DIGITAL FILTER NO. 1 
DIGITAL FILTER NO. 2 
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AMPLITUDE RESPONSE 
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Fig. 11.11. Amplitude response of two digital filters. 
(From Dailzell.55) 
11.5 APPLICATION OF THE HIGHER ORDER POLYSPECTRA 
When we have to take into account nonlinearities of higher order than quadratic, we 
need to consider the polyspectra of higher order than the bispectrum. For example, when 
Y(t) is expressed by 
oO 
Y(t) = | hy(r) X(t—r) drt | | hp(uy, U2) X(t—uy) X(t—Uuz)du,duz 
—co —_ 
oP | | | h3(q1, 92,93) X(t— qi) X(t— 92) X(t—9q3)dqidq2rdq3, (11.67) 
we have to consider the fourth moment correlation functions and their three—dimensional 
Fourier transforms, which are called trispectra. In this case, as we did for the quadratic 
case, we can calculate the statistical values as summarized in the following paragraph. 
There, the variables X;, X2 ,- - - X, are supposed to follow the joint Gaussian distri- 
bution, with £{X,} = 0, then as 
E[X] = 0, E[X,X2X3] = 0, E[XiX2-- -X5]=0, ---- (11.68) 
all of the odd moments are zero. Further, as shown in Eq. 11.39 
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