As was mentioned in the preceding subsections, s(w)/s(w) can be approximated by 
Ga distribution, with degree of freedom. Equivalent degree of freedom v is expressed 
by Eq. 2.98 when the window is of the parameter type. 
a 
Accordingly, ifa,(a) and b,(a) show the lower and upper 100( £) percentage 
points of ¥’ distribution, as shown in the key in Fig. 2.17, 
ie 
Prob. [y2 < a,(a)] = Prob. [y? = b,(a)] = =, (2.99) 
Probe (ay(@) ie ei (2.100) 
S(@) 
Therefore,a % confidence interval is 
Vv Vv 
S@) , —— S@) |. (2.101) 
b,(@) a,(@) 
This range is shown in Fig. 2.17 as a function of v, witha, the confidence level, of 
80%, 90%, or 95% as a parameter. 
2. Approximation by Normal Distribution; see Fig. 2.18. 
We know that when N tends to infinity, R(r) follows the normal distribution, and as 
$(@) is a linear combination of R(r), 8(@) also tends to follow the normal distribution. 
This relation can be assumed also from the fact that, when degrees of freedom vy > 00, 
as N— 0, x2—distribution tends to the normal distribution. Then, using the normal 
distribution, we can get the other expression for the confidence interval, as 
S(w) + c(x)V var. S(@). (2.102) 
Equations 2.100 and 2.101 give the confidence interval as, 
A 
1 1 A 
—— Ss@), —— S@) (2.102) 
1+ c(a) V2 1—c(a) V2 
37 
