CO Sy'x(M) _ Syx{@) 
= (2.173) 
Sxx(@)  Sxx(@) 
This shows that, if we take the cross spectrum of Y,’ with X,, the frequency response 
function G(@w) can be obtained as the ratio of Syx(@) to the spectrum of input sxx(w) as 
was the ideal case in Eq. 2.167, because syx(w) is not affected by the existence of noise 
N;. However, taking the spectrum from Eq. 2.169 gives 
sy'y'(w) = IG(w)Psxx(w) + syn(@) = syy(@) + syn). 
(2.174) 
Therefore, the coherency y?(w) is 
Syx(w) 2 
52 ey ees) bea 
ay ms ~  Spy(@) 
Sra) Sox) eer 
_ {G@){syy@) — sxn@)} 
[srr(@)-syo)} 
Syxx(@) X Syry(W) 
SNN() 
= 1-——_ = 1-7). (2.175) 
Sy'y(@) 
Here 
2 __ SNN() 
4(@)° = —— 2.176 
Sy'y'(@) ( ) 
is called the residual error. 
These results show that it is essential to calculate the cross spectrum to get a good 
estimate of the frequency response function. 
2.6.4 Multiple Inputs Multiple Outputs Case 
More generally for k multiple inputs and / multiple outputs, as in Fig. 2.30, the 
output is expressed by 
61 
