Table Of ContentRadar Equations for Modern Radar
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Radar Equations for Modern Radar
David K. Barton
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ISBN 13: 978-1-60807-521-8
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Chapter 1 Development of the Radar Equation . . . . . . . . . . . . . . . . 1
1.1 Radar Equation Fundamentals . . . . . . . . . . . . . . . . 1
1.1.1 Maximum Available Signal-to-Noise Ratio . . . . 2
1.1.2 Minimum Required Signal-to-Noise Ratio . . . . 4
1.1.3 Maximum Detection Range for Pulsed Radar . . . 5
1.2 The Original Radar Equation . . . . . . . . . . . . . . . . 5
1.3 Blake’s Radar Equation for Pulsed Radar . . . . . . . . . . 6
1.3.1 Significance of Terms in Blake’s Equation . . . . 7
1.3.2 Methods of Solving for Range . . . . . . . . . . . 9
1.3.3 Advantages of the Blake Chart . . . . . . . . . . 11
1.3.4 Blake’s Coherent Radar Equation . . . . . . . . 11
1.3.5 Blake’s Bistatic Range Equation . . . . . . . . . 12
1.4 Other Forms of the Radar Equation . . . . . . . . . . . . 13
1.4.1 Hall’s Radar Equations . . . . . . . . . . . . . . 13
1.4.2 Barton’s Radar Equations . . . . . . . . . . . . . 14
1.5 Avoiding Pitfalls in Range Calculation . . . . . . . . . . 16
1.5.1 System Noise Temperature T . . . . . . . . . . . 16
s
1.5.2 Use of Signal-to-Noise Energy Ratio . . . . . . . 17
1.5.3 Use of Average Power . . . . . . . . . . . . . . 18
1 5.4 Bandwidth Correction and Matching Factors . . 18
1.5.5 Detectability Factors for Arbitrary Targets . . . . 18
1.5.6 Pattern-Propagation Factor . . . . . . . . . . . . 19
1.5.7 Loss Factors . . . . . . . . . . . . . . . . . . . . 19
1.5.8 Summary of Pitfalls in Range Calculation . . . . 20
1.6 Radar Equation for Modern Radar Systems . . . . . . . . 20
1.6.1 Factors Requiring Modifications to the Radar
Equation . . . . . . . . . . . . . . . . . . . . 20
1.6.2 Equations Applicable to Modern Radars . . . . . 23
1.6.3 Method of Calculating Detection Range . . . . . 24
1.6.4 Vertical Coverage Charts . . . . . . . . . . . . . 27
1.6.5 Required Probability of Detection . . . . . . . . 28
v
vi Radar Equations for Modern Radar
1.7 Summary of Radar Equation Development . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Chapter 2 The Search Radar Equation . . . . . . . . . . . . . . . . . . . . 33
2.1 Derivation of the Search Radar Equation . . . . . . . . . 34
2.2 Search Sectors for Air Surveillance . . . . . . . . . . . . 37
2.2.1 Elevation Coverage in 2-D Surveillance . . . . . 37
2.2.2 Fan-Beam Pattern for 2-D Air Surveillance . . . 38
2.2.3 Cosecant-Squared Pattern for 2-D Surveillance . 39
2.2.4 Coverage to Constant Altitude . . . . . . . . . . 40
2.2.5 Enhanced Upper Coverage for 2-D Surveillance 40
2.2.6 Reflector Antenna design for 2-D Surveillance
Radar . . . . . . . . . . . . . . . . . . . . . . 41
2.2.7 Array Antennas for 2-D Surveillance Radar . . . 41
2.2.8 Example of Required Power-Aperture Product
for 2-D Radar . . . . . . . . . . . . . . . . . 42
2.3 Three-Dimensional Air Surveillance . . . . . . . . . . . 43
2.3.1 Stacked-Beam 3-D Surveillance Radar . . . . . 43
2.3.2 Scanning-beam 3-D Surveillance Radars . . . . 43
2.3.3 Search Losses for 3-D Surveillance Radar . . . . 44
2.4 Surveillance with Multifunction Array Radar . . . . . . . 44
2.4.1 Example of MFAR Search Sectors . . . . . . . . 45
2.4.2 Advantages and Disadvantages of MFAR Search 46
2.4.3 Example of Search Radar Equation for MFAR 47
2.5 The Search Fence . . . . . . . . . . . . . . . . . . . . . 48
2.5.1 Search Sector for the Fence . . . . . . . . . . . 49
2.5.2 Example ICBM Fence . . . . . . . . . . . . . . 50
2.6 Search Losses . . . . . . . . . . . . . . . . . . . . . . . 51
2.6.1 Reduction in Available Energy Ratio . . . . . . 51
2.6.2 Increase in Required Energy Ratio . . . . . . . . 52
2.6.3 Summary of Losses . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Chapter 3 Radar Equations for Clutter and Jamming . . . . . . . . . . . . 55
3.1 Signal-to-Interference Ratio . . . . . . . . . . . . . . . . 55
3.2 Clutter Effect on Detection Range . . . . . . . . . . . . . 57
3.2.1 Range-Ambiguous Clutter . . . . . . . . . . . . 57
3.2.2 Types of Radar Waveforms . . . . . . . . . . . 58
3.2.3 Clutter Detectability Factor . . . . . . . . . . . . 59
3.2.4 Effective Spectral Density of Clutter . . . . . . . 61
3.2.5 Detection Range with Clutter . . . . . . . . . . . 62
Contents vii
3.3 Detection in Surface Clutter . . . . . . . . . . . . . . . . 62
3.3.1 Clutter from a Flat Surface . . . . . . . . . . . . 62
3.3.2 Surface Clutter from the Spherical Earth . . . . . 65
3.3.3 Surface Clutter Cross Section . . . . . . . . . . 66
3.3.4 Input Energy of Surface Clutter . . . . . . . . . 68
3.3.5 Detection Range of Surface-Based CW and
HPRF Radars . . . . . . . . . . . . . . . . . . 73
3.3.6 Summary of Detection in Surface Clutter . . . . 76
3.4 Detection in Volume Clutter . . . . . . . . . . . . . . . . 77
3.4.1 Geometry of Volume Clutter . . . . . . . . . . 77
3.4.2 Volume Clutter Cross Section . . . . . . . . . . 78
3.4.3 Volume Clutter Energy . . . . . . . . . . . . . 79
3.4.4 Volume Clutter Detectability Factor . . . . . . . 80
3.4.5 Detection Range in Volume Clutter and Noise . 80
3.4.6 Volume Clutter in CW and PD Radars . . . . . 82
3.4.7 Summary of Detection in Volume Clutter . . . . 87
3.5 Effects of Discrete Clutter . . . . . . .. . . . . . . . . . 88
3.5.1 Effect of False Alarms . . . . . . . . . . . . . . 89
3.5.2 Required Noise False-Alarm Probability . . . . 89
3.5.3 Requirements for Rejection of Discrete Clutter . 90
3 5.4 Summary of Discrete Clutter Effects . . . . . . 91
3.6 Sidelobe Clutter . . . . . . . . . . . . . . . . . . . . . . 91
3.6.1 Surface Clutter in Sidelobes . . . . . . . . . . . 91
3.6.2 Volume Clutter in Sidelobes . . . . . . . . . . . 93
3.7 Detection in Noise Jamming . . . . . . . . . . . . . . . . 94
3.7.1 Objective and Methods of Noise Jamming . . . . 94
3.7.2 Radar Equations for Noise Jamming . . . . . . . 96
3.7.3 Examples of Noise Jamming . . . . . . . . . . . 98
3.8 Deceptive Jamming . . . . . . . . . . . . . . . . . . . . 101
3.8.1 Range Equations for Deceptive Jamming . . . . 102
3.9 Summary of Detection in Jamming . . . . . . . . . . . 106
3.9.1 Range with Noise Jamming . . . . . . . . . . . 106
3.9.2 Deceptive Jammer Equations . . . . . . . . . . 106
3.10 Detection in Combined Interference . . . . . . . . . . . 106
References . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Chapter 4 Detection Theory . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . 109
4.2 Steady-Target Detectability Factor . . . . . . . . . . . 110
4.2.1 Exact Steady-Target Detection Probability . . . 111
4.2.2 Threshold Level . . . . . . . . . . . . . . . . . 111
4.2.3 Exact Steady-Target Detectability Factor . . . . 114
viii Radar Equations for Modern Radar
4.2.4 Exact Single-Pulse, Steady-Target Detectability
Factor . . . . . . . . . . . . . . . . . . . . . 114
4.2.5 Approximations for Single-Pulse, Steady-
Target Detectability Factor . . . . . . . . . . 115
4.2.6 Approximations for n-Pulse, Steady-Target
Detectability Factor . . . . . . . . . . . . . . 116
4.3 Detectability Factors for Fluctuating Targets . . . . . 118
4.3.1 Generalized Chi-Square Target Fluctuation
Model . . . . . . . . . . . . . . . . . . . . . 118
4.3.2 Detection of Signals with Chi-Square Statistics 119
4.3.3 Swerling Case 1 . . . . . . . . . . . . . . . . . 120
4.3.4 Swerling Case 2 . . . . . . . . . . . . . . . . . 124
4.3.5 Swerling Case 3 . . . . . . . . . . . . . . . . . 125
4.3.6 Swerling Case 4 . . . . . . . . . . . . . . . . . 127
4.4 Equations Based on Detector Loss . . . . . . . . . . . 127
4.4.1 Coherent Detection . . . . . . . . . . . . . . . 127
4.4.2 Envelope Detection and Detector Loss . . . . . 129
4.4.3 Integration Loss . . . . . . . . . . . . . . . . . 129
4.4.4 Integration Gain . . . . . . . . . . . . . . . . . 131
4.4.5 Fluctuation Loss . . . . . . . . . . . . . . . . . 132
4.4.6 Case 1 Detectability Factor . . . . . . . . . . . 133
4.4.7 Detectability Factors for Other Fluctuating
Targets . . . . . . . . . . . . . . . . . . . . 134
4.5 Diversity in Radar . . . . . . . . . . . . . . . . . . . 134
4.5.1 Diversity Gain . . . . . . . . . . . . . . . . . . 134
4.5.2 Signal and Target Models with Diversity . . . . 135
4.6 Visibility Factor . . . . . . . . . . . . . . . . . . . . 138
4.7 Summary of Detection Theory . . . . . . . . . . . . . 140
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Chapter 5 Beamshape Loss . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . 143
5.1.1 Definition of Beamshape Loss . . . . . . . . . 143
5.1.2 Sampling in Angle Space . . . . . . . . . . . . 144
5.1.3 Literature on Beamshape Loss . . . . . . . . . 145
5.2 Beamshape Loss with Dense Sampling . . . . . . . . 146
5.2.1 Simple Beamshape Loss Model . . . . . . . . . 146
5.2.2 Antenna Patterns . . . . . . . . . . . . . . . . 147
5.2.3 Beamshape Loss for Different Patterns . . . . . 148
5.3 Sparse Sampling in 1-D Scan . . . . . . . . . . . . . 149
5.3.1 Method of Calculation for 1-D Scan . . . . . . 149
5.3.2 Steady Target Beamshape Loss for 1-D Scan . 151
5.3.3 Case 1 Beamshape Loss for 1-D Scan . . . . . . . . . 153
Contents ix
5.3.4 Case 2 Beamshape Loss for 1-D Scan . . . . . . . . . 155
5.3.5 Beamshape Loss Used in Search Radar Equation for
1-D Scan . . . . . . . . . . . . . . . . . . . 158
5.4 Sparse Sampling in 2-D Raster Scan . . . . . . . . . . 160
5.4.1 Method of Calculation for 2-D Scan . . . . . . 162
5.4.2 Steady Target Beamshape Loss for 2-D Scan . 162
5.4.3 Case 1 Beamshape Loss for 2-D Scan . . . . . 163
5.4.4 Case 2 Beamshape Loss for 2-D Scan . . . . . 165
5.4.5 Diversity Target Beamshape Loss for 2-D Scan 168
5.4.6 Beamshape Loss in the Search Radar Equation
for 2-D Raster Scan . . . . . . . . . . . . . . 171
5.5 Sparse Sampling Using a Triangular Grid . . . . . . . 174
5.5.1 Method of Calculation for Triangular Grid . . . 174
5.5.2 Steady Target Beamshape Loss for Triangular
Grid . . . . . . . . . . . . . . . . . . . . . . 175
5.5.3 Case 1 Beamshape Loss for Triangular Grid . . 175
5.5.4 Case 2 Beamshape Loss for Triangular Grid . . 176
5.5.5 Diversity Target Beamshape Loss for
Triangular Grid . . . . . . . . . . . . . . . . 178
5.5.6 Beamshape Loss in Search Radar Equation for
Triangular Grid . . . . . . . . . . . . . . . . 180
5.6 Summary of Beamshape Loss . . . . . . . . . . . . . 181
5.6.1 Beamshape Loss for Dense Sampling . . . . . 181
5.6.2 Beamshape Loss for Sparse Sampling . . . . . 182
5.6.3 Processing Methods . . . . . . . . . . . . . . . 184
5.6.4 Net Beamshape Loss for the Search Radar
Equation . . . . . . . . . . . . . . . . . . . 185
5.6.5 Beamshape Loss for Unequally Spaced 2-D
Scan . . . . . . . . . . . . . . . . . . . . . . 186
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Appendix 5A Analytical Approximations for Beamshape
Loss . . . . . . . . . . . . . . . . . . . . . . 188
5A.1 1-D Beamshape Loss . . . . . . . . . . . . . . 188
5A.2 2-D Beamshape Loss with Rectangular Grid . . 189
5A.3 2-D Beamshape Loss with Triangular Grid . . . 192
Chapter 6 System Noise Temperature . . . . . . . . . . . . . . . . . . . . 197
6.1 Noise in the Radar Bands . . . . . . . . . . . . . . . . 197
6.1.1 Noise Spectral Density . . . . . . . . . . . . . 197
6.1.2 Noise Statistics . . . . . . . . . . . . . . . . . 198
6.2 Sources of Noise in Radar Reception . . . . . . . . . . 200
