Probabilistic Methods of Signal and System Analysis

Preface 1. Introduction to Probability 1.1: Engineering Applications of Probability 1.2: Random Experiments and Events 1.3: Definitions of Probability 1.4: The Relative-Frequency Approach 1.5: Elementary Set Theory 1.6: The Axiomatic Approach 1.7: Conditional Probability 1.8: Independence 1.9: Combined Experiments 1.10: Bernoulli Trials 1.11: Applications of Bernoulli Trials 2. Random Variables 2.1: Concept of a Random Variable 2.2: Distribution Functions 2.3: Density Functions 2.4: Mean Values and Moments 2.5: The Gaussian Random Variable 2.6: Density Functions Related to Gaussian 2.7: Other Probability Density Functions 2.8: Conditional Probability Distribution and Density Functions 2.9: Examples and Applications 3. Several Random Variables 3.1: Two Random Variables 3.2: Conditional Probability--Revisited 3.3: Statistical Independence 3.4: Correlation between Random Variables 3.5: Density Function of the Sum of Two Random Variables 3.6: Probability Density Function of a Function of Two Random Variables 3.7: The Characteristic Function 4. Elements of Statistics 4.1: Introduction 4.2: Sampling Theory--The Sample Mean 4.3: Sampling Theory--The Sample Variance 4.4: Sampling Distributions and Confidence Intervals 4.5: Hypothesis Testing 4.6: Curve Fitting and Linear Regression 4.7: Correlation Between Two Sets of Data 5. Random Processes 5.1: Introduction 5.2: Continuous and Discrete Random Processes 5.3: Deterministic and Nondeterministic Random Processes 5.4: Stationary and Nonstationary Random Processes 5.5: Ergodic and Nonergodic Random Processes 5.6: Measurement of Process Parameters 5.7: Smoothing Data with a Moving Window Average 6. Correlation Functions 6.1: Introduction 6.2: Example: Autocorrelation Function of a Binary Process 6.3: Properties of Autocorrelation Functions 6.4: Measurement of Autocorrelation Functions 6.5: Examples of Autocorrelation Functions 6.6: Crosscorrelation Functions 6.7: Properties of Crosscorrelation Functions 6.8: Examples and Applications of Crosscorrelation Functions 6.9: Correlation Matrices For Sampled Functions 7. Spectral Density 7.1: Introduction 7.2: Relation of Spectral Density to the Fourier Transform 7.3: Properties of Spectral Density 7.4: Spectral Density and the Complex Frequency Plane 7.5: Mean-Square Values From Spectral Density 7.6: Relation of Spectral Density to the Autocorrelation Function 7.7: White Noise 7.8: Cross-Spectral Density 7.9: Autocorrelation Function Estimate of Spectral Density 7.10: Periodogram Estimate of Spectral Density 7.11: Examples and Applications of Spectral Density 8. Response of Linear Systems to Random Inputs 8.1: Introduction 8.2: Analysis in the Time Domain 8.3: Mean and Mean-Square Value of System Output 8,4: Autocorrelation Function of System Output 8.5: Crosscorrelation between Input and Output 8.6: Example of Time-Domain System Analysis 8.7: Analysis in the Frequency Domain 8.8: Spectral Density at the System Output 8.9: Cross-Spectral Densities between Input and Output 8.10: Examples of Frequency-Domain Analysis 8.11: Numerical Computation of System Output 9. Optimum Linear Systems 9.1: Introduction 9.2: Criteria of Optimality 9.3: Restrictions on the Optimum System 9.4: Optimization by Parameter Adjustment 9.5: Systems That Maximize Signal-to-Noise Ratio 9.6: Systems That Minimize Mean-Square Error Appendices A.: Mathematical Tables A.1: Trigonometric Identities A.2: Indefinite Integrals A.3: Definite Integrals A.4: Fourier Transform Operations A.5: Fourier Transforms A.6: One-Sided Laplace Transforms B.: Frequently Encountered Probability Distributions B.1: Discrete Probability Functions B.2: Continuous Distributions C.: Binomial Coefficients D.: Normal Probability Distribution Function E.: The Q-Function F.: Student's t Distribution Function G.: Computer Computations H.: Table of Correlation Function--Spectral Density Pairs I.: Contour Integration Index