Th T 
Elk) == | E[X(t) X(t+)] dt= | R(x)dt 
(T-Irl) (T-Ir1) 
| 
R(t)} 1 Cah Itl<T 
= OE : (2.45) 
0 Intl > T 
As T approaches infinity, 
T 
1 itl 
= lim) <—— 1-— | e™’R(t)d 
s(@) Jin {= | ar | ° (t)dt 
—T 
= idx | eat R(t)dt. (2.46) 
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For the existence of s(w), X(t) must have an autocovariance function R(r) that is 
continuous everywhere, including at t = 0, and 
R(t) = | eT dS(w) (2.47) 
o(t) = | eT dF(w). (2.48) 
This relation is called Wiener—Khinchine’s Theorem. Here, F(q) is the normalized 
integrated spectrum S(w)/o2, where if s(w) is continuous and smooth, 
S(@) = | soya , and F@)= | fede, (2.49) 
where f(w) = s(w)/o2. 
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