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Sec. 7–8 Output Signal-to-Noise Ratios for Analog Systems 541
n 0 (t) = a
K b
2pA c
dy n (t)
dt
(7–124b)
This result is valid only when the input signal is above the threshold [i.e., when (S>
N) in 1].
The derivative of the noise, Eq. (7–124b), makes the PSD of the FM output noise different
from that for the PM case. For FM, we have
or
n0 (f) = a
K b
2pA c
a K 2
b N
A 0 f 2 , ƒ f ƒ 6 B T >2
n0 (f) = c
L
0, f otherwise
2
ƒ j2pf ƒ 2 yn (f)
(7–125)
This shows that the PSD for the noise out of the FM detector has a parabolic shape, as illustrated
in Fig. 7–23b.
The receiver output consists of the low-pass filtered version of r 0 (t). The noise power
for the filtered noise is
[n ' 0(t)] 2 =
L
B
-B
n0 (f) df = 2 3 a K 2
b N
A 0 B 3
c
(7–126)
The output SNR is now easily evaluated using Eqs. (7–124a) and (7–126):
a S N b =
out
From Eqs. (5–44) and (5–48), it is realized that
2
s 0
[n ' 0] 2 = 3A c 2 [D f >(2pB)] 2 m 2
2N 0 B
where V p is the peak value of m(t). Then, the output SNR becomes
a S N b = 3A c 2 b 2 f (m/V p ) 2
out 2N 0 B
The input SNR is
a S 2
N b A c
=
in 4N 0 (b f + 1)B
Combining Eqs. (7–127) and (7–128), we obtain the ratio of output to input SNR
(S>N) out
(S>N) in
D f
2pB = b f
V p
= 6b 2 f (b f + 1) a m 2
b
V p
(7–127)
(7–128)
(7–129)
where b f is the FM index and V p is the peak value of the modulating signal m(t).