Page 449 - Fundamentals of Radar Signal Processing

interest: P , P , the noise power , and the constant m whose presence or
FA
D
absence is to be decided. The latter two are characteristics of the given signals,
while P is generally fixed as part of the system specifications at whatever
FA
level is deemed tolerable. Thus it is necessary only to determine P . The
D
approach is identical to that used for determining P : determine the probability
FA
density function of the sufficient statistic ϒ under the hypothesis H and integrate
0
the area under it from the threshold to + ∞.
Continuing the example, note that the only change under hypothesis H is
1
that the individual data samples y now each have mean m, so their sum ϒ has
n
mean Nm. Thus, and

(6.22)

Again applying the definition of the error function in Eq. (6.18) leads to

(6.23)

Since the primary concern is the relationship between the performance metrics
P and P , Eq. (6.21) can be used in Eq. (6.23) to eliminate the threshold T and
D
FA
arrive at

(6.24)

Nm is considered to be the signal voltage of interest in the sufficient
statistic ϒ(y). The corresponding signal power is (Nm) . The power of the noise
2
component of ϒ(y) is . Thus, the term is the square root of the
signal-to-noise ratio χ for this problem, and Eq. (6.24) can be rewritten as