Positioning Systems: A Unified Approach

167 Pages·1992·2.149 MB·English

by  Prof. Christopher R. Drane (auth.)

Lecture Notes
in Control and Information Sciences 181
W. and Thoma M. Editors: Wyner
C. R. Drane
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Advisory Board
L.D. Davisson • A.G.J. MacFarlane- H. Kwakernaak
J.L. Massey "Ya Z. Tsypkin-A.J. Viterbi
Author
Prof. Christopher R. Drane
School of Electrical Engineering
University of Technology, Sydney
P.O. Box 123
Broadway NSW 2007
Australia
ISBN 3-540-55850-0 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-55850-0 Springer-Verlag NewYork Berlin Heidelberg
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Dr. George Vorlicek
1948- 1988
Acknowledgements
I am very grateful to Professor Trevor Cole from University of Sydney for pro-
viding sanctuary to undertake the initial stages of this werk. I thank Professor
Rod Belcher and Professor Warren Yates from the University of Technology
for support during the latter stages of the study. Thanks also to Dr Godfrey
Lucas, Mr Mark Johnson, Dr Hong Yah, Dr Nicholas Birrell, Professor Hen-
ning Harmuth, Mr Craig Scott, Mr John Drane, Dr Greg Searl, Dr Doug Gray,
Mr Rick Jelliffe and the unknown reviewers for reading various versions of this
manuscript. Ms Carol Gibson is responsible for much of the detailed d~aXETSI
many of the pictures to be found in this document.
Preface
This monograph had an unusual genesis. In the 1970s Professor Harry Messel
had funded a research team at Sydney University to investigate novel techniques
for tracking crocodiles. After developing a crocodile tracking system, the team
turned their attention to tracking vehicles in urban areas. In the early 1980s,
Dr Michael Yerbury, Dr George Vorlicek, Mr Jorg Suchau and I invented a pro-
totype spread spectrum urban tracking system. By 1983 we had demonstrated
very accurate tracking in urban areas. Over the next few years this created con-
siderable commercial interest, both in Sydney and overseas. In 1987 1 became
the technical director of a company that was set up to develop a commercial
version of the spread spectrum vehicle tracking system.
Another group in Sydney was working on a system with similar capabilities
(lead by Dr Yerbury), and I was aware of several other such systems being
developed at other places around the world. As well, the Global Positioning
System was starting to come on-line.
It seemed to me that positioning systems were likely to move from the de-
fence sector to become of great commercial interest over the next twenty years.
The major area in which this would be likely to occur would be in the provision
of up-to-date position information for people, land-based vehicles, aeroplanes
and ships. This would coincide with the continued expansion of mobile commu-
nications systems. I believed that within twenty years, most mobile communi-
cations systems would also have a positioning capability.
Despite this likely scenario, it seemed that no-one was attempting to develop
a unified approach to the overall specification and design of such systems. I saw
such a unified approach providing a strongt heoretical underpinning to these new
positioning systems, as well as increasing understanding of more conventional
systems such as radars, sonars and radio telescopes.
I believed this to be an excellent area for academic inquiry. Accordingly I
spoke to research groups at a number of Universities with the aim of encour-
aging research in this area and had persuaded George Vorlicek to take up the
challenge. I had even persuaded the financial backer of the company to fund
University research of this nature.
All seemed set, but fate intervened. The financial backer ran out of money
and Dr Vorlicek died suddenly in tragic circumstances. The company became
insolvent and I became unemployed. I reconsidered the joys of University life
and decided to resume my academic career.
VIII Preface
Despite the failure of the company, I still believed that positioning systems
would have a bright commercial future. Accordingly I set out to develop the
theoretical framework which is described in this monograph. Initially I tried
an approach based on Estimation theory, however I found that an information
theoretic approach was much more rewarding. This was also a nice fit, as the
most likely industry sector to commercially exploit positioning systems will be
the communications industry.
Because the book is grounded in positioning systems rather than informa-
tion theory, researchers trained in information theory may find my notation
unorthodox. However, it is hoped that they can took beyond this. Rather than
fully solving the many problems posed by the need to unify work on position-
ing systems, I have endeavoured to formulate the problems in such a form that
researchers with expertise in information theory can become aware of the prob-
lems and perhaps solve them.
Despite the academic imperative of 'publish or perish', I found that the
comprehensive nature of the work did not allow publication in a serial manner.
I felt that all of the work would have to be published together in order for the
technical community to properly assess its worth. This is why I have chosen the
rather old-worldly method of first publishing my research in monograph form.
George Vorlicek looked forward to the day when vehicle tracking systems
would bring large economic, safety and security benefits to the community. I
hope this monograph can contribute to that vision.
Chris Drane
Foundation Professor of Computer Systems Engineering
University of Technology, Sydney
Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Wave-Based Systems . . . . . . . . . . . . . . . . . . . . . . . . 11
3 System Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Generalised Data Processing Theorem .............. 19
3.2 Upper Limit to Information Performance ............. 21
3.3 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.1 Device Invariance . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Estimator Invariance . . . . . . . . . . . . . . . . . . . . 30
3.4 Physical Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 System Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1 Optimal Configuration . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Comparison cf Different Systems ................. 41
4.3 Measurement Strategy . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Independent Error Approximation ................. 46
4.5 Classes of Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5 Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1 The Somce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.1 Stationary Gaussian Vector Sources ............ 59
5.1.2 Limits cn Source Information Rate ............ 62
5.2 Wave-Form Coding . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Physical Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6 Decodi~ g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1 Discrete Optimal Receiver . . . . . . . . . . . . . . . . . . . . . 73
6.1.1 Eit Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2 Continuous Optimal Receiver . . . . . . . . . . . . . . . . . . . 81
6.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
X Table of Contents
6.4 Noise Ambiguity .......................... 87
6.5 Signal Ambiguity .......................... 91
6.6 Performance cf Coding Schemes .................. ~4
6.7 Performance Enhancement Schemes ................ 102
6.8 Other Areas of Investigation .................... 105
6.8.1 Realistic Channels ..................... 105
6.8.2 Wave-Form Selection .................... 106
7 Esl imation ................................ 107
7.1 Derivation of Kalman Filter .................... 107
7.2 Kalman Algorithm ......................... 112
7.3 Information Flow During Estimation ............... 118
8 Conclusion ................................ 125
A Glossary of Commonly Used Symbols ................. 131
B Review of Information Theory ..................... 135
C Proof of Proposition 3.5 ......................... 137
D Pzoof cf Proposition 3.6 ......................... 141
E Example Performance Calculation ................... 145
F (lassification f c Systems ........................ 149
G Example cf a Class cf Systems 155
. . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
1. Introduction
Positioning systems measure the location of one or more objects. Examples in-
clude Loran [32, 82], Omega [60, 113], Global Positioning System (GPS) [83],
inertial guidance systems [68, 60], radars [109], sonars [118, ]36 and urban ve-
hicle tracking systems [48, 25, 24]. As well, most imaging systems inherently
perform a positioning function. Examples of imaging systems include: magnetic
resonance imaging, tomography, ultrasound [78], fluoroscopy, radiography ,]97[
optical telescopes and radio telescopes.
Positioning systems have very wide areas of application. For example GPS
will be applied to most forms of transport: general aviation [28], space naviga-
tion ]47[ and automobiles [21]. As well, various schemes for differential GPS will
be applied to geodetic surveying [30, ,57 70, 115]. Because of the wide area of
applicability of positioning and imaging systems, any general characterisation
could be used in many areas of science and engineering.
The aim of this monograph is to establish a general method for the math-
ematical analysis of positioning systems and to demonstrate the utility of this
characterisation. The basic approach uses information theory [103], differential
geometry [124], rate distortion theory ]8[ and chaos [104], as well as integrating
the more conventional methods of analysis based on estimation and detection
theory [95, 107].
The first step is to derive a general formulation. This formulation is then
applied to systems that use the propagation properties of waves to measure po-
sition. For example, systems that measure direction of a plane-wave arrival and
the finite propagation time of sound or electromagnetic waves. This restriction
to wave-based systems is not serious, as most important systems fall into this
category e.g. radars, sonars, radio-telescopes, GPS and Omega.
A wave-based positioning system consists of one or more references ites. The
position of remote objects are measured relative to these sites. Each reference
site may have a transmitter or a receiver or both. Each remote object may have
a transmitter, a receiver, reflective properties or some combination of these. For
example, in GPS each satellite transmitter is a reference site and a mobile vehicle
will have a GPS receiver which picks up the signals from the satellites. In a
simple radar system there is one reference site fitted with a transmitter/receiver
and the targets scatter the radio frequency energy back to the reference site.
Positioning systems can be divided into two categories: self-positioning and
remote-positioning. In self-positioning the remote objects measure where they