Continued Fractions and Signal Processing

Continued fractions and signal processing seem like very distant cousins in the world of mathematics. What do they really have to do with one another? The author tells about being asked by his thesis supervisor to devise an exercise demonstrating a connection between continued fractions and numerical analysis. In the course of doing this – and over several following years – the idea of this book took form. It offers a story relating continued fractions to signal processing and several other topics. It also suggests intriguing historical connections.

 

The author’s goal is to go back to the classical theory of continued fractions and identify a contemporary context for it in numerical analysis and signal processing. He begins with an elementary introduction to the classical theory of continued fractions of real numbers. This includes convergence, continuants, and approximation. 

 

The development then moves on to rational functions as continued fractions of polynomials. The key here is the applicability of the Euclidean division algorithm to rings of polynomials over a field. This enables the constructions of partial fractions that represent rational functions. Padé approximants - rational approximations of formal power series – are also discussed in this context. From here on in the book, continued fractions generally refer to continued fractions of rational functions.

 

Digital signal processing is treated in the second-to-last chapter, which includes a broader discussion of signals and filters. (In signal processing language, a filter is essentially an algorithm for selecting or removing components of a signal.) One application of continued fractions here is to rational filters (those whose impulse response function can be written as a rational function) and their stability. 

 

One notable feature is the author’s treatment of orthogonal polynomials and continued fractions. He notes that Gaussian quadrature (often used for numerical integration) is believed be have been developed by Gauss using orthogonal polynomials. Here the author argues that Gauss’ development really used polynomial interpolation and a continued-fraction expansion of a Laurent series.

 

According to the author, the book is essentially self-contained and requires only basic analysis and linear algebra. While this is perhaps strictly correct, even a well-prepared reader will need motivation and persistence to put all the pieces together. The author provides many references throughout the text, and readers might need to use those to fill in context and follow the development. While ”continued fractions” appears in the titles of nearly all the chapters, it is not always clear what the connection between continued fractions and the topics in numerical analysis that are the real subjects. 

 

One of the best uses of this book might be as a source for special projects for more advanced undergraduates. Of the individual topics discussed here, several would be of interest for further exploration and analysis.

---

Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent much of his career as a mathematician working in industry on a variety of applications ranging from network modeling and speech recognition to material science and optical films. He did his PhD work in dynamical systems and celestial mechanics.