or 
pé)=sa2 ; 3 (12.52’) 
0 <0 
At the other extreme, when € = 1, as when ripples are superimposed on slowly varying 
waves, 
(12.53) 
1 am) : 
p) = oe e “* = p(x), (23539) 
namely p(€) becomes Gaussian and it is the same as the Gaussian distribution that 
governs the original process x(t). 
When € is between 0 and 1, p(€) is given by Eq. 12.49 and is between the 
Rayleigh and Gaussian distributions, as shown in Fig. 12.1, and is now popular for us. 
2) 
o 
p(c) 
2 © 
how 
oOo 
-3 2 -1 0 Hv aeionve 3 
Fig. 12.1. Probability distribution density function of extremes. 
(From Cartwright, Longuet—Higgins.'%) 
From this distribution, the expected highest value of 1/n and the expected highest 
values of N independent samples have been derived as functions of €. The derivation will 
not be referred here, as they are well known. 
338 
