter computer simulations and first-order analysis of the GSCA objective
function are used to assess the merits of algorithms within the two spatial correlation sub­
classes. The algorithms are evaluated based on their performances in the presence of: (1)
element position errors and (2) element receiver noise. The evaluation is used to determine
the usefulness of the algorithms from one sub-class relative to algorithms in the other sub­
class. It is found that although the performance of the MMB is inferior to the performances
of the parametric algorithms, the lack of need for a priori information generally outweighs
the performance difference and makes the MMB widely applicable.
Section 4.1 presents the results of the element position error evaluation. Such
errors induce phase errors that are functions of angle of arrival and therefore cause a loss in
the main beam gain that depends on the angle to which the beam is scanned. The loss in
the main beam gain is also a function of several other variables, namely the size of the fieldof-view illuminated by the transmitting antenna, ufov, the rms position error in the locations
of the array elements, a Ax, and the number of correlation lags L incorporated into the
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calibration process. Consequently, the algorithms are evaluated by the maximum angle to
which the reconstructed beam can be steered without incurring a designated loss. The loss
is taken as 1 dB following Steinberg's evaluation of the DSA in [45].
Section 4.2 presents the results of the element receiver noise evaluation. The
algorithms were tested with two different source distributions; one corresponding to the
uniformly filled field-of-view used in the simulations of Chapters 3 and 4, and the other
corresponds to the uniformly filled field-of-view with a dominant scatterer present. The
latter source distribution corresponds to the dominant scatterer distribution also used in
Chapters 3.
Section 4.3 presents an analysis of the GSCA objective. The analysis is used to
explain the results obtained from the element receiver noise evaluation. At the same time,
the analysis connects the algorithms of the parametric sub-class with the MLSCA of [48]
when the receiver noise power is small. Under this condition the algorithm from the
parametric sub-class with weighting function w0( l ) yields the same solution as the
MLSCA with a weighting matrix G (discussed in Section 2.5.1) composed of the weights
of w0(/) .
4.1 Element Position Errors
4 .1 .1 In tro d u ctio n
If the exact relative locations of the elements of an array are known then appropriate
phase shifts can be calculated to form and scan a beam. If the exact element locations are
not known then only an estimate of the appropriate phase shifts can be made based on the
assumed relative locations of the array elements. Using the assumed locations instead of
the exact locations introduces phase errors into each channel that will degrade the quality of
the main beam and raise the sidelobes.
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The relative phase error introduced into the n1*1channel from a point source located
at an angle 0 measured from broad-side is calculated from the array geometry as [45]
cpn(0) = kAxn sin(0) + kAyn cos(0)
(4.1)
where Axn and Ayn are the differences in the x and y directions, respectively, between the
assumed relative location of the nth element and its actual location. The wavenumber, k,
converts the magnitude of the distance error to a phase error. When more than one point
source is present in a rangebin the total phase error (i.e., the difference between the phase
of the signal that would have been received if the element was located at the assumed
location and the phase of the signal that is actually received) in the n111channel is a nonlinear
function of the phase errors produced by the individual point sources (4.1) and the
dependence on 0 shown in (4.1) causes the total phase error to vary with 0 . If the
illuminated sector (i.e., the field-of-view) is sufficiently small, the cosine term in (4.1) will
be approximately unity for all point sources regardless of their angular location. The sine
term in (4.1) will vary approximately linearly with 0 with a slope of unity. Therefore, if 0
varies over a small range, sin(0) also varies over a small range and the phase error in the
nlh channel will appear as a constant phase error corrupted by noise. Therefore any of the
spatial correlation algorithms are able to calibrate the imaging system to yield an improved
image since an effective constant phase error exists in each channel. Also, because the
effective phase errors will appear embedded in noise, any of the multiple-lag algorithms
provide better performance than any algorithm employing only unit-lag correlation
measurements.
The effect of element position errors on the spatial correlation class of algorithms
has not been previously studied. Because of the nonlinear relationship between the element
position errors, Axn and Ayn, and the total phase error in the channel, mathematical
analysis is limited and does not yield useful design information. This section provides a
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quantitative description of the performances of the two algorithms of the Spatial Correlation
class through the use of extensive computer simulations.
In studying the effects of random element position errors on the performance of the
DSA, Steinberg [45] considered the expected relative loss in the main beam gain as a figure
of merit and derived a theoretical curve for this loss as a function of scan angle and the rms
element position error in the x-direction. Following [45], the expected loss in the main
beam is also taken as the figure of merit in this section for the spatial correlation algorithms.
With a dominant scatterer the performance of the parametric algorithm is identical to
the theoretical performance curve o f the DSA even when the unit-lag algorithm is used.
The non-parametric algorithm performance is inferior and requires the use of multiple lags
to achieve the DSA performance level. Without a dominant scatterer neither algorithm
achieves the DSA performance level but the performance curves of both algorithms can be
approximated by the DSA performance curve shifted by a constant loss term.
4 .1 .2 The E lem en t P o sitio n E rro r Sim ulation
The simulation used throughout Section 4.1 was used earlier in Chapter 3. The
elements formed a linear array with an interelement spacing of two wavelengths. Each
element was displaced from its assumed position by a random amount, Axn and Ayn . The
quantities Axn and Ayn are assumed to be independent random variables uniformly
distributed within
X X
2
’
2
Y Y
and
’
2 2
respectively, where X and Y are determined by
’
specifying the desired rms position errors, o Ax and o Ay, and using the relation
<JAxV l2 = X for zero mean, uniformly distributed random variables. A pair of iid random
variables (representing the x and y element position errors) were generated for each element
in the array and for each realization. Twenty-five realizations were generated in each
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simulation and an estimate of the expected relative gain curve was calculated by averaging
over the 25 realizations.
In each realization an algorithm from the parametric and non-parametric sub-classes
was used to estimate the phase errors present in the imaging system. The MMB algorithm
was used as the representative of the non-parametric sub-class. This algorithm was chosen
because it forms an unbiased estimate of wp(jt) and is a multiple-lag algorithm. Its
performance should be one of the best of the algorithms within this sub-class. From the
parametric sub-class a set of algorithms must be chosen since wp(jt) must change with the
intensity distribution. Since the field-of-view was filled with uniform scatterers and
sidelobes were not simulated on the transmitting antenna, the function p(u) was set equal to
a rectangle function with extent equal to the field-of-view. This corresponds to a sine
function for wp(Z). The estimates of the phase errors were then subtracted-out of the
respective data set and images were formed of a point source located at 10 angular positions
equally spaced between 0.0 radians and 0.1 radians. The peak of the main lobe in each
image was averaged over the 25 realizations and the estimate of the expected loss was taken
as the normalized averaged value.
To check the sufficiency of averaging over 25 realizations to approximate the
expected value, the simulation was first run using the DSA with parameter values
corresponding to those of an actual experiment performed at the VFRC field site in Valley
Forge, PA. The experiment, described in [45, pg. 193], was conducted at L-band (30 cm
wavelength). A 27 meter array composed of 15 randomly located elements was used as the
receiving aperture. The variance of the x-direction element position error (the y-direction
position errors affect the calibration process but the x-direction position errors predominate
the scanning process) was calculated to be 11.1 ft2. At 30 cm wavelength this corresponds
to an rms position error of approximately 3.33L The experimental results have been
presented in [45, pg.195] and are reprinted in Figure 4.1. As indicated on the graph, the
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solid line represents the theoretical normalized expected main beam relative gain predicted
from the equation [45]
AG= - 4 .3 k 2c ^ x62 dB
(4.2)
where 0 is the scan angle in radians and k represents the wavenumber. Figure 4.2 shows
the results of the simulation obtained by averaging 25 realizations of element position
errors. This figure is in very good agreement with both the theoretical curve predicted by
(4.2) and the experimental data of Figure 4.1. The average percent error is 5.06%.
Because of the agreement between Figure 4.1 and 4.2, 25 realizations were used to
approximate the expected main beam relative gain curves.
To obtain Figure 4.2 a dominant scatterer had to be created in the simulated data.
Therefore, to check the simulation using the DSA the amplitude of one of the simulated
point scatterers was increased to 10.9 amplitude units, which is 12.5 times larger than the
amplitude of the next strongest point source. For the evaluation of the spatial correlation
algorithm performances, the amplitude was diminished to 0.9 amplitude units, only 1.03
times larger than the next strongest source.
Equation (4.2) shows that the expected main beam relative gain for the DSA is a
function of two variables, (JAx and 0. However, the expected main beam relative gain for
the Spatial Correlation class of algorithms is a function of four independent variables; the
rms element position error, o Ax, the scan angle, 0, the number of correlation lags, L, used
in the calibration process, and the size of the field-of-view, ufov. Simulations were run for
various combinations of these parameters.
Table 4.1 lists the four parameters and the values for which simulations were run.
The variable 0 was varied over the interval [0.0,0.1] radians and was sampled at ten
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ELEMENT POSITION ERRORS
-2
s3 -3
e
3
P redicted loss
a.
f r o m th a o r y
-6
-7
S u n sngle (degrees)
Figure 4.1
Theoretical relative gain curve for the DSA with the results of four
experiments showing agreement with the theory.
(Reprinted from Steinberg [45])
Theoretical and Simulated DSA
Main Beam Loss Curves
Simulation R esults
Theoretical R esults
m
■o
_c
a
<3
o>
a
o
oc
-5 -
0.0
2.0
1.0
Scan
Figure 4.2
A n g le
3 .0
4 .0
(d e g re e s)
Simulated relative gain curve produced by the simulation of Section 4.1.2
showing very good agreement with the theoretical DSA curve and the
experimental results of Figure 4.1.
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equally spaced intervals yielding a total of 11 sample points. The rms element position
error, c Ax, was varied over one wavelength and sampled at four points corresponding to
X X X
—, —, —, and X. The field-of-view was varied from 17 mrads to 167 mrads; the latter
8 4 2
represents a field-of-view of almost 9.6 degrees which is very large for practical high
resolution radar imaging systems. For the variable L, only L = 1 and L = 29 were
simulated representing the use of a small number of lags and a large number of lags. These
two cases represent the extremes for the simulated thirty element array. Performance
between these two cases is expected to follow the
Table 4.1
Parameters of Main Beam Loss and the Values Used in the Simulations
Ax
0.017
0.04
0.08
0.12
0.167
0 rads
performance of "constant-phase-error-in-noise" model studied in Section 4.2. This model
assumes that the phase error in the nlh channel is composed of a time invariant term plus a
time varying term. The simulations of Section 4.1.3 verify this by exhibiting the same
characteristics as the noise performance curves of Section 4.2.
Although L = 29 is the largest set of lags simulated, this does not limit the use of
the performance data since most practical systems will not use more lags for two reasons.
First, recall that the measurement of high-order lags introduces a redundancy into the
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GSCA objective function.
As shown in Section 4.3 the weight this redundant
measurement carries is proportional to the squared magnitude of the measurement (for the
non-parametric sub-class) or the theoretical value (for the parametric sub-class). In either
case the magnitude will always decrease with increasing lag number and the contribution of
the measurement to the solution will become negligible. Consequently, adding additional
array elements to increase L will, at some point, become useless. Figures 4.3 and 4.4
indicate that L < 5 is needed when a dominant scatterer is present and that 5 < L < 10 is
useful when no dominant scatterer is present but beyond 10 lags the performance
improvement vanishes. Table 4.2 summarizes these results.
Table 4.2 Largest CoiTelation Lag Necessary to Achieve Peak Performance
Dominant Scatterer
Parametric Algorithm
Non-Parametric Algorithm
Present
1
1 -5
Absent
5 -1 0
5 -1 0
Secondly, practical considerations must also be taken into account when considering
implementation of an algorithm from the Spatial Correlation class. The computation time
required to calibrate using 29 correlation lags is significantly greater than the time required
using only a low-order, multiple-lag algorithm. Consequently, any system employing an
array containing more than thirty elements will most likely be limited to the use of a lowerorder multiple-lag calibration algorithm due to processing time constraints. Therefore the
need for performance curves with L > 29 is seen as a rare situation.
4 .1 .3 M a in B ea m L o ss D u e to E lem en t P o sitio n E rro rs
Figures 4.3 and 4.4 show relative gain curves for both the parametric and nonparametric sub-classes with o Ax =-g and ufov = 167 mrads and a dominant scatterer
present. Figure 4.3(a) contains several curves from the non-parametric sub-class
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R e la tiv e
0.0
(a)
G a in
C u rv e s
N on-P aram etric
DS Class
MMB Lag 29
MMB Lag 10
MMB Lag 5
MMB Lag 1
O
•0.2 «
-0.3
0 .0 0
0 .0 2
0 .0 4
Scan
0.00*
to r th e
A lgorithm
0 .0 6
Angle
0 .0 8
0 .1 0
0 .1 2
(radians)
R elative Gain C u r v e s for the
N on-P aram etric
A lgorithm
•0.01 .
c
(b)
3 -0.02
o>
<3
Ea> -0.03
-0.04
0 .0 0
DS Class
MMB Lag 29
MMB Lag 10
MMB Lag 5
0 .0 2
0 .0 4
Scan
0.00
0 .0 6
A ng le
0 .0 8
0 .1 0
0 .1 2
(radians)
R elative Gain C u r v e s for the
P a r a m e t r i c A lg o r i t h m
CO
(c)
■u -0.01 •
c
<o
u
a>
« -0.02 •
c®
c
-0.03
0 .0 0
DS C lass
ECA Lag 29
ECA Lag 10
ECA Lag 5
ECA Lag 1
0 .0 2
0 .0 4
Scan
Figure 4.3
0 .0 6
Angle
0 .0 8
0 .1 0
0 .1 2
(radians)
(a) Relative gain curves for the non-parametric sub-class for different values of L with a
dominant scatterer in the simulated data set.
(b) Same as (a) with ordinate scale expanded.
(c) Relative gain curves for the parametric sub-class for the same case as (a).
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Relative Gain Curves for the
Non-Parametric Algorithm
0.2
0.0
CD
U
-0.2-
Domscat
MMB Lag 29
MMB Lag 10
MMB Lag 5
MMB Lag 1
_c
(3
o
o
>
n
o
-0.4-
-0.6 -
CC
-0.8 -
0.00
0.02
0.04
Scan
0.06
A n g le
0.08
0.10
0.12
(radians)
Relative Gain Curves for the
Parametric Algorithm
0.0 -O D— Q D1 ] F =e— a
— r - f i — ^ =3 ;
CD
■8
c
Domscat
ECA Lag 29
ECA Lag 10
ECA Lag 5
ECA Lag 1
-0.2
(S
(b)
(5
©
>
0
cc
-0.3 -
-0.4
-0.5
0.00
-a— b— h
0.02
0.04
Scan
0.06
A ng le
0.08
— i---- r~
0.10
0.12
(radians)
Figure 4.4 (a) Relative gain curves for the non-parametric sub-class for different values of L with no
dominant scatterer in the simulated data set
(b) Relative gain curve for the parametric sub-class for the same case as (a)
-1 0 0 -
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corresponding to the different number of correlation lags used into the calibration process.
Part (b) of Figure 4.3 shows the same curves as part (a) with the L = 1 curve removed to
show the curvature of the remaining curves. Part (c) shows the corresponding curves from
the parametric sub-class. Note that this graph contains the L = 1 curve. The curves of (b)
and (c) are virtually identical and the performance variation as a function of L is the same as
the performance with additive noise shown later in Section 4.2. Here, as in Section 4.2, a
significant improvement occurs in the non-parametric algorithm when the number of
correlation lags is increased from L = l .
Also, the parametric algorithm yields
performance nearly identical to the DSA performance, even for L = 1. This is also the
case in Section 4.2.
Figures 4.4(a) and (b) show the performance curves for the same parameter values
as Figure 4.3 when a dominant scatterer is not present. Again, the performance follows the
results of Section 4.2 where a significant improvement in the performance is obtained
around L = 10 in both sub-classes. The performance in the presence of element position
errors as a function of L follows the same characteristics as the additive receiver noise case
between L = 1 and L = 29.
Like the DSA, all of the algorithms belonging to the Spatial Correlation class are
phase correcting algorithms. Therefore it is expected that the SCA main beam gain curves
will possess a dependency on the scan angle, 0, similar to the DSA gain curve.
Figures 4.5(a,b) and 4.6(a,b) make the comparison for two different FOV's. SCA and
DSA gain curves are plotted. The DSA gain curve is used as a benchmark for the spatial
correlation algorithms because it represents an ideal calibration process. The parametric
curve is always plotted using squares for plot symbols and is listed as "P Sub-Class" in
the legends. Similarly, the nonparametric sub-class is listed as "NP Sub-Class " in the
legends and its curve is plotted using a diamond shaped plot symbol.
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Main Beam Gain Curves
o.o
P Sub-Class
NP Sub-Class
DS Class
m -0.5 ■o
c
eg
C3
ID
>
«
ID
DC -1.5 -
-2.0
0.00
0.02
0.04
0.06
S can
A ng le
0.08
0.10
0.12
(ra d ia n s )
Main Beam Gain Curves
o.o
P Sub-Class
NP Sub-Class
DS Class
ca -0.5 u
©
>
R>
■*-
S>
CC
-2.0
0.00
Figure 4.5
0.02
0.04
0.06
Scan
A ng le
0.08
0.10
0.12
(ra d ia n s )
Main beam gain curves from simulated data for ufov = 17 mrads., o Ax = A.
(a) L = 1
(b) L = 29
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Following Steinberg [45], 1 dB is taken as an acceptable main beam loss and the
angle, 9, at which the relative main beam gain is -1.0 dB is the maximum allowable scan
angle, 0max, of the calibrated system. Since the beam can be steered in either the positive
or negative theta direction, the scan range of the calibrated system is 20max.
Figure 4.5 shows the relative gain curves for a small field-of-view ,
u fov = ^ torads, G ^x = X meters, and L = 1 and 29 lags.
Such a field-of-view
represents a single point source being illuminated and seen through a diffraction limited
system or a data set containing a dominant scatterer. The performance difference of the
parametric algorithm is negligible between (a) and (b) and equal to the DSA performance.
This is true of all the parameter combinations from Table 6.1 using ufov = 17 mrads (see
Appendix B). Because of the complexity of L = 29 relative to L = 1, the parametric
unit-lag algorithm is the preferred algorithm in this situation.
To achieve DSA performance with a non-parametric algorithm, a multiple-lag
algorithm must be used. Part (a) shows 17.2% of the total allowed main beam loss is lost
simply by using the non-parametric, unit-lag algorithm.
The loss in 0 max due to
implementing the non-parametric unit-lag algorithm instead of the parametric unit-lag
algorithm is 15 mrads or a loss of 30 mrads in scan range. The performance loss can be
overcome by incorporating more correlation lags into the calibration process. Figure 4.5(b)
shows that for L = 29, the performances of the two algorithms are essentially identical to
each other and to the DSA. Therefore DSA performance can be achieved when a dominant
scatterer is present by implementing a multiple-lag spatial correlation algorithm or the
parametric unit-lag algorithm. When a multiple-lag algorithm is used the non-parametric
algorithm is preferred since it does not require the additional step of testing the data set for
the presence of the dominant scatterer before calibration. Overall, the parametric unit-lag is
the preferred algorithm with a dominant scatterer.
- 103 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The curves of Figure 4.6 indicate that the performance characteristics remain
unchanged by variations of the field-of-view. However, with increasing field-of-view size
the performance difference between the two unit-lag algorithms also becomes increasingly
important. For example, two imaging systems operating with the performance curves of
Figure 4.5(a) have scan ranges that differ by 16.05 mrads while the same two systems
operating with the performance curves of Figure 4.6(a) have scan ranges that differ by 100
mrads. Although the scan range difference of 16.05 mrads may be acceptable, the 100
mrads difference in scan range is most likely unacceptable. In this situation, again, the
parametric unit-lag algorithm is the preferred algorithm.
Another consequence of increasing the field-of-view is the number o f correlation
lags needed to achieve DSA performance in a non-DSA environment. With a small fieldof-view or in the presence of a dominant scatterer only 1 lag is needed with the parametric
algorithm. However, as the field-of-view is increased, the unit-lag algorithm is no longer
sufficient to achieve DSA performance (Figure 4.6(a)). With a large field-of-view the 29lag parametric algorithm still does not achieve DSA performance (Figure 4.6(b)).
However, the scan range difference between the parametric algorithm and the DSA has
been reduced from 57.6 mrads to 17.5 mrads by including the higher-order lags. The 29lag non-parametric algorithm is competitive with the parametric algorithm since the scan
ranges differ by 12.5 mrads. Provided this difference is acceptable, the non-parametric
multiple-lag algorithm is now the preferred algorithm since it does not require assumptions
to be made regarding the intensity distribution.
Table 4.3 summarizes the utility of the spatial correlation algorithms.
All
algorithms listed have a non-zero scan range relative to 1 dB expected relative gain loss for
the particular case of interest. Algorithms in bold italics represent the preferred algorithms
as discussed in this section.
- 104 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 4.3
Situation
Dominant Scatterer
Applicable Algorithm
Parametric }
xr
• r
Non - ParametncJ
Parametric
1
N on-Param etncJ
No Dominant Scatterer
Large FOV = 167 mrads
U n it-L ag
Multiple-Lag
Parametric Unit-Lag
Parametric
1
^ M u ltip le-L a g
Non - ParametricJ
-1 0 5 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Main Beam Gain Curves
o.o
P Sub-Class
NP Sub-Class
DS Class
-0.5
CD
5. -1.0
c
«
O
©
>
-2.0 -2.5 -3.0
0.00
0.02
0.04
0.06
Scan
Angle
0.08
0.10
0.12
(radians)
Main Beam Gain Curves
P Sub-Class
NP Sub-Class
DS Class
(dB)
-0.5 -
c
(b)
■
-1.0 -
CO
O
0)
> -1.5 *-*
n
0)
DC
-2.0 -
—i 1---- 1---- 1--------1-1-------- 1-1-------- 1-1---- 1-2.5
0.00
0.02
0.04 0.06
0.08
0.10
0.12
Scan
Figure 4.6
Angle
(radians)
Main beam gain curves from simulated data for ufov = 83.5 mrads., g Ax =
X
(a) L = 1
(b) L = 29
-106-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4 .1 .4
A n A p p ro x im a tio n to th e S C A P erfo rm a n ce C u rves
The data of Figures 4.5, 4.6, and Appendix B indicate that wheno Ax < X or ufov <
120 mrads, the relative gain curves can be approximated by the DSA curve shifted by a
small constant downward. Furthermore, the differences are vanishingly small (< 0.1 dB).
Figure 4.7 plots this shift for all of the parameter combinations used in Figures 4.5, 4.6,
and Appendix B. Therefore, the relative gain curve changes from (4.2) to
AG = - 4 .3 k 2 o 2 02 - C
Ax
dB
(4.3)
where the constant C is obtained experimentally or through simulation by a single
measurement on boresight.
Beyond one wavelength rms the difference is better
approximated by a linear function and (4.3) must be modified to
AG = - 4 . 3 k 2 o 2 62 - C ,9 - C„ dB
Ax
1
2
(4.4)
where two measurements must be made to determine Cj and C2.
-107 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Difference Between DSA and SCA
Relative Gain Curves
0.1 0
CD
0.08 i
o
o
c
ok .
0)
0 .0 6 -
Q
0 .0 4 -
Field-of-view (radians)
0.167
0.12
0.08
0.04
0.017
_c
a
O
ra
QC
0.00
"3
-0.02
0 .0 0
0.0 2
0 .0 4
Scan
0.06
A ngle
0.08
0 .1 0
0.12
(ra d ia n s )
Difference Between DSA and SCA
Relative Gain Curves
0.10'
~
o
o
0.08
0)
0.0 6 -
c
Field-of-view (radians)
0.0 4 -
(b)
0.167
-*—*—-*—ft—ft—ft—a_ -a—ft—*
i
0.12
0.08
0.04
0.017
c
(9
o
0)
>
I
0)
cc
0.02 J
0.00
-0.02 H— i— i— >— i— ■— i— ■— i— ■— i— r0 .0 0
0 .0 2
0 .04
Scan
Figure 4.7 (a) and (b)
0 .0 6
A ngle
0.08
0 .1 0
0 .1 2
(radians)
Difference between the DSA and SCA relative gain curves for fields-of-view
X
X
o
4
simulated and for (a) ^ rms position error and (b) 7 rms position error
-108 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Difference Between DSA and SCA
Relative Gain Curves
0.20
m
•o
o
o
c
0
0
Q
c
0
o
0
>
2.
0
cc
0.15Field-of-view (radians)
0.10-
0.167
0.12
0.05-
0.08
0.04
0.017
0.00 -0.05 -
-0.10
0 .0 0
0 .0 2
0 .0 4
Scan
0 .0 6
A ngle
0 .0 8
0.10
0.12
(radians)
Difference Between DSA and SCA
Relative Gain Curves
0.6
*D
0
o
c
0
i.
0
0 .4 Field-of-view (radians)
0.2-
0.167
0.12
D
c
a
0.08
0.04
0.017
0.0
(3
0
>
0
0
-0.2-
DC
-0.4
0 .0 0
0 .0 2
0 .04
Scan
Figure 4.7 (c) and (d)
0.06
A ngle
0 .0 8
0.10
0.12
(radians)
Difference between the DSA and SCA relative gain curves for fields-of-view
X
simulated and for (c) - rms position error and (d)
\ rms position error
-1 0 9 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.2 Receiver Noise Analysis
The simulation of Section 4.1 was modified to study the effect of element receiver
noise on the spatial correlation algorithms. A spatially random element phase-error profile
x
with an rms value of 0.78 radians ~ g wavelengths was added (Figure 4.8). Zero mean
complex Gaussian noise was added to the received signal at each element from each of the
10 rangebins. The real and imaginary components of the noise were independent from
element to element and from rangebin to rangebin with equal variances inversely
proportional to the desired signal-to-noise ratio. Twenty-five noise realizations were
generated and the array was calibrated using one algorithm from each sub-class. The
residual phase error on each element was calculated as the difference between the known
phase error at that element (Figure 4.8) and the estimate of the phase error as determined by
the two algorithms. Each set of residual phase errors was recorded for each
Phase Error Profile Across the Array
1 .5
0 .5
S3
os
- 0 .5
- 1 .5
10
15
20
25
30
Array Element Number
Figure 4.8
The phase error profile across the array used in the simulations of Section 4.2
- 110-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
noise realization. The rms residual phase error, the expected mainbeam relative gain, and
the expected average and peak sidelobe levels were estimated from the 25 realizations.
The plots of Figures 4.9 through 4.15 illustrate the performances of algorithms
of the two sub-classes in terms of the residual phase error, mainbeam gain, average and
peak sidelobe levels as a function of correlation lag. The number of elements is N = 30
and the SNR per element is 3, 13, and 23 dB. Figures 4.9 - 4.12 exclude a dominant
scatterer; in Figures 4.13 - 4.15 one is included.
The validity of the simulations can be checked by comparing the measured
mainbeam relative gain and average sidelobe level to the values predicted from theory. The
mainbeam relative gain is related to the phase error variance by [45]
AG = - 4 .3 ap dB
(4.5)
where o^ represents the variance of the residual phase errors, (3n, n = 2 , . . . , N . The
values shown in Figures 4.10 and 4.14 are within 2.6% of the values predicted by (4.5).
The average power pattern of a uniformly weighted array distorted by phase errors
only is given by [46]
2
e|
f(u)f*(u) j = | E je-^j
1 r
i
l E{l e jP}
Jpl l2 )
f Q(u)f*(u) + — af l((ll--|E
N
(4.6)
where a is the magnitude of the element weights and f0(u) is the undistorted radiation
pattern. Assuming a = 1, the average sidelobe level is then given by
ASL=l k ) I E(f<u)f*(u)}du
i2
E{ejfJ}|
l- |E { e jP]
L - L J L j f o(u) f o(u)du - — l - l - i
XM i 0
0
N
(4.7)
- Ill -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where U denotes the sidelobe region (all u-space exclusive of the mainlobe) and xCV)
represents the size o f the sidelobe region. The first term of (4.7) is the ASL of the
normalized undistorted power pattern (ASL0) scaled by the quantity |E |e ^ J j
.
The
second term represents an increase in the ASL due to phase errors. For small phase errors
such that sin (5 = (5 and cos (3 = 1 - ^ p 2 , |E |e ^ || can be approximated by
lE{cil f " ’ - i 6 ! 152} =
(4-8)
and (4.7) can be written as
ASL »
f
1
1
1 ^
2 °P
_2
°
ASL + - 1
o
N
(4.9)
For a 30 element uniformly weighted array with a = 1, ASLQ =0.0035 or -24.56 dB
and Op of Figures 4.9 and 4.13, the ASL's of Figures 4.10 and 4.14 are within 6.0% of
the values predicted by (4.9).
For medium and high SNR's ( > 13 dB, Figures 4.9(a,b)) there is little difference
between the performances of the two algorithms of the two sub-classes until L > 25 lags.
Both algorithms require L > 10 to substantially increase their performances over the
respective unit-lag performance levels. As L is increased beyond 10 the MMB exhibits a
constant performance level while the ECA performance continues to improve. At L = 29
the MMB rms residual phase error remains unchanged from its value o f 0.13 radians at
L = 10 while the ECA rms residual phase error is reduced to 0.035 radians.
Consequently, the parametric algorithm is the preferred algorithm when a large number of
lags are available for use.
- 112-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The difference in peak sidelobe performance is also important (Figure 4.12). At
L = 29 the peak sidelobe level of the ECA is 2 dB below that of the MMB. This is due to
the leveling-off of the performance of the MMB for L > 10 lags.
A trade-off exists when using the MMB algorithm; the price paid for not
knowing characteristics of the source distribution is a lower limit on the performance of the
algorithm. This is evident by the constant performance level exhibited by the MMB
algorithm at lags > 10 in Figures 4.9(a,b). The performance limit is reached within 10 lags
and the use of any more lags, regardless of their strengths, does not improve performance.
Figure 4.9(c) shows that at low SNR's (= 3 dB) the gap between the ULSCA and
the ECA or MMB narrows and the preferred algorithm is less definitive. Table 4.4
illustrates the weak performances of both algorithms. The multiple-lag algorithms do
provide some improvement in performance which the designer may or may not find
adequate for the increased complexity. Quantitatively the preferred algorithm is still the
parametric algorithm since this algorithm provides the greatest improvement over the
ULSCA performance level. However, the need to model the source distribution can
arguably out-weigh the marginally superior performance provided by the parametric
algorithm.
Table 4.4 Performance of the ECA and MMB
Residual Phase Error
Relative to the ULSCA
Level for 3 dB SNR
Change From 23 dB SNR
Parametric Algorithm
(ECA)
61.3 %
46.3 %
Non-Parametric Algorithm
(MMB)
80.4 %
32.3 %
-1 1 3 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
With a dominant scatterer present (Figures 4.13 - 4.15) all of the multiple-lag
algorithms perform equally well and yield performances equal to that of the ULSCA. The
unit-lag algorithm corresponding to L = 1 in the MMB objective function yields
considerably worse performance. Therefore the ULSCA (the simplest) is the preferred
unit-lag algorithm and the MMB (non-model based) algorithm is the preferred multiple-lag
algorithm.
Table 4.5 summarizes the algorithm preferences for the various cases discussed
in this section.
Table 4.5
Situation
Preferred Algorithm
Dominant Scatterer
Parametric Unit-Lag
No Dominant Scatterer
Non-Parametric Multiple-Lag
-114 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
RM S R esid u al Phase Error (SN R = 23 dB )
0.4
0 .3 -
•3
SCA
ECA
MMB
0.2-
(a)
0.0
0
10
20
30
Maximum Correlation Lag
RMS Residual Phase Error (SNR = 13 dB)
0.4
0.3-
-* —
-* —
-O
0.2(b)
0.0
0
10
20
SCA
ECA
MMB
30
Maximum Correlation Lag
RMS Residual Phase Error (SNR = 3 dB)
0.4
0 .3 -
SCA
0.2-
ECA
MMB
(c)
o.o
0
10
20
30
Maximum Correlation Lag
Figure 4.9
(a) rms residual phase error with a 23 dB SNR
(b) rms residual phase error with a 13 dB SNR
(c) rms residual phase error with a 3 dB SNR
- 115 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M ain Beam Gain (S N R = 23 dB)
0.0
c
•a
-
0.1
-
0.2
o
SCA
ECA
MMB
-0.3
(a)
CQ
c
-0.4
-0.5
-
0.6
0
10
20
30
Maximum Correlation Lag
Main Beam Gain (SNR = 13 dB)
o.o
-0.1
03
3
c
■a
o
-0.2
SCA
ECA
MMB
-0.3
(b)
PQ
c
•a
-0.4
-0.5
-
0.6
10
0
20
30
Maximum Correlation Lag
Main Beam Gain (SNR = 3 dB)
-0.1
-0.2
a
•a
a
(c)
cq
c
I
-0.3
-K—
-tr
-0.4
SCA
ECA
MMB
-0.5
-0.6
0
10
20
30
Maximum Correlation Lag
Figure 4.10
(a) Main beam gain with a 23 dB SNR
(b) Main beam gain with a 13 dB SNR
(c) Main beam gain with a 3 dB SNR
116 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A verage Sidelobe Level (SN R = 2 3 dB)
•21
O
5
£o
(a)
73
.*
2
c/3
o
o2>
<
-22
SCA
ECA
-23
MMB
•24
0
10
20
30
Maximum Correlation Lag
Average Sidelobe Level (SNR = 13 dB)
-21
CQ
T3.
C
>J
(b)
£O
73
2
c/3
0bo
a
-22-
SCA
ECA
MMB
-2 3 -
-2 4 -
1
-25
0
10
20
30
Maximum Correlation Lag
Average Sidelobe Level (SNR = 3 dB)
-21
CQ
■o
o>
(C)
£o
"53
•o
c?3
«
S2
>
<
-2
2-
SCA
ECA
-2 3 -
MMB
-2 4 -
-25
0
10
20
30
Maximum Correlation Lag
Figure 4.11 (a) Average sidelobe level with a 23 dB SNR
(b) Average sidelobe level with a 13 dB SNR
(c) Average sidelobe level with a 3 dB SNR
- 117 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Peak S id elo b e L evel (SN R = 23 dB)
-80
%
(a)
SCA
ECA
MMB
.”2
c/3
1
Dm
-12•13
0
20
10
30
Maximum Correlation Lag
Peak Sidelobe Level (SNR = 13 dB)
OQ
S'
15
>
(b)
$
££
■o
C/3
I
Cl.
SCA
ECA
-10
MMB
•11-
-13
0
10
20
30
Maximum Correlation Lag
Peak Sidelobe Level (SNR = 3 dB)
CQ
(c)
£0
SC A
ECA
MMB
-10
c/3
1
Oh
-12
-13
0
20
10
30
Maximum Correlation Lag
Figure 4.12 (a) Peak sidelobe level with a 23 dB SNR
(b) Peak sidelobe level with a 13 dB SNR
(c) Peak sidelobe level with a 3 dB SNR
-
118
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
RM S R esid u al Phase Error (S N R = 23 dB)
0.08
0 .0 6 -
0 .0 4 -
0 .0 2 -
0.00
0
10
20
30
Maximum Correlation Lag
RMS Residual Phase Error (SNR = 13 dB)
0.08
0 .0 6 -
vi
s
(b)
’•CQ
§
SCA
ECA
MMB
0.04 -
OS
0 .0 2 -
0.00
10
20
30
Maximum Correlation Lag
Figure 4.13
(a) rms residual phase error with a 23 dB SNR and a dominant scatterer
(b) rms residual phase error with a 13 dB SNR and a dominant scatterer
- 119 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
M ain Beam Gain (S N R = 23 dB )
0.005
i
0.000
-0.005
•I
o
- 0 .010
-
0.020
-0.025
0
20
10
30
Maximum Correlation Lag
Main Beam Gain (SNR = 13 dB)
0.005
0.000
e
•a
-0.005
a
(b)
SCA
ECA
MMB
-
0.010
-
0.020
-0.025
0
20
10
30
Maximum Correlation Lag
Figure 4.14
(a) Main beam gain with a 23 dB SNR and a dominant scatterer
(b)Main beam gain with a 13 dB SNR and a dominant scatterer
-
120
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A verage Sid elob e Level (SN R = 23 dB)
-24.00
O
-24.50 ■
-24.75 -
-25.00
0
10
20
30
Maximum Correlation Lag
Average Sidelobe Level (SNR = 23 dB)
-24.00
—
-24.25 -
-M
-A —
-o —
-24.50 -
(b)
SCA
ECA
MMB
t/5
|
-2 4 .7 5 -
-25.00
0
20
10
30
Maximum Correlation Lag
Figure 4.15
(a) Average sidelobe level with a 23 dB SNR and a dominant scatterer
(b) Average sidelobe level with a 13 dB SNR and a dominant scatterer
-121
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.3 Analysis of the Solutions of the GSCA
and the MMB
The GSCA objective function is a highly nonlinear function of the variables
and
consequently general closed-form analysis is intractable. However considerable insight
regarding several of the results of the previous two sections can be obtained through firstorder analysis of the GSCA objective function. Such analysis is made in this section and is
valid provided good initial estimates of the phase errors are used and the variance of the
phase noise,
n+^, is small. It is shown that linearization leads to a weighted linear least-
squares problem and that the two sub-classes are asymptotically equivalent as the
measurement noise goes to zero, but otherwise the parametric sub-class is superior. In
addition, it is shown that the MMB does not degenerate to the ULSCA when only the unit
lag measurements are used as claimed in [29].
4 .3 .1 L in ea riza tio n o f the M M B an d th e P a ra m e tric S u b -C la ss
To demonstrate the linearization of the parametric sub-class of Spatial Correlation
algorithms, consider a 5 element array example. The GSCA objective function can be
expressed as
4
•>GSCA=
L= - 4
= |wp( l) |2cos(p2+512-ft2) + |wp( l) |2cos(p3-P2+52 3-^3+^2)
+ |w p( l ) |2COs(p4-p3+53)4-$4+$3)+ |w p( l ) |2COS(p5-p4+54j5-$ 5+p4)
+ Iwp(2) |2cos(P3+8] 3-^3) + |wp(2)|2cos(p4-p2+52)4-$4+$2)
+ |wp(2)|2cos(p5-p3+53)5-ft5+(33)+ |wp(3)|2cos(p4+51)4-$4)
-122
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
+ |w p(3 )|2cos(p5-P2+82)5-$5',$ 2 )+ |w p(4)|2cos(p5+51)5-$ 5)
(4.10)
Following the notation of Chapter 2, 8n n+/, is the phase noise in the measurement of the
correlation between the n111 and the (n+j£)th elements. The weights wp(Z) are complex in
general. The phase of wp(jt) ideally represents the true phase,
, of the true spatial
correlation function, R^. Consequently, the true correlation phase can be assumed
canceled and any mismatch resulting from incorrect modeling of the source distribution can
be absorbed into the phase noise term, 5njn+ /,. This is the reason that only the magnitude
o f Wp(Z) is included in (4.10). The magnitudes are squared because the magnitude of the
product [wpOO R "(>t ) ] is |w (Z)|l Rz | = |w pOO|2 provided the weighting function is
selected correctly. Equation (4.10) can be written more compactly in the form
= |w p( l ) |2cos(r12) + |w p( l ) |2cos(r2 3 ) + | wp( l ) |2cos(r34) + |w p( l ) |2cos(r4 5 )
+ |w p(2 )|2C0S(r13) + |wp(2)|2cos(r24) + | wp(2)|2cos(r3)5)
+ |w p(3 )|2cos(r14) + |w p(3)|2cos(r2 5 ) + |w p(4 )|2cos(r15)
(4.11)
where rn n+jt represents the phase residual of the product formed between the nth and
(n+Z)th elements. If the algorithm is initialized with the unit-lag phases then the residuals
in (4.11) will be relatively small, the cosine terms in (4.11) can be approximated by the
first two terms of its Taylor series expansion
«*<w
> " 1 - 2 w
<4 J 2 )
and (4.11) can be approximated by
-123 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
^
w pO t)R * (/)
1= - 4
X ( 5- ^ ) | w p W | 2 - \
( I Wp (1) | Z l 2 + | Wp(l) | Z £3+ I Wp (1) | Z l 4 + | Wp(l)
22
r4.5
^=1
Wp(2)
+ Wp(3)
fU +
r M+
Wp (2 )
Wp(3)
'
2
,
r 2,4
r 2,5+
wp(2)
wp(4)
3,5
1,5
(4.13)
and generalized to an N-element array by
J GSCA=
X
f=-N+l
2
N -l
» £ ( N - 0 wp(/)
f=l
N -l N - f
-X X
wpW
f=l m=l
m,m+f
(4.14)
Since all of the quantities in the double summation are positive, (4.14) will be maximized
when the weighted sum of the squares of the residuals is minimized. This implies that the
GSCA objective function yields a weighted least-squares solution and the various
algorithms characterized by the GSCA by their different weighting functions of wp(Z) can
be analyzed using least-squares theory.
Least-squares theory indicates that the measurements should be weighted inversely
proportional to their variances [18], [30], [34]. Appendix A derives the variance of 5n n+jt
under an assumption of strong wavefront correlation.
The variance is inversely
2
proportional to | R(^) | . Consequently, multiplying the residuals of lag L by | R(^) |
properly weights them.
- 124 -
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2
It is easy to show that a unique the least-squares solution exists. Define matrices
Ap and Gp for the 5 element array example
w
<1)1
<1)1
1
0
0
O'
w
U)|
-1
1
0
0
-1
0
1
0
0
0
-1
1
0
-1
1
0
1
0
0
1
w
w P(i)|
wp(2)
G P = Diag
wp(2)
0
0
0
-1
0
wp(3)
-1
0
0
0
0
0
wp(2)
wp(3)
0
1
(4.15)
0
1
1
wp(4)
and two vectors, one containing the phase error estimates and another containing the
measured correlation phases
P2 + 8 ,i2
P3 ~ P2 + ^2,3
P4 - P3 + 834
P5 ~ P4 + ^ 4,5
P3 + $1,3
P2
A
Pp =
P3
P4 - P2 + 82j4
bp -
P4
P5 - P 3 + 53,5
(4J6)
P4+814
LP5
P5 ~P2 + ^2,5
P5 + ^1,5
and find the solution to the least-squares problem
A
A
GpApPp ~ bp
(4.17)
-125
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It is easy to verify that the first four rows of A are linearly independent; consequently Ap is
full rank and a unique least-squares solution exists. In fact, the first N -l rows of Ap for
an N-element array represent unit-lag correlation measurements of exactly the same form as
the ULSCA. Since the ULSCA induces a full rank mapping, the addition o f more
equations to the ULSCA system cannot decrease the rank of A p . Additionally, the
product, G PAP, is also full rank since G P is square and full rank [32]. Consequently, a
unique least-squares solution will always exist even for an N-element array of arbitrary
size.
The linearization of the MMB algorithm is accomplished in much the same manner
as with the parametric sub-class. The MMB objective function of a 5-element array is
4
I A
X
|o
A *
A
A * A
A*
A
A*
A
I ROOT = 2R ( l) R ( l) + 2R (2)R(2) + 2R (3)R(3) + 2R (4)R(4)
1=
-
(4.18)
1
A *
A
A*
A
A
where the factor of 2 arises because R (-1 )R(-1) = R (1 )R(1) and R(jt) is an average over
the N-jt lag L measurements available. For the 5-element example the terms of (4.18)
evaluate to
2 R * ( l ) R ( l ) = 4 | R 1| 2 + 2 | R 1| 2c o s (p 1- 2 p 2+ p 3 + 5 2)3 - 5 1)2- ^ 1+ 2 ^ 2- ^ 3)
+ 2 | r J cos(P2-2P3+P4+83)4-82)3-$2+2^3-$4)
+ 2 |R j| COS(P3-2P4+p5+S4)g-83)4-P3+2P4-P5)
+ 2 | r J COS(Pj -P2-P4+P5+84 g-8j 2-Pj+ p2+P4-p5)
+ 2 | r J COS(P]-P2-P3+P4+83j4-S ^ 2-$ |+ $ 2+$3-$4)
+ 2 | r J cos(P2-P3-p4+P5+84)5-S2)3-P2+P3+$4-P5)
-126
(4.19)
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2R*(2)R(2) = 4 |R 2| 2 + 2 |R 2| 2cos(pr p2-P3+P4+52)4-5 1)3 ^ 14 24 3 ^ 4 )
+ 2 | r 2| cos(Pj-2P3+P3+83 3-5jj3-pj+2P3-p3)
2 R (3)R (3) = 4 | r 3|
+ 2 | r 2| cos(P2-P3-P4+P5+53 5-§24-^2+^3+^4-^5)
(4.20)
+ 2 | r 3| c o s t p ^ ^ + p s + S ^ - S ^ p ^ + V f e )
<4-21)
Following (4.12) and (4.13), the cosines are expanded into the first two terms of
their Taylor series and the residual terms are collected. Maximization of the objective
function (4.18) results in weighted least-squares solution. In this case the matrix problem
A
A
(4.22)
G NpA Np P Np ~ b NP
is characterized by matrices GNP and ANP, and vectors PNp and b Np of the form
lR i
-2
1
0
O'
1
-2
1
0
0
1
-2
1
-1
0
-1
1
-1
-1
1
0
1
-1
-1
1
R2
-1
-1
1
0
|r 3
0
-2
0
1
1
-1
-1
1
-1
0
-1
1
lR i
lR i
lRi
lR:
G Np - D i a g
R2
Am
NP
lR3
(4.23)
|r 4
and
- 127 -
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-2 P 2+ P3+52,3'5 1,2
P2-2P3+P4+53)4-82>3
p3-2p4+p5+64)5-83j4
P2
A
(3
_
HNP “
%
A
P4
-P2-P4+P5+54,5'81,2
-P2-P3+P4+834-812
A
b NP =
(A
n
A\
P2 - P3 - P4 + P5 + 84 5 - 82)3
- P2 - P3 + P4 + 82 4 - S1>3
^5
~2p3 + p5 + S3j5 - S1>3
P2 "P3 ‘ P4 + P5 + ^3,5 ' &2,4
- p2 - P4 + p5 + 82>5 - S1>4
The generalization of (4.24) to an N-element array is straightforward but cumbersome.
Note that the matrix ANP is rank deficient by one. For the five-element array ANP
in (4.23) has rank 3 although it has 4 columns. In general, ANp will have N -l columns
and a rank of N-2. Because ANP is rank deficient an infinite number of solutions to the
least-squares problem exist. The particular solution obtained by maximizing the MMB
objective function cannot be determined without more information.
The two vectors, 6 P and 6 NP, can each be decomposed into the sum of two
vectors. One vector in each decomposition will lie in the range space of the appropriate
mapping matrix (Ap for ftp and ANp for 6 NP), and the other vector will lie outside of it.
A
A
The vectors bNP and b p can be expressed as
b NP “ b NP + ^ b NP
(4.25)
bp — b p + 8bP
(4.26)
where the vectors bNP and b p lie within the range space of ANP and Ap, respectively,
and the vectors 8bNP and 8 b p do not lie within the range space of their respective A
-1 2 8 -
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matrix. If there is no system noise, both systems o f equations are consistent and the
resulting vectors lie within the range space of the respective A matrix. Therefore, the
vectors bNP and bp can be readily identified as
-2p2+p3
p2-2p3+p4
-p2-p4+p5
P2
P3’ P2
P 4'P3
-P 2 ‘ P4+ P5
P 5'P4
' P 3 ' P4 + P 5
P2 - P3 ' P4 + P5
P2
bp
P3
—
- P2 - P3 + P4
-2P3 + p5
P2
(4.27)
P4'P2
P5‘ P3
P4
‘ P 3 ' P4 + P 5
P 5'P 2
- P2 - P4 + P5
-
and the vectors lying outside of the range spaces of ANp and Ap are, respectively,
52 ,3'5 1,2
8 1,2
83,4‘ 82,3
S2,3
84,5’ 83,4
5 3,4
84,5‘ 8 1,2
S4,5
5 3,4"5 1,2
5 1,3
5bp =
84,5 ' 82,3
52,4
52,4 " 5 1,3
83,5
8 3,5 ' 8 1,3
8 1,4
8 3,5 ' 82,4
82,5
(4.28)
82,5 ' 5 1,4
Likewise the vectors f ^ P and $ pcan also be decomposed as
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&NP “ P + % P
(4.29)
ftp = P + e P
(4.30)
where (3 is a vector of the actual element phase errors and £NP and £p are error vectors.
From equations (4.25) through (4.27) it can be seen that
G NpA NpP = G Np b N p
(4.31)
G pA pp = G pbp
(4.32)
and consequently the least-squares problems of (4.179) and (4.22) become
G NpANp6Np = G NpSb^ p
(4.33)
G p A p £ p ~ G NP8 b P
(4.34)
At this point the hinderance of the rank deficiency of G NPANP8NP to the analysis of the
estimation errors in each sub-class becomes apparent.
Because ANp is rank deficient,
G NpANp is also rank deficient and, therefore, the pseudoinverse does not exist and a
generalized inverse must be used to solve (4.33). However, without more information, the
generalized inverse that yields the same solution as the maximization of the MMB objective
function, cannot be determined.
The superior performance of the parametric algorithm over the MMB can be
explained from this analysis. The measurement vector, 8bNp, has twice the variance of the
measurement vector Sbp. Thus the advantage of having a priori knowledge is that the
phase error estimates of the parametric algorithm will generally be better than the estimates
provided by the MMB algorithm.
Further consider (4.10) and (4.18) with the lag index, I , running between -1
and +1. Both objective functions now incorporate only the unit-lag measurements into the
-1 3 0 -
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calibration procedure. In [29] the claim is incorrectly made that the MMB degenerates to
the ULSCA. However the ULSCA equations are obtained from (4.17) by retaining only
the unit-lag terms. Equation (4.17) derives from the parametric sub-class. Since the MMB
belongs to the non-parametric sub-class its unit-lag algorithm is not the ULSCA and will
yield performance inferior to that of the ULSCA.
4 .3 .2 The G SC A a n d the M LSC A
The analysis o f this section reveals a connection between the algorithm
characterized by the GSCA objective function and the MLSCA. The connection is revealed
through the linearization of the parametric sub-class. The linearization shows that the phase
error estimates provided by the parametric algorithms are the same, within a first-order
approximation, as the estimates provided by the MLSCA. The importance of this is three­
fold. First, the performance of the MLSCA can be obtained through the parametric sub­
class of algorithms and because of the commonality of the maximization procedures,
discussed in Section 3.4, this performance can be achieved at will by simply altering the
weighting function wp(Z) in software. Secondly, the MLSCA performance can be
obtained without the burden of phase-unwrapping. Thirdly, the MLSCA is a parametric
algorithm and is therefore not scene independent. For best results, the measurements in the
least-squares problem should be weighted inversely to their variances.
Therefore,
knowledge of the intensity distribution must known a priori .
The first and second conclusions are the most significant. The least squares
problem that results from linearizing the parametric sub-class is the same problem solved
by the MLSCA. However because of the cosinusoidal nature of the parametric objective
function, modulo 2k ambiguities do not affect the parametric algorithms. Since the
parametric and non-parametric algorithms can be implemented with one efficient
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maximization procedure, the MLSCA can also be implemented with the same maximization
procedure.
The conditions for which the parametric algorithms yield the MLSCA solution must
be recognized. It is important that the initial guesses of the phase errors used to initialize
the maximization procedure be good guesses and that the variances of the random
variables, 5n n+^ ,be small. Figure 4.16 shows
the average ratio of the 2-norm of the
error vector to the 2-norm of the least-squares solution averaged over 50 realizations.
Therefore % is
P g sc a " P m lsca
£ = ---------
(4.35)
MLSCA
where PGSCA is the vector of phase error estimates determined from the GSCA algorithm
and PMLSCA is the vector determined from the MLSCA. The results of Figure 4.16 were
obtained from a 5 element and a 7 element array and the cosine terms in the GSCA
objective function were given equal weights. Even if 5n
( has variance of ^ radians, the
parametric solution will differ from the MLSCA solution by 18%. The ULSCA provides
good initial estimates for the pn's and can be used to satisfy the first condition. The
second condition is less controllable. Excluding lags with small theoretical amplitudes
from the calibration process will help to meet the second condition.
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Comparison of GSCA and MLSCA Solutions
0.20
0.15-
7 Elements
0.10-
5 Elements
0.05-
O
0.00
0.0
0. 2
0 .4
0. 6
0 .3
1.0
Noise Variance
Figure 4.16
Relative difference between the GSCA and the MLSCA solutions.
4 .4 S u m m a ry
The set of expected relative gain curves presented in this chapter and Appendix B is
the first published study of element position error tolerances of spatial correlation
algorithms and the performance plots for the additive noise case are the first to study the
performance over various SNR's.
Element position tolerances of the spatial correlation algorithms have never been
studied.
The relative gain curves compiled in Appendix B are the basis for the
mathematical approximation of the relative gain curves presented in Section 4.1.4 and
provide quantitative tolerances for system designers.
When a dominant scatterer is present the relative gain curve should be approximated
by the DSA relative gain curve given by (4.2). When there is no dominant scatterer present
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and a Ax < X meters or ufov < 0.12 radians, (4.3) approximates the relative gain curve to
within 10% error. Since the 0 dependence of the relative gain is known through (4.2) and
(4.3) there is no need to make measurements of the relative gain curves at multiple scan
angles.
For L < 10 the parametric algorithm is more desirable to implement since it
exhibits twice the relative gain (in dB) of the non-parametric algorithm. As L increases
both algorithms experience a large improvement in performance between L = 5 and
L = 10. For L > 10 the performance difference is negligible and in this case the nonparametric algorithm should be implemented.
Processing time constraints and computation complexity must also be assessed
versus the required performance of the main beam gain or scan range. If processing time
and computation complexity are not severe constraints then a high-order multiple-lag nonparametric algorithm should be implemented to achieve the desired performance with no
a priori knowledge regarding the intensity distribution, However, if processing time and
computation complexity are severe constraints then it is worth obtaining a priori
knowledge regarding the intensity distribution to allow the use of a low-order multiple-lag
parametric algorithm.
Quantitatively, the parametric algorithm out-performs the non-parametric algorithm
in the presence of element receiver noise. The performance difference is small and if this
difference is tolerable the non-parametric algorithm is preferred. However, for L > 10 the
performance of the parametric algorithm continues to improve with increasing lag while the
performance of the non-parametric algorithm remains constant. If this performance is
required, then the parametric algorithm with L > 10 must be used.
-1 3 4 -
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Chapter 5
Self-Calibrating with the GSCA Using
Near-Field Data
The GSCA has been derived for a far-field source distribution for which the
L o rr
distance is nominally > - ~ L where Larr represents the length of the array. In general the
A
high-resolution array aperture is so large that the source distribution is in the near field
instead, meaning that the wavefront from a point source appears spherical rather than
planar.
Since the GSCA is derived under far-field assumptions, the near-field phase
curvature has not been taken into account. It is shown in the first section of this chapter
that the GSCA objective function can be modified to self-calibrate the array using near-field
data. It also has the ability to eliminate the quadratic terms of the polynomial expansion of
the phase curvature by treating such terms as system phase errors.
The second section mathematically characterizes the near-field effects on the spatial
correlation algorithms and proposes a method of eliminating them by adding another
-1 3 5 -
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weighting function termed the "near-field weighting function". The inclusion of this
weighting function into the GSCA objective function makes the GSCA useful for near-field
self-calibration. Section 5.2.3 discusses the necessity of this weighting function in general
and under certain specific conditions. The third section o f this chapter discusses the
performance of the various spatial correlation algorithms using near-field data and the last
section provides a summary of the chapter.
5.1 The Near Field of a Linear Array
Beginning with Section 2.1, the received radiation field as given by (2.2) is
e(x) =
fe 'jkP (x )l
J (0 )
{
Expanding p(x) = '\J
de
( 2 .2 )
j
PM
+ x2 -2xposin(0) in a binomial expansion and preserving only
the linear and quadratic terms, (2.2) can be written as
-lk(ux e(x) =
J(u) e
*2/1_n2\
— L)
2Po
du
(5.1)
Au
where, as before, u=sin8 and constant multipliers are ignored. If the source distribution is
concentrated at u0 (i.e., J(u) = 8(u-u0)), the radiation field has the form
x2(l-u 2)
2(1-Uq)
)
xr
x
'jk
(
u
x
'
~
2
^
~
)
a
'
jk(uox
'
1
T
o(u-u0) e
zPo
du = e
zPo
e(x) =
(5.2)
Au
and is no longer the Fourier transform of the source distribution because of the quadratic
term in the phase. The quadratic term is the focusing term. It is necessary to eliminate it to
-1 3 6 -
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focus the array. It is also necessary to eliminate it to apply the GSCA, which invokes a
Fourier relationship between source distribution and the radiation field. If the quadratic
phase term were be eliminated then the curvature of the phase-front is removed and the
Fourier relationship between the radiation field and the source distribution is restored.
Furthermore, when J(u) is a random function, then elimination of the quadratic term makes
the radiation field a stationary random process and its spatial correlation function is
independent of the measurement location in the array. Consequently, the Fourier
relationship between the spatial correlation function and the intensity distribution is
restored.
If Au is small then the quadratic kernel becomes approximately independent of u
and (5.1) reduces to
(5.3)
J
Au
which reduces to
(5.4)
for a point source. The second term is a plane wave propagating from u = uQ. The first
term can be viewed as a deterministic phase distortion term A(|)(x) = k
X
. It can be
2p0
treated as an error that appears in the aperture. This is exactly the type of error the self­
calibration procedure is designed to eliminate. Consequently, the GSCA removes the
phase curvature of near-field data. Therefore only standard Fourier beamforming weights
are necessary to scan the beam and image the source distribution after self-calibration has
taken place. The requirement of Au being small is not restrictive since most high-resolution
systems operate with a small field-of-view.
-137 -
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For example, both the synthetic-on-receive and ISAR systems used to collect the
experimental data presented in this document employed a 4 ft parabolic dish for the
transmitting antenna. The beamwidth was 1.75 degrees and the total variation of the (1u2) = cos20 term in (5.1) is less than 0.05% of its maximum value of unity.
5.2 Elimination of the Effects of Near-Field Phase
Curvature on the GSCA
5 .2 .1
M a th em a tica l C h a ra cteriza tio n o f the N e a r-F ie ld E ffects o n the
G SC A O b je c tiv e F unction
The quadratic phase distortion term makes the radiation field a nonstationary
random process and the corresponding autocorrelation function dependent on the
measurement position in the array. Let
Ku) = X aiSC u-U i)
(5.5)
Then
x2 ( l - u ? )
-jk
e(x) = £
a; e
X U . -------------------- —
1
2p
ro
(5.6)
x2 ( l - u 2 )
jk
XU
0
-
o
Let the array be focused by multiplying (5.6) with a focusing term e
Then
-jk x (u .-u
E[e(x)e*(x + X)} = ]T a 2e
i
)— - — (u 2 - u ? )
o' 2o
*0
0
1
jk ( x + X ) ( u . - u ) - ( x + X )
i
o'
2p
o
(u 2 - u 2 )
0
1
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which reduces to
jk
£d(u.-u
' 1 o')-
f2d2(u2-u?)
v 0 1'
2P„
-jk
nW2(u2-u?)
v 0 1'
(5.7)
=X*i
where the discrete nature of the array has been accounted for by substituting x = nd and
X = (n^ - npd = PA with d representing the inter-element spacing, n the element index, and
I the correlation lag index. The nonstationarity appears in the second exponential of (5.7).
The phase of this term is a random quantity due to the presence of the random variable u;.
However, some lags are approximately stationary if the expected value of the nonstationary
_
U.
phase term is less than J q radians [29]. If Uj is uniformly distributed between uq —
u
and u + —^
0
2
then
k n & i V - u 2)1
o
1
kn£d2u?fov
12pr
(5.8)
For a particular lag, t , the largest that (5.8) can be is
kn^d2(u2 - u 2) 1
o
I
k ( N - l ) l d 2u2Qv
_____
(5.9)
max
12p„
Forcing (5.9) to be less than or equal to ^ radians establishes a quadratic inequality in the
lag variable, L, given by
2
(0.6)?ipo
Z2 -(N )(* )+ p° > 0
2 j2
(5.10)
ufo v d
where
ufov = an§ular extent half of the field of view
d = spacing between array elements
p0 = range to which the array is focused
-139
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N = total number of elements in the array
X = operating wavelength of the imaging system
for the largest approximately stationary lag to be used in the calibration process. The
solution of the strict equality of (5.10) yields the highest-order lag that has a nonstationary
term no greater than ^ radians. This result suggests that an additional weighting function
should be included in (3.26). The weighting function should be zero for all lags beyond
the largest allowed lag determined from the strict equality of (5.10) and may be rectangular
or tapered for smaller lags. Therefore the objective function of (3.26) should be modified
to
L
J4 =
Z wnf (*)w p(/)R (Z )
i =-L
(5.11)
where wNF0O represents the necessary weighting of the correlation coefficients to account
for the nonstationarity of the received radiation field.
It is interesting to examine Tsao's ECA in the near field. The ECA was founded on
Parseval's Theorem, which describes a Fourier Transform relationship, implying that the
received data set has originated in the far field of the array. The objective function (3.7)
can be Fourier transformed to the form of equation (3.26) in which the dependence on the
spatial correlation lags is explicitly shown. Equation (3.26) requires that all the measured
correlation lags be calculated and summed.
To test the effects of a nonstationary correlation function on the objective function
of (3.26), the simulation of Tsao and Patrick was modified to include a quadratic phase
curvature in the data to simulate near-field conditions. The results showed that the image
improved when the highly nonstationary lags were removed from the calculation of the
objective function; see Figure 5.1. Each plot is an image of a point source located at 0
radians 8 meters away from a 1.74 meter array putting the source in the near field of the
-140-
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array. This is a very severe test. The array was calibrated using two algorithms; one that
made use of all of the available lags and another that used only the first three lags. After
calibration with each algorithm a point source located at zero radians was imaged. The top
plot is the image obtained when all the lags available in the array are used; in the bottom
plot only lags 1, 2, and 3 are used. Although neither image is a high quality, the lower
image is superior in its definition of the mainlobe, the level of its first sidelobe, and in the
asymptotic sidelobe level.
-141 -
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ECA Performance In New Field with 29 Correction Lags
(a)
S
•20
*40.
• 0.3
•
0.2
•
0.1
0.2
0.3
ECA Performance In Netr Field with 3 Correlation Lagt
(b )
8
•
Figure 5.1
0.2
•
0.3
0.1
Illustration of the effect of sever ncar-field phase curvature, (a) calibration using all 29
lags. Mainlobe hardly evident. First sidelobe - 4 dB. (b) only the first 3 lags. Mainlobe
is evident. First sidelobe i s -10 dB.
- 142-
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The experimental data set originates in the near field of the array. When the phase
curvature of the near-field data is approximated even by a simple quadratic approximation,
the spatial correlation function is no longer stationary in the array aperture. The value of a
particular correlation lag now depends on the location at which it was measured within the
aperture. Therefore, averaging a particular lag over locations is meaningless. Because the
GSCA is a multiple lag correlation algorithm, the stationarity of the radiation field must be a
concern. This concern is not encountered in the Unit-Lag Spatial Correlation Algorithm [2]
because generally the unit-lag correlation exhibits a negligibly small nonstationary term
even in near-field conditions. This property is discussed further in the following section.
5 . 2 . 2 R o b u stn ess o f the U nit-L ag S p a tia l C o rrela tio n A lgorith m
The unit-lag algorithm emerges from the generalized objective function by
maximizing (3.26) with L = l. As discussed in Chapter 3, this situation can occur in one of
three different ways. First, the algorithm can be arbitrarily truncated to include only the
unit-lag measurements. This is strictly done to achieve minimal computational complexity.
Second, the weighting function, wp(Z), may fall-off very rapidly such that it is essentially
zero for all lags of order higher than one. This situation is solely a result of the scene being
imaged, the characteristics of which may dictate that such a weighting function is
appropriate. The third manner in which the unit-lag algorithm manifests itself is when the
imaging geometry and the parameters of the imaging system dictate that a near-field
weighting function must be applied to the general objective function. If the imaging is deep
enough into the near field that all correlation lags above the unit-lag must be eliminated
from the objective function, then the unit-lag algorithm becomes the appropriate algorithm
for calibration.
Very little about the properties of the Spatial Correlation class can be learned from
the first two situations just described. In the first case the lags are arbitrarily truncated
-143 -
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without any consideration given to the GSCA theory. The manifestation of the unit-lag in
the second case is due only to the characteristics of the scene being imaged. However, the
third situation does reveal an important property of the unit-lag algorithm. As the source
distribution comes further into the near field more of the higher-order correlation lags have
to be eliminated from the objective function. Only the lower-order lags will be sufficiently
"stationary" to remain in the objective function. Obviously, at a certain point only the unitlag will remain included in the objective function and beyond this none of the correlation
lags are suitable to calibrate the array. The unit lag is the last correlation lag to breakdown
and, consequently, the unit-lag algorithms are the most robust near-field spatial correlation
algorithms within their respective sub-classes with respect to near-field calibration
capability.
5 .2 .3 The N e a r-F ie ld W eigh tin g Function
The near-field weighting function wNF(Z) arising in (5.11) as a result of the
nonstationarity of the correlation function turns out to be unnecessary in most highresolution microwave imaging scenarios. The reason for this lies in the scaling and duality
properties of the Fourier Transform and the structure of the data. Consider an arbitrary
GSCA objective function of the form
(5.12)
When (3.4) is substituted into (5.12) only the functions p(u) and e"^kuxn^ are functions of
u. Therefore the integral over u degenerates to
(5.13)
Recognizing that xm - xn = Zd, it is easily seen that (5.13) is a Fourier transformation of
the function p(u) from the variable u to the variable Z. The scaling property of the Fourier
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Transform tells us that as the spatial extent of the p(u) in u-space increases, the width of the
"mainlobe" of the corresponding transformed function in lag-space decreases. Since this
function weights the correlation values of various lags, only the lags within the mainlobe
contribute significantly to the objective function. Alternately when the extent of the source
distribution is small in u-space, and therefore the integration interval is narrow, the
weighting function in lag-space expands and more correlation lags can contribute
significantly to the objective function.
In summary, when the field-of-view is large the mainlobe of the weighting function
extends for only a few lags and the objective function is minimally affected by near-field
phasefront curvature. The higher-order lags, which are the most nonstationary, get
multiplied by very small weights. Only the low-order lags are given significant weights.
Provided that these lags have small nonstationary components no significant deterioration
in the image will be noticeable. If more lags than just the unit lags have negligibly
nonstationary components then the resulting final image will be superior to the image
produced by the ULSCA. However, at extremely short range such that even the unit lags
have significant nonstationary components, the GSCA will not produce a superior image
over the ULSCA since any contribution o f the higher-order lags introduces more
nonstationarity into the objective function.
For example, p(u) = rect(ui,u2) yields the ECA for a rectangular unaberrated
intensity distribution located between Uj and u2 in u-space. Following (5.13) the lagdomain weighting function is
w (£) = sine
(5.14)
of which the mainlobe extends
i
(U2-Uj )d
(5.15)
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If the solution to (5.10) is greater than (5.15) there is no need for the near-field weighting
function in (5.14).
5 . 2 . 4 The E ffect o f a D o m in a n t S c a tte re r on the N e a r-F ie ld W eigh tin g
Function
The situation changes when either a dominant scatterer is in the data set or only a
narrow field of scatterers exists. An example is an aircraft target free of clutter. When a
dominant scatterer dominates the averaging process, the error-free averaged image closely
resembles the error-free image of the dominant scatterer rangebin. In this case the most
appropriate integration interval is a very narrow interval on the order of one receive array
beamwidth (discussed in Section 3.3.2). A second interesting case is a narrow field of
scatterers, typified, for example, also by an airplane target free of clutter. In this case a
narrow integration interval is also the most appropriate. Due to the Fourier relation of
(5.12), the correlation-domain weighting function will have a broad mainlobe. This will
cause many of the higher-order correlation lags to carry significant weight in the calculation
of the objective function. This is a potentially dangerous situation when self-calibrating
with near-field data since the higher-order lags are the most nonstationary lags.
This problem is remedied when a dominant scatterer is in the field of view. When
phase errors are present the correlation value between the signals at the nlh array element
and the (n+Z)1*1 array element after focusing the array to a distance p0 and angle u0 is
R „.„« = e‘ i<li»w " li“ > E = '? e i<T® t V
i
(5.16)
which is (5.7) modified by the phase error term e\s\up5(-j(P^s\do4(n+jt) - p\s\do4(n))). The
phase terms T§ and T ^ § represent the stationary and nonstationary terms, respectively,
-1 4 6 -
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-k{d/.(Uj - u ) - d2( l 2 - 2/.)(u? - u q )}
T s = -------------!— 5----------------------!------- 5—
(5.18)
2 Po
and Uj is the direction of the ith point source in the source distribution. From (5.17) we can
see two alternative ways to make TNS negligible. One is to keep the product nI small,
( 2
2\
which was the objective of section 4.2.1. Another way is to make the difference vuj ' u o '
very small. This is the situation when a dominant scatterer is present.
Let a dominant scatterer located at u = u;
‘d s
be in the data set and let the ULSCA
phases be used to initialize the GSCA. Then the array will always be approximately
focused and steered toward the dominant scatterer, i.e., u„ = u;
0
. In this case the term
‘d s
i = iD§ in (5.16) dominates the summation. Both T§ and Tj^§ are approximately zero
2
2
because each term has as a factor either (u; - uQ) or (u; - uQ) and the range o f u is small, by
definition, for this case.
Therefore, the undistorted correlation values of all lags of the
dominant scatterer rangebin should be zero phase. Additionally, the undistorted rangebin
averaged correlation values used in (3.27) will also be approximately zero phase since the
dominant scatterer bin will dominate the averaging process.
Consequently, no
nonstationary effects will appear in the objective function and no near-field weighting
function is needed. In this case the higher-order lags now included in the objective
function through the wide mainlobe of the sine weighting function can be used to provide a
better estimate of the system phase errors than would be obtained by the unit lag SCA, and
a superior image will result.
This predicted effect is verified by the results of the application of the GSCA, using
the rectangular approximation, to the experimental data. Figures 5.2 and 5.3 show the
- 147 -
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error-corrected phases of most of the lags from rangebin #39 (the dominant scatterer bin of
the ISAR data set). It is evident that all of the phases of the correlation lags have
approximately zero slope as a function of array element number, n. The deviations from
zero of the individual phases are caused by the residual phase errors after phase correction
and also by the fact that uQ only approximately equals ujDS. Even the higher-order
correlation lags have nearly zero phase slope. Figure 5.4 shows the phase of the lag 1
correlation value as a function of array element number averaged over 30 rangebins. Due
to the influence of the dominant scatterer the average correlation value exhibits zero phase
slope. No significant nonstationary effects are observed. Consequently, no near-field
weighting function is required when applying the GSCA with the rectangular
approximation to this data set. The image produced by incorporating all of the correlation
lags into the objective function is verified in Figure 3.6(c) to be superior to the one
produced by incorporating only the unit lag (Figure 3.6(a)).
-148 -
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Phases of Lags 1 -10
Phases of Lags 11 -20
Radians
M
§
3d
a
Array Element Number
Array Ebmcnt Number
Phases of Lags 31 -40
Radians
Phases of Lags 21 -30
§
39
Array Element Number
Figure 5.2
Array Elemenl Number
Error-corrected phases of the correlation lags 1 through 40. These lags were
computed from die dominant scatterer rangebin of the experimental data after
focusing had been applied.
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Phases of Lags 51 -60
4
3
2
I
}
0
0
■2
•3
-4
0
Array Element Number
20
40
60
80
100
120
140
A m y Element Number
Phases of Lags 100 -110
Phases of Lags 61 -70
(0
§
60
100
Array Elcmmt Number
Amy Element Number
Figure 5.3
80
Error-corrected phases of the correlation lags 41 through 70 and 100 through
110. These lags were computed from the dominant scatterer rangebin of the
experimental data after focusing had been applied.
- 150-
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4
Phase o f Lag 1 of Rangebin Averaged Correlation Coefficient
3
2
1
0
1
2
3
■4
0
20
40
60
80
100
120
140
Array Element Number
Figure 5.4
Phase of Averaged Correlation Coefficient of Lag 1. Averaged over
rangebins 30 - 60. The zero slope indicates the dominant scatterer bin
dominates the averaging and nonstationary effects are eliminated.
The same results are observed in the simulated near-field data. Rangebin #1
contains only one point source at u = 0. Figure 5.5 shows the phases of the first ten
correlation values computed from the data before the simulated phase errors were added to
the data. The non-zero slopes of the curves in this figure illustrate the nonstationarity of the
correlation phases. Figure 5.6 shows the same correlation phases computed from the
error-corrected data when the point source in rangebin #1 is a dominant scatterer. Figure
5.7 shows the same correlation phases also computed from error-corrected data when there
is no dominant scatterer present in the data set. The zero slope can be observed in the case
of a dominant scatterer (Figure 5.6) but not in the case when a dominant scatterer is absent
(Figure 5.7). The correlation function is clearly stationary when a dominant scatterer is
present and nonstationary when a dominant scatterer is not present.
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Radians
Unwrapped Phases of Correlation Lags 1 - 1 0
-10
Array Element Number
Figure 5.5
Unwrapped undistorted phases of correlation lags 1 -10 of the simulated data
without a dominant scatterer present. The non-zero slopes indicate the
nonstationarity of the correlation function. Slopes increase with lag
number.
Unwrapped Phases of Correlation Lags 1 -1 0
3-
2
-
1
-
-1
-
-2
-
-3 -
4
. ------- .------ .-------------- ,---------------,-------------- ,--------------0
5
10
15
20
25
30
Array Element Number
Figure 5.6
Unwrapped error-corrected phases of the correlation lags 1 - 10 of the
simulated data with a dominant scatterer present. The zero slopes indicate
the slationarity of the correlation function.
-1 5 2 -
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U nw rapped P hases o f Correlation L ags 1 - 1 0
C
/3
£
os
- 10.
Array Element Number
Figure 5.7
5 .2 .5
Unwrapped error-corrected phases of lags 1 -10 of the simulated data with a
dominant scatterer not present. The non-zero slopes indicate that the
correlation function is still nonstationary.
U se o f the N e a r-F ie ld W eighting F unction w ith P r a c tic a l
H igh -R esolu tion Im agin g System s
The solution of the strict equality of (5.10) yields the largest correlation lag that
should be included in the GSCA objective function when a near-field data set is being
calibrated. It turns out that for most practical high-resolution imaging systems the solution
of (5.10) is greater than the largest lag measurable in the array aperture. Consequently
there is generally no need to apply a near-field weighting function.
This can be verified by considering the parabola defined by
"fov d2
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= (x -C !)2 + c2
where Ci = -z ,
c2 =
(5.19)
/
ctXp0
N2 \
6 0
-=— r - -r, and a = — when the upper bound on
V uf o v d
)
3
(5 .9 ) is — (C3 = 10 yields equation (5.10)). The minimum of this parabola occurs at
c3
N
L = c j = ^ and the minimum value is ymin = c2. If c2 > 0 then the equality o f (5.10)
is satisfied for all lags and there is no need to include the near-field weighting function in
the GSCA objective function. When c2 < 0 equation (5.19) will have two roots and the
smallest root represents the maximum lag that should be included in the objective function.
However, for most practical high-resolution imaging systems c2 is positive. Since c2 is
dependent on several parameters, it can become negative in many different ways.
However, these cases generally represent impractical situations.
Consider the dependence of c2 on p0. For c2 to be positive requires
“ 1 Po
~ 2
ur
fov
N2
„
~
7
L' T > 0
d
<5 -20>
and, therefore, p0 to satisfy
P o >v4f^
(f)“L ^ NA
aJ
V4 « y
where q = ^ represents the number of inter-element wavelengths.
Consider a
1000-element linear array operating at 1 cm wavelength, an interelement spacing of 10 cm,
and
a field-of-view
o f 20
mrad,
as
Po > (^ )(0 .2 5 )(1 0 6)(10)(4x10’4)(10-1) = ^
listed
in Table
5.1.
T herefore,
meters = ( ^ ) c 3 » 17c3. Choosing a
nominal value of 10 for c3 yields p0 > 170 meters. For targets beyond this distance no
near-field weighting is required.
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Table 5.1 Order of Magnitude A jproximation of System Parameters
Parameter
X
Order of Magnitude
3
10 elements
_2
10 meters
d
10'1 meters
ufov
2x1O'2
Po
170 meters
N
These values are order-of-magnitude parameters for most VFRC experiments.
Tables 4.2 and 4.3 list the parameter values and p0 for the X-band ISAR system and the
83-meter X-band phased array at VFRC.
Table 5.2 System Parameters for VFRC ISAR Imaging Systei
1
Parameter
|
Parameter Value
N
128 elements
P
11.84 wavelengths
d
0.37 meters
ufov
0.0305 radians
Po
27 meters
Table 5.3 System Parameters for the VFRC
_________ 83-Meter Imaging System_______
Parameter
Parameter Value
N
330 elements
P
8.04 wavelengths
d
0.2515 meters
Ufov
0.0305 radians
Po
85 meters
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All VFRC targets observed with these systems are at ranges greater than p 0 and
consequently there is no concern for adverse effects from the near-field phase curvature.
This prediction is verified by the image shown in Figure 3.6(c), which is a high
quality image obtained by calibrating a near-field data set using the GSCA with no near­
field weighting function, and p(u) set to a very narrow rectangle function as described in
Chapter 3, Section 3.3.2. This causes wp(jt) to be a sine function with a very wide main
lobe as shown by the dashed line in Figure 5.8. In this case all of the measurable
correlation lags are within the main lobe and all contribute significantly to the GSCA
objective function.
C o r r e l a t i o n F u n c tio n E s t im a te o f E x p erim en tal D a ta
o
Z
0.2
0
20
40
60
80
100
120
140
Lag
Figure 5.8
Correlation function and weighting function taken from experimental data
(Run 3) with a dominant scatterer and integration interval of 0.66 mrads.
Solid line shows correlation function from experimental data set. Dashed
line shows weighting function with integration interval of 0.66 mrads.
Dashed-dotted line shows weighting function with integration interval of
30.5 mrads.
- 156 -
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5.3 Performance of the GSCA with Near-Field
Data
The problem of the nonstationarity of near-field data has been shown to disappear
when the backscattered radiation arises from a compact source such as a point radiator (a
dominant scatterer) or when a second weighting function is applied to the objective function
as shown in (5.11). What has not been discussed is the strength of the correlation values.
The radiation field of a source distribution must be highly correlated in space for the
sampled correlation function to have large values at the higher order lags. Because of the
inverse relationship between lag-space and u-space, a complex source distribution which
radiates a highly correlated electric field must be very narrow in extent in u-space. These
are the characteristics of a dominant scatterer. They are also the characteristics of a data set
obtained from a system with a narrow transmit antenna pattern. However as the field-ofview increases, the extent of the source distribution in u-space increases and the extent of
the spatial correlation function in lag-space decreases. This means fewer correlation lags
have strong values and are reliable enough to use to estimate the system phase errors. One
can easily think of the limiting case where the field-of-view becomes large enough that only
the unit lag is significant. In such a case the GSCA degenerates to the ULSCA and exhibits
no superiority over the ULSCA. Because of this the GSCA will yield its best performance
when a dominant scatterer is present in the data set or when a narrow transmitting pattern is
employed, thus allowing multiple correlation lags to be used. For this reason the GSCA
proves to be a superior algorithm to the Unit-Lag SCA in cases where a dominant scatterer
is present, a narrow field-of-view exists, or the system parameters are such that near-field
effects are not significant.
These characteristics are demonstrated in Figure 5.9 and Figures 5.10 through
5.13. Figures 5.9 and 5.10 are prime examples of the ability of the GSCA to produce
images superior to the ULSCA when there are higher order correlation lags of significant
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strength in the spatial correlation function. Figure 5.9 shows the error-corrected image of
the first rangebin of the simulated data set, as obtained from the GSCA, with p(u) set to a
rectangle function. There are no dominant scatterers present in the data set. Figure 5.10
shows the ULSCA image of the same rangebin. The ULSCA image is fair, while the
GSCA image is close to a diffraction-limited image. The quantitative superiority of the
GSCA image is measured by the peak and average sidelobe levels, shown in Table 5.4,
which are 4 and 2 dB better respectively.
Two factors contribute to the GSCA's ability to remove the artifacts of the ULSCA
image. First, the number of correlation lags is large. Another important factor is that the
GSCA possesses more information. This information is imbedded in the weighting
function wp(jt). As shown in
Table 5.4
Results of Calibration with the GSCA and Unit-Lag SCA with no Dominant
Scatterer and a Data Set Originating in the Near Field
Main Beam Gain Average Sidelobe Level
(dB)
(dB)
Peak Sidelobe Level
(dB)
GSCA Image
(Figure 5.8)
-0.011
-24.5
-12.38
Unit-Lag SCA Image
(Figure 5.9)
-0.32
-22.5
-8.15
Section 3.3.2, to obtain a weighting function p(u) that is narrower than the field-of-view,
an assumption must be made on the shape of the expected intensity distribution.
Consequently, the GSCA is being implicitly told what to expect the mean intensity
distribution to look like whereas the ULSCA, and also the multiple-lag MMB, are given no
such information.
This effect is analogous to the experience in the spectral estimation field where
parametric spectral estimation techniques perform well when the parametric model and its
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G S C A Im age o f R an geb in #1
-10
-15
-20
-25
-30
-
0.05
0.05
0.15
0.2
0.25
u = sin ( 0 )
Figure 5.9
GSCA Image with rectangular weighting function of extent Au = 0.167.
There is no dominant scatterer present and the data set originates in the near
field
Unit-Lag SCA Image of Rangebin #1
-10
.8
-15
-20
-25
■301.11 fl. I l l n II I III I I II LLI U I
-
0.25
-
0.2
-
0.15
-
0.1
-
I
0.05
0
0.05
0.1
0.15
0.2
0.25
u = sin ( 0 )
Figure 5.10 Unit-Lag SCA image with no dominant scatterer and a near-field data set.
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parameters are chosen correctly. These methods have been called "super-resolution"
techniques based on their ability to resolve very closely spaced frequencies. However, if
the model and/or parameters are not chosen correctly, the techniques fail badly.
When assumptions are made on the expected intensity distribution p(u) the
procedure effectively becomes a model-based self-calibration technique. When the
assumptions are correct, the calibration is better than ULSCA or the MMB. However,
when the assumptions are incorrect, the resulting calibration is virtually useless.
Figure 5.11 shows an estimate of the correlation function obtained from the GSCA
error-corrected data set of the simulation. Also shown (dashed line) is the sine weighting
function used in (3.26) corresponding to a rectangular p(u). The weighting function
applies large weights to the correlation function where the function is large and applies
small weights to the function where the correlation is weak. It is apparent that at least two
and perhaps as many as 5 or 6 lags contribute to the GSCA objective. Thus the GSCA has
Amplitude of Correlation Function and Sine W eighting Function
’o0
3
1
0.6
1
0.4
0.2
\/
Lag
Figure 5.11 Estimated Correlation Function and the Sine Weighting Function when no
Dominant Scatterer is present.
-1 6 0 -
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access to better estimates of the system phase errors than the SCA, provided that strong
correlation exists between array elements at higher-order lags.
Figures 5.12(a) and (b) show that the same effect occurs when the point source in
rangebin #1 is just barely dominant (the amplitude has been increased to 1.9). The GSCA
image has lower sidelobes than the ULSCA in the neighborhood of the main lobe. This is
demonstrated by the measurements shown in Table 5.5 that show the disparity between the
main beam gains and average sidelobe levels have largely vanished. However, the peak
sidelobe levels are still significantly different and because of this the GSCA image is still
the preferred image.
Table 5.5
Results of Calibration with the GSCA and Unit-Lag SCA with a Small
Dominant Scatterer and a Data Set Originating in the Near Field
Main Beam Gain Average Sidelobe Level
(dB)
(dB)
Peak Sidelobe Level
(dB)
GSCA Image
(Figure 5.10(a))
-0.016
-24.5
- 12.88
Unit-Lag SCA Image
(Figure 5.10(b))
-0.13
-23.6
-9.73
As the strength of the point source grows, the superior performance of the GSCA over the
Unit-Lag SCA becomes less noticeable. Figure 5.13 shows that the weighting function
Wp(Z) (dashed line), "over-weights" the correlation lags (solid line) since the higher-order
lags have not substantially increased in strength. However, as the point source increases in
strength, the normalized correlation function flattens and it levels off at unity as the point
source amplitude grows without bound.
As the strength of the source grows, the
weighting function better matches the correlation function.
Similar results are obtained with the experimental data.
The solid line of
Figure 5.8 illustrates the estimated correlation function obtained from the phase corrected
data after application of the GSCA. The dashed line represents the sine weighting function
- 161 -
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G S C A Im a g e ( E C A )
■10
(a)
-20
'8Q
-O.OS
0
0 05
0.1S
0.2
0.25
0.2
025
u = sin (0 )
T he U n k -L ag SCA Im age
•10
•15
(b)
-20
•30
-35
•0.05
0
0.05
0.15
u = sin (8 )
Figure 5.12 (a) GSCA image with ECA approximation. Dominant Scatterer
amp(l)=1.9 integration interval 17.24 mrads (b) Unit-Lag SCA Image with
Dominant Scatterer amp(l)=1.9
C o r r e la ti o n E s t im a te S in e W e i g h t in g F u n c tio n
0.8
■auo
f 0.6
■§
Z
0.2
0
5
10
20
15
25
30
Lag
Figure 5.13 Estimated Correlation Function and the Sine Weighting Function when a
Dominant Scatterer is present.
-
162
-
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obtained when an integration interval the width of one array beamwidth is applied. The
dashed-dotted line represents the sine weights obtained from an integration interval with a
width of the entire field-of-view. Knowing that this data set contains a dominant scatterer,
we expect the sine weights corresponding to an interval the width of one array beamwidth
to be a more appropriate weighting function than the weighting function corresponding to
the entire field-of-view. This hypothesis is clearly verified by Figure 5.13. Figures 3.6(a)
and 3.6(c) illustrate the improvement in image quality (even when the ECA approximation
is used) of the GSCA image over the Unit-lag SCA image.
5.4 Summary
This chapter extends the GSCA to the near field. It is shown that the algorithm can
successfully self-calibrate an array imaging system using data acquired in the near field of
the array provided that the nonstationarity of the received radiation field is compensated by
a near-field weighting function in the GSCA objective function. This weighting function
either removes or tapers higher-order correlation lags, thereby removing those lags that
exhibit nonstationarity.
The Unit-Lag Spatial Correlation Algorithm emerges from the GSCA class when
the near-field weighting function has such a form that only the unit-lag weight is non-zero,
at which point the stationarity constraint of Section 4.2 is enforced only on the unit-lag
measurements. Since the non-zero extent of the near-field weighting function will diminish
as the source distribution is moved further into the near field of the array, the unit-lag
weight is the last weight to become zero. Consequently, the unit-lag algorithms are the
most robust to near-field effects in the Spatial Correlation class since it is the last lag to
suffer the effects of the nonstationarity of the received radiation field.
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The effects of the nonstationarity of the received radiation field are not seen in most
practical applications. Generally, most high-resolution imaging systems illuminate small
enough sectors at large enough ranges that all of the measurable correlation lags in the
aperture are not functions of measurement position in the array aperture. Additionally, a
dominant scatterer in the field of view has been shown to eliminate the near-field effects on
the algorithms.
Inclusion of the higher-order correlation lags is desirable provided that the
redundancy they offer smoothes the phase-error estimations more than the noise they
introduce disrupts the estimation process. The GSCA with an appropriate weighting
function produces superior images than the ULSCA even when the calibration is done
using near-field data.
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Chapter 6
Conclusions
The failure of the ECA to successfully self-calibrate an array imaging system using
experimental data provides the motivation for this work. The search for a modification of
the ECA results in the development of the GSCA. The GSCA explains why the ECA failed
to calibrate the imaging system with the experimental data and provides a solution to
achieve successful calibration with that data. Performance characteristics of the two spatial
correlation sub-classes revealed by the GSCA are studied in the presence of element
position errors and element receiver noise through simulations and first-order analysis of
the GSCA objective function. Additionally, the GSCA, which is developed under far-field
assumptions, is extended to calibrate in near-field conditions. The following is a list of the
conclusions of this research.
C onclusions
(From Chapter 3)
The ECA has been shown to be inadequate to successfully self-calibrate the
experimental data obtained at VFRC. The search for a modification has led to the
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development of the GSCA. The GSCA has been used to explain the unsuccessful results
of the ECA with the VFRC experimental data and has also been used to determine the
appropriate change in the ECA to obtain successful calibration.
In general, the weighting function should match the theoretical spatial correlation
function. This requires a priori knowledge regarding the intensity distribution and spatial
correlation algorithms requiring such knowledge have been shown to define a sub-class of
the Spatial Correlation class termed the parametric sub-class. It has been shown that the
ECA is a spatial correlation algorithm and, in particular, belongs to the parametric sub­
class. The presence of a dominant scatterer has adverse effects on the ECA. The
assumptions implicit in the ECA theory are not appropriate when a dominant scatterer is
present. In such a case the weighting function determined from the ECA does not match
the theoretical correlation function and calibration is unsuccessful. It has been shown from
GSCA theory that when such a scatterer is present the mainlobe of the weighting function
must be narrowed to the size of one receive-array beamwidth.
If a priori knowledge regarding the intensity distribution is not available then
algorithms from the non-parametric sub-class must be used to self-calibrate. These
algorithms estimate the theoretical correlation function from the data and use these values
for the weighting function; consequently, they are scene independent. The MMB algorithm
has been shown to belong to this sub-class.
Algorithms belonging to both the parametric and non-parametric sub-classes are all
characterized by the GSCA objective function. The manner in which the weighting
function is determined separates the two sub-classes and the particular weighting functions
separate the individual algorithms. Because the GSCA objective function is the general
form of all of the algorithms, only a single, efficient maximization procedure needs to be
implemented to maximize the objective function. This means that an imaging system can
have one efficient maximization procedure hardwired and still have access to all of the
-
166
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spatial correlation algorithms, both parametric and non-parametric, by simply changing the
weighting function in software.
(From Chapter 4)
The discovery of two sub-classes prompted a performance comparison between
algorithm s from the sub-classes.
Simulations were established to compare the
performances of a parametric algorithm and a non-parametric algorithm with a particular
source distribution in the presence of element position errors and element receiver noise. In
both cases the parametric algorithms performed better than the non-parametric algorithms.
However the performance difference is generally minimal (1 to 2 dB) and in many cases the
need for a priori
knowledge of the intensity distribution outweighs the superior
performance of the parametric algorithm.
In these cases the non-parametric MMB
algorithm is the preferred algorithm.
Element position error performance curves measured as a function of scan angle for
the ECA, the MMB, and the ULSCA exhibit the same shape as the performance curves
measured and derived by Steinberg [45] for the DSA. The curves for the spatial correlation
algorithms are shifted vertically by a constant (over scan angle) relative mainbeam loss
term. From this a quick and easy approximation to the performance curves of the spatial
correlation algorithms has been presented. This approximation requires the measurement
of the relative mainbeam gain after calibration at only a single scan angle. For simplicity,
this can be taken on boresight. This measurement is used to vertically shift the DSA
performance curve, yielding an approximation to the performance curve of the particular
spatial correlation algorithms used for the calibration.
It has been shown that this
approximation is accurate to within a tenth of a decibel provided o Ax < X wavelengths or
ufov < 120 mrads.
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When a dominant scatterer is present the performance curves of the parametric and
non-parametric multiple-lag algorithms are essentially identical to the DSA performance
curve. Consequently, there is no loss from DSA performance by implementing a multiplelag spatial correlation algorithm instead of the DSA when a dominant scatterer is present. It
has also been shown that DSA performance is attained by the ULSCA when a dominant
scatterer is present.
In this case the ULSCA is the preferred algorithm since it is
computationally simple and does not depend on the shape of the intensity distribution.
First-order analysis o f the GSCA objective function for both sub-classes has
provided evidence that the ULSCA belongs to the parametric sub-class and is not
equivalent to the MMB algorithm with L = 1. This explains the performance difference
between the ULSCA and the MMB with L = 1 and the superior performance of the
ULSCA over the unit-lag MMB illustrated by the performance curves for these algorithms.
The first-order analysis has shown that phase error estimates from non-parametric
algorithms have more noise than estimates from parametric algorithms.
The poor
performance of the unit-lag MMB is explained by this analysis since the unit-lag MMB
belongs to the non-parametric sub-class.
Lastly, the first-order analysis has shown that algorithms from the parametric sub­
class solve nearly the identical least-squares problem solved by the MLSCA when the
phase noise variance is small. Therefore, essentially the same solution obtained by the
MLSCA can be obtained from a parametric algorithm. The significance is that because of
the cosinusoidal nature of the parametric algorithm objective functions modulo 2k
ambiguities in phase measurements are not important.
However, the same phase
measurements must be unwrapped before the MLSCA can be used.
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168
-
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(From Chapter 5)
When the source distribution lies in the near field of the array imaging system, the
received radiation field is a nonstationary random process. Consequently, the theoretical
spatial correlation function will depend on the position in the array at which it is measured.
This is a problem for the spatial correlation algorithms since they all assume any
nonstationarity is due solely to the presence of phase errors.
It has been shown that the GSCA objective function must be modified to account
for the nonstationarity that occurs with near-field source distributions. The solution is the
inclusion of a near-field weighting into the GSCA objective function that truncates, or re­
weights, the correlation lags used for calibration.
Analysis o f the origin of the
nonstationarity has shown that the higher-order lags are the most severely nonstationary
while the lower-order lags can generally still be used for calibration. Consequently, the
unit-lag algorithms are the most robust to near-field effects of all of the spatial correlation
algorithms, since the unit-lag is the last lag to become nonstationary.
The analysis also indicates that the nonstationarity problem is not a serious as it was
previously thought to be. The reason for this is two-fold. First, the analysis has shown
that the presence of a dominant scatterer eliminates the nonstationarity and the need for the
near-field weighting function. Second, because of the parameters of most practical highresolution microwave imaging systems the nonstationarity o f the radiation field is not
perceivable and the near-field weighting function is, again, not needed. However, because
the GSCA is inherently a multiple-lag algorithm, the effects of near-field source
distributions must be considered and the need for the near-field weighting function must be
checked.
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6.2 Suggestions for Future Work
The intention of this research was to "close the book" on the Spatial Correlation
class of algorithms. As mentioned in Chapter 1, to a large extent, this goal has been
accomplished. However, some loose ends still exist and these are cited in this section as
suggestions for future work. There are four such topics and these are discussed briefly
below.
1)
Completion o f the Linearization o f the GSCA Objective Function
Chapter 5 shows that a linearization of the GSCA objective function leads to a
weighted least-squares problem. However, the rank deficiency of the non-parametric
mapping currently prevents the derivation of any mathematical analysis useful in comparing
the performances of the two sub-classes. The least-squares theory is exhaustive, as is the
theory of generalized inverses for matrices. At this time, it appears as though one piece of
information is missing from the least-squares problem of the non-parametric sub-class.
The crucial piece of information may be found by re-visiting the non-parametric objective
function and extracting some information describing the manner in which the phase
estimates are obtained through maximization of the function. This information needs to be
incorporated into the non-parametric system of equations to boost the mapping matrix to
full-rank status. At that point the comprehensive theory of least-squares could be used to
compare the variances of the phase estimates from the two sub-classes. The theory of
generalized inverses of matrices may provide the solution to this problem.
Another route to a mathematical comparison of the performance of the two sub­
classes may be the development of upper and lower bounds on the performance measures.
Many upper and lower bounds exist for various matrix norms (for example, see [50]).
Additionally, the mapping matrices of both sub-classes possess an immense amount of
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structure.
Possibly, this structure, combined with the appropriate bounds on an
appropriate matrix norm, could yield a useful mathematical performance measure.
2)
The Performance Limits o f the Two Spatial Correlation Sub-Classes
It is very possible that this topic goes hand-in-hand with the previous topic. In
Chapter 6 the observation was made that the performance of the non-parametric sub-class
appeared to level-off as the number of incorporated into the calibration process was
increased. At the same time the parametric sub-class appeared to continue to approach the
diffraction limit. It would certainly by useful to know what are the performance limits of
the two sub-classes and if the performance limit of the non-parametric sub-class is below
that of the parametric sub-class. It seems very possible that if the linearization problem
discussed in 1) is solved, this question may also be simultaneously answered.
Consequently, the best attack on this problem may be to attack the problem discussed in 1).
3)
Experimental Verification o f the Tolerance Curves
As mentioned in Chapter 6, until this research no extensive tolerance study had
been published and no research at all had been published regarding the tolerances of the
element position errors. The tolerance curves presented in Chapter 6 and Appendix B
should be good approximations to the actual curves. However, experimental verification
would certainly be gratifying. Such verification would allow the system designer to
unhesitatingly employ the approximations (7.1) and (7.2) to the relative gain curves as a
function of scan angle, q, suggested from the simulation results of Chapter 6 (and
Appendix B).
4)
Expanding the Spatial Correlation Class to Include the DSA
This last suggestion is probably the most challenging, and exciting, topics of the
four listed in this section. Now that the Spatial Correlation Class has been expanded and
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its scope appears to be established, the question of further expansion arises: namely, "Can
the Dominant Scatterer Algorithm be incorporated into this class"? There appears to be
several things in favor of this pursuit. First, it can be argued that since the DSA is also a
phase-correcting algorithm it might also be subsumed by another expansion of the Spatial
Correlation Class. Second, the tolerance curves presented in Chapter 6 and Appendix B
indicate that the algorithms of the parametric sub-class and the multiple-lag algorithms of
the non-parametric sub-class yield relative gain curves nearly identical to the DSA element
position error tolerance curve. A breakthrough in this area would certainly be exciting and
probably carry with it many ramifications.
-1 7 2 -
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Appendix A
Derivation of the Variance of the Correlation Phase Noise
2
The quantity Cj can be evaluated by considering a signal-plus-noise model for the
received complex field at each array element.
T
^n,n+jt = T
S
^et,ne^^n + Tlt)n)(et,n+£e^ n+* + Tlt,n + ^
t=l
3
=
^n,n+jt
+
S
^Pn n+jt^
Pn,n+jt
(A .2 )
^.n+ Z
The sum of the terms containing noise in (A.l) is represented in (A.2) as the sum of three
complex numbers with each complex number defined as
T
•^1
^ r i n,n+£ ~ T ^
n,n+j£
rlt,ne t,n+jde
( A . 3)
T
%2
.e^r2n.n+^ = T ^ ^ t ,n + jd et,ne^ n
n,n+i
’
t=l
(A -4)
T
•^ 3
n,n+£
^
3n,n+Z = j ^
^t.n^n+Z
(A-5)
t=l
»
and Rn n+jt the noiseless estimate of the phase distorted true correlation value
=k£
et,nHn+1£<=K|in ' Iin^ )
(A.6 )
t=l
In this model complex Gaussian noise is assumed to be present at each array element. The
noise observations are assumed to be independent both spatially and temporally.
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Additionally, the I and Q components of the noise are also assumed to be independent.
Each component (I and Q) is zero mean and has a variance of o^.
The phase of
n+ji can be found from (A.2) as
+ Pn - P n + * ]
+
^
p
V
. = T an' 1
j?Pn,n+ZS in ( r Pn,n+jt)
= 1
n.n+t
n ,n + £
| r J c 0s [ v < + f n - | W ]
+
£
(A.7)
J’P n ^ i C0S(rf » , +i )
Pn,n+jt1
p
Tan' 1
= v * + Pn - Pn+Jt +
= 1
n.n+fc
R- t i +
(A.8)
5 > p „ , t i cos(!w
Pn fl+t
where £p = r p -
+ Pn - Pn+jtl- T°r elements possessing a strong correlation with a
moderate signal-to-noise ratio the inverse tangent function in (A.8) is approximately linear
with its argument. Therefore
Pn - P n + i + i
|
(A.9)
£
^
P
=
p n,n+jt
= /
1
and the error, 5n n+^, due to noise is approximately
3
X
5n’n+* ~ | R I
1
p
^Pn,n+Jt Sln(^Pn,n+i)
=
(A. 10)
1
n,n+jt
The quantity ^
n,n+jt
sin(4,
a,
n,n+£
n,n+i
A
) can be considered as the imaginary part of the quantity
,»♦/ =
t=l
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t=l
Likewise, the quantities %
, sin (t2
n ,n + t
,) and A-i
n,n+j i
. sin(E2
n,n+X.
,) represent the
n ,n+ Z
imaginary parts of
*
A
, .
-
n,n+i
t=i
T
t=l
and
* 3
respectively.
n,n+i
e j 6 3 „ ,„ + i
=
i
X
—
’ 1 . , n 1l . t n t f ' i [ V ' + |J " ‘ |5 n ^
’
’
1
<A ' 1 3 >
Express the noise in terms of inphase and quadrature components as
Tl»IjlIn = nhIjlln + j n<fil^lln where two components are independent of each other and also
between rangebins and elements. Similarly express the data in terms of inphase and
quadrature components as e( n = eit n + je q t n .
Equation (A.l 1) can now be expressed as
T
= T - S ^nit,n + j ntH ,n^eH,n+j6 + j e<^t,n+Z^ [cos(x*//.+ Pn) ' j sin('*fJt +Pn)]
(A. 14)
1=1
= T
X
[Ci+jC^]leosCVjt+Pn) - jsin(Vz+pn)]
(A.15)
t=l
T
= T
X
C 1cos(Vjt+pn)+C2sin(Vjt+pn) + j(C 2cos(Vjt+pn)-C 1sin(Vjt+pn))
(A.16)
t=l
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Therefore the imaginary part is
T
Im a g U j
e ^ 1n,n+Z} =
( C2cos(Vji+pn)-C 1sin(Vjt+pn) )
n,n+jt
(A.17)
j
where
C 1 = n*t,neit,n+/ - nV .,eV + £
(A. 18)
C 2 = " V 'Y n + t + ni.,ne V t Z
(A. 19)
By defining C, = nit n+jte it „ - n q ^ e q , „ and C2 = nql n+jteil n + n y n+l e\ n the
imaginary part of (A. 12) evaluates to
T
I m a g l^
n,n+jt
eJ^2n,n+jt} = ^ T , ( C2Cos(Vjt-pn+jt)-C’1sin(Vjt-pn+^))
t=l
(A.20)
The imaginary part of (A. 13) is more easily determined to be
T
Im a g U 3
e \ n+t | 4 l (
n,n+jt
t~l
ncU,nnW
+ nit,nnCW
^
(A.21)
Following equation (A. 10)
8 n ,n +1
“ Ima§ t
e^2n,n+1 } +
eJ^n,n+Z) + Imag {^
n,n+jl
n,n+jt
Im ag{^3
eJ^3n,n+jt}
(A.22)
n,n+jt
The variance of 8nin+J£ can now be computed as
E[8„Vi] E[lmag2{^,
L
e^Y.n+z} + e [~Imag2] ^
n,n+jt
L
[
n,n+jt
2
Imag [M-,
jEo
e Vn+jt}
(A.23)
n,n+Z
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Since the noise components are independent from element to element, all of the cross-terms
are zero under the expectation operator. Since each term in (A.22) represents a sample
mean averaged over T rangebins, each term in (A.23) represents the variance of the
correspondng estimator in (A.22). Consider the variance of the first term
E [im ag2 {.3,
L
e^n.n+jl}!
n,n+jt
T
= ^2 X
J
T
I
K
ti=l t2=l
cos(Vjt+pn)-C1 sin(Vjt+pn) ) ( c 2t c o sO ^ + P n )-^ s i n ^ + p j )
1
1
2
2
(A.24)
T
= ^2 S
E [ C 2cos2(Vjt+pn)]+ E [C isin 2(Vj6+pn) ] - E [ C 1C2cos(Vjt+pn)sin(Vjt+pn) ]
t=l
(A.25)
It is tedious, but fairly simple to verify that
E [ c 2cos2(Vj£+pn) ] =
eit,n+/ E [ nClu n ]COs2^ +Pn) +
eClt,n2^ E [ nit!n]COs2^ +Pn) +
2eit,n+Ze(lt,„+/ E[ nit,n]E[nclt,n]cos2^ +Pn)
(A-26)
2
If the I and Q components of the noise are zero mean and have equal variances, o^, then
(A.26) becomes
E [ c 2cos2(Vjt+pn) ] = 0 2 Icl n+jL l2cos2(Vjt+pn)
(A.27)
Following the same approach it can be shown that
E [ c 2sin2('l/2+Pn)] =
letn+jt P s i n ^ + p ,,)
(A.28)
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and
e
(A.29)
[ C 1C2cos(Vjt+pn)sin(Vjt+pn) ] = 0
Therefore (A.25) evaluates to
Imag2{ ^
y \ , n +*}]
let,n+/
n,n+Z
t=l
= T 1( t Z
l e t,n+J 2 )
t=l
(A.30)
4 < k J 2>
where < • > denotes the spatial (rangebin) average.
The same approach can be followed to evaluate E
Im a g 2 } ^
e J^2n , n + / ) l .
n,n+jt
J
All
that is required is to interchange the positions of the n and n+ i subscripts and the
manipulations remain the same. This yields
e|" Imag2 {^
L
e ^ 2n,n+jt}
n,n+jt
- ^ < 1 l2>
(A.31)
The last term in (A.23) evaluates to
r
2 ,
,
Imag [A,
e Jn,n+z}
n.n+jt
_
-
AT
(A.32)
Substituting (A.30), (A.31), and (A.32) into (A.23) yields
n+jt|2> +
<1 et,n
2
2 an
> + “t
(A.33)
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^ [ < K ,J 2>+<I e,,nP>+2o^]
(A.34)
The terms | e t n+l | 2 and | e t n |2 are estimates of the zero-lag correlation value, R(0), and
should nearly equal provided a sufficiently large number of rangebins are averaged.
Thereofore (A.34) can be approximated by
:[ 5n ,n .i] ”
°T| ( R(0)
T
|R /P
V
[1+ 0n]
(A.35)
clearly showing that E ^ 8 2)n+Jj J is inversely proportional to | R ^ |
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Appendix B
Main Beam Relative Gain Curves For Array Element
Position Errors
The plots presented in this appendix represent approximations to the expected main
beam relative gain for all combinations of the variable values listed in Table 4.1. Figures
4.5 and 4.6 have been taken from this set as representative plots to illustrate the general
characteristics exhibited by all of the 40 plots generated.
In addition to displaying the general characteristics discussed in Chapter 4, this set
can, and should, be used quantitatively as very good approximations to the expected main
beam relative gain curves as a function of scan angle, 0, for various conbinations of system
specifications.
The plots are presented here mostly for this purpose in addition to
supporting the claims made in Chapter 4 ffom based on the plots of Figures 4.5 and 4.6.
The appendix is organized in the following manner. It is assumed that the design
engineer will either be given an existing radar sytem possessing a specific transmitting
antenna or the size o f the field-of-view will be set by some higher-priority system
specification. In either case the size of the field-of-view is assumed to be pre-determined
and canno be altered. The remaining free variables then are L, the number of correlation
lags incorporated into the calibration process and o ^ , the rms element position error in the
x-direction. Therefore, the plots are organized so that plots deriving from the same size
field-of-view are clustered. Each page contains two plots corresponding to the same fieldof-view and rms element position error, but different number of correlation lags. The
succession of plots begins with the smallest field-of-view simulated,
U fo v
= 0.017
radians, and ends with the largest, Ufov = 0.167 radians. In all of the following plots, the
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field-of-view values represent the full field-of-view. In some plots the dashed line
representing the DSA curve is not visible. This is because it is overlaid with a solid curve
from one of the sub-classes o f the Spatial Correlation Class. The DSA curve should
always be one of the upper curves in the plots since it always passes through the point
(AG=0.0,0=0.0).
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Main Beam Gain Curves
m
■a
P Sub-Class
NP Sub-Class
DS Class
o
>
ra
o
cc
•*—
-
0.2
-
-0.3
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Scan Angle (radians)
Main Beam Gain Curves
0.00
-
0.01
-
-
0.02
-
a
oc
P Sub-Class
NP Sub-Class
DS Class
-0.03
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Scan Angle (radians)
Figure B.l
Main beam gain curves for ufov = 0.017 and a Ax = g
(a) L = 1
(b) L = 29
-
182
-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Main Beam Gain Curves
0 . 0 -O '
m
•o
c
cs
(5
-
0.2
P Sub-Class
NP Sub-Class
DS Class
-
0)
>
■*->
o
v -0.3 CC
-0.4
0.00
0.02
0.04
0.06
Scan
0.08
0.10
0.12
Angle (radians)
Main Beam Gain Curves
0.00
0.02
-
ffi
2.
-0.04
C
0
0)
>
«=
-0.06
-0.08
fl>
P Sub-Class
NP Sub-Class
DS Class
CC
-0.10
-
0.12
0.00
Figure B.2
0.02
0.04
0.06
Scan
Angle
0.08
0.10
0.12
(radians)
Main beam gain curves for ufov = 0.017 and ctAx = (a)
L = 1
(b) L = 29
-
183-
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Main Beam Gain Curves
OQ
2.
-0.5
c
‘5
(5
P Sub-Class
NP Sub-Class
DS Class
0 .0 0
0 .0 2
0.04
0.0 6
0.08
0 .1 0
0.12
Scan Angle (radians)
Main Beam Gain Curves
o.o
«> -0.3 J5
0)
P Sub-Class
NP Sub-Class
DS Class
-0 .4 -
-0.5
0.00
0 .0 2
0.04
0.0 6
Scan Angle
Figure B.3
0.08
0 .1 0
0.12
(radians)
Main beam gain curves for ufov = 0.017 and a Ax = ^
(a) L = 1
(b) L = 29
-
184-
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REFERENCES
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-2 0 5 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
IN D E X
angle of arrival 89
necessity of 143
use w/ practical imaging systems
152-156
relationship to existing algorithms 67,
et seq.
weighting function of 64-65
average image intensity 56, 87
beam width, see resolution
bistatic radar 9
calibration 6
contributions 2-5
image sharpness 20
incoherent source distribution 19,23
Dominant Scatterer Algorithm 1,7,92,
ISAR 12
100, 102
main beam relative gain curve 94
least squares
see GSCA , linearization of objective
function
ECA87
linearization
of objective function
of non-parametric sub-class 125127
of parametric sub-class 121-125
element position error analysis 9 0 ,, et
seq.
phase errors due to 91
simulation of 92-100
Energy Conservation Algorithm 2, 7,13,
MMB algorithm 87
139
Modified Muller-Buffington 86
experimental
data, description of 12
imaging system, description of 12
Modified Muller-Buffington Algorithm 7,
13, 84-85, 121, 129, 157, 159
multiple-lag 29
far-field 18
of linear array 17
multiple-lag algorithm
full 36
iterative 33
least-squares 29-33
partial 36
field-of-view 9,10, 14
Fourier relationship
of radiation field and intensity
distribution 18
near field
of linear array 135-136
future work 167-170
Generalized Spatial Correlation Algorithm
noise analysis 109,, et seq.
2, 42, 57, et seq.
development of 60,64
linearization of objective function
121,, et seq.
of non-parametric sub-class 125127
of parametric sub-class 121-125
near-field weighting function 135,
142, 162
effect of dominant scatterer on
145-152
non-parametric sub-class see Spatial
Correlation Class of algorithms,sub­
classes of
parametric sub-class
see Spatial Correlation Class of
algorithms, sub-classes of
Parseval's Theorem 43
radar cross section 1
-2 0 6 -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
radio camera 7, 8
relative gain curves
of main beam
non-parametric sub-class 9 7 ,, et
seq.
parametric sub-class 9 7 ,, et seq.
resolution 8
scan angle 92, 94, 100, 102
self-calibration
algorithms 1
history of 6
spatial correlation
algorithms, class of 7
function 7, 59, 65, 84, 142
derivation of 17-19
Spatial Correlation Class of algorithms
sub-classes of
non-parametric sub-class 65
parametric sub-class 65
transmitter location diversity 8
Van-Cittert Zemike Theorem 19
-207-
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