Fourier Series, Fourier Transforms and the DFT

Classical Fourier methods such as the Fourier series and the Fourier integral are used for continuous time (CT) signals and systems. A more recently developed set of Fourier methods, including the discrete time Fourier transform and the discrete Fourier transform, are extensions of basic Fourier concepts that apply to discrete time (DT) signals. This chapter presents many different Fourier transform concepts for both CT and DT signals and systems. It illustrates how these various forms of the Fourier transform relate to one another, and how they are all derived from more general complex transforms, the complex Fourier transform for CT, and the bilateral z-transform for DT. The chapter shows that many of these transforms have similar properties which are inherited from their parent forms, and that a parallel hierarchy exists among Fourier transform concepts in the CT and the DT worlds.