The Fourier Transform

Abstract

The definition of a Fourier transform will differ depending on the signal type. The definitions all have a common form, however, and all can be thought of as a means of mapping a signal g = g(t);t ∈ τ, which depends on a parameter t in some time index set τ, into another signal G = G(f); f ∈ S, which depends on a new parameter or independent variable f, which we shall call frequency. As does t, the independent variable f takes values in a domain of definition or index set, denoted S. We will eventually show that there are natural frequency domains of definitions for use with each of the basic signal types, as summarized in Table 2.1.