A rigorous and concise introduction to Kalman filtering is presented in this well-written book. It is suitable for graduate studies, as well as refresher courses and self-study. It is not truly self-contained, but an introductory chapter helps make it less dependent on external references.

The book begins with a chapter that presents essential linear algebra topics as related to Kalman filtering, followed by relevant probability and estimation basics. The remaining ten chapters deal with Kalman filtering and some of its variants.

One of the strong features of the book is that Kalman filtering is presented from a few different viewpoints. An elementary introduction, based on a few idealizations and the least squares approach, is given in chapter 2. Chapter 3 removes the idealized assumptions and presents Kalman filtering more rigorously, within an orthogonal projection framework. Chapter 4 introduces Kalman derivations for the case of correlated noise processes, using as an example aircraft inertial navigation. Chapter 5 is devoted to colored noise and the study and derivation of Kalman filtering within that framework. A practical approach to save computations when applying Kalman filtering to time-invariant stochastic systems--replacing gain matrix continuous computations with the so-called constant gain matrix--is studied in chapter 6. Sequential and square root algorithms that permit fairly substantial computational savings are studied in chapter 7. Application to nonlinear systems via linearization procedures is the topic of chapter 8. Chapter 9 presents an algorithm to decouple filtering equations in the case of very large state vectors, with only a few components of interest. These discussions relate the procedures involved with the material from chapter 6, for a high-efficiency computational structure. Chapter 10 discusses Kalman filtering through interval arithmetic, a topic that is rarely addressed in this way in other signal processing courses. It is of real practical importance, since it assumes system elements are not well known at the time of computation; thus, some robustness must be built into Kalman algorithms. The last chapter is devoted to Kalman filtering in subbands, particularly in a wavelet type of multi-resolution framework.

Some interesting applications that couldn’t be covered due to size restrictions are briefly presented in an appendix. These are the Kalman smoother, the alpha tracker, adaptive Kalman filtering, Kalman-Bucy, and a brief mention of systolic array computation for the square root algorithm.

The many end-of-chapters exercises--and the section at the end of the book with solutions and hints to several of them--are another strong feature of the book.