dP 1 dQ(E[N.(a)]) 
a) = = Ba ee Sis ee eee 12.47 
Pp (a) da E[N,.(0)] da ‘ 
When X(t) and X(t) are independent, 
dp(x) 
dx |x=a 
(12.48) 
p(0) 
Py (a) = 
Expanding this kind of relation, Longuet-Higgins, * and Cartwright and Longuet— 
Higgins,!> developed the well-known results for the distribution function of extreme 
values of a general process with various bandwidths. 
The probability distribution density function for extreme values of a process X(t), 
or &),&>, .. . normalized by the standard variation of the original process 0,, as 
E/mo = &/o, = was derived as, 
raed? 
Sa r 
Ge Sees | cea (12.49) 
Dp Sire €e ~ +(1—€%~)*Ce e u : 
~—  —& 
as shown in Fig. 12.1, where € is the so—called bandwidth parameter of the spectrum 
<= moma—ms) 
moms, 
(12.50) 
> 
m,, being the moment of spectrum 
Mm, = | wrsoyde. (25D) 
—-o 
When ¢€ = 0, as for a narrow banded spectrum, p(¢) becomes a Rayleigh distribution as 
is well known, 
ase, 
po) = ie c= 0 (12.52) 
or 
331 
