Page 493 - Fundamentals of Radar Signal Processing

Assuming that P 1, Eq. (6.114) can be approximated as
FA

(6.115)

where the binomial series expansion was used to obtain the second line.
Equation (6.115) shows that the “1 of N” rule increases P by a factor of
FA
approximately N, an undesirable result. What is needed is a binary integration
rule that increases P compared to P , while leaving P BFA equal to or less than
BD
D
P . An “M of N” strategy provides better results.
FA
Consider the binary integrated probability P of M successes in N trials
B
when the probability of success on a single trial is p; it is

(6.116)

where

(6.117)

Equation (6.116) can be applied to the probability of false alarm by letting p =
P , and to the probability of detection by letting p = P . In the former case, a
FA
D
“success” is a false alarm, i.e., the event that has a probability of p; in the latter
case, a “success” is a correct detection. Consider the specific example of a “2

of 4” rule, that is, N = 4 and M = 2. Using these parameters in Eq. (6.116) gives

(6.118)

To determine the effect of this rule on the probability of false alarm, let p =
P . Assuming that P 1, Eq. (6.118) can be approximated by its first term
FA
FA
and simplified to obtain