Since there is no periodicity, no Fourier series exists, and because of stational- 
ity, | LX(1)ldt < © does not hold, so X(t) does not possess a Fourier integral. Thus, 
2x lim LX(w)!? will not exist and may become infinity. So here 
X(t) -T<t<T 
Xt) = 2.38 
rt) 0 otherwise Gee) 
and the following process is introduced: 
X7(t) = | X7(w)e™'dw 
) T 
1 ; 1 ; 
xX =— | XndeMdt=— | X(Ne dt. 2.39 
TO) = | mt)e = | (the (2.39) 
—0 —T 
Then, since the power is 27IX7(w)I*/T instead of 27lX7(w)I?, which makes T > ©, 
_ 2nIXr(w)? . 
lim Ton is for the total process. The expected value of this power is 
2X 1(w)!* 
2nXro)I" (2.40) 
E [Power] = lim E 
Th 
and is defined as s(w). By manipulation, 
aiXHw yh LL 
Ge OE [2xX7(w)] [2xX7(w)] 
CE ne | X7(t)e"at | Xr(t—t)e"™ at 
Oe Ye 
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