Radar Equations for Modern Radar

by David K. Barton

Overview

Based on the classic Radar Range-Performance Analysis from 1980, this practical

volume extends that work to ensure applicability of radar equations to

the design and analysis of modern radars. This unique book helps engineers

identify what information on the radar and its environment is needed to predict

detection range. Moreover, it provides equations and data needed to improve

the accuracy of range calculations. Professionals find detailed information on

propagation effects and methods of range calculation in environments that

include clutter, jamming, and thermal noise, as well as loss factors that reduce

radar performance. This invaluable book is supported with nearly 200 illustrations

and over 430 equations.

Product Details

ISBN-13:	9781608075218
Publisher:	Artech House, Incorporated
Publication date:	11/30/2012
Pages:	264
Sales rank:	447,215
Product dimensions:	6.30(w) x 9.20(h) x 1.10(d)

About the Author

David K. Barton is a consultant on radar systems, retired from ANRO Engineering, Inc., of Hudson, MA. Since 1975, he has been series editor of Artech House's highly successful Radar series. Holder of the IEEE's Centennial Medal, Millennium Medal, and the Dennis J. Picard Medal, he is widely regarded throughout the world as a leading authority of radar technology. He is the author of Radar System Analysis and Modeling (Artech House, 2005) and coeditor of Radar Technology Encyclopedia, CD-ROM Edition (Artech House, 1999), among other titles.

Table of 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_s 16

1.5.2 Use of Signal-to-Noise Energy Ratio 17

1.5.3 Use of Average Power 18

15.4 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

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

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

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

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

6.3 Antenna Noise Temperature 201

6.3.1 Sources of Antenna Noise Temperature 201

6.3.2 Sky Noise Temperature 204

6.3.3 Noise Temperature from the Surface 209

6.3.4 Noise Temperature from Antenna Ohmic Loss 211

6.3.5 Noise Temperature from Antenna Mismatch 212

6.3.6 Approximation for Antenna Noise Temperature 215

6.4 Receiving Line Temperature 217

6.5 Receiver Noise Temperature 217

6.5.1 Noise in Cascaded Receiver Stages 217

6.5.2 Input and Output Levels 219

6.5.3 Quantizing Noise 220

6.6 Summary of Receiving System Noise 221

6.6.1 Thermal Noise Dependence on Carrier Frequency 221

6.6.2 Applicability of Blake's Method 222

6.6.3 Refined Method for Modem Radar 222

6.6.4 Receiver and Quantization Noise Temperature 223

References 223

Chapter 7 Atmospheric Effects 225

7.1 Tropospheric Refraction 225

7.1.1 Refractive Index of Air 226

7.1.2 Standard Atmosphere 227

7.1.3 Inclusion of Water Vapor 228

7.1.4 Vertical Profile of Refractivity 229

7.1.5 Ray Paths in the Troposphere 231

7.2 Attenuation in the Troposphere 232

7.2.1 Sea-Level Attenuation Coefficients of Atmospheric Gases 233

7.2.2 Variation of Attenuation Coefficients with Altitude 237

7.2.3 Attenuation Through the Troposphere 237

7.2.4 Attenuation to Ranged 238

7.2.5 Attenuation for Dry and Moist Atmospheres 244

7.3 Attenuation from Precipitation 246

7.3.1 Rain Attenuation Coefficient at 293K 246

7.3.2 Temperature Dependence of Rain Attenuation 247

7.3.3 Rainfall Rate Statistics 249

7.3.4 Attenuation in Snow 251

7.3.5 Attenuation in Clouds 253

7.3.6 Weather Effects on System Noise Temperature 255

7.4 Tropospheric Lens Loss 255

7.5 Ionospheric Effects 257

7.5.1 Geometry of Ray in Ionosphere 258

7.5.2 Ionospheric Structure 258

7.5.3 Total Electron Count 260

7.5.4 Faraday Rotation 260

7.5.5 Dispersion Across Signal Spectrum 264

7.6 Summary of Atmospheric Effects 269

References 270

Chapter 8 The Pattern-Propagation Factor 273

8.1 Equations for the F-Factor 274

8.1.1 Derivation of the F-Factor 274

8.1.2 Application of the F-Factor 276

8.2 Geometrical Models of the Ray Paths 277

8.2.1 Method 1: Flat-Earth Approximation for Distant Target 278

8.2.2 Method 2: Flat Earth Approximation with Target at Arbitrary Range 279

8.2.3 Method 3: First-Order Approximation for Spherical Earth 280

8.2.4 Method 4: Approximation for Spherical Earth with Distant Target R 282

8.2.5 Method 5: Approximation for Spherical Earth with Target at Arbitrary Range 283

8.2.6 Method 6: Exact Expressions for Spherical Earth with Target at Arbitrary Range 285

8.2.7 Comparison of Approximate Methods 286

8.3 Reflection Coefficient 287

8.3.1 Fresnel Reflection Coefficient 288

8.3.2 Reflection from Rough Surfaces 292

8.3.3 Land Surfaces with Vegetation 295

8.3.4 The Divergence Factor 295

8.4 Diffraction 296

8.4.1 Smooth-Sphere Diffraction 296

8.4.2 Knife-Edge Diffraction 299

8.5 The Interference Region 302

8.6 The Intermediate Region 303

8.6.1 F-Factor as a Function of Range 303

8.6.2 F-Factor as a Function of Altitude 305

8.6.3 Vertical-Plane Coverage Charts 306

8.7 Summary of Propagation Factors 309

References 310

Chapter 9 Clutter and Signal Processing 311

9.1 Modes of Surface Clutter 311

9.1.1 Clutter Cross Section and Reflectivity 311

9.1.2 Surface Clutter Pattern-Propagation Factor 313

9.1.3 Spectral Properties of Surface Clutter 318

9.1.4 Amplitude Distributions of Surface Clutter 321

9.2 Models of Sea Clutter 323

9.2.1 Physical Properties of the Sea Surface 323

9.2.2 Reflectivity of Sea Clutter 324

9.2.3 Power Spectrum of Sea Clutter 326

9.2.4 Amplitude Distribution of Sea Clutter 327

9.3 Models of Land Clutter 327

9.3.1 Reflectivity of Land Clutter 329

9.3.2 Power Spectrum of Land Clutter 331

9.3.3 Amplitude Distribution of Land Clutter 332

9.4 Discrete Clutter 333

9.4.1 Discrete Land Features 333

9.4.2 Birds and Insects 333

9.4.3 Land Vehicles 334

9.4.4 Wind Turbines 335

9.5 Models of Volume Clutter 335

9.5.1 Volume Clutter Cross Section and Reflectivity 336

9.5.2 Volume Clutter Pattern-Propagation Factor 337

9.5.3 Spectral Properties of Volume Clutter 338

9.5.4 Amplitude Distribution of Volume Clutter 340

9.5.5 Precipitation Clutter Models 340

9.5.6 Chaff Models 343

9.6 Clutter Improvement Factor 344

9.6.1 Coherent MTI improvement Factors 345

9.6.2 Noncoherent MTI Improvement Factors 347

9.6.3 Other MTI Considerations 347

9.6.4 Pulsed Doppler Processing 348

9.6.5 Clutter Maps 352

9.7 Summary of Clutter and Signal Processing 352

References 353

Chapter 10 Loss Factors in the Radar Equation 357

10.1 Reduction in Received Signal Energy 358

10.1.1 Terms Specified in the Radar Equation 358

10.1.2 Components of Range-Dependent Response Factor F_rdr 361

10.1.3 Losses Included in System Noise Temperature 364

10.1.4 Losses in Search Radar Equation 364

10.1.5 Losses Included in Antenna Gain 367

10.2 Increases in Required Signal Energy 370

10.2.1 Statistical Losses 370

10.2.2 Losses in Basic Detectability Factor 374

10.2.3 Matching and Bandwidth Losses 375

10.2.4 Beamshape Loss L_p 378

10.2.5 Signal Processing Losses 379

10.2.6 Losses in Clutter Detectability Factor 388

10.3 Losses in Visual Detection 394

10.3.1 Losses in the Visibility Factor 394

10.3.2 Collapsing Loss on the Display 394

10.3.3 Bandwidth Correction Factor C_b 395

10.3.4 Operator Loss L_o 395

10.4 Summary of Loss Factors 396

References 397

List of Symbols 399

Appendix Analysis Tools on DVD 411

About the Author 419

Index 421