ond-order characterization of the random process. In a second- order characterization, you provide two things: (1) the mean of x (t,,A) = X1: this number (which may be different at different times t1) tells you the average value of x (t,A) at t=t. This value can be generated using the equation (3.7) Chapter Three (2) the autocovariance, (x,(t),: this number (which may be different for different t1 and t2 2 values) describes the relationship between the random variable x(t,,A) = X1 and the random variable x(t2,A) = X2. The larger this number, the more closely related x(t1,A) is to x(t,A). This value can be generated mathematically through the equation (3.8) 3.2.4 And in Telecommunications Now, you know what a random process is, and you know how to fully describe it and how to partially describe it. In most communication systems, random processes have a rather nice property that makes them simple to deal with. In most telecommunication applications, random processes are wide sense stationary (WSS). This means just two things: (1) the mean of x(t,,A) = X1: this value is the same as the mean of the random variable x(t,,A) = x , and the same as the mean of the random variable x(t,A) = X3, and SO on. Mathematically, that means that the value Xm (t1) xm(t2)xm(t3)..xm = (a single number) (3.9) (2) the autocovariance, (x,t2): this value depends only on the time difference between t1 and t2. That is, mathematically, (3.10) Example 3.2 Determine the mean and the autocovariance of a random process described by (E3.12) for all times t,and p(asA)p p = = (E3.13) A Review of Some Important Math, Stats, and Systems Solution: To solve for the mean, we test out equation (3.7): (E3.14) (E3.15) (E3.16) (E3.17) (E3.18) To figure out the autocovariance, we plug and play with equation (3.8) like this: (E3.19) (E3.20) (E3.21) (E3.22) Chapter Three 3.3 Signals and Systems: A Quick Peek 3.3.1 A Few Signals As a way to begin our brief review of signals and systems, I'll describe two signals that we'll use throughout the upcoming chapters. F