H\(@) =-o7H\@) 
H>(@1, 2) = —(@1 + @2)"H(W1, 02) (12.140) 
H3(@1,@2, @3) = —(@1 + @2 + ©3)"H3(@1, @2, 3). 
In addition, the characteristics of the products of white noise were fully utilized as 
follows: 
Wi (t—t)W(t—t) . . . Wit—ty) = 0 for N odd, 
W(t—1)W(t—72) = 0(t1 —T2) = O12, 
W(t=1)W(t—12)W(t— 73) W(t — 4) = 012034 + 613024 + 6 14623, 
W(t—1)W(t—12)W(t—73)W(t — T4) W(t — 75) W(t — 7.) 
= 612034056 + 012035046 + 012036045 (4 141) 
+ 613024056 + 013025046 + 013026045 
+ 614023056 + 614025036 + 014026035 
+ 615034026 + 015032046 + 015036024 
+ 016034052 + 016035042 + 016032045, 
where 6;; = 0(t; —T;). 
Eq. 12.141 is the special case of Eqs. 11.68, 11.69, and 11.70 in Section 11.5. 
If we do not assume narrow bandedness of the output spectrum, the expected 
number of maxima of the response Y greater than Y = € per unit time is approximated as 
«2 0 
Nz = | | IYI p(Y,0,¥) d¥ dy. (12.142) 
— -w 
Similarly, the expected number of minima of Y less than Y = € per unit time is 
approximated as 
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