co eo io) 
G(@,, 2, = Ure On) = yy y Pate a » Su, Uy: ++ Uy EqOreir@ au aa +0 Hn) 
= =o u,=0 
pee (11.9) 
cL It 
1 . 
Sujuy O97 = (27r)” | oa | G(@, OD a Wn) ellOilrt@2ot- + +O Un) 
dw\dw.----- dw, (11.10) 
For convenience the expressions for a continuous process will be used in this 
chapter unless otherwise stated. 
11.3 SECOND ORDER NONLINEAR PROCESS 
BISPECTRUM, CROSS BISPECTRUM 
For example in Eq. 11.1, the quadratic nonlinear characteristics of Y(t) is expressed 
by n = 2, and the terms n > 3 can be neglected. Then Eq. 11.1 becomes 
Y(t) = | hy(t)X(t—T)dt + | | h2(t1,T2) X(t—7T1) X(t—T2) dtydtz, (11.11) 
which corresponds to Eq. 11.2 for a linear process. As the expected values of the second 
moment, the auto or cross covariance functions Rxx(T), Ryy(t), and Ryy(t), and their 
Fourier transforms, the auto or cross spectra syx(), syy(w), and syx(w) played big roles 
in the analysis of the responses of linear systems, the expected values of the third 
moments Ryxxx(T}1,7T2), Ryyy(t1,T2), Ryxy(t},T2), and their double Fourier transforms 
Sxxx(@1,@2), Syyy(@1,@2), Syxx(@),@2), are important in the analysis of second order 
nonlinear processes. The third moment correlation functions are, for example, 
Ryyy(1, T2) = E[{¥(t + 71) — my|[¥(¢ + t2) — my|[¥(¢) — my} (11.12) 
Ryxy(t,T2) = E[X(t +1) X(t + t2)[Y(0) — my]]. (11.13) 
Here E[X(t)] = 0 is assumed. Their double Fourier transforms are called the bispectrum 
Syyy(@ 1, @2) or the cross bispectrum syxy(@ 1, @2), 
Syyy(@ 1,2) = Ryyy( 1, T2) exp[— i(@ jT1 + W2T2)|dt,dt2 ~— (11.14) 
a 
SXx¥(W1,@2) = Ryxy(T1,T2) exp[- i(@ 171 + W2T2)) dtydt2. (11.15) 
Bat 
fl 
fi 
The inverse transforms are 
297 
