Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

8.2 CORRELATION REVISITED

In Chapter 7, we investigated the DE:

(8.2)

where {a1, … , an} are fixed coefficients, random process Y(t) is the input of the linear system represented by the DE, and X(t) is the output (note that the roles of X(t) and Y(t) are reversed in Chapter 7 and (8.2)) . Since Y(t) is random, it is generally not possible to write an expression for each realization of X(t). We first considered evaluating the mean:

Table 8.1 Mean-Square Results for a Random Process

Continuity:	RXX(t1, t2) is continuous at t1 = t2 = t or RXX(τ) is continuous at τ = 0
Derivative:	exists at t1 = t2 = t or exists at τ = 0
Integral:	exists or exists

Figure 8.1 Input/output frequency-domain characterization of LTI systems for deterministic signals and wide-sense stationary random signals.

(8.3)

for t>0, where it is assumed that the derivatives and expectations can be interchanged. ...