The properties of F(w) and f(w) correspond to those of the probability density and 
probability density distribution functions, P(w) and p(w), for continuous distribution. 
2.4.2 Spectral Representation of the Stationary Random Process 
Instead of using the Fourier integral, the stationary process is more generally ex- 
pressed by the Fourier Stieltjes integral as 
X(t) = | e! dZ(w), (2.50) 
where dZ(w) is in the order of O(Vdw) and is larger than dw. dZ(w) and dZw ) are un- 
correlated, that is they are orthogonal to each other 
5 [iaz(w y?] = dS). (2.51) 
When the process has a purely continuous spectrum, from Eq. 2.49, rewritten as 
dS(w) = s@@) da, (2.49°) 
E [iaz(wy?] = s(w)dw = AS(w). (2.52) 
For a real valued process, 
R(-T) = R(t), (2.53) 
and 
1 1 
s(w@) = — | coswt R(t) dt =— | cos@t R(t) dt. (2.54) 
20 a 
ae 0 
s(@) is an even function, and 
S(—@) = s(@). (2.55) 
Therefore, 
R@)= | coswt s(w) dw = 2 | coswt s(@) do. (2.56) 
9 0 
19 
