Probability theory is fundamental to communication system design, especially for digital communication. Not only are there uncontrolled sources of uncertainty such as noise, interference, and other channel impairments that are amenable only to statistical modeling, but also the very notion of information underlying digital communication is based on uncertainty. In particular, the receiver in a communication system does not know a priori what the transmitter is sending (otherwise the transmission would be pointless), hence the receiver designer must employ statistical models for the transmitted signal. In this chapter, we review basic concepts of probability and random variables with examples motivated by communications applications. We also introduce the concept of random processes, which are used to model both signals and noise in communication systems.

Chapter plan

The goal of this chapter is to develop the statistical modeling tools required in later chapters. For readers who are already comfortable with probability and random processes, the shortest path to Chapter 6 is to review the material on Gaussian random variables in Section 5.6 and noise modeling in Section 5.8. Sections 5.1 through 5.5 provide a review of background material on probability and random variables. Section 5.1 discusses basic concepts of probability: the most important of these for our purpose are the concepts of conditional probability and Bayes’ rule. Sections 5.2 and 5.4 discuss random variables and functions of random variables. Multiple random variables, or random vectors, are discussed in Section 5.3. Section 5.5 discusses various statistical averages and their computation.