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1 AN INTRODUCTION TO PROBABILITY THEORY 1

1.1 The Rules of Probability 1

1.2 BERNOULLI'S Theorem 2

1.3 Moments and Generating Functions 4

1.4 Convolution 7

1.5 Events at Random in Time 10

1.6 Probability Density and the Convolution Integral 12

1.7 The Delta-function 14

1.8 Characteristic Functions and the Normal Distribution 16

1.9 The Rayleigh Distribution 20

1.10 Entropy as a Measure of Spread 21

2 WAVEFORM ANALYSIS AND NOISE 26

2.1 The Complex Spectrum of Positive and Negative Frequencies 26

2.2 The Rectangular Function and its Spectrum 29

2.3 PARSEVAL's Theorem 31

2.4 Sampling Analysis 31

2.5 Sampling of High-frequency Waveforms 34

2.6 PoIssoN's Formula 35

2.7 Vector Representation of Waveforms 36

2.8 Uniform Gaussian Noise 37

2.9 Complex Representation of Real Waveforms 40

3 INFORMATION THEORY 43

3.1 HARTLEY's Measure of Information Capacity 43

3.2 SHANNoN's Measure of Information Content 45

3.3 Information Gain 49

3.4 The Symmetrical Formulation of Information Transfer 52

3.5 Average Expressions 54

3.6 The Capacity of a Noisy Communication Channel 56

3.7 Information-destroying Processes 58

3.8 Guesswork 60

4 THE STATISTICAL PROBLEM OF RECEPTION 62

4.1 The Ideal Receiver 62

4.2 Inverse Probability 63

4.3 Reception of a Steady Voltage in Gaussian Noise 65

4.4 Sufficiency and Reversibility 67

4.5 Correlation Reception 68

4.6 Signals with Unknown Parameters 72

4.7 Observation Systems and the a priori Difficulty 73

4.8 Reception of High-frequency Signals 76

4.9 The Complex Formulation 79

5 SIMPLE THEORY OF RADAR RECEPTION 81

5.1 The Measurement of Delay 81

5.2 Threshold Effects 85

5.3 Continuous Observation and Filtering 91

5.4 Signals of Doubtful Strength or Existence 94

5.5 Detection 96

5.6 Conclusion 98

6 THE MATHEMATICAL ANALYSIS OF RADAR INF OR-MATION 100

6.1 Introduction 100

6.2 Complex Signal and Noise Functions 102

6.3 Range Accuracy 104

6.4 Noise Ambiguity 105

6.5 Information Gain 109

6.6 Discussion of the Threshold Effect 112

7 THE TRANSMITTED RADAR SIGNAL 115

7.1 Accuracy Resolution and Signal Ambiguity 115

7.2 Ambiguity in Range and Velocity 118

7.3 The Gaussian Pulse-train 121

7.4 Frequency Modulation 123

7.5 Conclusion 125

REFERENCES 125

INDEX 127