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.

Digital modulation is the process of translating bits to analog waveforms that can be sent over a physical channel. Figure 4.1 shows an example of a baseband digitally modulated waveform, where bits that take values in {0, 1} are mapped to symbols in {+1, −1}, which are then used to modulate translates of a rectangular pulse, where the translation corresponding to successive symbols is the symbol interval T. The modulated waveform can be represented as a sequence of symbols (taking values ±1 in the example) multiplying translates of a pulse (rectangular in the example). This is an example of a widely used form of digital modulation termed linear modulation, where the transmitted signal depends linearly on the symbols to be sent. Our treatment of linear modulation in this chapter generalizes this example in several ways. The modulated signal in Figure 4.1 is a baseband signal, but what if we are constrained to use a passband channel (e.g., a wireless cellular system operating at 900 MHz)? One way to handle this to simply translate this baseband waveform to passband by upconversion; that is, send up(t) = u(t)cos(2πfct), where the carrier frequency fc lies in the desired frequency band. However, what if the frequency occupancy of the passband signal is strictly constrained? (Such constraints are often the result of guidelines from standards or regulatory bodies, and serve to limit interference between users operating in adjacent channels.) Clearly, the timelimited modulation pulse used in Figure 4.1 spreads out significantly in frequency.

We conclude with a brief discussion of research and development frontiers in communication systems. This discussion is speculative by its very nature (it is difficult to predict progress in science and technology) and is significantly biased by the author's own research experience. There is no attempt to be comprehensive. The goal is to highlight a few of the exciting challenges in communication systems in order to stimulate the reader to explore further.

The continuing wireless story

The growth of content on the Internet continues unabated, driven by applications such as video on demand, online social networks, and online learning. At the same time, there have been significant advances in the sophistication of mobile devices such as smart phones and tablet computers, which greatly enhance the quality of the content these devices can support (e.g., smart phones today provide high-quality displays for video on demand). As a result, users increasingly expect Internet content to be ubiquitously and seamlessly Available on their mobile device. This means that, even after the runaway growth of cellular And WiFi starting in the 1990s, wireless remains the big technology story. Mobile operators today face the daunting task of evolving networks originally designed to support voice into broadband networks supplying data rates of the order of tens of Mbps or more to their users.