Is21(@)| = {Co3w) + Qui(w)} (2.156) 
1] Co2(@) (2.157) 
Arg [s21(@)] = tan” Quz(w) 
and 
dolso(w)| eAtels)] = 5,,(w)dw 
= E[dZ\(w) dZ>(w)). (2.158) 
In this case the coherency function 3;(w) is defined as 
@) 21) cov|dZi(w) dZx(w)| mate 
o) = ——— =. : 
oes (s11@)s2x(w)}/?_— [var{aZ1(w)} var[dZ2(w)}}!/? 
Thus 72;(@) is the correlation coefficient of dZ}(w) and dZ>(w). Here, 
0<l2(@)l <1 at alla. (2.160) 
Usually ty2;(@)l or ty2;(w)I? is called the coherency function and shows the extent 
to which X2,t and Xj,t are linearly related. 
2.6.2 Linear Responses 
Suppose there is a linear system, X, being the input, Y, the output as in Fig. 2.28. If 
Y= >) a Xen. (2.161) 
T=-0O 
{g:| is called the impulse response function. If this is physically realizable, 
g:=0 for tT <0. 
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