A canonical problem formulation needed for the analysis of the performance of a communication systems design is given in Figure 9.1. The thermal noise generated within the receiver is denoted W(t). This noise is then processed in the receiver and will experience some level of filtering, represented with the transfer function HR(f ). The simplest analysis problem is examining a particular point in time, ts and characterizing the resulting noise sample, N(ts), to extract a parameter of interest (e.g., average signal–to–noise ratio (SNR)). Additionally, we might want to characterize two or more samples, e.g., N(t1) and N(t2), output from this filter.

To accomplish this analysis task this chapter first characterizes the thermal noise, W(t). It turns out that W(t) is accurately characterized as a stationary, Gaussian, and white random process. Consequently, our first task is to define a random process (Section 9.1). The exposition of the characteristics of a Gaussian random process (Section 9.2) and a stationary Gaussian random process (Section 9.3) then will follow. A brief discussion of the characteristics of thermal noise is then followed by an analysis of stationary random processes An additive noise is characteristic of almost all communication systems. This additive noise typically arises from thermal noise generated by random motion of electrons in the conductors comprising the receiver. In a communication system the thermal noise having the greatest effect on system performance is generated at and before the first stage of amplification. This point in a communication system is where the desired signal takes the lowest power level and consequently the thermal noise has the greatest impact on the performance. This characteristic is discussed in more detail in Chapter 10. This chapter’s goal is to introduce the mathematical techniques used by communication system engineers to characterize and predict the performance of communication systems in the presence of this additive noise. The characterization of noise in electrical systems could comprise a course in itself and often does at the graduate level. Textbooks that provide more detailed characterization of noise in electrical systems are [DR87, Hel91, LG89, Pap84, YG98].

A canonical problem formulation needed for the analysis of the performance of a communication systems design is given in Figure 9.1. The thermal noise generated within the receiver is denoted W(t). This noise is then processed in the receiver and will experience some level of filtering, represented with the transfer function HR(f ). The simplest analysis problem is examining a particular point in time, ts and characterizing the resulting noise sample, N(ts), to extract a parameter of interest (e.g., average signal–to–noise ratio (SNR)). Additionally, we might want to characterize two or more samples, e.g., N(t1) and N(t2), output from this filter.

To accomplish...