Fundamentals of Fourier Analysis

Abstract

This chapter contains foundational material for modelling of signals and systems. Section 2.2 introduces classes of functions useful in signal processing and analysis. The Fourier transform, in Section 2.3, begins with the Fourier integral and develops the Fourier series, the discrete time Fourier transform and the discrete Fourier transform as special cases. The following material in this chapter can be skipped on a first reading. † denotes material relevant to multidimensional signals in Chapters 8 and 11. ‡ denotes material relevant to probability and stochastic processes in Chapter 4. ¶ denotes material used in continuous sampling in Chapter 10. There are a number of signal classes to which we will make common reference. Continuous time signals are denoted with their arguments in parentheses, e.g., x(t). Discrete time signals will be bracketed, e.g., x[n]. A continuous time signal, x(t), is periodic if there exists a T such that x(t) = x(t − T) for all t. The function x(t) = constant is periodic. A discrete time signal, x[n], is periodic if there exists a positive integer N such that x[n] = x[n − N] for all n. The function x[n] = constant is periodic.