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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
2.3. Multifunction Ra