is used and the spectrum syy(w) becomes a white specttum syy(w) = 0}, then as will be 
shown in Section 5.4.2 of Part II, 
Syy(@) = a; = lap + aye + ane +... + ape syy(w). (2.135) 
Then 
o} 
Sxx(@) = - —. (2.136) 
xn lag + aye7? + age +... tape HO? 
If we can find the filter that will make this spectrum completely white, that filter is 
called a complete pre—whitening filter. Then from Eq. 2.136, we can find syy(w) from 
the variance o¥ of the filtered process (that is uniform), and the frequency function 
IG) = lag + aye + ane 2 +... tape. (2.137) 
This tells us that finding the pre—whitening filter G(w) is the same procedure as fit- 
ting a model to the X; process expressed as 
€;= 0X; + a4X;1+ 66.6 0 QyX 14+ 050 o>” (2.138) 
where €, is a completely random process with variance ay. This is the problem of AR— 
model fitting to the process X(t), which will be discussed in detail in Chapter 5, Part II of 
this paper. 
2.6 MULTI-VARIATE SPECTRAL ANALYSIS; SPECTRAL 
ANALYSIS OF FREQUENCY RESPONSE 
2.6.1 Two Variate Spectral Analysis 
If there are two stochastic processes {X14} {Xo 4} C= 00-2 se eeach' 
weakly stationary, then Cov. {x Lh X2..| is a function of (t; — rt) only. 
For this stationary bivariate process, the correlation matrix is defined as 
Ris(r) ~—-Ryar) 
; 2.139 
Ror) Roar) Ce 
R(r) = 
where 
53 
