0, o za’ 
E[dZ3(w) dZ\(w')] = as (2.150) 
5y2(@)dw, w=O 
Inverting Eq. 2.144 gives for the two variate process 
7 4 
Ro(r) = | e*saywrde, R12(r) = | esio(orde), (2.151) 
I I 
as was for the single variate process 
i4 P14 
Rii(r) = | e's) \(@)dw, Ror) = | e syo(w do). (2.152) 
It TT 
When X;, leads X2, Eq. 2.144 gives 
een 
SG) ime IR ems. 
r=-@ 
Here, since FR (r) is not symmetrical on r = 0, the cross spectrum 52;(m) is a complex 
function, 
= Co21(@) + iQu21(@), (2.153) 
Co21(@) = = y 5 (Rast) + Ro\(- r)} cosrao, (2.154) 
1 ed 
Qu21(@) Oe ») 5 Rai) + Ra r)} sinra. (2.155) 
Co2;(@) and Quz;(@) are the co— and quadrature—spectra of X;(t) and X(t) when 
X(t) leads X2(t). 
Thus the cross spectrum is expressed by its absolute value and argument as 
56 
