Introduction to Random Signals and Noise

1 Introduction.

1.1 Random Signals and Noise.

1.2 Modelling.

1.3 The Concept of a Stochastic Process.

1.4 Summary.

2 Stochastic Processes.

2.1 Stationary Processes.

2.2 Correlation Functions.

2.3 Gaussian Processes.

2.4 Complex Processes.

2.5 Discrete-Time Processes.

2.6 Summary.

2.7 Problems.

3 Spectra of Stochastic Processes.

3.1 The Power Spectrum.

3.2 The Bandwidth of a Stochastic Process.

3.3 The Cross-Power Spectrum.

3.4 Modulation of Stochastic Processes.

3.5 Sampling and Analogue-To-Digital Conversion.

3.6 Spectrum of Discrete-Time Processes.

3.7 Summary.

3.8 Problems.

4. Linear Filtering of Stochastic Processes.

4.1 Basics of Linear Time-Invariant Filtering.

4.2 Time Domain Description of Filtering of Stochastic Processes.

4.3 Spectra of the Filter Output.

4.4 Noise Bandwidth.

4.5 Spectrum of a Random Data Signal.

4.6 Principles of Discrete-Time Signals and Systems.

4.7 Discrete-Time Filtering of Random Sequences.

4.8 Summary.

4.9 Problems.

5 Bandpass Processes.

5.1 Description of Deterministic Bandpass Signals.

5.2 Quadrature Components of Bandpass Processes.

5.3 Probability Density Functions of the Envelope and Phase of Bandpass Noise.

5.4 Measurement of Spectra.

5.5 Sampling of Bandpass Processes.

5.6 Summary.

5.7 Problems.

6 Noise in Networks and Systems.

6.1 White and Coloured Noise.

6.2 Thermal Noise in Resistors.

6.3 Thermal Noise in Passive Networks.

6.4 System Noise.

6.5 Summary.

6.6 Problems.

7 Detection and Optimal Filtering.

7.1 Signal Detection.

7.2 Filters that Maximize the Signal-to-Noise Ratio.

7.3 The Correlation Receiver.

7.4 Filters that Minimize the Mean-Squared Error.

7.5 Summary.

7.6 Problems.

8 Poisson Processes and Shot Noise.

8.1 Introduction.

8.2 The Poisson Distribution.

8.3 The Homogeneous Poisson Process.

8.4 Inhomogeneous Poisson Processes.

8.5 The Random-Pulse Process.

8.6 Summary.

8.7 Problems.

References.

Further Reading.

Appendices.

A. Representation of Signals in a Signal Space.

A.1 Linear Vector Spaces.

A.2 The Signal Space Concept.

A.3 Gram–Schmidt Orthogonalization.

A.4 The Representation of Noise in Signal Space.

A.5 Signal Constellations.

A.6 Problems.

B. Attenuation, Phase Shift and Decibels.

C. Mathematical Relations.

C.1 Trigonometric Relations.

C.2 Derivatives.

C.3 Indefinite Integrals.

C.4 Definite Integrals.

C.5 Series.

C.6 Logarithms.

D. Summary of Probability Theory.

E. Definition of a Few Special Functions.

F. The Q(.) and erfc Function.

G. Fourier Transforms.

H. Mathematical and Physical Constants.

Index.