Introduction

The Power Spectral Density of a stochastic process tells us more about the SP than just its power. It tells us something about how this power is distributed among the different frequencies that the SP occupies. The purpose of this chapter is to clarify this statement and to derive the PSD of PAM signals. Most of this chapter is written informally with an emphasis on ideas and intuition as opposed to mathematical rigor. The mathematically-inclined readers will find precise statements of the key results of this chapter in Section 15.5. We emphasize that this chapter only deals with real continuous-time stochastic processes.

The classical definition of the PSD of continuous-time stochastic processes (Definition 25.7.2 ahead) is only applicable to wide-sense stationary stochastic processes, and PAM signals are not WSS. Consequently, we shall have to introduce a new concept, which we call the operational power spectral density, or the operationalPSD for short. This new concept is applicable to a large family of stochastic processes that includes most WSS processes and most PAM signals. For WSS stochastic processes, the operational PSD and the classical PSD coincide (Section 25.14). In addition to being more general, the operational PSD is more intuitive in that it clarifies the origin of the words “power spectral density.” Moreover, it gives an operational meaning to the concept.

Section 15.2 provides some motivation for our definition of the operational PSD. Readers in a rush to get to the definition can skip it and jump directly to Section 15.3. Section 15.4 derives the operational PSD of PAM signals, and Section 15.5 does so a bit more rigorously. Sections 15.6–15.8 expand on the operational PSD. Section 15.6 explores its relationship to the average autocovariance function; Section 15.7 discusses the effect of filtering on the operational PSD; and Section 15.8 explores the technical assumptions that are required for the power to equal the integral of the operational PSD over all the frequencies.