Phil Gustafson

Elementary Fourier Analysis Phil Gustafson

This book was written as a textbook for a one semester undergraduate mathematics course in Fourier analysis, with an emphasis on applications to basic signal processing. This course was developed by the author and first taught in 2007, using an early version of the book manuscript. The course has been taught annually since then, and both the course and book manuscript have evolved significantly over the years. The concepts in the book are illustrated with many MATLAB graphs and output, using programs written by the author for use by the students. The course has been a popular addition to a traditional undergraduate mathematics program at the author's university, using a hands-on approach in a computer classroom with MATLAB. For a wider audience, the book can be valuable for those interested in learning basics of Fourier analysis and signal processing, with or without MATLAB. The book does not explore advanced topics in signal processing and harmonic analysis, nor the application of Fourier series to the solution of separable partial differential equations - there are already many books written for these purposes. However, there are not many books that comprehensively address Fourier analysis and signal processing at the introductory level presented in this book. Prerequisite knowledge for this book is a year of single variable calculus, familiarity with vectors and matrices, and some mathematical maturity. The author has successfully used this book with students at the junior and senior level in mathematics, mathematics secondary education, computer science, statistics, physics, chemistry, geology and mass communication. Fourier analysis has something of significance to offer many areas. The content of the book is organized as follows. Chapter 1 provides an introduction to the discrete nature of digital signals. Vectors represent sampled functions, and matrices represent pixel value colorations for digital images. After working with basic characteristics of signals, the need for a more advanced theoretical understanding of signals becomes clear. Chapters 2 and 3 develop the required continuous-time theory, including orthogonal function expansions, Haar wavelet expansions, Fourier expansions, Fourier transforms and inverse Fourier transforms, time and frequency domain settings, and the Cosine Transform of Type II. Chapters 4 and 5 apply these ideas to the discrete-time setting by way of vector and matrix expansions, discrete Fourier and Cosine transforms, spectrograms and JPEG compression. Chapter 6 provides additional material on MP3 compression.