UN EAR THEORY OF FLUCTUATIONS 139 



By Parseval's theorem we obtain 



y'{t, t) dl = Itt I I yico, t) I" Joj, 



00 *'— 00 



which, when combined with (I-l), gives finally the desired result (using 

 the fact that | y(co, r) |^ is an even function of co) 



Til) = f F(a,) d^ (1-3) 



Jo 



where 



F(co) = 47r Lim - | y(a), r) |' (1-4) 



T-»oo T 



is the spectral density. 



By a procedure not very different from the preceding, one can show that 



yiOyil + m) = / Y(ui) cos COM dw, (1-5) 



Jo 



Y(u) = — I y{l)y{l + u) cos co?^ du. (1-6) 



TT J 



The quantity (y(/)y(/ -|- «) is called the self-covariance. 



Now let us suppose that we have a set of random variables yiit) which are 

 in general complex and whose time averages vanish. We are then led to 

 consider, instead of (1-3), relations of the form 



y.{i)y*{i) = /" I\;(C0) 6fcO (1-7) 



Jo 



where now 



riy(co) = 27r Lim - [y,(co, T)y*(w, t) + Jz(-co, r)y*(-aj, r)] (1-8) 



T->00 T 



in which 



J /. + (t/2) 



>'i(w, r) = — \ yXl)e"^ dt. 



iTT J-(rl2) 



Instead of self-covariances like y{l)y{L -{• n) we have to consider a covari- 

 ance matrix of the form y,(/)y; (/ + u). Since we shall not have occasion in 

 this paper to consider the relation between the spectral density matrix and 

 the covariance matrix we will not consider the derivation of the analogue of 

 Eq.(I-5). 



