Foundations of Signal Processing

Our main goals in this book and its companion volume, Fourier and Wavelet Signal Processing (FWSP) [57], are to enable an understanding of state-of-the-art signal processing methods and techniques, as well as to provide a solid foundation for those hoping to advance the theory and practice of signal processing. We believe that the best way to grasp and internalize the fundamental concepts in signal processing is through the geometry of Hilbert spaces, as this leverages the great innate human capacity for spatial reasoning. While using geometry should ultimately simplify the subject, the connection between signals and geometry is not innate. The reader will have to invest effort to see signals as vectors in Hilbert spaces before reaping the benefits of this view; we believe that effort to be well placed.

Many of the results and techniques presented in the two volumes, while rooted in classic Fourier techniques for signal representation, first appeared during a flurry of activity in the 1980s and 1990s. New constructions of local Fourier transforms and orthonormal wavelet bases during that period were motivated both by theoretical interest and by applications, multimedia communications in particular. New bases with specified time – frequency behavior were found, with impact well beyond the original fields of application. Areas as diverse as computer graphics and numerical analysis embraced some of the new constructions – no surprise given the pervasive role of Fourier analysis in science and engineering.