Introduction to Random Signals and Noise

Introduction to Random Signals and Noise......Page 3
Contents......Page 9
Preface......Page 13
1.2 Modelling......Page 17
1.3 The Concept of a Stochastic Process......Page 18
1.3.1 Continuous Stochastic Processes......Page 20
1.3.2 Discrete-Time Processes (Continuous Random Sequences)......Page 21
1.3.3 Discrete Stochastic Processes......Page 22
1.3.4 Discrete Random Sequences......Page 23
1.4 Summary......Page 24
2.1.1 Cumulative Distribution Function and Probability Density Function......Page 25
2.1.2 First-Order Stationary Processes......Page 26
2.2.1 The Autocorrelation Function, Wide-Sense Stationary Processes and Ergodic Processes......Page 27
2.2.2 Cyclo-Stationary Processes......Page 32
2.2.3 The Cross-Correlation Function......Page 35
2.2.4 Measuring Correlation Functions......Page 40
2.2.5 Covariance Functions......Page 42
2.3 Gaussian Processes......Page 43
2.4 Complex Processes......Page 46
2.5.1 Mean, Correlation Functions and Covariance Functions......Page 47
2.6 Summary......Page 49
2.7 Problems......Page 50
3.1 The Power Spectrum......Page 55
3.2 The Bandwidth of a Stochastic Process......Page 59
3.3 The Cross-Power Spectrum......Page 61
3.4 Modulation of Stochastic Processes......Page 63
3.4.1 Modulation by a Random Carrier......Page 65
3.5 Sampling and Analogue-To-Digital Conversion......Page 66
3.5.1 Sampling Theorems......Page 67
3.5.2 A/D Conversion......Page 70
3.6 Spectrum of Discrete-Time Processes......Page 73
3.7 Summary......Page 74
3.8 Problems......Page 75
4.1 Basics of Linear Time-Invariant Filtering......Page 81
4.2.1 The Mean Value of the Filter Output......Page 84
4.2.2 The Autocorrelations Function of the Output......Page 85
4.2.3 Cross-Correlation of the Input and Output......Page 86
4.3 Spectra of the Filter Output......Page 87
4.4.1 Band-Limited Processes and Systems......Page 90
4.4.2 Equivalent Noise Bandwidth......Page 91
4.5 Spectrum of a Random Data Signal......Page 93
4.6.1 The Discrete Fourier Transform......Page 98
4.6.2 The z-Transform......Page 102
4.7.1 Time Domain Description of the Filtering......Page 106
4.7.2 Frequency Domain Description of the Filtering......Page 107
4.8 Summary......Page 109
4.9 Problems......Page 110
5.1 Description of Deterministic Bandpass Signals......Page 117
5.2 Quadrature Components of Bandpass Processes......Page 122
5.3 Probability Density Functions of the Envelope and Phase of Bandpass Noise......Page 127
5.4.1 The Spectrum Analyser......Page 131
5.4.2 Measurement of the Quadrature Components......Page 134
5.5.2 Direct Sampling......Page 135
5.7 Problems......Page 137
6.1 White and Coloured Noise......Page 145
6.2 Thermal Noise in Resistors......Page 146
6.3 Thermal Noise in Passive Networks......Page 147
6.4 System Noise......Page 153
6.4.1 Noise in Amplifiers......Page 154
6.4.2 The Noise Figure......Page 156
6.4.3 Noise in Cascaded systems......Page 158
6.6 Problems......Page 162
7 Detection and Optimal Filtering......Page 169
7.1.1 Binary Signals in Noise......Page 170
7.1.2 Detection of Binary Signals in White Gaussian Noise......Page 174
7.1.3 Detection of M-ary Signals in White Gaussian Noise......Page 177
7.2 Filters that Maximize the Signal-to-Noise Ratio......Page 181
7.3 The Correlation Receiver......Page 187
7.4.1 The Wiener Filter Problem......Page 191
7.4.2 Smoothing......Page 192
7.4.3 Prediction......Page 195
7.4.4 Discrete-Time Wiener Filtering......Page 199
7.6 Problems......Page 201
8.1 Introduction......Page 209
8.2.1 The Characteristic Function......Page 210
8.2.2 Cumulants......Page 212
8.2.3 Interarrival Time and Waiting Time......Page 213
8.3 The Homogeneous Poisson Process......Page 214
8.3.1 Filtering of Homogeneous Poisson Processes and Shot Noise......Page 215
8.4 Inhomogeneous Poisson Processes......Page 220
8.5 The Random-Pulse Process......Page 221
8.6 Summary......Page 223
8.7 Problems......Page 224
References......Page 227
Further Reading......Page 229
A.1 Linear Vector Spaces......Page 231
A.2 The Signal Space Concept......Page 232
A.3 Gram–Schmidt Orthogonalization......Page 234
A.4 The Representation of Noise in Signal Space......Page 235
A.4.1 Relevant and Irrelevant Noise......Page 237
A.5.1 Binary Antipodal Signals......Page 238
A.5.2 Binary Orthogonal Signals......Page 239
A.5.4 Multiamplitude Signals......Page 240
A.5.7 Biorthogronal Signals......Page 241
A.5.8 Simplex Signals......Page 242
A.6 Problems......Page 243
B. Attenuation, Phase Shift and Decibels......Page 245
C.1 Trigonometric Relations......Page 247
C.2.1 Chain Rule......Page 248
C.3.1 Basic Integrals......Page 249
C.3.3 Rational Algebraic Functions......Page 250
C.3.4 Trigonometric Functions......Page 251
C.4 Definite Integrals......Page 252
C.5 Series......Page 253
C.6 Logarithms......Page 254
D. Summary of Probability Theory......Page 255
E. Definition of a Few Special Functions......Page 257
F. The Q(.) and erfc Function......Page 259
G. Fourier Transforms......Page 261
H. Mathematical and Physical Constants......Page 263
Index......Page 265