Page 459 - Fundamentals of Radar Signal Processing

(6.33)

Thus, under hypothesis H , . Similarly, under hypothesis H y = m
0
1
+ w and . Note that the mean of z is real in both cases. The power
of the complex Gaussian noise splits evenly between the real and imaginary
H
parts of z. Since ϒ = Re{m y}, it follows that under H and
0
under H . Following the procedure used in Sec. 6.1.2, it can be
1
seen that

(6.34)

Repeating the development of Eqs. (6.22) to (6.24) gives the probability of
detection

(6.35)

Note again that the last term in Eq. (6.35) is the square root of the energy in the

signal m, divided this time by the noise power , i.e., the signal-to-noise
ratio. Thus Eq. (6.35) can be written as

(6.36)

Finally, in the equal means case when m = m1 , Eq. (6.35) is similar (but not
N
identical) to Eq. (6.24). The coherent case includes the term
because all of the signal energy competes with only
half of the noise power.

Figure 6.5 shows the receiver operating characteristic for this example. It