Principles of Interferometric and Polarimetric Radiometry
portes grátis
Principles of Interferometric and Polarimetric Radiometry
Corbella, Ignasi
John Wiley & Sons Inc
12/2024
320
Dura
9781394255108
15 a 20 dias
Descrição não disponível.
Foreword xiii
About the Author xix
Preface xxi
Acknowledgments xxv
1 Signals, Receivers, and Antennas 1
1.1 Random Variables, Real and Complex 1
1.1.1 Definitions 2
1.1.2 Operations 4
1.1.3 Normal Random Variables 6
1.1.4 The Arc Sine Law 8
1.2 Stochastic Processes 9
1.2.1 Stationarity 10
1.2.2 Correlation and Power 11
1.2.3 Jointly Normal Processes 14
1.2.4 Spectral Densities 17
1.2.5 Linear Systems 19
1.2.6 Time Averaging or Integration 20
1.3 Analytic Signals 23
1.3.1 Hilbert Transform and Quadrature Filter 23
1.3.2 Complex Envelope 25
1.3.3 Correlation and Spectra 26
1.4 Phasors of Random Signals 28
1.4.1 Concept 29
1.4.2 Power and Cross-correlation 30
1.4.3 Linear Systems 31
1.5 Microwave Networks 32
1.5.1 Voltage and Current 32
1.5.2 Normalized Voltage Waves 34
1.5.3 Available Power 35
1.5.4 S-parameters and Power Gains 36
1.5.5 Noise Waves and Temperature 38
1.5.6 Interconnection 40
1.5.7 Two-port Networks 42
1.5.8 Cascade 45
1.5.9 High Gain Receiver 46
1.5.10 The Bosma Theorem 48
1.6 Antennas 50
1.6.1 Radiated Electric Field and Power Density 50
1.6.2 Antenna Pattern and Directivity 51
1.6.3 Antenna Polarization 53
1.6.4 Thermal Noise Radiation 56
1.6.5 Received Signal 57
1.6.6 Phase Center 59
1.6.7 Polarization Misalignment 60
1.6.8 Transmission Link 61
1.6.9 Reciprocity 62
1.6.10 Other Definitions 64
1.6.11 Antenna Loss 65
References 66
2 Microwave Radiometry 69
2.1 Thermal Emission 69
2.1.1 Emissivity and Brightness Temperature 70
2.1.2 Planck and Rayleigh-Jeans Laws 73
2.2 Polarization 74
2.2.1 Stokes Parameters and Polarimetric Brightness Temperature 74
2.2.2 Change of Polarization Frame 76
2.2.3 Linear Axis Rotation 77
2.2.4 Horizontal and Vertical Polarization 78
2.2.5 Circular Polarization 80
2.3 Antenna Temperature 81
2.3.1 Concept 81
2.3.2 Flat Target 83
2.3.3 Point Source 84
2.3.4 Extended Source 85
2.3.5 Angular Resolution 85
2.4 Total Power Radiometers 86
2.4.1 Received Signal 87
2.4.2 Power Measurement and Sensitivity 90
2.4.3 Square Law Device 94
2.4.4 Quadratic Detector 97
References 100
3 Interferometry and Polarimetry 101
3.1 Historical Perspective 101
3.1.1 The Proposed Formulation 105
3.2 A Single Baseline 106
3.2.1 Visibility 106
3.2.2 Single Polarization 109
3.2.3 Polarimetric Radiometry: Ideal Case 110
3.2.4 Full Polarimetric Case 111
3.2.5 Receivers Interaction 114
3.2.6 The "?T r " Term 119
3.3 The Visibility Equation 120
3.3.1 Complex Correlation 121
3.3.2 The Fringe Washing Function 123
3.3.3 Director Cosines 125
3.3.4 Fourier Relation 127
3.4 Correlation Measurement 127
3.4.1 Sensitivity 128
3.4.2 Four Signal Multipliers 131
3.4.3 Two Signal Multipliers 133
3.4.4 Analog Multipliers 137
3.4.5 Signal Clipping and Normalized Correlation 139
References 146
4 Aperture Synthesis 149
4.1 Synthetic Beam 150
4.1.1 Hexagonal Sampling 153
4.2 Radiometric Sensitivity 155
4.2.1 Variance of the Modified Brightness Temperature 155
4.2.2 Uncorrelated Visibility Samples 157
4.2.3 Correlation of Visibility Samples 159
4.3 Spatial Sampling 163
4.3.1 Visibility Coverage 163
4.3.2 Reciprocal Grids 164
4.3.3 Aliasing 166
4.3.4 Field of View 168
4.3.5 Hexagonal Grids: Y-shape Instrument 168
4.3.6 Hexagonal Instrument 171
4.4 Imaging 175
4.4.1 System of Equations 175
4.4.2 Conjugate Extension and Redundant Baselines Averaging 178
4.4.3 Fourier Image Reconstruction 179
4.4.4 G-matrix Image Reconstruction 181
4.4.5 Polarimetric Retrieval: Ideal Case 184
4.4.6 Full Polarimetric Case 184
4.4.7 Spatial Frequency Components 186
4.4.8 Reconstruction Error and Alias Mitigation 189
References 191
5 Instrument Techniques 193
5.1 Frequency Conversion 193
5.1.1 Frequency Bands 193
5.1.2 Mixer Operation 195
5.1.3 Image Rejection Mixer 196
5.2 In-phase and Quadrature (IQ) Mixer 197
5.2.1 Concept 197
5.2.2 General Analysis 198
5.2.3 Quadrature Error 200
5.2.4 Correction of Phase Errors 201
5.2.5 Normalized Correlations 202
5.3 Quarter Period Delay 203
5.3.1 Concept 203
5.3.2 Center Frequency Error 205
5.3.3 Normalized Correlations 206
5.4 Digital Techniques 207
5.4.1 Sampling 207
5.4.2 Impact on Measurement Uncertainty 215
5.4.3 Low-frequency Spectrum 217
5.4.4 Spectrum with High-frequency Content 221
5.4.5 I/Q Alternate Sampling 225
5.4.6 Nyquist Zones 227
5.4.7 Correlation in the Frequency Domain 231
References 235
6 Calibration and Characterization 237
6.1 Calibration Standards 237
6.1.1 Antenna and Calibration Planes 238
6.1.2 Plane Change in Total Power Radiometers 240
6.1.3 External Passive Targets 243
6.1.4 Probe Antenna 245
6.1.5 Internal Load 247
6.1.6 Noise Distribution 248
6.2 Parameter Retrieval 250
6.2.1 Correlator Gain 251
6.2.2 Inter-element Phase and Amplitude 253
6.2.3 Correlator Offset 255
6.2.4 Flat Target Response 257
6.2.5 Fringe Washing Function Shape 259
6.2.6 Receiver Gain and Offset 261
6.2.7 Instrumental Offset 263
6.3 Nonlinearity 265
6.3.1 Deflection Ratio 266
6.3.2 Impact on Instrumental Offset 267
6.4 Calibration Rate 268
6.4.1 Averaging and Interpolation 268
6.4.2 Temperature Correction 269
References 270
A Definitions and Concepts 273
A. 1 Complex Vectors 273
A. 2 Useful Complex Number Identities 274
A. 3 Energy Conservation and Unitary Matrix 274
A. 4 Spherical Coordinates and Solid Angle 275
A.4. 1 Differential Surface 275
A.4. 2 Solid Angle 275
A. 5 Quadrature Equation Inversion 276
A. 6 Special Functions 277
A. 7 Fourier Transform 278
A.7. 1 Convolution 278
A.7. 2 Properties 279
A.7. 3 Transform Pairs 279
A.7. 4 Real Signals 279
A.7. 5 Two-dimensional Fourier Transform 280
A. 8 Discrete Fourier Transform 281
A.8. 1 Correlation in Time and in Frequency 283
A.8. 2 Random Signals 283
A.8. 3 Two-dimensional Case 284
Reference 284
Index 285
About the Author xix
Preface xxi
Acknowledgments xxv
1 Signals, Receivers, and Antennas 1
1.1 Random Variables, Real and Complex 1
1.1.1 Definitions 2
1.1.2 Operations 4
1.1.3 Normal Random Variables 6
1.1.4 The Arc Sine Law 8
1.2 Stochastic Processes 9
1.2.1 Stationarity 10
1.2.2 Correlation and Power 11
1.2.3 Jointly Normal Processes 14
1.2.4 Spectral Densities 17
1.2.5 Linear Systems 19
1.2.6 Time Averaging or Integration 20
1.3 Analytic Signals 23
1.3.1 Hilbert Transform and Quadrature Filter 23
1.3.2 Complex Envelope 25
1.3.3 Correlation and Spectra 26
1.4 Phasors of Random Signals 28
1.4.1 Concept 29
1.4.2 Power and Cross-correlation 30
1.4.3 Linear Systems 31
1.5 Microwave Networks 32
1.5.1 Voltage and Current 32
1.5.2 Normalized Voltage Waves 34
1.5.3 Available Power 35
1.5.4 S-parameters and Power Gains 36
1.5.5 Noise Waves and Temperature 38
1.5.6 Interconnection 40
1.5.7 Two-port Networks 42
1.5.8 Cascade 45
1.5.9 High Gain Receiver 46
1.5.10 The Bosma Theorem 48
1.6 Antennas 50
1.6.1 Radiated Electric Field and Power Density 50
1.6.2 Antenna Pattern and Directivity 51
1.6.3 Antenna Polarization 53
1.6.4 Thermal Noise Radiation 56
1.6.5 Received Signal 57
1.6.6 Phase Center 59
1.6.7 Polarization Misalignment 60
1.6.8 Transmission Link 61
1.6.9 Reciprocity 62
1.6.10 Other Definitions 64
1.6.11 Antenna Loss 65
References 66
2 Microwave Radiometry 69
2.1 Thermal Emission 69
2.1.1 Emissivity and Brightness Temperature 70
2.1.2 Planck and Rayleigh-Jeans Laws 73
2.2 Polarization 74
2.2.1 Stokes Parameters and Polarimetric Brightness Temperature 74
2.2.2 Change of Polarization Frame 76
2.2.3 Linear Axis Rotation 77
2.2.4 Horizontal and Vertical Polarization 78
2.2.5 Circular Polarization 80
2.3 Antenna Temperature 81
2.3.1 Concept 81
2.3.2 Flat Target 83
2.3.3 Point Source 84
2.3.4 Extended Source 85
2.3.5 Angular Resolution 85
2.4 Total Power Radiometers 86
2.4.1 Received Signal 87
2.4.2 Power Measurement and Sensitivity 90
2.4.3 Square Law Device 94
2.4.4 Quadratic Detector 97
References 100
3 Interferometry and Polarimetry 101
3.1 Historical Perspective 101
3.1.1 The Proposed Formulation 105
3.2 A Single Baseline 106
3.2.1 Visibility 106
3.2.2 Single Polarization 109
3.2.3 Polarimetric Radiometry: Ideal Case 110
3.2.4 Full Polarimetric Case 111
3.2.5 Receivers Interaction 114
3.2.6 The "?T r " Term 119
3.3 The Visibility Equation 120
3.3.1 Complex Correlation 121
3.3.2 The Fringe Washing Function 123
3.3.3 Director Cosines 125
3.3.4 Fourier Relation 127
3.4 Correlation Measurement 127
3.4.1 Sensitivity 128
3.4.2 Four Signal Multipliers 131
3.4.3 Two Signal Multipliers 133
3.4.4 Analog Multipliers 137
3.4.5 Signal Clipping and Normalized Correlation 139
References 146
4 Aperture Synthesis 149
4.1 Synthetic Beam 150
4.1.1 Hexagonal Sampling 153
4.2 Radiometric Sensitivity 155
4.2.1 Variance of the Modified Brightness Temperature 155
4.2.2 Uncorrelated Visibility Samples 157
4.2.3 Correlation of Visibility Samples 159
4.3 Spatial Sampling 163
4.3.1 Visibility Coverage 163
4.3.2 Reciprocal Grids 164
4.3.3 Aliasing 166
4.3.4 Field of View 168
4.3.5 Hexagonal Grids: Y-shape Instrument 168
4.3.6 Hexagonal Instrument 171
4.4 Imaging 175
4.4.1 System of Equations 175
4.4.2 Conjugate Extension and Redundant Baselines Averaging 178
4.4.3 Fourier Image Reconstruction 179
4.4.4 G-matrix Image Reconstruction 181
4.4.5 Polarimetric Retrieval: Ideal Case 184
4.4.6 Full Polarimetric Case 184
4.4.7 Spatial Frequency Components 186
4.4.8 Reconstruction Error and Alias Mitigation 189
References 191
5 Instrument Techniques 193
5.1 Frequency Conversion 193
5.1.1 Frequency Bands 193
5.1.2 Mixer Operation 195
5.1.3 Image Rejection Mixer 196
5.2 In-phase and Quadrature (IQ) Mixer 197
5.2.1 Concept 197
5.2.2 General Analysis 198
5.2.3 Quadrature Error 200
5.2.4 Correction of Phase Errors 201
5.2.5 Normalized Correlations 202
5.3 Quarter Period Delay 203
5.3.1 Concept 203
5.3.2 Center Frequency Error 205
5.3.3 Normalized Correlations 206
5.4 Digital Techniques 207
5.4.1 Sampling 207
5.4.2 Impact on Measurement Uncertainty 215
5.4.3 Low-frequency Spectrum 217
5.4.4 Spectrum with High-frequency Content 221
5.4.5 I/Q Alternate Sampling 225
5.4.6 Nyquist Zones 227
5.4.7 Correlation in the Frequency Domain 231
References 235
6 Calibration and Characterization 237
6.1 Calibration Standards 237
6.1.1 Antenna and Calibration Planes 238
6.1.2 Plane Change in Total Power Radiometers 240
6.1.3 External Passive Targets 243
6.1.4 Probe Antenna 245
6.1.5 Internal Load 247
6.1.6 Noise Distribution 248
6.2 Parameter Retrieval 250
6.2.1 Correlator Gain 251
6.2.2 Inter-element Phase and Amplitude 253
6.2.3 Correlator Offset 255
6.2.4 Flat Target Response 257
6.2.5 Fringe Washing Function Shape 259
6.2.6 Receiver Gain and Offset 261
6.2.7 Instrumental Offset 263
6.3 Nonlinearity 265
6.3.1 Deflection Ratio 266
6.3.2 Impact on Instrumental Offset 267
6.4 Calibration Rate 268
6.4.1 Averaging and Interpolation 268
6.4.2 Temperature Correction 269
References 270
A Definitions and Concepts 273
A. 1 Complex Vectors 273
A. 2 Useful Complex Number Identities 274
A. 3 Energy Conservation and Unitary Matrix 274
A. 4 Spherical Coordinates and Solid Angle 275
A.4. 1 Differential Surface 275
A.4. 2 Solid Angle 275
A. 5 Quadrature Equation Inversion 276
A. 6 Special Functions 277
A. 7 Fourier Transform 278
A.7. 1 Convolution 278
A.7. 2 Properties 279
A.7. 3 Transform Pairs 279
A.7. 4 Real Signals 279
A.7. 5 Two-dimensional Fourier Transform 280
A. 8 Discrete Fourier Transform 281
A.8. 1 Correlation in Time and in Frequency 283
A.8. 2 Random Signals 283
A.8. 3 Two-dimensional Case 284
Reference 284
Index 285
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Passive remote sensing; aperture synthesis; Earth observation; microwave systems; microwave radiometry; antennas; antenna arrays; polarization; Fourier inversion; image reconstruction; calibration; noise waves; stochastic processes
Foreword xiii
About the Author xix
Preface xxi
Acknowledgments xxv
1 Signals, Receivers, and Antennas 1
1.1 Random Variables, Real and Complex 1
1.1.1 Definitions 2
1.1.2 Operations 4
1.1.3 Normal Random Variables 6
1.1.4 The Arc Sine Law 8
1.2 Stochastic Processes 9
1.2.1 Stationarity 10
1.2.2 Correlation and Power 11
1.2.3 Jointly Normal Processes 14
1.2.4 Spectral Densities 17
1.2.5 Linear Systems 19
1.2.6 Time Averaging or Integration 20
1.3 Analytic Signals 23
1.3.1 Hilbert Transform and Quadrature Filter 23
1.3.2 Complex Envelope 25
1.3.3 Correlation and Spectra 26
1.4 Phasors of Random Signals 28
1.4.1 Concept 29
1.4.2 Power and Cross-correlation 30
1.4.3 Linear Systems 31
1.5 Microwave Networks 32
1.5.1 Voltage and Current 32
1.5.2 Normalized Voltage Waves 34
1.5.3 Available Power 35
1.5.4 S-parameters and Power Gains 36
1.5.5 Noise Waves and Temperature 38
1.5.6 Interconnection 40
1.5.7 Two-port Networks 42
1.5.8 Cascade 45
1.5.9 High Gain Receiver 46
1.5.10 The Bosma Theorem 48
1.6 Antennas 50
1.6.1 Radiated Electric Field and Power Density 50
1.6.2 Antenna Pattern and Directivity 51
1.6.3 Antenna Polarization 53
1.6.4 Thermal Noise Radiation 56
1.6.5 Received Signal 57
1.6.6 Phase Center 59
1.6.7 Polarization Misalignment 60
1.6.8 Transmission Link 61
1.6.9 Reciprocity 62
1.6.10 Other Definitions 64
1.6.11 Antenna Loss 65
References 66
2 Microwave Radiometry 69
2.1 Thermal Emission 69
2.1.1 Emissivity and Brightness Temperature 70
2.1.2 Planck and Rayleigh-Jeans Laws 73
2.2 Polarization 74
2.2.1 Stokes Parameters and Polarimetric Brightness Temperature 74
2.2.2 Change of Polarization Frame 76
2.2.3 Linear Axis Rotation 77
2.2.4 Horizontal and Vertical Polarization 78
2.2.5 Circular Polarization 80
2.3 Antenna Temperature 81
2.3.1 Concept 81
2.3.2 Flat Target 83
2.3.3 Point Source 84
2.3.4 Extended Source 85
2.3.5 Angular Resolution 85
2.4 Total Power Radiometers 86
2.4.1 Received Signal 87
2.4.2 Power Measurement and Sensitivity 90
2.4.3 Square Law Device 94
2.4.4 Quadratic Detector 97
References 100
3 Interferometry and Polarimetry 101
3.1 Historical Perspective 101
3.1.1 The Proposed Formulation 105
3.2 A Single Baseline 106
3.2.1 Visibility 106
3.2.2 Single Polarization 109
3.2.3 Polarimetric Radiometry: Ideal Case 110
3.2.4 Full Polarimetric Case 111
3.2.5 Receivers Interaction 114
3.2.6 The "?T r " Term 119
3.3 The Visibility Equation 120
3.3.1 Complex Correlation 121
3.3.2 The Fringe Washing Function 123
3.3.3 Director Cosines 125
3.3.4 Fourier Relation 127
3.4 Correlation Measurement 127
3.4.1 Sensitivity 128
3.4.2 Four Signal Multipliers 131
3.4.3 Two Signal Multipliers 133
3.4.4 Analog Multipliers 137
3.4.5 Signal Clipping and Normalized Correlation 139
References 146
4 Aperture Synthesis 149
4.1 Synthetic Beam 150
4.1.1 Hexagonal Sampling 153
4.2 Radiometric Sensitivity 155
4.2.1 Variance of the Modified Brightness Temperature 155
4.2.2 Uncorrelated Visibility Samples 157
4.2.3 Correlation of Visibility Samples 159
4.3 Spatial Sampling 163
4.3.1 Visibility Coverage 163
4.3.2 Reciprocal Grids 164
4.3.3 Aliasing 166
4.3.4 Field of View 168
4.3.5 Hexagonal Grids: Y-shape Instrument 168
4.3.6 Hexagonal Instrument 171
4.4 Imaging 175
4.4.1 System of Equations 175
4.4.2 Conjugate Extension and Redundant Baselines Averaging 178
4.4.3 Fourier Image Reconstruction 179
4.4.4 G-matrix Image Reconstruction 181
4.4.5 Polarimetric Retrieval: Ideal Case 184
4.4.6 Full Polarimetric Case 184
4.4.7 Spatial Frequency Components 186
4.4.8 Reconstruction Error and Alias Mitigation 189
References 191
5 Instrument Techniques 193
5.1 Frequency Conversion 193
5.1.1 Frequency Bands 193
5.1.2 Mixer Operation 195
5.1.3 Image Rejection Mixer 196
5.2 In-phase and Quadrature (IQ) Mixer 197
5.2.1 Concept 197
5.2.2 General Analysis 198
5.2.3 Quadrature Error 200
5.2.4 Correction of Phase Errors 201
5.2.5 Normalized Correlations 202
5.3 Quarter Period Delay 203
5.3.1 Concept 203
5.3.2 Center Frequency Error 205
5.3.3 Normalized Correlations 206
5.4 Digital Techniques 207
5.4.1 Sampling 207
5.4.2 Impact on Measurement Uncertainty 215
5.4.3 Low-frequency Spectrum 217
5.4.4 Spectrum with High-frequency Content 221
5.4.5 I/Q Alternate Sampling 225
5.4.6 Nyquist Zones 227
5.4.7 Correlation in the Frequency Domain 231
References 235
6 Calibration and Characterization 237
6.1 Calibration Standards 237
6.1.1 Antenna and Calibration Planes 238
6.1.2 Plane Change in Total Power Radiometers 240
6.1.3 External Passive Targets 243
6.1.4 Probe Antenna 245
6.1.5 Internal Load 247
6.1.6 Noise Distribution 248
6.2 Parameter Retrieval 250
6.2.1 Correlator Gain 251
6.2.2 Inter-element Phase and Amplitude 253
6.2.3 Correlator Offset 255
6.2.4 Flat Target Response 257
6.2.5 Fringe Washing Function Shape 259
6.2.6 Receiver Gain and Offset 261
6.2.7 Instrumental Offset 263
6.3 Nonlinearity 265
6.3.1 Deflection Ratio 266
6.3.2 Impact on Instrumental Offset 267
6.4 Calibration Rate 268
6.4.1 Averaging and Interpolation 268
6.4.2 Temperature Correction 269
References 270
A Definitions and Concepts 273
A. 1 Complex Vectors 273
A. 2 Useful Complex Number Identities 274
A. 3 Energy Conservation and Unitary Matrix 274
A. 4 Spherical Coordinates and Solid Angle 275
A.4. 1 Differential Surface 275
A.4. 2 Solid Angle 275
A. 5 Quadrature Equation Inversion 276
A. 6 Special Functions 277
A. 7 Fourier Transform 278
A.7. 1 Convolution 278
A.7. 2 Properties 279
A.7. 3 Transform Pairs 279
A.7. 4 Real Signals 279
A.7. 5 Two-dimensional Fourier Transform 280
A. 8 Discrete Fourier Transform 281
A.8. 1 Correlation in Time and in Frequency 283
A.8. 2 Random Signals 283
A.8. 3 Two-dimensional Case 284
Reference 284
Index 285
About the Author xix
Preface xxi
Acknowledgments xxv
1 Signals, Receivers, and Antennas 1
1.1 Random Variables, Real and Complex 1
1.1.1 Definitions 2
1.1.2 Operations 4
1.1.3 Normal Random Variables 6
1.1.4 The Arc Sine Law 8
1.2 Stochastic Processes 9
1.2.1 Stationarity 10
1.2.2 Correlation and Power 11
1.2.3 Jointly Normal Processes 14
1.2.4 Spectral Densities 17
1.2.5 Linear Systems 19
1.2.6 Time Averaging or Integration 20
1.3 Analytic Signals 23
1.3.1 Hilbert Transform and Quadrature Filter 23
1.3.2 Complex Envelope 25
1.3.3 Correlation and Spectra 26
1.4 Phasors of Random Signals 28
1.4.1 Concept 29
1.4.2 Power and Cross-correlation 30
1.4.3 Linear Systems 31
1.5 Microwave Networks 32
1.5.1 Voltage and Current 32
1.5.2 Normalized Voltage Waves 34
1.5.3 Available Power 35
1.5.4 S-parameters and Power Gains 36
1.5.5 Noise Waves and Temperature 38
1.5.6 Interconnection 40
1.5.7 Two-port Networks 42
1.5.8 Cascade 45
1.5.9 High Gain Receiver 46
1.5.10 The Bosma Theorem 48
1.6 Antennas 50
1.6.1 Radiated Electric Field and Power Density 50
1.6.2 Antenna Pattern and Directivity 51
1.6.3 Antenna Polarization 53
1.6.4 Thermal Noise Radiation 56
1.6.5 Received Signal 57
1.6.6 Phase Center 59
1.6.7 Polarization Misalignment 60
1.6.8 Transmission Link 61
1.6.9 Reciprocity 62
1.6.10 Other Definitions 64
1.6.11 Antenna Loss 65
References 66
2 Microwave Radiometry 69
2.1 Thermal Emission 69
2.1.1 Emissivity and Brightness Temperature 70
2.1.2 Planck and Rayleigh-Jeans Laws 73
2.2 Polarization 74
2.2.1 Stokes Parameters and Polarimetric Brightness Temperature 74
2.2.2 Change of Polarization Frame 76
2.2.3 Linear Axis Rotation 77
2.2.4 Horizontal and Vertical Polarization 78
2.2.5 Circular Polarization 80
2.3 Antenna Temperature 81
2.3.1 Concept 81
2.3.2 Flat Target 83
2.3.3 Point Source 84
2.3.4 Extended Source 85
2.3.5 Angular Resolution 85
2.4 Total Power Radiometers 86
2.4.1 Received Signal 87
2.4.2 Power Measurement and Sensitivity 90
2.4.3 Square Law Device 94
2.4.4 Quadratic Detector 97
References 100
3 Interferometry and Polarimetry 101
3.1 Historical Perspective 101
3.1.1 The Proposed Formulation 105
3.2 A Single Baseline 106
3.2.1 Visibility 106
3.2.2 Single Polarization 109
3.2.3 Polarimetric Radiometry: Ideal Case 110
3.2.4 Full Polarimetric Case 111
3.2.5 Receivers Interaction 114
3.2.6 The "?T r " Term 119
3.3 The Visibility Equation 120
3.3.1 Complex Correlation 121
3.3.2 The Fringe Washing Function 123
3.3.3 Director Cosines 125
3.3.4 Fourier Relation 127
3.4 Correlation Measurement 127
3.4.1 Sensitivity 128
3.4.2 Four Signal Multipliers 131
3.4.3 Two Signal Multipliers 133
3.4.4 Analog Multipliers 137
3.4.5 Signal Clipping and Normalized Correlation 139
References 146
4 Aperture Synthesis 149
4.1 Synthetic Beam 150
4.1.1 Hexagonal Sampling 153
4.2 Radiometric Sensitivity 155
4.2.1 Variance of the Modified Brightness Temperature 155
4.2.2 Uncorrelated Visibility Samples 157
4.2.3 Correlation of Visibility Samples 159
4.3 Spatial Sampling 163
4.3.1 Visibility Coverage 163
4.3.2 Reciprocal Grids 164
4.3.3 Aliasing 166
4.3.4 Field of View 168
4.3.5 Hexagonal Grids: Y-shape Instrument 168
4.3.6 Hexagonal Instrument 171
4.4 Imaging 175
4.4.1 System of Equations 175
4.4.2 Conjugate Extension and Redundant Baselines Averaging 178
4.4.3 Fourier Image Reconstruction 179
4.4.4 G-matrix Image Reconstruction 181
4.4.5 Polarimetric Retrieval: Ideal Case 184
4.4.6 Full Polarimetric Case 184
4.4.7 Spatial Frequency Components 186
4.4.8 Reconstruction Error and Alias Mitigation 189
References 191
5 Instrument Techniques 193
5.1 Frequency Conversion 193
5.1.1 Frequency Bands 193
5.1.2 Mixer Operation 195
5.1.3 Image Rejection Mixer 196
5.2 In-phase and Quadrature (IQ) Mixer 197
5.2.1 Concept 197
5.2.2 General Analysis 198
5.2.3 Quadrature Error 200
5.2.4 Correction of Phase Errors 201
5.2.5 Normalized Correlations 202
5.3 Quarter Period Delay 203
5.3.1 Concept 203
5.3.2 Center Frequency Error 205
5.3.3 Normalized Correlations 206
5.4 Digital Techniques 207
5.4.1 Sampling 207
5.4.2 Impact on Measurement Uncertainty 215
5.4.3 Low-frequency Spectrum 217
5.4.4 Spectrum with High-frequency Content 221
5.4.5 I/Q Alternate Sampling 225
5.4.6 Nyquist Zones 227
5.4.7 Correlation in the Frequency Domain 231
References 235
6 Calibration and Characterization 237
6.1 Calibration Standards 237
6.1.1 Antenna and Calibration Planes 238
6.1.2 Plane Change in Total Power Radiometers 240
6.1.3 External Passive Targets 243
6.1.4 Probe Antenna 245
6.1.5 Internal Load 247
6.1.6 Noise Distribution 248
6.2 Parameter Retrieval 250
6.2.1 Correlator Gain 251
6.2.2 Inter-element Phase and Amplitude 253
6.2.3 Correlator Offset 255
6.2.4 Flat Target Response 257
6.2.5 Fringe Washing Function Shape 259
6.2.6 Receiver Gain and Offset 261
6.2.7 Instrumental Offset 263
6.3 Nonlinearity 265
6.3.1 Deflection Ratio 266
6.3.2 Impact on Instrumental Offset 267
6.4 Calibration Rate 268
6.4.1 Averaging and Interpolation 268
6.4.2 Temperature Correction 269
References 270
A Definitions and Concepts 273
A. 1 Complex Vectors 273
A. 2 Useful Complex Number Identities 274
A. 3 Energy Conservation and Unitary Matrix 274
A. 4 Spherical Coordinates and Solid Angle 275
A.4. 1 Differential Surface 275
A.4. 2 Solid Angle 275
A. 5 Quadrature Equation Inversion 276
A. 6 Special Functions 277
A. 7 Fourier Transform 278
A.7. 1 Convolution 278
A.7. 2 Properties 279
A.7. 3 Transform Pairs 279
A.7. 4 Real Signals 279
A.7. 5 Two-dimensional Fourier Transform 280
A. 8 Discrete Fourier Transform 281
A.8. 1 Correlation in Time and in Frequency 283
A.8. 2 Random Signals 283
A.8. 3 Two-dimensional Case 284
Reference 284
Index 285
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