Passband Signals and Their Representation

Introduction The signals encountered in wireless communications are typically real passband signals . In this chapter we shall define such signals and define their bandwidth around a carrier frequency . We shall then explain how such signals can be represented using their complex baseband representation. We shall emphasize two relationships: that between the energy in the passband signal and in its baseband representation, and that between the bandwidth of the passband signal around the carrier frequency and the bandwidth of its baseband representation. We ask the reader to pay special attention to the fact that only real passband signals have a baseband representation. Most of the chapter deals with the family of integrable passband signals. As we shall see in Corollary 7.2.4, an integrable passband signal must have finite energy, and this family is thus a subset of the family of energy-limited passband signals. Restricting ourselves to integrable signals—while reducing the generality of some of the results—simplifies the exposition because we can discuss the Fourier Transform without having to resort to the L 2 -Fourier Transform, which requires all statements to be phrased in terms of equivalence classes. But most of the derived results will also hold for the more general family of energy-limited passband signals with only slight modifications. The required modifications are discussed in Section 7.7.