Ryx(r) = E [XY] = > 8 ELK; Xue] 
T=-@ 
= Sy &, Rxx(r—-T), (2.165) 
T=-@ 
and therefore 
1 ce 
syx(@) => > e@'Ryx(r) 
=G(@) Sxx(@). (2.166) 
Accordingly, 
Cy ae, (2.167) 
Sxx(@ ) 
From Eq. 2.163 
IG@)? = 2) 2.163” 
G@) eG) (2.163”) 
These equations show that, from the spectra of output and input, we get the IG(w)!’, the 
response amplitude function, but if we need the complete response function including the 
phase relation, we have to use the cross spectrum syx(w) as shown in Eq. 2.167. 
In this ideal linear case 
yo)? = IO) aan _ IG@)? 
| 
=——, = 1. (2.168) 
511(@) son(o)| Fecal IG(w)!* 
The coherency ty(w)/? should be 1. 
2.6.3 Linear Response in the Presence of Noise 
When noise is added to the output Y;, as in Fig. 2.29, where we assume N, is real— 
valued, zero mean, uncorrelated with Y, and with X,. 
59 
