This introduction to random variables and signals is intended to provide engineering students with the analytical and computational tools for processing random signals using linear systems. It presents the underlying theory as well as examples and applications using computational aids throughout, in particular, computer-based symbolic computation programs are used for performing the analytical manipulations and the numerical calculations. Intended for a one-semester course for advanced undergraduates or beginning graduate students, the book covers such topics as: set theory and an introduction…mehr
This introduction to random variables and signals is intended to provide engineering students with the analytical and computational tools for processing random signals using linear systems. It presents the underlying theory as well as examples and applications using computational aids throughout, in particular, computer-based symbolic computation programs are used for performing the analytical manipulations and the numerical calculations. Intended for a one-semester course for advanced undergraduates or beginning graduate students, the book covers such topics as: set theory and an introduction to probability; random variables, distributions, and processes; deterministic signals, spectral properties, and transformations; and filtering, and detection theory. The large number of worked examples together with the programming aids provided on the CD make the book eminently suited for self study as well as classroom use.
1 Introduction to Sets and Probability.- 1.1 Introduction to Nondeterministic Signals.- 1.2 Introduction to Sets.- 1.3 Operations on Sets.- 1.4 Combined Operations on Sets.- 1.5 Notion of Probability.- 1.6 Relative Frequency and Probability.- 1.7 Conditional Probability.- 1.8 Total Probability.- 1.9 Independence.- 1.10 Summary.- Problems for Chapter 1.- 2 One-Dimensional Random Variables.- 2.1 Concept of Distributions.- 2.2 Random Variables.- 2.3 Distribution Functions.- 2.4 Density Functions.- 2.5 Continuous Density Functions.- 2.6 Conditional Distribution and Density Functions.- 2.7 Generation of Random Numbers.- 2.8 Summary.- Problems for Chapter 2.- 3 Operations on Random Numbers.- 3.1 Concept of Expectation.- 3.2 Moments and Functions.- 3.3 Moment Generating Functions.- 3.4 Transformation of Random Variables.- 3.5 Random Variables with Prescribed Distributions.- 3.6 Summary.- Problems for Chapter 3.- 4 Two-Dimensional Random Variables.- 4.1 Joint Distribution and Density Functions.- 4.2 Conditional Density Functions.- 4.3 Expectation and Joint Moments.- 4.4 Transformations and Joint Characteristic Functions.- 4.5 Independence.- 4.6 Sum of Independent Random Variables.- 4.7 Generation of Correlated Gaussian Random Sequences.- 4.8 Summary.- Problems for Chapter 4.- 5 Introduction to Random Processes.- 5.1 Methods of Generation of Random Processes.- 5.2 IID Random Variables.- 5.3 Distribution Functions for a Random Process.- 5.4 Properties of Expectation Operators.- 5.5 Properties of the Correlation Functions.- 5.6 Numerical Computation of the Correlation Function.- 5.7 Summary.- Problems for Chapter 5.- 6 Introduction to Transformations.- 6.1 Function Transformation of Random Processes.- 6.2 Transformation by Integration.- 6.3 Transformation by Differentiation.- 6.4Linear Systems.- 6.5 Power Spectrum Functions.- 6.6 Transforms of Linear Systems.- 6.7 Calculation of Power Density Spectrum.- 6.8 Summary.- Problems for Chapter 6.- 7 Introduction to Applications.- 7.1 Matched Filtering.- 7.2 Mean Square Filtering.- 7.3 Detection Theory.- 7.4 Radar Systems.- 7.5 Noise in Control Systems.- Problems for Chapter 7.- Appendix A.- A.l Signals and Spectra.- A.2 Singularity Functions.- A.3 Linear-Time-Invariant Systems.- A.4 Correlation Functions.- Appendix B.- B.1 Matlab and Mathcad.- B.2 Matlab.- B.3 Mathcad.- B.4 Contents of the CD-ROM.- References.
1 Introduction to Sets and Probability.- 1.1 Introduction to Nondeterministic Signals.- 1.2 Introduction to Sets.- 1.3 Operations on Sets.- 1.4 Combined Operations on Sets.- 1.5 Notion of Probability.- 1.6 Relative Frequency and Probability.- 1.7 Conditional Probability.- 1.8 Total Probability.- 1.9 Independence.- 1.10 Summary.- Problems for Chapter 1.- 2 One-Dimensional Random Variables.- 2.1 Concept of Distributions.- 2.2 Random Variables.- 2.3 Distribution Functions.- 2.4 Density Functions.- 2.5 Continuous Density Functions.- 2.6 Conditional Distribution and Density Functions.- 2.7 Generation of Random Numbers.- 2.8 Summary.- Problems for Chapter 2.- 3 Operations on Random Numbers.- 3.1 Concept of Expectation.- 3.2 Moments and Functions.- 3.3 Moment Generating Functions.- 3.4 Transformation of Random Variables.- 3.5 Random Variables with Prescribed Distributions.- 3.6 Summary.- Problems for Chapter 3.- 4 Two-Dimensional Random Variables.- 4.1 Joint Distribution and Density Functions.- 4.2 Conditional Density Functions.- 4.3 Expectation and Joint Moments.- 4.4 Transformations and Joint Characteristic Functions.- 4.5 Independence.- 4.6 Sum of Independent Random Variables.- 4.7 Generation of Correlated Gaussian Random Sequences.- 4.8 Summary.- Problems for Chapter 4.- 5 Introduction to Random Processes.- 5.1 Methods of Generation of Random Processes.- 5.2 IID Random Variables.- 5.3 Distribution Functions for a Random Process.- 5.4 Properties of Expectation Operators.- 5.5 Properties of the Correlation Functions.- 5.6 Numerical Computation of the Correlation Function.- 5.7 Summary.- Problems for Chapter 5.- 6 Introduction to Transformations.- 6.1 Function Transformation of Random Processes.- 6.2 Transformation by Integration.- 6.3 Transformation by Differentiation.- 6.4Linear Systems.- 6.5 Power Spectrum Functions.- 6.6 Transforms of Linear Systems.- 6.7 Calculation of Power Density Spectrum.- 6.8 Summary.- Problems for Chapter 6.- 7 Introduction to Applications.- 7.1 Matched Filtering.- 7.2 Mean Square Filtering.- 7.3 Detection Theory.- 7.4 Radar Systems.- 7.5 Noise in Control Systems.- Problems for Chapter 7.- Appendix A.- A.l Signals and Spectra.- A.2 Singularity Functions.- A.3 Linear-Time-Invariant Systems.- A.4 Correlation Functions.- Appendix B.- B.1 Matlab and Mathcad.- B.2 Matlab.- B.3 Mathcad.- B.4 Contents of the CD-ROM.- References.
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