2 
N (o) 
6D Gan - (6.3) 
n=] u=0 
However, €; is not necessarily white, and moreover it may sometimes have some 
feedback effect on the input. 
As already discussed in Section 2.6.3, even when noise exists in the output Y, of 
input X,, if there is no feedback effect as in Fig. 6.1, we can get rid of the effect of noise 
in computing the real spectrum of the output by finding the real response function of this 
system from the cross spectrum of the output to the input. 
‘LINEAR SYSTEM 
INPUT OUTPUT OBSERVED 
Fig. 6.1. Linear system without feedback. 
However, if somehow this output is fed back to the input, as in Fig. 6.2, the input X, 
is no longer uncorrelated with N,, and N; is included in X; as a part of the input. Accord- 
ingly, taking the cross correlation does not help in getting the real response characteristic 
of this system. 
LINEAR SYSTEM 
INPUT OBSERVED 
Fig. 6.2. Linear system with feedback. 
By a parametric method different from the nonparametric method referred to above, 
we can obtain the real relation by fitting an appropriate model to this system, getting rid 
of the effect of noise by following Akaike.>! 
Under these circumstances, we first transform the noise €; into pure white noise. 
Generally €; can be expressed by an AR model of arbitrary order /, 
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