thinking of the symmetry character of 92(t),T2) = 22(T2,7)). On the basis of this expres- 
sion T. Vinje®’ formulated a general method for getting the probability distribution 
function of the extreme values, under the assumption that the response z(t) is narrow 
banded and the input x(r) is Gaussian with variance o*. His method is also based on the 
assumption that the probability distribution function of extreme values can be obtained by 
the joint distribution function p(z, z) only. 
Now, we start from the statistical moment generating function to get p(z), 22) 
[here z) = z(t), 22 = 2(t) z], that is, from the double Fourier transform of p(zj, 22), 
(01, 62) = Elexpli(@1z1 + 6222)}). 
= | | exp(i6 Zz) + 10222)p(z1, 22)dzdz2. (12.102) 
Then expanding exp (i612; + 10222) into a series gives 
oo 
Zi 25 
exp (16121 + 1022) =1+ >. = (i6)"(62)", (12.103) 
nO min. 
assuming m,n are positive integers whose sum is greater than zero. 
Inserting Eq. 12.103 into Eq. 12.102 gives 
(01,02) =1+>, ao (iO 1)"(i02)". (12.104) 
Here 
Emn = E[z7, 25] = | | 2725 P(2, 22)dz dz. (12.105) 
From the Fourier inverse transform of Eq. 12.102, 
co 
| | (61,02) exp{-i( 121 + 8222)}d01d82. (12.106) 
—o —0 
1 
P(21, 22) = Ome 
Putting Eq. 12.104 in Eq. 12.106 
355 
