SIGNALS, NOISE, AND ACTIVE SENSORS Radar, Sonar, Laser Radar

JOHN MINKOFF Distinguished Member of the Technical Staff AT&T Bell Laboratories Whippany, New Jersey

A Wiley-Interscience Publication

JOHN WILEY & SONS, INC. New York • Chichester • Brisbane . Toronto • Singapore

CONTENTS

Preface

1. Introduction—Fundamentals of Receivers

2. Review of Probability 2.1 Bernoulli Trials—The Binomial Distribution, 13 2.2 The Poisson Distribution, 15 2.3 The Exponential Distribution, 16 2.4 The Gaussian Distribution, 16 2.5 The Rayleigh and Rice Distributions, 16 2.6 Joint Distributions, Conditional Distributions, and

Bayes Theorem, 17 2.7 Characteristic Functions, 18 2.8 The Law of Large Numbers, 20 2.9 The Central Limit Theorem, 21 2.10 Approximations to the Gaussian Distribution, 23 2.11 Functions of a Random Variable, 25

3. Review of Noise s d̂ Random Processes

3.1 Introduction—Correlation Functions and Power Spectral Densities, 28

3.2 Types of Noise, 31 3.3 Power Spectral Density of Thermal Noise—

Nyquist's Theorem, 33 3.4 Power Spectral Density of Shot Noise, 35 3.5 Shot Noise and Optical Receivers—

The Quantum Limit, 39

vi CONTENTS

3.6 Noise Statistics—Shot Noise in Radar and Laser Radar, 42

3.7 Noise Figure and Noise Temperature, 48 3.8 Noise Figure of an Attenuator, 50 3.9 Applications: Noise Power Measurements, 51 3.10 Connections with Statistical Physics, 55

4. Continuous and Discrete-Time Signals

4.1 The Sampling Theorem and Oversampling, 61 4.1.1 Application of the Sampling Theorem to

Delay of Discrete-Time Signals, 69 4.2 The Sampling Theorem for Bandpass Carrier Signals, 70 4.3 Signal Duration and Bandwidth, 72 4.4 The Analytic Signal, 74 4.5 Processing of Continuous and Discrete-Time Signals, 77

5. Detection of Signals in Noise

5.1 Statistical Decision Theory: The Likelihood Ratio Test, 87

5.2 Decision Criteria—Bayes, Maximum Likelihood, Neyman-Pearson, 90

5.3 Implementation of Decision Criteria, 93 5.3.1 Gaussian Noise, 93 5.3.2 Shot Noise—Poisson Distribution, 96

5.4 Correlation Detection: The Matched Filter—I, 99 5.4.1 The Gaussian Channel, 99 5.4.2 The Shot Noise Channel, 106

5.5 The Matched Filter—II, 107

6. Coherent and Noncoherent Detection and Processing,

6.1 Ideal Noncoherent Detection of a Single Pulse, 118 6.2 Comparison of Coherent and Noncoherent Detection,

of a Single Pulse, 124 6.3 Improvement in Signal-to-Noise Ratio by Coherent and

Noncoherent Integration, 129 6.3.1 Noncoherent Integration, 129 6.3.2 Coherent Integration, 135

6.4 Performance of Coherent and Noncoherent Integration, 6.4.1 Noncoherent Integration, 140 6.4.2 Coherent Integration, 142

6.5 Summary of Coherent and Noncoherent Detection and Processing, 144

CONTENTS

7. Parameter Estimation and Applications 148

7.1 Estimation of Range to a Target, 149 7.2 Generalized Parameter Estimation, 158

7.2.1 The Cramer-Rao Lower Bound on the Variance of an Estimator, 159

7.2.2 Maximum Likelihood Estimation, 161 7.3 Applications of Maximum Likelihood Estimation to

Sensor Measurements, 162 7.3.1. Calculation of the Cramer-Rao Bound for

Coherent and Noncoherent Observations, 164 7.4 Application of Parameter Estimation to Tracking

and Prediction, 169

8. Waveform Analysis, Range-Doppler Resolution and Ambiguity 178

8.1 Waveform Analysis, 179 8.2 Range-Doppler Resolution and Ambiguity—The

Generalized Ambiguity Function, 183

9. Large Time-Bandwidth Waveforms 191

9.1 Chirp Waveforms and Pulse Compression, 193 9.2 Doppler Invariant Properties of Chirp Waveforms, 198 9.3 Hyperbolic Frequency Modulation, 202 9.4 Ambiguity Function for Large BT Waveforms, 205 9.5 Coded Waveforms, 207

10. Generalized Coherent and Noncoherent Detection and Processing 212

10.1 Noncoherent Detection of a Single Pulse, 213 10.2 Coherent and Noncoherent Integration, 215

11. Systems Considerations 218

11.1 Beampatterns and Gain of Antennas and Arrays, 218 11.2 The Radar and Sonar Equations, 224 11.3 The Search Problem, 228 11.4 Specification of the False-Alarm Probability, P f a , 229

Appendix Table of Values the Error Function 233

References 239

Index 241