Page 478 - Fundamentals of Radar Signal Processing

This reduces to the exponential PDF when N = 1 as would be expected since in

that case z′ is the magnitude squared of a single sample of complex Gaussian
noise.
The probability of false alarm is obtained by integrating Eq. (6.78) from
the threshold value to + ∞. The result is (Olver et al., 2010)

(6.79)

where

(6.80)

12
is Pearson’s form of the incomplete gamma function. For a single sample (N =
1), Eq. (6.79) reduces to the especially simple result

(6.81)

This value of the threshold applies to the normalized statistic . The
corresponding result for the unnormalized statistic z is

(6.82)

Equation (6.79) can be used to determine the probability of false alarm P for a
FA
given threshold T or, more likely, the required value of T for a desired P .
FA
Now the probability of detection P corresponding to the same threshold
D
must be determined. Start by finding the PDF of the normalized, integrated, and
square-law-detected samples under hypothesis H . Each individual data sample
1
r has a generalized noncentral chi-squared PDF [Eq. (6.74)]; the corresponding
n
characteristic function is

(6.83)

The CF of the sum z′ of N such samples is