Therefore from Eq. 12.40 
E[X°@)| = of -€ LX’). (12.42) 
For the Gaussian process E[X“(t)] = 30,*. Therefore in this case, the order is also in 
E[X*(1)] = 02 —3e0%. (12.43) 
It is interesting to find that this approximation is the same type as Eq. 10.17 
obtained through the equivalent linearization method and Eq. 10.50 obtained through 
the perturbation method. 
12.5 PROBABILITY CHARACTERISTICS OF AMPLITUDES, 
MAXIMA AND MINIMA 
S. O. Rice® has shown that, when the joint probability density function of x and 
x, the values of a stochastic process X(t) at time t, and its derivative (d/dt)X(f), i.e., 
p(x,X), are known, the frequency of occurrence of any threshold value crossing or zero 
crossing is easily derived. 
For example, the expected value of the frequency of occurrence of the process X(t) 
for the upward crossing of a threshold of level a, E[N,(a)], is 
E[N,(a)] = | xp(a, x)dx. (12.44) 
0 
If x(t) and x(t) are independent, p(x, x) = p(x) - p(x), therefore 
EIN+(@)] _ [p@)]x=a 
= (12.45) 
FIN.()] [p@)].=0 
When the power spectrum is narrow banded, the amplitude of the sample process 
varies slowly, just like the envelope of sine waves, and as the result, the process is 
assumed to have an extreme positive value a, at each cycle of this sine wave. 
Then the probability distribution is 
_ E[N.(@)] 
ee 2) TENA 
=1-P(@p, S a). (12.46) 
Therefore, the probability density distribution is 
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