Fourier Transforms

PREFACE

The mathematical techniques known as “transform methods” have long been a basic tool in several branches of engineering and science, and no wonder. Fourier's simple idea, radical in its time, that a function can be expressed as a sum of sine waves, is ubiquitous. It underlies fields as diverse as communications, signal and image processing, control theory, electromagnetics, and acoustics. Electrical engineers typically encounter the rudiments of Fourier transforms in undergraduate systems and circuits courses, for modeling the spectral content of signals and designing frequency selective circuits (filters). The Laplace transform, a close cousin of the Fourier transform, enables the efficient analytical solution of ordinary differential equations and leads to the popular “S plane” and “root locus” methods for analyzing linear systems and designing feedback controllers. Discrete-time versions of the Fourier and Laplace transforms model spectra and frequency responses for digital signal processing and communications. Physics and engineering students meet the Fourier series when learning about harmonic motion or solving partial differential equations, for example, for waves and diffusion. The Fourier transform also models wave propagation from acoustics to radio frequencies to optics to X-ray diffraction. The widespread dissemination of the fast Fourier transform algorithm following its publication in 1965 added a computational dimension to all of these applications, from ...