134
6 Detection of Radio Signals and Its Parameters Measuring
Maximum of output signal max{y out } corresponds to coincidence of delays t s and
t dly and Doppler frequencies f s = f Dp . Then for an output signal, we write down
the following:
y out (t s ; f s ) = a
t n
0
S
t − t dly ; f d + f Dp
S
∗
(t − t s ; f d + f s )dt
+
t n
0
n(t)S
∗
(t − t s ; f d + f s )dt
(6.11)
Determine statistical moments of conditional probabilities distribution for the
second hypothesis. An average value M{y out } equals to (at maximum):
M{max(y out )} = a
t n
0
S
2
t − t dly
dt +
t n
0
¯
n(t)S(t − t dly ; f d + f Dp )dt (6.12)
Considering that the first integral in (6.13) corresponds to signal “shape energy”
E 0 and ¯
n(t) = 0, then we obtain:
M{max(y out )} = ¯
y out = a E 0
(6.13)
Dispersion of output signal y out for the second hypothesis equals to dispersion
y out , related to the first hypothesis (6.9), since a dispersion of random value sum
in (6.11), represented by the second summand, and constant (nonrandom) value
(first summand in (6.11)), equals to dispersion of a random value. Consequently, by
substituting expressions (6.13) and (6.9) for an average value and dispersion of output
signal into probabilities density distribution formula (6.6), we obtain a conditional
density for the second hypothesis—presence of a target—in the following form:
p(y out |S t = 0) =
1
2πσ 2
y out
exp
−
(y out − a E 0 )
2
N 0 E 0
2
(6.14)
In Fig. 6.6, distributions of (6.10) and (6.14) form are performed. Obviously, that
they have a similar dispersion, however dispersion (6.14) is shifted along y out axis
at a value, proportional to signal amplitude a, consequently, at a value, proportional
to signal energy E = a(a E 0 ).
Find a probability of false alarm P f a . For this purpose, integrate a conditional
density (7.10) from a threshold level y thr (Fig. 6.6) ad infinitum:
P f a =
1
2πσ 2
y out
exp
−
y
2
out
2
N 0 E 0
2
dy out
(6.15)
6 Detection of Radio Signals and Its Parameters Measuring
Maximum of output signal max{y out } corresponds to coincidence of delays t s and
t dly and Doppler frequencies f s = f Dp . Then for an output signal, we write down
the following:
y out (t s ; f s ) = a
t n
0
S
t − t dly ; f d + f Dp
S
∗
(t − t s ; f d + f s )dt
+
t n
0
n(t)S
∗
(t − t s ; f d + f s )dt
(6.11)
Determine statistical moments of conditional probabilities distribution for the
second hypothesis. An average value M{y out } equals to (at maximum):
M{max(y out )} = a
t n
0
S
2
t − t dly
dt +
t n
0
¯
n(t)S(t − t dly ; f d + f Dp )dt (6.12)
Considering that the first integral in (6.13) corresponds to signal “shape energy”
E 0 and ¯
n(t) = 0, then we obtain:
M{max(y out )} = ¯
y out = a E 0
(6.13)
Dispersion of output signal y out for the second hypothesis equals to dispersion
y out , related to the first hypothesis (6.9), since a dispersion of random value sum
in (6.11), represented by the second summand, and constant (nonrandom) value
(first summand in (6.11)), equals to dispersion of a random value. Consequently, by
substituting expressions (6.13) and (6.9) for an average value and dispersion of output
signal into probabilities density distribution formula (6.6), we obtain a conditional
density for the second hypothesis—presence of a target—in the following form:
p(y out |S t = 0) =
1
2πσ 2
y out
exp
−
(y out − a E 0 )
2
N 0 E 0
2
(6.14)
In Fig. 6.6, distributions of (6.10) and (6.14) form are performed. Obviously, that
they have a similar dispersion, however dispersion (6.14) is shifted along y out axis
at a value, proportional to signal amplitude a, consequently, at a value, proportional
to signal energy E = a(a E 0 ).
Find a probability of false alarm P f a . For this purpose, integrate a conditional
density (7.10) from a threshold level y thr (Fig. 6.6) ad infinitum:
P f a =
1
2πσ 2
y out
exp
−
y
2
out
2
N 0 E 0
2
dy out
(6.15)