Stochastic Processes and Filtering Theory (Dover Books on Electrical Engineering)
Taking the state-space approach to filtering, this text models dynamical systems by finite-dimensional Markov processes, outputs of stochastic difference, and differential equations. Starting with background material on probability theory and stochastic processes, the author introduces and defines the problems of filtering, prediction, and smoothing. He presents the mathematical solutions to nonlinear filtering problems, and he specializes the nonlinear theory to linear problems. The final chapters deal with applications, addressing the development of approximate nonlinear filters, and presenting a critical analysis of their performance.
Why Read This Book
You should read this book if you want a rigorous, unified foundation in filtering theory that covers both linear (Kalman) and nonlinear filters from first principles. It trains you to model dynamical systems with stochastic differential equations and to derive and understand the mathematical filtering, prediction, and smoothing equations used in modern DSP and estimation.
Who Will Benefit
Advanced graduate students, researchers, and experienced DSP/estimation engineers who need a deep theoretical grounding in stochastic processes and filtering for algorithm development or analysis.
Level: Advanced — Prerequisites: Advanced calculus, ordinary differential equations, matrix analysis, and basic probability and stochastic-process familiarity; prior exposure to linear systems and signals is highly recommended.
Key Takeaways
- Derive and understand the Kalman–Bucy and discrete-time Kalman filter from the state-space viewpoint.
- Formulate nonlinear filtering problems and derive the fundamental filtering equations (e.g., Kushner–Stratonovich and related formulations).
- Model dynamical systems using finite-dimensional Markov processes and stochastic differential equations (Itô calculus).
- Perform optimal prediction and smoothing for stochastic systems and relate these to practical estimation tasks.
- Analyze existence, uniqueness, and properties of filtering solutions and understand conditions for finite-dimensional filters.
- Apply theory to a range of engineering examples that connect abstract results to practical estimation problems.
Topics Covered
- Preface and notation; overview of filtering problems
- Probability theory and stochastic processes (review)
- Markov processes and basic properties
- Wiener processes and Brownian motion
- Itô calculus and stochastic differential equations
- State-space models for stochastic systems
- Linear filtering: Kalman–Bucy and discrete Kalman filter
- Nonlinear filtering: formulation and fundamental equations
- Finite-dimensional approximations and special finite filters
- Prediction, smoothing, and estimation theory
- Stability, convergence, and uniqueness results
- Selected applications and examples
How It Compares
More mathematical and theory-focused than Dan Simon's 'Optimal State Estimation' (which is more application- and algorithm-oriented); covers nonlinear filtering in greater depth than Anderson & Moore's 'Optimal Filtering', which emphasizes linear theory.
