Content uploaded by Alexander Charlish

Author content

All content in this area was uploaded by Alexander Charlish on Dec 19, 2015

Content may be subject to copyright.

Autonomous Agents for Multi-Function Radar

Resource Management

Alexander Charlish

A thesis submitted for the degree of

Doctor of Philosophy

of the

University College London.

Electronic and Electrical Engineering

2011

I, Alexander Charlish, conﬁrm that the work presented in this thesis is my own. Where information has

been derived from other sources, I conﬁrm that this has been indicated in the thesis.

Signature.......................................................

2

Abstract

The multifunction radar, aided by advances in electronically steered phased array technology,is capable

of supporting numerous, differing and potentially conﬂicting tasks. However, the full potential of the

radar system is only realised through its ability to automatically manage and conﬁgure the ﬁnite resource

it has available. This thesis details the novel application of agent systems to this multifunction radar

resource management problem. Agent systems are computational societies where the synergy of local

interactions between agents produces emergent, global desirable behaviour.

In this thesis the measures and models which can be used to allocate radar resource is explored; this

choice of objective function is crucial as it determines which attribute is allocated resource and conse-

quently constitutes a description of the problem to be solved. A variety of task speciﬁc and information

theoretic measures are derived and compared. It is shown that by utilising as wide a variety of measures

and models as possible the radar’s multifunction capability is enhanced.

An agent based radar resource manager is developed using the JADE Framework which is used

to apply the sequential ﬁrst price auction and continuous double auctions to the multifunction radar

resource management problem. The application of the sequential ﬁrst price auction leads to the devel-

opment of the Sequential First Price Auction Resource Management algorithm from which numerous

novel conclusions on radar resource management algorithm design are drawn. The application of the

continuous double auction leads to the development of the Continuous Double Auction Parameter Se-

lection (CDAPS) algorithm. The CDAPS algorithm improves the current state of the art by producing

an improved allocation with low computational burden. The algorithm is shown to give worthwhile

improvements in task performance over a conventional rule based approach for the tracking and surveil-

lance functions as well as exhibiting graceful degradation and adaptation to a dynamic environment.

To Mum and Dad

Acknowledgements

I would like to thank my supervisors Karl Woodbridge, Hugh Grifﬁths and Chris Baker, for the contri-

butions and guidance which has made this work possible. My thanks go to Alastair Thomas, Chris Dent

and Colin Bell from BAE Insyte for their support through the CASE award. I would also like to thank

Glen Davidson, Andy Stove and Simon Watts from Thales UK for the numerous valuable discussions

and continued interest throughout the work.

I am deeply indebted to my family Barbara, Jeremy and Lorna whose continued support has made

this work achievable. Their contribution is without doubt the greatest as it started long before my time

at UCL.

My thanks also go to everyone who gave me much needed distractions during my Ph.D. This in-

cludes all those at London University Swimming Team but especially Katy Rose, Stuart Ventham and

Louise Oliver for their particularly distinguished yet refreshing lack of interest in radar systems. Finally I

would like to thank everyone from the UCL radar group, who made such a pleasant environment to work

in; particularly Marc Thomas, Daniel O’Hagan and Emanuela Cerrone at the start and Matt Ritchie, Matt

Ash and Waddah Al-Ashwal at the end.

Contents

1 Introduction 20

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 Novel Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Multifunction Radar 25

2.1 Radar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Radar Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.2 Radar Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.1.3 Electronically Steered Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 Measurement Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.1 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.3 Kinematic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2.4 Data Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.5 Track Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3 Multifunction Radar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.1 Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.2 Operational Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.3 Operational Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Multifunction Radar Resource Management 51

3.1 Parameter Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.1 Surveillance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Resource Management Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Methodologies for Resource Management . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.1 Rules and Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.2 Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Contents

3.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.1 Local Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.2 Best First . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Priority Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5.1 Function Ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5.2 Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Resource Allocation Measures and Models 71

4.1 Task Speciﬁc Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.1.1 Surveillance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 Information Theoretic Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.1 General Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2.2 Information in Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2.3 Information in Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3 Analysis of Measures for Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.1 Parameter Selection Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.2 Tracking in Clutter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3.3 Comparison of Information Theoretic and Task Speciﬁc Measures . . . . . . . . 94

4.4 Task Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4.1 Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4.2 Logarithmic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5 Resource Manager Performance Assessment . . . . . . . . . . . . . . . . . . . . . . . . 99

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Agent Systems in Multifunction Radar Resource Management 101

5.1 Agent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1.1 Intelligent Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.1.2 Multi-Agent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2 Mechanism Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2.1 Auction Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.2 Social Choice Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3 Java Agent Development Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4 Agent Based Resource Management Testbed . . . . . . . . . . . . . . . . . . . . . . . . 108

5.4.1 System Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7

Contents

5.4.2 Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4.3 Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Sequential First Price Auction 115

6.1 Sequential First Price Auction Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2 Sequential First Price Resource Management Algorithm . . . . . . . . . . . . . . . . . 116

6.2.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.2.2 Task Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2.3 Auctioneer Agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3 Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.3.1 Single Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.3.2 Multiple Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7 Continuous Double Auction 130

7.1 Continuous Double Auction Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.1.1 Market Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.1.2 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.1.3 New York Stock Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.2 Continuous Double Auction Parameter Selection Algorithm . . . . . . . . . . . . . . . 134

7.2.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.2.2 Task Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.2.3 Auctioneer Agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.3.1 Mechanism Efﬁciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.3.2 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.4 Simulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.4.1 Tracking Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.4.2 Surveillance Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8 Conclusions 159

8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.2 Key Research Achievements and Contributions . . . . . . . . . . . . . . . . . . . . . . 161

8.3 Future Work and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Bibliography 164

8

List of Figures

1.1 Typical maritime air defence multifunction radar scenario [Butler, 1998]. . . . . . . . . 21

2.1 Process of monopulse measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Linear electronically steered array, steering at angle θ0[Wirth, 2001] . . . . . . . . . . . 32

2.3 Linear array radiation patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4 Linear array radiation pattern steered at θ0=−π/2. . . . . . . . . . . . . . . . . . . . 34

2.5 Probability density functions for the envelope of noise and target plus noise . . . . . . . 36

2.6 Cell averaging constant false alarm rate detector . . . . . . . . . . . . . . . . . . . . . . 38

2.7 Target dynamic system, measurement system and sequential state estimation . . . . . . . 39

2.8 MESAR2 and SAMPSON multifunction radars [BAE Systems Insyte]. . . . . . . . . . . 49

3.1 Interleaved search beam pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Track sharpness adaptive revisit strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Typical multifunction radar resource management architecture [Miranda et al., 2006] . . 57

3.4 Markov decision problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5 Resource utility space for an example task . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Local optimum/brick packing scheduler . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.7 Queue/best ﬁrst scheduler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 Radar loading for dwell length and revisit interval parameters . . . . . . . . . . . . . . . 72

4.2 Single look 0.8 detection probability range as a function of coherent dwell length . . . . 73

4.3 Single look and cumulative detection probabilities as a function of range . . . . . . . . . 74

4.4 Cumulative detection range as a function of dwell length and revisit interval parameters . 75

4.5 Measures of predicted track estimation error . . . . . . . . . . . . . . . . . . . . . . . . 77

4.6 Tracking loading using various models for differing signal to noise ratios . . . . . . . . . 78

4.7 Tracking performance using various models for differing signal to noise ratios . . . . . . 79

4.8 Likelihood ratio and track existence examples for track initiation . . . . . . . . . . . . . 83

4.9 Number of updates required to release track . . . . . . . . . . . . . . . . . . . . . . . . 84

4.10 Mutual information and Kullback-Leiber divergence . . . . . . . . . . . . . . . . . . . 87

4.11 Kullback-Leibler divergence as expectation of likelihood ratio . . . . . . . . . . . . . . 89

4.12 Interpretation of discrimination information through Neyman-Pearson lemma . . . . . . 90

4.13 Optimal setting of track sharpness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

List of Figures

4.14 Tracking loading using track sharpness method in varying clutter density . . . . . . . . . 93

4.15 Minimum loading track sharpness setting for varying false target density . . . . . . . . . 94

4.16 Mutual information gain of measurement as function of revisit interval for varying false

target density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.17 Loading for mutual information and track sharpness . . . . . . . . . . . . . . . . . . . . 96

4.18 Information rate for mutual information and track sharpness . . . . . . . . . . . . . . . 97

4.19 RMSE for mutual information and track sharpness . . . . . . . . . . . . . . . . . . . . 97

4.20 Example of linear and logarithmic utility functions for tracking and surveillance functions 99

5.1 Architecture of an agent [Jennings and Wooldridge, 1998] . . . . . . . . . . . . . . . . 102

5.2 A Multi-Agent System (MAS) [Wooldridge, 2002] . . . . . . . . . . . . . . . . . . . . 104

5.3 Agent platform deﬁned by FIPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4 Agent based resource management architecture . . . . . . . . . . . . . . . . . . . . . . 110

5.5 Inheritance structure for task agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.6 Inheritance structure for auctioneer agents . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.7 Inheritance structure for schedulers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.8 Inheritance structure for functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.1 Sequential ﬁrst price auction cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2 Multifunction radar discrete time decision process . . . . . . . . . . . . . . . . . . . . . 118

6.3 Valuations against time for three SFPRAM variants . . . . . . . . . . . . . . . . . . . . 122

6.4 Mean track angular estimation error standard deviation for SFPARM types. . . . . . . . 125

6.5 Number of active targets for SFPARM variants . . . . . . . . . . . . . . . . . . . . . . 126

6.6 Utility production for SFPARM variants . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.1 Example supply and demand curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.2 New York Stock Exchange system architecture . . . . . . . . . . . . . . . . . . . . . . 133

7.3 Resource utility space with example gradient between parameter selections marked . . . 137

7.4 Auction clearing decision process in CDAPS . . . . . . . . . . . . . . . . . . . . . . . 139

7.5 Possible parameter selections for three example surveillance tasks . . . . . . . . . . . . 142

7.6 Concave majorant for three example surveillance tasks . . . . . . . . . . . . . . . . . . 143

7.7 Equal gradient points for three example surveillance tasks . . . . . . . . . . . . . . . . . 144

7.8 Time delay inserted by the scheduler for CDAPS and RBPS . . . . . . . . . . . . . . . 146

7.9 Mean track angular estimation error standard deviation for CDAPS and RBPS . . . . . . 147

7.10 Time delay inserted by the scheduler for CDAPS and RBPS with a dynamic target scenario.148

7.11 Market equilibrium prices for CDAPS with a dynamic target scenario. . . . . . . . . . . 149

7.12 Mean track angular estimation error standard deviation for CDAPS and RBPS with dif-

fering task priorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.13 Comparison of mean utility for CDAPS and SFPARM types . . . . . . . . . . . . . . . 152

7.14 Average utility per task for CDAPS and RBPS allocation in a static environment . . . . . 153

10

List of Figures

7.15 Dynamic scenario over dynamic simulation . . . . . . . . . . . . . . . . . . . . . . . . 154

7.16 Cumulative detection range for CDAPS and RBPS allocation. . . . . . . . . . . . . . . . 155

7.17 Average utility per task for CDAPS and ﬁxed (RBPS) allocation in a dynamic environment156

7.18 Transaction prices in CDAPS market for varying resource availability over simulation . . 157

11

List of Tables

2.1 Swerling target ﬂuctuation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Multifunction radar task parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1 Priority structure for MESAR radar resource manager . . . . . . . . . . . . . . . . . . . 68

6.1 Fixed radar parameters for SFPARM multiple target simulation . . . . . . . . . . . . . . 123

6.2 Target environmental parameters for SFPARM simulations . . . . . . . . . . . . . . . . 124

7.1 Example of an orderbook which is able to clear . . . . . . . . . . . . . . . . . . . . . . 138

7.2 Example of an orderbook which is unable to clear . . . . . . . . . . . . . . . . . . . . . 140

7.3 Fixed radar parameters for CDAPS simulations . . . . . . . . . . . . . . . . . . . . . . 145

7.4 Target environmental parameters for static scenario . . . . . . . . . . . . . . . . . . . . 146

7.5 Target environmental parameters for dynamic scenario . . . . . . . . . . . . . . . . . . 148

7.6 Target environmental parameters for priority simulation . . . . . . . . . . . . . . . . . . 150

7.7 Task parameters in simulation for dynamic environment . . . . . . . . . . . . . . . . . . 154

List of Symbols

αFalse track conﬁrmation probability

βTrue track deletion probability

ΓParameter space

γkParameter selection

ˆγ(Ik)Policy using information accumulated up to time k

∆az Extent of search volume in azimuth

∆(θ)Difference response in angle for two beams

ǫT, ǫPTrue and predicted target angle in elevation

ηAperture efﬁciency of array antenna

ηT, ηPTrue and predicted target angle in azimuth

ΘSinger target manoeuvre time

θ0Angle of steered beam

θB3dB width of beam

θfOff-boresight measurement angle

θelmin Minimum elevation angle of search volume

θelmax Maximum elevation angle of search volume

θgGrating lobe angle

θsBeam separation in angle

λWavelength of carrier

µFalse target density

˜νWhite noise process

ξArbitrary position coordinate

σRadar cross section

¯σMean radar cross section

σθMeasurement thermal noise error standard deviation

σmMeasurement error standard deviation

σpAngular estimation error standard deviation

σu, σvState estimation error standard deviation in u,v coordinates

˜σθOff-boresight measurement error standard deviation

Σ(θ)Sum response in angle for two beams

List of Symbols

τPulse width

τAAlert dwell length

τCConﬁrm dwell length

τdDwell length

τsTime required to survey a search volume

φ(t)Phase modulation

χ(td, fd)Autocorrelation function

Ψ(td, fd)Ambiguity function

ΩSinger target manoeuvre standard deviation

ΩsSearch volume in steraradians

ACoherent signal amplitude

AASet of currently active asks

AeArray antenna effective aperture area

AkAction at time step k

ArAttenuation factor

ATActual antenna aperture area

atAsk announced by agent t

BSignal bandwidth

BASet of currently active bids

b1(θ),b2(θ)Angle response of two adjacent beams

btBid announced by agent t

cSpeed of electromagnetic wave propagation

d2Statistical distance squared

dcTarget closure range between scans

dtk(o)Delay function for task agent tkunder outcome o

D={d1, ...dn}Data vector

DKL (P||Q)Kullback-Leibler divergence between P and Q

DPBeam spacing for volume search

ESignal envelope

E(θ)Complete array antenna radiation pattern

Eα(θ)Array antenna radiation pattern with isotropic elements

Ee(θ)Array antenna element factor

ErReceived bandpass signal energy

FkState transition matrix at time k

FnReceiver noise ﬁgure

fcFrequency of carrier

fdDoppler frequency

fk(xk, vk)State transition function

14

List of Symbols

f(TA)Social choice function for agent set TA

GMajor axis of uncertainty ellipse

GtTransmitter gain

GrReceiver gain

gValidation gate size

g(t)Amplitude modulation

gk(γ)Resource function

H(X)Entropy of random variable X

Hf(f)Frequency response

HkMeasurement observation matrix at time k

HNTarget not present hypothesis

HTTarget present hypothesis

h(t)Impulse response

hk(xk, wk)Observation function

ISubset of asks

I0Modiﬁed Bessel function of ﬁrst kind with zero order

I(X;Y)Mutual information between Xand Y

itk(o)Information function for agent tkunder outcome o

JSubset of bids

JzFisher information for measurement z

kBoltzmann’s Constant

ˆ

k k-pricing rule weight

kcComplex constant

kmGradient of measurement slope on measurement axis

ksNormalised error signal response

lLength of array antenna

ldResource loading

LLog Likelihood ratio

LR Likelihood ratio

LpLosses

mMeasurement dimension

NMean noise power

nNumber of elements in antenna array

n1,n2Zeros mean unit standard deviation random variables

NBNumber of detection bins

N0Noise spectral density

OSet of possible outcomes for event

oOutcome of an event

15

List of Symbols

PDProbability of detection

PD0Probability of detection at the beam centre

PDc Cumulative probability of detection

PF A Probability of false alarm

Pk|kCovariance of state estimate at time kgiven the measurements up to time k

PNTarget not present probability density function

PrReceived power

PtTransmit power

PTTarget present probability density function

P RF Pulse repetition frequency

piPriority of task i

pnPrice of offer n

p∗Agent’s true price valuation

˜pTransaction price

q2Measurement origin uncertainty

qk(γ)Quality function

QkCovariance of target dynamic noise at time k

ˆ

QkQuality space

˜

QkPOMDP Q-value

QIQuantity of ask subset I

QJQuantity of bid subset J

qnQuantity of offer n

˜qProcess noise intensity

˜

Rk(s, a)Reward function for action afrom state s

RkCovariance of measurement noise at time k

RmMaximum detection range

RrRange resolution

RtRange between the target and the antenna

RuMaximum unambiguous range

R0Range at which signal to noise ratio is unity

R85 Range at which cumulative detection probability is 0.85

R90 Range at which cumulative detection probability is 0.9

rkResource held by task agent k

rTTotal resource available for set of agents T

SkPredicted covariance of track residual

Smin Minimum detectable signal

SN R Signal to noise ratio

SN RmSignal to noise ratio given a detection

16

List of Symbols

SN R0Signal to noise ratio at beam center

sAntenna element spacing

sr(t)Received signal

st(t)Transmitted signal

TThreshold

T1,T2Likelihood ratio thresholds

TSTotal time for search dwell

TA={t1, ..., tk}Set of k task agents

T0Temperature of ideal noise source

TkLength of one time step

TsEffective temperature

tdRound trip time from antenna to target to antenna

tfRevisit interval

tkTask agent k

tpTime interval between pulses

UVariance reduction ratio

utk(o)Utility function for agent tkunder outcome o

uR(t)Complex envelope of the received signal

uT(t)Complex envelope of the transmit signal

VkValidation gate volume

VITotal value of ask subset I

VJTotal value of bid subset J

v0Track sharpness

vkTarget dynamic noise

vrRelative radial velocity between target and radar

wkMeasurement noise

WkFilter gain matrix

XkState vector

xkTarget state at time k

ˆxk|kState estimate at time kgiven measurements up to time k

ZkMeasurement series up to time k

zSignal intensity

zkMeasurement at time k

ˆzkMeasurement prediction

˜zkMeasurement residual

17

List of Acronyms

(CA)-CFAR (Cell Averaging) - Constant False Alarm Rate

CDA Continuous Double Auction

CDAPS Continuous Double Auction Parameter Selection

ECM Electronic Counter Measures

EDF Earliest Deadline First

ESA Electronically Steered Array

FIPA Foundation for Intelligent Physical Agents

FOV Field of View

GIF Greatest Information First

GMTI Ground Moving Target Indicator

GNN Global Nearest Neighbour

HPF Highest Priority First

IMM Interacting Multiple Model

JADE Java Agent Development

JPDA Joint Probabilistic Data Association

KKT Karush Kuhn Tucker

KLD Kullback-Leibler Divergence

LFM Linear Frequency Modulation

LQF Lowest Quality First

MAS Multi-Agent System

MDP Markov Decision Process

MESAR Multi-Function Electronically Scanned Adaptive Radar

MFR Multi-Function Radar

MHT Multi-Hypothesis Tracking

MI Mutual Information

NCA Nearly Constant Acceleration

NCV Nearly Constant Velocity

NYSE New York Stock Exchange

OARS Opening Automated Report Service

POMDP Partially Observable Markov Decision Process

List of Acronyms

(L/M/H)-PRF (Low/Medium/High) - Pulse Repetition Frequency

PDA Probabilistic Data Association

QoS Quality of Service

Q-RAM Qos Resource Allocation Method

RaDAR Radio Detection and Ranging

RB-EDF Rule Based Earliest Deadline First

RBPS Rule Based Parameter Selection

RCS Radar Cross Section

RGPO Range Gate Pull Off

RMSE Root Mean Squared Error

RM Resource Management

RRM Radar Resource Manager

SAR Synthetic Aperture Radar

ISAR Inverse Synthetic Aperture Radar

SFPARM Sequential First Price Auction Resource Management

SNR Signal to Noise Ratio

SOJ Stand Off Jammer

STAP Space Time Adaptive Processing

TWS Track While Scan

19

Chapter 1

Introduction

1.1 Motivation

Advances in modern electronic components have driven the commercialisation of electronically steered

phased array antenna technology. In contrast to the traditional mechanically scanned antenna, the elec-

tronically steered phased array has signiﬁcantly increased beam agility which allows dynamic allocation

of the time-energy resource. This has led to a new generation of multifunction radar systems, where

multifunction can be deﬁned as the ability to sequentially execute numerous, differing and potential

conﬂicting tasks which support a variety of different radar functions.

Requirements of multifunction radar according to the maritime, airborne and land domains vary

greatly. However,a typical system is required to search a volume for new targets and once detected fuse

the information from multiple scans into target tracks. The system may also be required to perform addi-

tional functions depending on the application domain such as data link, weapons support, identiﬁcation

or classiﬁcation. A typical scenario is shown in Fig. 1.1 for a maritime air defence type application. This

ﬁgure shows the potential operational complexity for the multifunction radar as the ﬁnite radar resource

is required to be distributed between the wide variety of modes which may need to be deployed. The

ultimate performance of the system is dependent on how well the numerous tasks which support the

differing modes are able to fulﬁl the requirements of the system.

Multifunction radars have increasing appeal, which can be attributed to several key beneﬁts:

•Flexibility - Flexibility over allocation in space and time, including variable update rates, dwell

times and surveillance coverage, tailored to each application or role.

•Adaptability - Multifunction radar performance speciﬁcation can be dynamically adjusted to match

the dynamic and uncertain scenario and environment.

•Efﬁciency- Increased efﬁciency in terms of space, time, energy, production and maintenance effort.

The overall beneﬁt is the potential to vary nearly instantaneously an array of radar parameters to achieve

1.1. Motivation

Figure 1.1: Typical maritime air defence multifunction radar scenario [Butler, 1998].

a desired goal. This thesis concentrates on the exploitation of beam agility and development of methods

to divide the ﬁnite time-energy resource.

The control and conﬁguration of the multifunction radar is beyond the response capability of the

human operator and so an automated Radar Resource Manager (RRM), most likely with operator super-

vision, is required. Consequently, the full potential of the multifunction radar system is only realised

through the RRM’s ability to automatically allocate and conﬁgure the ﬁnite resource it has available. In

addition, the RRM has access to all the information in the system, which exceeds the information that

is able to be displayed to an operator in the loop. The RRM can therefore theoretically achieve superior

decision making at a rate faster than the human operator. These factors have created a strong desire to

maximise the potential of the hardware by intelligently adapting to dynamic scenarios, environments and

missions.

Agent systems are computational societies where the synergy of local interactions between agents

produces emergent, globally desirable behaviour. Typically, agent systems are governed by distributed

and decentralised mechanisms which are inherently computationally efﬁcient and scalable. The automa-

tion of human interaction mechanisms in agents systems, such as economic paradigms, can replicate the

ability to achieve robust behaviour in dynamic and uncertain environments. This provides the motivation

for their application to multifunction radar resource management.

Economic paradigms and market mechanisms have evolved over centuries in human societies, as

efﬁcient, trusted and highly developed methods of distributing goods and commodities. Free markets

tend to competitive equilibrium which maximises participant proﬁt and optimises social welfare. This

desirable characteristic can be harnessed in resource allocation problems, such as multifunction radar

resource management, to produce emergent intelligent and desirable behaviour.

21

1.2. Thesis Layout

The primary aim of this research has been to investigate for the ﬁrst time the application of agent

systems and economic paradigms to multifunction RRM. This research also had the following secondary

aims:

•Provide a thorough review of existing work, to identify where agent techniques can be most bene-

ﬁcially applied.

•Investigate the role of information theory in multifunction radar resource management.

•Explore suitable objective functions and measures which guide the resource allocation.

•Develop agent based resource allocation mechanisms utilising suitable choices of objective func-

tions.

•Create a radar simulator testbed upon which differing agent systems can be applied.

•Demonstrate and quantify enhanced multifunction capability of resulting allocation mechanism.

This thesis is organised according to the layout described in the following section.

1.2 Thesis Layout

This thesis contains eight chapters which detail the aspects of the application of agent systems to multi-

function radar resource management.

An overview of the fundamental operation of a multifunction radar system is given in Chapter 2.

This includes aspects of radar signal processing, measurement data processing and the electronically

steered array antenna. Operational multifunction radar systems are discussed alongside the parameter

and mode view of multifunction operation. This collection describes what is automatically managed.

Chapter 3 gives an overview of current radar resource management techniques. This includes the af-

fect of parameter selection on performance, a discussion of resource management architectures, method-

ologies for resource management and approaches to scheduling and prioritisation. Gaps in completed

research are identiﬁed, to which the research in this thesis is targeted. This chapter details how the

resource is automatically managed.

An exploration of the critical choice of objective function is given in Chapter 4 for the surveillance

and tracking applications. The objective function is crucial for the development of a resource allocation

mechanism as it provides the interface to the task function and so determines which attribute of the prob-

lem is allocated resource. This includes a discussion of task speciﬁc measures, derivation of information

theoretic measures and an investigation into the suitability of the derived measures for control.

Agent systems are introduced in Chapter 5 along with some relevant theory for resource alloca-

tion problems. The Java Agent Development (JADE) framework is referenced, which is an agent based

22

1.3. Novel Aspects

extension to the Java platform. Then, development of an agent based multifunction radar resource man-

agement testbed using JADE is described. Details of the design and structure of the testbed are also

detailed.

The sequential ﬁrst price sealed bid auction mechanism is applied to the multifunction radar re-

source management problem in Chapter 6, using the agent based testbed described in Chapter 5. Com-

parisons are drawn with existing resource management methods using a complex multi-target tracking

scenario and with reference to the conclusions from Chapter 4.

In Chapter 7 the continuous double auction mechanism is applied to the multifunction radar resource

management problem, which leads into the development of the continuous double auction parameter

selection algorithm (CDAPS). Desirable characteristics of the mechanism are demonstrated on multi-

target tracking and surveillance scenarios.

Finally the conclusions of the research are presented in Chapter 8 including a discussion of possible

future extensions to the work.

1.3 Novel Aspects

The aspects of this work believed to be novel are contained in Chapter. 4-7. Speciﬁcally:

•Information theoretic measures for multifunction radar resource management have been derived

and developed for estimation and discrimination problems. This has led to an improved under-

standing of the role of information theoretic measures for multifunction radar resource manage-

ment and sensor management in general. [Chapter 4, pages 84-90, 94-97]

•The Modiﬁed Riccati Equation has been successfully applied to tracking control under signiﬁcant

measurement origin uncertainty. [Chapter 4, pages 93-94]

•An agent based multifunction radar resource management architecture using the JADE framework

has been developed. This has provided the basis for a better understanding of agent based resource

management architecture designs which allow rapid upgrades and maximum code re-use. [Chapter

5, pages 108-114]

•The sequential ﬁrst price sealed bid auction mechanism has been applied to multifunction radar

resource management including development and analysis of lowest quality ﬁrst and greatest in-

formation ﬁrst schedulers. This provides a detailed insight into radar resource manager design and

selection of appropriate objective functions. [Chapter 6, pages 115-129]

•The continuous double auction mechanism has been applied to multifunction radar resource man-

agement leading to the development and assessment of the Continuous Double Auction Parame-

ter Selection (CDAPS) algorithm which generates high performance radar resource management.

23

1.4. Publications

[Chapter 7, pages 130-158]

The culmination of these individual aspects constitutes the ﬁrst application of agent systems to multi-

function radar resource management.

1.4 Publications

The following publications are a result of the work in this thesis:

•Charlish, A., Woodbridge, K. and Grifﬁths, H.: Economic paradigms in cognitive sensor signal

processing, Defence Applications of Signal Processing, Coolum, Australia, July 2011. (by invita-

tion)

•Charlish, A., Woodbridge, K. and Grifﬁths, H.: Agent based multifunction radar surveillance

control, IEEE International Radar Conference, Kansas City, USA, May 2011.

•Ritchie, M., Charlish, A., Woodbridge, K. and Stove, A.: Application of Kullback-Leibler diver-

gence to sea clutter estimation, IEEE International Radar Conference, Kansas City, USA, May

2011.

•Charlish, A., Woodbridge, K. and Grifﬁths, H.: Auction mechanisms in multi-function radar re-

source management, Proc. Cognitive Systems with Interactive Sensors, Crawley, UK, Nov. 2010.

•Charlish, A., Woodbridge, K. and Grifﬁths, H.: Information theoretic measures for MFR tracking

control, Proc. IEEE International Radar Conference, pp. 987-992, Washington, USA, May 2010.

24

Chapter 2

Multifunction Radar

A multifunction radar system is capable of supporting numerous tasks which in turn support differing

radar functions. The multi-functionality is primarily enabled by some degree of beam agility, which is

predominantly attributable to the use of an electronically steered, phased array antenna. In contrast to a

non-agile system where ﬁxed behaviour and hence performance is speciﬁed at design time, beam agility

allows the performance of the radar system to be adapted during operational deployment. Additionally,

as the execution of differing tasks is separable, the signal and data processing applied can be controlled

and optimised given the objectives of each speciﬁc task.

This chapter describes the theoretical principles of the signal and data processing that can be applied

in a multifunction radar. The automatic radar resource manager is required to optimise this processing,

which in this thesis is taken as the selection of parameters for all supported tasks, which controls the

processing applied. Included in this chapter is fundamental radar theory, the production and processing

of radar measurements, and the Electronically Steered Array (ESA) antenna. Finally, an overview of the

system’s multifunction capability in terms of the variety of task parameters and modes under control is

given, alongside examples of operational systems.

2.1 Radar Systems

The Radio Detection And Ranging (RaDAR) system has matured over a period exceeding half a century

in a range of civilian and military applications for the ground, airborne and maritime domains. As the

name suggests, radar systems provide detection and accurate range measurement of distant or otherwise

unobservable objects. The following radar system theory has provided the basis upon which multifunc-

tion radar systems are built.

2.1.1 Radar Fundamentals

A radar operates by emitting electromagnetic energy from an antenna, the energy is scattered by the

environment, with some of the scattered energy being re-intercepted by the receiving antenna. In the

monostatic case, which is assumed throughout this thesis, the transmit and receive antennas are co-

2.1. Radar Systems

located and potentially a single antenna is used for both transmission and reception. The received signal

can be processed to retrieve information on the environment such as the presence and state of a target.

Target range can be found by measuring the time taken for a pulse to make the round trip from the

antenna, to the target, and back to the antenna. The round trip time (td) is proportional to target range

(Rt):

td=2Rt

c(2.1)

where cis the speed of the electromagnetic wave propagation. Successive pulses are transmitted at time

intervals dictated by the pulse repetition frequency (PRF). Range ambiguities occur, which depend on

the PRF, when it is not clear from which of the recently transmitted pulses the received pulse originated.

The maximum unambiguous range Ruis proportional to the time interval between pulses tp:

Ru=ctp

2(2.2)

and the time interval between pulses is inversely proportional to the pulse repetition frequency (tp=

1

P RF ). The radar range resolution (Rr), which is the minimum separation between two targets which are

individually resolvable, is inversely proportional to the signal bandwidth B:

Rr=c

2B(2.3)

For an uncompressed pulse B=1

τ, where τis the pulse width, in which case the range resolution can

be visualised as the two way distance travelled during one pulse duration. Longer pulses allow for an

increase in the average transmitted power given a ﬁxed PRF, and so result in a greater detection range.

However, as longer pulses result in poorer range resolution, pulse compression is used which increases

the bandwidth of the signal to improve range resolution while maintaining the larger energy of a longer

pulse.

Assuming a coherent pulse train, target radial velocity, or range-rate, can be found by measuring

the Doppler shift on the returned pulses. The doppler shift is proportional to the relative radial velocity

vrbetween the radar and the target according to:

fd=2vrfc

c(2.4)

where fdis the Doppler frequency and fcis the frequency of the carrier. The spectrum of the pulsed sig-

nal contains spikes above and below the carrier frequency at multiples of the pulse repetition frequency.

Therefore, doppler ambiguities occur for low PRFs when it is not clear how many multiples of the PRF

are contained in the measured doppler shift. The width of each spike determines the doppler resolution,

26

2.1. Radar Systems

which is inversely proportional to the duration of the coherent pulse train. High-PRF (HPRF) radar is

conventionally deﬁned to give unambiguous doppler measurement, Low-PRF (LPRF) radar is conven-

tionally deﬁned to give unambiguous range measurements and Medium-PRF (MPRF) is conventionally

deﬁned to give both ambiguous range and doppler measurement.

Simple analysis of the monostatic radar-target geometry yields the widely used radar range equation

which provides an indication of the received power from which the maximum detection range for a given

target can be deduced. The geometry is modelled as an antenna radiating power Ptwith directional gain

Gt, which is intercepted and isotropically reradiated by the target before being re-intercepted by an

antenna with receiver gain Gr. Assuming free space with no losses, the received single pulse power Pr

can be calculated as the product of three terms [Skolnik, 2008]:

Pr=PtGt

4πR2

t

.σ

4πR2

t

.Grλ2

4π(2.5)

where σis the target radar cross section and λis the wavelength of the carrier. The ﬁrst term is the power

density at range Rtgiven a transmit power Ptand transmit antenna gain Gt. The second term is the

power per unit area at the receiver given a target of radar cross section σ. The ﬁnal term is the receiver

antenna effective area Aewhich intercepts the return. Given the minimum detectable signal is Smin and

incorporating losses Lpthis can be rearranged to give the maximum detectable range Rmas:

Rm=4

sPtGtGrλ2σ

(4π)3SminLp(2.6)

Smin is often limited by thermal noise in the receiver, which can be represented as a termination resistor

at the receiver antenna output. This equivalent resistor has an effective temperature Ts(Kelvins) which

passes noise with spectral density N0=kTswhere k= 1.38 ×10−23 J/K is Boltzmann’s constant. The

temperature of the noise source is expressed in terms of an ideal source with temperature T0= 290K,

multiplied by a noise factor Fnto account for the non-ideal nature of the receiver. The noise power

N=N0Bwhere Bis the bandwidth of the receiver can be combined with the required signal to noise

ratio SN R for detection, to give the minimum detectable signal:

Smin =kT0BFnSN R (2.7)

where the quantity kT0has a convenient round value of 4×10−21. Substitution of Eq. 2.7 into Eq. 2.6

with SN R = 1 yields the maximum instrumental range R0, or the range at which the SNR is unity for

an expected radar cross section. Calculation of R0allows the radar range equation to be conveniently

27

2.1. Radar Systems

expressed as [Blackman and Popoli, 1999]:

SN R =R0

Rt4(2.8)

giving the SN R as a function of range.

2.1.2 Radar Signal Processing

Given the pulsed operation described in the preceding section, it is required to process the received signal

to extract measurement information. It is possible to produce measurement data of target range and radial

velocity as well as separable measurements of azimuth and elevation.

2.1.2.1 Range and Radial Velocity

Knowledge of the transmitted signal enables the received signal to be processed to detect and measure

a potential time delay and doppler shift which is related to range and radial velocity according to Eq.

2.1 and Eq. 2.4 respectively [Skolnik, 2008]. The transmitted signal st(t)is comprised of a sinusoidal

carrier, of frequency fc, which is modulated by a comparatively slowly varyingwaveform:

st(t) = g(t) cos(2πfct+φ(t)) (2.9)

where g(t)is the amplitude modulation and φ(t)is the phase modulation of the waveform. This signal is

known as a narrow bandpass signal as the signal bandwidth is small compared to the carrier frequency.

The complex envelope of this transmit signal uT(t)is expressed as:

uT(t) = g(t)ejφ(t)(2.10)

which undergoes an additional modulation by the environment, which implants the information to be

extracted. Speciﬁcally, the transmit signal undergoes a time delay td, a shift in frequency fdand an

attenuation in amplitude Ar. The received signal sr(t)can therefore be expressed as:

sr(t) = Arg(t−td) cos[2π(fc+fd)(t−td) + φ(t−td)] (2.11)

If a target is present a delayed and potentially frequency shifted replica of the complex envelope of the

transmit signal uT(t−td)is received. Additionally, the time delay shifts the phase of the received signal

by −2πfctddue to the oscillations of the carrier and the doppler shift applies a linear phase modulation

of ej2πfd(t−td). Combining these elements gives the complex envelope of the received signal uR(t)as:

uR(t) = Are−j2πfctdu(t−td)ej2πfd(t−td)(2.12)

28

2.1. Radar Systems

To preserve the phase information the received signal is demodulated by two channels in the receiver.

The in-phase (I) channel demodulates by cos(2πfct)and the π/2out of phase quadrature channel (Q)

demodulates by −sin(2πfct).

After demodulation the received signal is ﬁltered by a matched ﬁlter [North, 1963] which maximises

signal to noise ratio by exploiting the knowledge of the transmit signal. Maximum signal to noise ratio

in the presence of white noise is achieved when the ﬁlter has a frequency response:

Hf(f) = kcU∗

T(f)e−j2πf t0(2.13)

where ∗denotes the conjugate, kcis a complex constant and t0is a time delay required to maintain a

casual impulse response. This matches the frequency response of the ﬁlter to the expected spectrum of

the signal given the known transmit signal. The corresponding impulse response is expressed:

h(t) = kcu∗

T(t0−t)(2.14)

which is the conjugate of the transmit signal delayed in time. This matched ﬁlter produces a maximum

possible output SNR [North, 1963] depending on the received bandpass signal energy Erand noise

power spectrum at the ﬁlter input N0:

SN R =2Er

N0(2.15)

however, when incorrectly matched the maximum SNR is not achieved. The autocorrelation function

describes the output of a speciﬁc matched ﬁlter for varying time delay and doppler shifts and can be

expressed as [Skolnik, 2008]:

χ(td, fd) = Z∞

∞

uT(t)u∗

T(t+td)ej2πfdt.dt (2.16)

Woodward’s [Woodward, 1980] ambiguity function follows as the squared magnitude of the autocor-

relation function Ψ(td, fd) = |χ(td, fd)|2. The autocorrelation and ambiguity function describe the

fundamental measurement capability of waveform and matched ﬁlter by demonstrating the resolution

and sidelobe properties as well as allowing measures such as Fisher information to be extracted. Mea-

surement data for range and range rate is produced from the responses of a bank of matched ﬁlters within

the unambiguous range and doppler limits.

By matching the ﬁlter over the pulse train duration, coherent integration is achieved. The phase

coherence ensures that the amplitude and phase of target returns are correlated whereas noise returns are

uncorrelated. As such, noise returns cancel and target returns combine which gives an improvement in

signal to noise ratio (SNR). Maximum practical coherent integration time is limited by target movement,

29

2.1. Radar Systems

as the target returns must be integrated in one ﬁlter.

Incoherent integration sums the magnitude of the received signal after envelope detection when the

phase information is removed. Noise integrates in the same way as target returns and an improvement

in SNR is not achieved. Although incoherent integration is less efﬁcient than coherent integration, it

is required to integrate the multiple PRF dwells from a single burst, which are used to mitigate range-

Doppler blind zones. Also, by averaging the returns over the integration period the signal is low passed

ﬁltered and the ﬂuctuation in the noise amplitude reduced. This improves detection sensitivity as the

detection threshold multiplier can be lowered without increasing the false alarm probability.

2.1.2.2 Bearing

Estimates of the target’s angular location can be produced with sub-beamwidth accuracy by comparing

the signals from two or more beams. This can be achieved using sequential beams, where measurement

accuracy is hindered by scintillation errors, or preferably using simultaneous beams on a single (mono)

pulse.

Amplitude comparison monopulse interpolates using the difference in amplitude between beams

slightly separated in angle. Fig 2.1(a) shows the response of two beams b1(θ)and b2(θ)with 1◦

beamwidth separated by 0.7◦and Fig. 2.1(b) shows the sum and difference response of the beams.

As the magnitude of the difference depends on the target signal amplitude, the difference response ∆(θ)

is normalised by the sum of the beams Σ(θ)to give the error signal response:

ks(θ) = ∆(θ)

Σ(θ)=b1(θ)−b2(θ)

b1(θ) + b2(θ)(2.17)

which is shown in Fig 2.1(c).The gradient of this discrimination slope k′

s(θ)determines the sensitivity of

the measurement which is quantiﬁed at the point where the measurement slope crosses the measurement

axis km=k′

s(0).

Thermal noise creates an error in the monopulse measurements as a function of signal to noise ratio

as derived by Barton [2004] which can be modelled by the thermal noise error standard deviation σθ:

σθ=θB

km√2SN R ≈θB

2√SN R (2.18)

where θBis the 3dB beamwidth. It is common to assume the measurement error standard deviation σm

is solely due to thermal noise. When the SN R becomes large the measurement error becomes hardware

limited and does not continue to reduce.

Targets which are offset from the beam centre experience a loss of gain relative to the maximum

antenna gain which, can be approximately modelled by reducing the target SNR according to a Gaussian

30

2.1. Radar Systems

−1.5 −1 −0.5 0 0.5 1 1.5

Angle (degress)

Voltage Response

Reponse of two Gaussian Beams Seperated in Angle

b1(θ)b2(Θ)

(a) Voltage response of beams b1and b2

−1.5 −1 −0.5 0 0.5 1 1.5

Angle (degress)

Voltage Response

Sum and Difference Beams

Sum Beam

Difference Beam

(b) Voltage response of sum and difference beams

−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Angle (degress)

Voltage Ratio

Normalised Error Signal

Slope = −km

(c) Monopulse error slope

Figure 2.1: Process of monopulse measurement

loss function [Blackman and Popoli, 1999]:

SN R =SNR0exp −CL[(ηT−ηP)2+ (ǫT−ǫP)2]

θ2

B(2.19)

where SN R0is the beam centre SNR, ηTand ηPare true and predicted azimuth and ǫTand ǫPare

the true and predicted elevations. CLcan be taken as 2.77 which is found by substituting SN R

SN R0= 0.5

when the angle off boresight (p(ηT−ηP)2+ (ǫT−ǫP)2) is equal to half the half power beamwidth

[Blackman and Popoli, 1999]. This loss in SNR affects the measurement accuracy as deﬁned by Eq.

2.18.

In addition to the loss in SNR, the accuracy of the monopulse measurement degrades as the target is

off the centre of the measurement axis, as evident by the reduction in sensitivity visible in Fig. 2.1(c). The

off-boresight measurement accuracy ˜σθcan be modelled by including a second component of thermal

noise error, which causes a scaling of the on-boresight measurement accuracy :

˜σθ=σθr1 + (km

θf

θB

)2(2.20)

31

2.1. Radar Systems

where θfis the offset angle.

2.1.3 Electronically Steered Array

The Electronically Steered Array (ESA) [Stimson, 1998; Wirth, 2001; Skolnik, 2008] is able to provide

the multifunction radar system’s requirement for beam agility. The ESA is an antenna with an array of

radiating elements which have controllable phase and amplitude as shown in Fig. 2.2. Modern electronic

components allow for the array control to be rapidly applied which enables an agile and ﬂexible beam.

s s

RF

-(n-1)/2 (n-1)/2

θ0

0 1-1

r

….. …..

Figure 2.2: Linear electronically steered array, steering at angle θ0[Wirth, 2001]

An array of nelements with linear spacing seach isotropically radiating equal amplitude and phase

produces a radiation pattern which can be found by summing the vector contributions of all the elements.

The subsequent radiation pattern Eα(θ)is [Skolnik, 2008]:

Eα(θ) = sin[nπ(s/λ) sin θ]

nsin[π(s/λ) sin θ](2.21)

and is plotted in Fig. 2.3(a) for 10 and 20 elements with a spacing of λ/2. The main lobe is clearly

identiﬁable at θ= 0 with additional side lobes. The 3dB beamwidth in radians is a function of the

wavelength λand the length of the aperture lin the relevant dimension:

θB=0.886λ

l(2.22)

which is evident in Fig. 2.3(a) where increasing the number of elements to 20 creates a longer aperture

which reduces the beamwidth.

Fig. 2.3(b) shows the radiation pattern for 10 elements with a 1.5λspacing. Additional main beams

called grating lobes can be seen at ±0.2323π. Grating lobes occur, due to spatial under-sampling, at

angles θgdetermined by the element spacing in relation to the wavelength:

sin θg=±mλ

s(2.23)

32

2.1. Radar Systems

where mis an integer m= 1,2,3.......

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−30

−25

−20

−15

−10

−5

0

Angle (π radians)

Radiation Pattern (dB)

Radiation Pattern

10 Elements

20 Elements

(a) 10 and 20 elements spaced at λ/2

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−30

−25

−20

−15

−10

−5

0

Angle (π radians)

Radiation Pattern (dB)

Radiation Pattern

Grating Lobe Grating Lobe

(b) 10 elements spaced at 1.5λ

Figure 2.3: Linear array radiation patterns

For a rectangular array, the gain of the main beam is a function of the wavelength and effective

aperture area and assuming the same transmit and receive antennas are used:

Gt=Gr= 4πAe

λ2(2.24)

where the effective aperture area is related to the true area by the aperture efﬁciency η, i.e Ae=ηAT

where ATis the actual aperture area.

In reality each element has a non-isotropic radiation pattern which necessitates the inclusion of the

element factor Eeto produce the complete radiation pattern E(θ):

E(θ) = Ee(θ)Eα(θ) = Ee(θ)sin[nπ(s/λ) sin θ]

nsin[π(s/λ) sin θ](2.25)

The pointing angle of the main beam can be steered, as shown in Fig. 2.2, by applying linear phase

increments △ϕbetween each element. It can be seen that the phase difference required to steer the beam

at angle θ0is the difference in phase over the distance △r. As there are 2πradians in one wavelength

and △r=ssin θ0the difference in phase between elements required to steer at an angle θ0is:

△ϕ=2πs sin θ0

λ(2.26)

The linear array radiation pattern in Eq. 2.21 can be adapted to include the effect of beam steering

at angle θ0to give:

E(θ) = Ee(θ)sin[nπ(s/λ)(sin θ−sin θ0)]

nsin[π(s/λ)(sin θ−sin θ0)] (2.27)

this radiation pattern is shown in Fig. 2.4 for a beam steered at θ0=−π/4in comparison to a linear array

33

2.2. Measurement Data Processing

where no beam steering is applied. When scanning off the radar boresight the effectiveaperture length l

is reduced by cos θ0which increases the beamwidth according to Eq. 2.22. However, mutual coupling

and the non-isotropic nature of the array elements causes the one way gain to drop off by approximately

cos1.5θ[Sabatini and Tarantino, 1994].

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−30

−25

−20

−15

−10

−5

0

Angle (π radians)

Radiation Pattern (dB)

Radiation Pattern

θ0 = −π /4

θ0 = 0

Figure 2.4: Linear array radiation pattern steered at θ0=−π/2

When radiating elements are closely spaced, energy is coupled between elements which affects the

each element’s radiation pattern. The magnitude of the coupling depends on the distance between the

elements and the distribution pattern of the elements in the array. For an ESA where there are numerous

closely spaced elements the effect of mutual coupling can be strong and can result in loss of the main

beam, the magnitude of the loss depending on the coherent combinations of the coupling signals between

elements in the array.

The spacing between elements is dictated by the desire to avoid grating lobes, which must not

appear within the ﬁeld of view (FOV) when the array is steered to the maximum scan angle. Grating

lobes are avoided if:

s

λ<1

1 + |sin θ0|(2.28)

which gives a maximum spacing of λ

2for a ±90◦FOV, 0.536λfor a ±60◦FOV and 0.586λfor a ±45◦

FOV. Loss of gain due to off boresight scanning typically limits the ﬁeld of view to ±60◦or ±45◦and

so several arrays or rotating arrays must be used for full 360◦coverage.

2.2 Measurement Data Processing

The fundamental radar system described in the preceding section produces range, doppler and angle

measurement data. However, the multifunction radar system must apply signiﬁcant data processing

34

2.2. Measurement Data Processing

before it can be presented to the operator in a meaningful way. This processing involves the automatic

detection and tracking of targets within the measurement data. The multifunction radar system is required

to optimise the data processing given the objectives of each individual task.

2.2.1 Detection

Detection is a binary hypothesis testing problem to differentiate between the target present hypothesis

HTand target not present hypothesis HN. The Neyman-Pearson lemma deﬁnes the optimal decision

region for a ﬁxed probability of false alarm PF A as a threshold Ton likelihood ratio LR for data vector

D={d1, ..., dn}:

LR(d1, ..., dn) = p(d1, ..., dn|HT)

p(d1, ..., dn|HN)

>HT

<HNT(2.29)

where Tis chosen so that p(d1, ..., dn> T |HN) = PF A. The optimality condition ensures the proba-

bility of detection PDis maximised for the ﬁxed false alarm probability.

When the receiver is dominated by thermal noise, the target not present hypothesis relates to an

output of the I and Q channels according to a complex Gaussian probability density function [Ward

et al., 2006]. The corresponding envelope of the signal E=qE2

I+E2

Q, which is the output from a

linear envelope detector,is characterised by a Rayleigh probability density function:

PN(E) = 2E

¯zn

exp −E2

¯zn(2.30)

where ¯znis the mean noise intensity. The target present hypothesis can be assumed to be a coherent

signal embedded in the thermal noise, which produces a signal with envelope characterised by a Rician

probability density function:

PT(E|A) = 2E

¯zn

exp −E2+A2

¯znI02EA

¯zn(2.31)

where Ais the amplitude of the signal and I0is the modiﬁed Bessel function in the ﬁrst kind with zero

order. Fig. 2.5 shows the probability density functions for the envelope of thermal noise and target plus

thermal noise. It can be shown through the Neyman-Pearson lemma that thresholding on the envelope

of the measurement data is optimal for large signals [Skolnik, 2008]. An example of such a threshold on

the signal envelope is marked in Fig. 2.5.

Alternatively, a square law envelope detector can be used which produces an output proportional to

the intensity z=E2of the signal. In this case the target not present hypothesis, which corresponds to

thermal noise, has an exponential probability density function:

PN(z) = 1

¯zn

exp −z

¯zn(2.32)

35

2.2. Measurement Data Processing

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Envelope E

Probablity Density P(E)

Envelope Probability Density for Noise and Target Plus Noise

Noise

Target Plus Noise

Threshold

Figure 2.5: Probability density functions for the envelope of noise and target plus noise

and the target present hypothesis, which can be assumed as a coherent signal embedded in thermal noise,

is characterised by:

PT(z|A) = 1

¯zn

exp −z+A2

¯znI02A√z

¯zn(2.33)

It can be shown through the Neyman-Pearson lemma that thresholding on the intensity of the measure-

ment data is optimal for small signals. However, the linear and square law detectors exhibit similar

detection performance.

2.2.1.1 Calculation of Detection Probability

The probability of detection and false alarm for threshold Tcan be seen with reference to Fig. 2.5 as

the integral of the respective probability density functions above the threshold. Marcum [Marcum, 1947,

1948] investigated this statistical nature of radar measurement data and produced functions for calculat-

ing the probability of detection and probability of false alarm for a number of incoherently integrated

pulses.

Marcum gave the probability of detection of a single normalised pulse as:

PD(A, T ) = Z∞

T

e−(z+A2)I02√zA.dz =Q(√2A, √2T)(2.34)

PF A =e−T(2.35)

where Qis Marcum’s Q-function.

36

2.2. Measurement Data Processing

2.2.1.2 Target Fluctuation

Swerling extended Marcum’s work to consider the non-ﬂuctuating or Swerling 0 case and four different

cases of ﬂuctuating target radar cross section and hence signal to noise ratio. In case one and two the

target is modelled as a number of independent scatters, no one of which is dominant, and is used to

describe large complex targets. The radar cross section ﬂuctuations follow a Rayleigh, or chi-squared

with two degrees of freedom, probability density function:

w(σ, ¯σ) = 1

¯σexp −σ

¯σ(2.36)

where ¯σis the mean radar cross section. For case one the ﬂuctuations occur between scans and for case

two the ﬂuctuations occur between pulses. The probability of detection for cases one and two can be

calculated as a function of the false alarm probability and the SNR:

PD=P1/(1+SN R)

F A (2.37)

Cases three and four model the target as a single large dominant scatterer surrounded by a number

of smaller scatters, which is assumed to be characterised by a Rician probability density function:

w(σ, ¯σ) = 4σ

¯σ2exp −2σ

¯σ(2.38)

for case three the ﬂuctuations occur between scans and for case four the ﬂuctuations occur between

pulses. These four cases are summarised in Table 2.1.

Table 2.1: Swerling target ﬂuctuation models

Scan to Scan Pulse to Pulse

Many Small Case 1 Case 2

One Large Case 3 Case 4

2.2.1.3 False Alarm Control

In reality a global homogenous background is rarely faced and so the use of a ﬁxed global threshold

would produce local regions of excessive false alarms which overload the data processor and tracker. To

avoid this it is necessary to estimate the statistics of the local background to apply a dynamic threshold.

Typically a model of the probability density of the background is known, with potentially unknown

parameters. A constant false alarm rate detector uses a set of local background reference cells to estimate

the unknown parameters of the model. The test and reference cells are separated by a number of guard

cells to ensure the target is not present in the reference.

For a Rayleigh background it is sufﬁcient to estimate the mean in order to set the correct local

37

2.2. Measurement Data Processing

threshold to maintain a speciﬁed probability of false alarm. This can be implemented through a Cell-

Averaging (CA)-CFAR which is shown in Figure 2.6. In the CA-CFAR a number of reference cells,

seperated from the cell under test by guard cells, are used to estimate the mean intensity of the local

background. This mean is multiplied by the threshold multiplier, to produce the intensity threshold

which must be exceeded to declare the target presence.

Guard

Cell

Guard

Cell

Test

Cell ................

Reference Cells Reference Cells

1/nΣzi

zt > Tzm

z1zn

zt

zm

Figure 2.6: Cell averaging constant false alarm rate detector

2.2.2 Tracking

Tracking is the process of fusing sequences of detected measurements to estimate the kinematics of the

underlying targets. It is required to estimate the state of the target as a kinematic parameter vector X,

such as the position and velocity in cartesian coordinates, i.e. X= (x, x′, y, y′)T. The kinematic state

of the target xkis assumed to evolve as a potentially non-linear discrete time stochastic system described

[Ristic et al., 2004] by the dynamic equation:

xk=fk−1(xk−1, vk−1)(2.39)

where fk−1describes the predictable disturbances to motion, xk−1is the previous state and vk−1is

a noise sequence which allows for unpredicted disturbances to motion. Measurements zkare used to

estimate the target state xkwhich are received corrupted by measurement noise wkand so are modelled

by the measurement equation:

zk=hk(xk, wk)(2.40)

where hkis the observation function and both the process noise vkand measurement noise wkare

assumed known. Fig. 2.7 shows a block diagram of the discrete time sequential state estimation process,

adapted from Bar-Shalom et al. [2001]. Bayes’ theorem provides the framework for sequential state

estimation, enabling new measurements to be fused with estimates from previous time steps:

p(x|Zk) = p(zk|xk)p(xk|Zk−1)

p(zk|Zk−1)(2.41)

38

2.2. Measurement Data Processing

where p(zk|xk)is the likelihood function, p(xk|Zk−1)is the state estimate at time k−1and p(zk|Zk−1)

is a normalising constant. This can be re-arranged to give the optimal recursive Bayesian estimator

[Ristic and Hernandez, 2008]:

Prediction: p(xk+1|Zk) = Zp(xk+1|xk)p(xk|Zk)(2.42)

Update: p(xk+1|Zk+1) = p(xk+1|xk)p(xk|Zk)

Rp(zk+1|xk+1 )p(xk+1|Zk).dxk+1 (2.43)

The state of the underlying system can be estimated as the minimum mean square estimate, which is the

conditional mean of the state estimate:

ˆxMM SE

k|k=E[xk|zk] = Zxk.p(xk|Zk).dxk(2.44)

Dynamic

System

Measurement

System

State

Estimator

Target State Measurements

Target Process

Noise

Measurement

Noise

vk-1 wk

xk = fk-1(xk-1,vk-1)zk = hk(xk,wk)xkzk

State Estimate

p(xk|zk)

Figure 2.7: Target dynamic system, measurement system and sequential state estimation

This optimal estimator requires the propagation of the entire posterior probability density which is

a potentially inﬁnite data vector. As a result it is necessary to ﬁnd closed form or sub-optimal solutions.

The Kalman ﬁlter is a closed form solution which can be used if the dynamic and measurement models

are linear and the process and measurement noises are Gaussian, with covariances denoted E[vkv′

k] =

Qkand E[wkw′

k] = Rk. As the state estimate is Gaussian it is completely described by its ﬁrst two

moments, the conditional vector mean Eq. 2.44 and the covariance matrix:

Pk|k=E[[xk−ˆxk|k][xk−ˆxk|k]′|Zk](2.45)

The Kalman ﬁlter breaks the sequential estimation of the mean and covariance into prediction and

update stages as follows:

39

2.2. Measurement Data Processing

Stage 1 - Prediction

State estimate prediction: ˆxk|k−1=Fk−1ˆxk−1|k−1(2.46)

State covariance prediction: Pk|k−1=Fk−1Pk−1|k−1F′

k−1+Qk−1(2.47)

Prediction of next measurement: ˆzk|k−1=Hkˆxk|k−1(2.48)

Innovation covariance calculation: Sk=Rk+HkPk|k−1H′

k(2.49)

Stage 2 -Update (On receiving measurement zk)

Measurement residual calculation: ˜zk=zk−ˆzk|k−1(2.50)

Filter Gains Calculation: Wk=Pk|k−1H′

kS−1

k(2.51)

State estimate update: ˆxk|k= ˆxk|k−1+Wk˜zk(2.52)

State covariance update: Pk|k=Pk|k−1−WkSkW′

k(2.53)

The linear-Gaussian assumptions at the heart of the Kalman ﬁlter rarely hold in reality and so

it is often necessary to use sub-optimal methods. Non-linear models can be incorporated using the

extended Kalman ﬁlter, which approximates the non-linear functions, or the unscented Kalman ﬁlter,

which approximates the posterior distribution as a Gaussian sum.

2.2.3 Kinematic Models

The implementation of the Kalman ﬁlter requires the speciﬁcation of models for the target dynamics

and the measurement system. The measurement system model can readily be derived from Sec 2.1.2,

however,the choice of target process noise covariance and dynamic system transition matrix are not so

apparent. Various models exist which are suited to differing target dynamics and the subsequent choice

can have signiﬁcant effect on tracking performance.

2.2.3.1 Continuous White Noise Models

Continuous white noise models assume that deviations to predictable motion enter the system as zero

mean white noise process ˜ν(t), i.e. E[˜ν(t)] = 0 and E[˜ν(t)˜ν(τ)] = ˜qδ(t−τ). The process noise

intensity ˜qrepresents the strength of the deviations from predictable motion. For an arbitrary position

coordinate ξ, the white noise can enter the system as a white noise acceleration, ¨

ξ(t) = ˜ν(t)or white

noise jerk (derivative of acceleration) ...

ξ(t) = ˜ν(t). As derived by Bar-Shalom et al. [2001] this gives

the following system transition and process noise covariance matrices:

40

2.2. Measurement Data Processing

Continuous white noise acceleration

Transition matrix: Fk=

1Tk

0 1

(2.54)

Process Noise Covariance: Qk=E[vkv′

k] =

T3

k

3

T2

k

2

T2

k

2Tk

˜q(2.55)

Continuous white noise jerk

Transition matrix: Fk=

1Tk1

2T2

k

0 1 Tk

0 0 1

(2.56)

Process Noise Covariance: Qk=E[vkv′

k] =

T5

k

20

T4

k

8

T3

k

6

T4

k

8

T3

k

3

T2

k

2

T3

k

6

T2

k

2Tk

˜q(2.57)

where Tkis the time between time steps. When the process noise intensity ˜qis small these represent a

nearly constant velocity (NCV) and nearly constant acceleration (NCA) model respectively.

An alternative family of models called discrete white noise models allow for deviations to motion

to enter the system as a zero mean white noise sequence. This manifests itself as a constant acceleration

or constant jerk over the sampling period, which is uncorrelated to the previous time step. In this work

the previous continuous white noise models are preferred as the same amount of process noise enters the

system regardless of the length of the sampling interval [Blackman and Popoli, 1999], i.e.:

Fk.Qk(Tk).Fk+Qk(Tk) = Qk(2Tk)(2.58)

which is a useful property for adaptive update rate tracking.

2.2.3.2 Singer

Singer [Singer, 1970] provides a more realistic model of a correlated acceleration sequence between time

steps represented as a Markov process:

...

ξ(k+ 1) = ρm

...

ξ(k) + p1−ρ2

mΩn1(2.59)

41

2.2. Measurement Data Processing

where ρm=e−βmTk,βm=1

Θand Θis the target manoeuvre time constant, Ωis the target manoevure

standard deviation and n1is a zero mean unit standard deviation Gaussian distributed random variable.

In the limit where the sampling interval is much less than the manoeuvre time constant, the Singer model

tends to the continuous white noise jerk model in Eq. 2.56 and Eq. 2.57. In the opposite case where the

sampling interval is much greater than the manoeuvre time constant then estimates of the acceleration

are not possible and so the Singer model tends to the continuous white noise acceleration model in Eq.

2.54 and Eq. 2.55.

2.2.3.3 Adaptive Filtering

As target dynamics are likely to change over the track duration, it is necessary to implement adaptive

ﬁltering methods which change the model of the target dynamic upon manoeuvre to ensure the ﬁlter is

matched to the current target dynamic.

Reactive Adaptation - The residual vector from the tracking ﬁlter can be monitored to detect ma-

noeuvres. If the residual becomes large, as deﬁned by some rule of thumb, then the process noise can be

increased to reduce the smoothing applied by the ﬁlter and to apply more weight to new measurements.

Variable Dimension Filtering - When the manoeuvre detection logic indicates a manoeuvre, the

dimension of the ﬁlter state can be changed. For example in periods of benign motion a NCV can be

adopted, which can be changed to a NCA upon manoeuvre.

Multiple Model Filtering - Kalman ﬁlters with differing models are run in parallel, the residual is

monitored to determine the probability of each of the models being correct. The output is each of the

ﬁlter outputs is merged by the ﬁlter probability.

From the available adaptive ﬁltering techniques Interacting Multiple Model (IMM) has emerged as

the best performer but with the greatest complexity and computational cost.

2.2.4 Data Association

It was previously assumed that the Kalman ﬁlter was updated with a measurement that was known to be

from the target in question. In reality the measurement could also have originated from some form of

interference or from a different, nearby target. As such data association techniques are used to improve

correct measurement to track assignment.

2.2.4.1 Gating

To reduce the complexity of data association a gate is applied to discard unlikely target to track pairings.

The gate is centered on the tracks predicted state, and only detections falling within this gate are consid-

ered for assignment. Rectangular and ellipsoidal gates can be used, the size of which is determined by

the residual vector in the track. An ellipsoid gate speciﬁes a valid association region within the statistical

distance d2:

d2= ˜zk′S−1

k˜zk≤g(2.60)

42

2.2. Measurement Data Processing

where ˜zis the measurement residual from Eq. 2.50 and gis the gate size. The volume Vkof the validation

gate is given by:

Vk=πg2|Sk|1/2(2.61)

where Skis the innovation covariance from Eq. 2.49.

2.2.4.2 Data Association Methods

Measurements falling within the validation gate are eligible for track update. Common methods for data

association are global nearest neighbour, probabilistic data association and multi-hypothesis tracking,

which have increasing complexity and effectiveness.

Global Nearest Neighbour (GNN) - Nearest neighbour assigns an observation to a track whereby the

subsequent assignment minimises the statistical distance of all possible observations to that track. GNN

performs this process for all tracks in the system and hence minimises the global statistical distance for

all observation to track assignments. GNN is the simplest approach to data association but performs

poorly in high clutter or dense target scenarios.

Probabilistic Data Association and Joint Probabilistic Data Association - Probabilistic Data Asso-

ciation (PDA) forms hypotheses on all possible observation to track assignments falling in the gate. The

probability of each of these being the correct assignment is calculated and the hypotheses are merged,

weighted by the respective probabilities. Joint Probabilistic Data Association (JPDA) extends PDA by

calculating the global probabilities of all observations and all tracks. PDA performs better in clutter than

GNN and JPDA performs better than PDA in multi-target situations. Both have extra computational cost

over GNN.

Multi-Hypothesis Tracking - Multi-Hypothesis Tracking (MHT) forms hypotheses for observation

to track assignments which are not merged at each scan as in JPDA. Hypotheses are propagated so that

future scans resolve the uncertainty in previous time steps. This produces a branching tree of hypotheses,

each with a probability of being correct. This tree is managed so that unlikely hypothesis branches are

pruned to manage computation. MHT performs better in clutter and dense target regions but at an added

computational cost.

2.2.5 Track Management

Track life cycles must be monitored for tracks to be correctly started, terminated and maintained in the

tracking system. Stages in the life cycle of a track can include alert, conﬁrmation, initiation, tentative

track, established track or deleted track. Additional events may also occur over the duration of the

track life such as track splitting and track merging. Policies for handling track life cycles are given in

Blackman and Popoli [1999].

Two methods for determining the track status are:

43

2.3. Multifunction Radar Systems

Logic - In logic based track management systems the status of a track is determined by some pre-

deﬁned logic. Based on this logic, rules are designed to determine the status of tracks. For example, a

simple rule to trigger track initiation may be two detections out of three, or an example condition for

track deletion may be three missed detections. More complicated Markov chains can be constructed to

deﬁne rules for statuses such as tentative, preliminary or conﬁrmed track.

Track Scoring - Track scoring methods calculate the likelihood ratio of the hypothesis that a true

target is present against the hypothesis the returns are due to interference [Blackman and Popoli, 1999]:

LR =p(D|HT)P0(HT)

p(D|HN)P0(HN)≡PT

PF A (2.62)

Where HTand HNare the presence of true target and false alarm respectively, given data D. This is

discussed further in Sec 4.1.2.4.

2.3 Multifunction Radar Systems

The multifunction radar described in the preceding section is required to control and optimise the nu-

merous tasks which support differing radar functions. This control can be applied as task parameter and

mode selection which is described in this section. Example operational systems are also described.

2.3.1 Control Parameters

Optimising the conﬁguration of the multifunction radar for each task involves the selection of a set of

radar control parameters. There is a large number of parameter dimensions under control in a typical

multifunction radar system, which are listed in Table 2.2

Table 2.2: Multifunction radar task parameters

Parameter Description Impact

RF Frequency Frequency of the carrier •Choice of frequency motivated by utilising fre-

quency diversity which allows mitigation of in-

terference and environmental losses.

Continued on next page

44

2.3. Multifunction Radar Systems

Table 2.2 - Continued from previous page.

Parameter Description Impact

PRF (Hz) Frequency of pulses in

burst •Increasing PRF increases energy on target up

to the maximum allowable duty cycle but also

increases eclipsing loss for a ﬁxed pulse width.

•Increasing PRF increases unambiguous doppler

range but decreases unambiguous range.

•Multiple PRFs selected in a burst to mitigate

range-Doppler blind zones.

Pulse Width τ(secs) Width of modulating

pulse. •Increasing pulse width increases energy on tar-

get up to the maximum allowable duty cycle but

also increases eclipsing loss for a ﬁxed PRF.

Pulse Compression ρRatio of compressed to

uncompressed pulse. •Increasing pulse compression increases the sig-

nal bandwidth.

Coherent Integration

(secs)

Duration of coherent in-

tegration period. •Increasing coherent integration time improves

frequency resolution and SNR.

•Practical integration time limited by target

movement.

Non-coherent Inte-

gration

Number of non-coherent

integrations. •Increasing non-coherent integrations enables

multiple dwells per burst, improving detection

probability for diverse targets.

•Increasing non-coherent intregrations reduces

noise amplitude ﬂuctuations which improves

detection sensitivity.

Time on Target (secs) Coherent and non-

coherent dwell duration

(secs).

•Increasing time on target improves detection

performance.

Continued on next page

45