Stochastic Processes and Filtering Theory (Volume 64) (Mathematics in Science and Engineering, Volume 64)
This book presents a unified treatment of linear and nonlinear filtering theory for engineers, with sufficient emphasis on applications to enable the reader to use the theory. The need for this book is twofold. First, although linear estimation theory is relatively well known, it is largely scattered in the journal literature and has not been collected in a single source. Second, available literature on the continuous nonlinear theory is quite esoteric and controversial, and thus inaccessible to engineers uninitiated in measure theory and stochastic differential equations. Furthermore, it is not clear from the available literature whether the nonlinear theory can be applied to practical engineering problems. In attempting to fill the stated needs, the author has retained as much mathematical rigor as he felt was consistent with the prime objective-to explain the theory to engineers. Thus, the author has avoided measure theory in this book by using mean square convergence, on the premise that everyone knows how to average. As a result, the author only requires of the reader background in advanced calculus, theory of ordinary differential equations, and matrix analysis.
Why Read This Book
You should read this book if you want a rigorous, unified foundation for both linear and nonlinear filtering that connects stochastic-process theory to practical engineering problems. You will learn how core ideas—innovations, Wiener/Kalman filtering, stochastic differential equations, and nonlinear continuous-time filters—fit together so you can apply them in communications, radar, audio, and other signal-processing systems.
Who Will Benefit
Practicing engineers, graduate students, and researchers with a solid mathematical background who need a deep, unified treatment of stochastic processes and filtering for applications in communications, radar, DSP, and control.
Level: Advanced — Prerequisites: Undergraduate probability and statistics, linear algebra, ordinary differential equations, and basic measure-theoretic or at least graduate-level probability (familiarity with stochastic processes and calculus will help).
Key Takeaways
- Formulate stochastic-process models for signals and noise and use spectral representations for stationary processes.
- Derive and implement optimal linear estimators (Wiener, discrete Kalman, Kalman–Bucy) in both discrete and continuous time.
- Develop and interpret nonlinear continuous-time filtering frameworks, including the Kushner–Stratonovich and Zakai formulations.
- Analyze filter performance and stability through innovations, Riccati equations, and martingale methods.
- Apply filtering theory to practical signal-processing problems in radar, communications, and audio/speech contexts.
- Translate theoretical filters into numerical approximations and understand their limitations and convergence properties.
Topics Covered
- Introduction and motivation
- Probability and stochastic-process preliminaries
- Stationary processes and spectral representation
- Linear estimation and Wiener filtering
- Innovations approach and discrete-time filtering
- Continuous-time stochastic processes and the Wiener process
- Stochastic differential equations and martingales
- Continuous-time linear filtering and the Kalman–Bucy filter
- Nonlinear filtering: Kushner–Stratonovich and Zakai equations
- Riccati equations, stability, and performance analysis
- Applications to communications, radar, and signal processing
- Numerical methods, approximations, and implementation issues
Languages, Platforms & Tools
How It Compares
Compared with Kay's 'Fundamentals of Statistical Signal Processing' or Kailath/Sayed-style treatments, Jazwinski is older and more mathematically thorough on continuous-time and nonlinear filtering (stochastic calculus), whereas newer texts may provide more computational examples and modern implementation details.
