PREFACE AND OVERVIEW
Fourier series and the Fourier transform have been around since the 1800s, and many research articles and books (at both the graduate and undergraduate levels) have been written about these topics. By contrast, the development of wavelets has been much more recent. While its origins go back many decades, the subject of wavelets has become a popular tool in signal analysis and other areas of applications only within the last decade or two, partly as a result of Daubechies’ celebrated work on the construction of compactly supported, orthonormal wavelets. Consequently, most of the articles and reference materials on wavelets require a sophisticated mathematical background (a good first-year real analysis course at the graduate level). Our goal with this book is to present many of the essential ideas behind Fourier analysis and wavelets, along with some of their applications to signal analysis to an audience of advanced undergraduate science, engineering, and mathematics majors. The only prerequisites are a good calculus background and some exposure to linear algebra (a course that covers matrices, vector spaces, linear independence, linear maps, and inner product spaces should suffice). The applications to signal processing are kept elementary, without much use of the technical jargon of the subject, in order for this material to be accessible to a wide audience.
The basic goal of Fourier series is to take a signal, which will be considered as ...