Applied Methods of the Theory of Random Functions

Publisher's NoteChapter I. General Properties of Random Functions § 1. The Theory of Random Functions as a Branch of the Theory of Probability § 2. The Basic Notations and Formulae of the Theory of Probability § 3. Random Functions and Methods of Describing Them § 4. Typical Problems, Solved by Means of the Theory of Random Functions § 5. Properties of the Correlation Function § 6. The Differentiation and Integration of Random Functions § 7. The Action of a Linear Operator on a Random Function § 8. A System of Random Functions. The Cross-Correlation Function § 9. Problems on Overshoots: The Mean Number of Overshoots of a Random Function Above a Given Level, the Mean Duration of an OvershootChapter II. The Spectral Theory of Stationary Random Functions § 10. The Spectral Representation of Stationary Random Functions § 11. Examples of the Calculation of the Spectral Density of a Stationary Random Process § 12. The Spectral Density of a Linear Combination of a Stationary Random Function and its Derivatives. The Stationary Solution of a Linear Differential Equation With Constant Coefficients § 13. Examples of Spectral Densities and Correlation Functions in More Complicated Cases § 14. Determination of the Correlation Function of the Solution of a Non-Homogeneous Differential Equation with Constant Coefficients When the Right-Hand Side is Non-Stationary § 15. Linear Differential Equations with Variable Coefficients § 16. The Probability Characteristics of the Solutions of a System of Linear Equations § 17. The Density Distribution of the Solution of a Linear Differential EquationChapter III. Determination of Optimal Dynamical Systems § 18. Statement of the Problem of the Determination of Optimal Systems § 19. The General Solution of the Problem of Determining (Given the Infinite Past of the Process) the Optimal Dynamical System Representing the Operations of Smoothing (Filtering), Extrapolation and Differentiation § 20. Formulae For the Determination of the Optimal Transfer Function of a Dynamical System Performing Filtration, Extrapolation and Differentiation in the Case of Rational Spectral Densities of Signal and Noise § 21. Practical Formulae for the Optimal Transfer Function of a Dynamical System with Delay § 22. Optimal Smoothing, Extrapolation and Differentiation for a Finite Observation Time § 23. Examples of the Determination of Optimal Dynamical Systems for a Finite Observation TimeChapter IV. Experimental Methods for the Determination of Characteristics of Random Functions § 24. The General Principles of the Determination of Characteristics of Random Functions From Experimental Data § 25. The Principles of the Construction of Correlators — Devices for the Determination of the Correlation Functions of Stationary Random Processes § 26. Determination of the Estimated Value of the Spectral DensityChapter V. The Method of Envelopes § 27. The Method of Envelopes and the Derivation of General Formulae § 28. Application of the Method of Envelopes in the Case of a Narrow-Band SpectrumChapter VI. Some Supplementary Problems of the Theory of Random Functions § 29. Random Sequences § 30. Random Functions of Many Variables § 31. Canonical Expansions of Random FunctionsReferencesIndexOther Titles in this Series