For the existence and convergence of this expansion, conditions 
| IX(t)l dt < 0, | x(n}? dt < © (2.36) 
are necessary. In this case, Parseval’s Relation is 
| X°(t)dt = 21 | X(w)*dw. (2.37) 
Here, as in Fig. 2.5 271X(w)|7dw is the contribution to the total energy of those 
components in X(t) whose frequency lies between w and w + dw. Accordingly, 
27lX(w)I? denotes the energy density function. In this case determining the power makes 
no sense, because when T — ©, from Eq. 2.36, power tends to 0. 
| 2n|X(co)|* 
Fig. 2.5. Energy spectrum of nonperiodic function. 
c. For Stationary Stochastic Processes 
For a single realization x(t) of X(t), as in Fig. 2.6 assume E[X(1)] = 0; X(t) is sto- 
chastically continuous. [R(0) = ¢2,0(0) = 1] 
X (t) 
“SP e) T 
Fig. 2.6. Stationary stochastic process with finite power. 
15 
