As has been formulated in Eqs. 11.12 and 11.6 
E[¥(t)— my|[¥(¢ + 11) — my][¥ (ri + 72) — my] = Ryyy(t1, 72) 
S(@1, @2) exp[+ 1(@1T1 + W2T2)] dw dw 2 
8 
8 
thus 
oc co 
E|[¥@)—my}*| = Ryvv(0, 0) = | | stco:,2ddo da (11.28) 
—-O —-O 
The bispectrum shows the contribution of the third moment of frequency components at 
three frequencies @),@2, and 3, where w; + W2 + W3 = O are satisfied. When square 
mean, cubic mean, and fourth power mean are expressed as 
El [ys - my} | Oe El{y(e) = my}*| =, El{y(e) - my|*| =u, 
the statistical values called skewness s, and peakedness p are defined as 
3 
s=—_, A (11.29) 
(o?) 
4 
u 
=—., (11.30 
PU G2) 
When Y(t) is a Gaussian (not nonlinear) process, “> =0, «4 = 30%. Therefore, 
s=0, p=3. By the values of s and p, we can show the extent of nonlinearity of the 
process. ”? 
11.4 CHARACTERS OF QUADRATIC RESPONSE TO 
GAUSSIAN INPUT PROCESS 
As already was shown in Eq. 11.2, when X(t), the input process is a Gaussian linear 
process with E[X(rt)] = 0, 
Ree Exe +1) X(t+™) x(o] = (11.31) 
SXxx(@1,@2) = 0. (11.32) 
Several statistical relations of the quadratic nonlinear output Y(t) to input X(r) and their 
derivations will be summarized as follows: 
1. Mean of Y(t), E[Y(t)] = my. 
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