Discrete Fourier and Wavelet Transforms: An Introduction through Linear Algebra with Applications to Signal Processing

Undergraduate students in mathematics are typically introduced to the Fourier series as a tool in the solution of partial differential equation boundary value problems. They might also be introduced to the use of the Fourier transform in a more advanced course on PDE. The discrete Fourier transform is seldom introduced in the undergraduate curriculum in mathematics, although it is commonly taught to undergraduate students in engineering for its applications in digital signal processing. The wavelet transform, in its discrete and continuous forms, is typically not taught to undergraduate students but is a popular topic for graduate students in mathematics and electrical engineering.

A rigorous presentation of the continuous wavelet transform requires significant background in analysis that most undergraduate mathematics students — and nearly all undergraduate engineering students — lack. In comparison, the mathematical prerequisite for the discrete wavelet transform is simply linear algebra. In their textbook on the discrete wavelet transform, Arne Jensen and Anders la Cour-Harbo adopted this approach with great success in their textbook Ripples in Mathematics: The Discrete Wavelet Transform (Springer, 2001). Goodman’s book follows the same general approach.

The book begins with a chapter reviewing linear algebra. Most of the material in this chapter should be familiar to students who have taken an introductory course in linear algebra. However, the book makes extensive use of complex vectors, block matrices, and unitary matrices, so these receive special attention. Fourier series are introduced in their complex exponential form. This serves as an example of orthogonal projection in a complex vector space.

Goodman develops the discrete Fourier transform in the second chapter. The DFT is presented in matrix form as a change of basis operation. This chapter includes discussion of circulant matrices, circular convolution, Parseval’s theorem, and a brief discussion of the fast Fourier transform.

There are two common approaches to deriving discrete wavelet transforms. In chapter four, the author uses the lifting scheme to derive wavelet transforms in one and two dimensions. Haar wavelets and the fourth order Debauchies wavelets are used in examples. MATLAB based computer explorations help to make the transformation extremely concrete to the reader. In chapter five, the author derives wavelet transforms using the filter bank approach. The book concludes with a chapter introducing continuous wavelet transforms. In comparison with the earlier chapters this chapter is not as in-depth and feels somewhat incomplete.

This book is suitable as a textbook for an introductory undergraduate mathematics course on discrete Fourier and wavelet transforms for students with background in calculus and linear algebra. The particular strength of this book is its accessibility to students with no background in analysis. The exercises and computer explorations provide the reader with many opportunities for active learning. Studying from this text will also help students strengthen their background in linear algebra. However, it is not as strong in its coverage of applications to signal processing and would not be suitable for an engineering course focused on applications in signal processing.

Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.