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Sec. 7–8 Output Signal-to-Noise Ratios for Analog Systems 537
When the input is Gaussian noise (only), R n (t) is Rayleigh distributed and u n (t) is uniformly
distributed, as demonstrated in Example 6–13.
The phase detector output is proportional to u T (t):
r 0 (t) = Klg T (t) = Ku T(t)
Here K is the gain constant of the detector. For large (S/N) in , the phase angle of g T (t) can be
approximated with the help of a vector diagram for g T = g s + g n . This is shown in Fig. 7–22.
Then, for A c R n (t), the composite phase angle is approximated by
r 0 (t) = Ku T (t) L Ke u s (t) + R n(t)
sin [u (7–108)
A n (t) - u s (t)] f
c
For the case of no phase modulation (the carrier signal is still present), the equation reduces to
r 0 (t) L K A c
y n (t), u s (t) = 0
(7–109)
where, using Eq. (6–127e), we find that y n (t) = R n (t) sin u n (t). This shows that the presence of
an unmodulated carrier (at the input to the PM receiver) suppresses the noise on the output.
This is called the quieting effect, and it occurs when the input signal power is above the
threshold [i.e., when (S>N) in 1]. Furthermore, when phase modulation is present and
(S>N) in 1, we can replace R n (t) sin [u n (t) - u s (t)] by R n (t) sin u n (t). This can be done
because u s (t) can be considered to be deterministic, and, consequently, u s (t) is a constant for a
given value of t. Then, u n (t) - u s (t) will be uniformly distributed over some 2p interval, since,
from Eq. (6–153), u n (t) is uniformly distributed over (0, 2p). That is, cos [u n (t) - u s (t)] will
have the same PDF as cos [u n (t)], so that the replacement can be made. Thus, for large
(S/N) in , the relevant part of the PM detector output is approximated by
where
r 0 (t) L s 0 (t) + n 0 (t)
s 0 (t) = Ku s (t) = KD p m(t)
(7–110)
(7–111a)
g n (t)=R n (t)e j¨n(t)
Imaginary
R n
R n sin(¨n-¨s )
g s (t)=A c e j¨s (t) ¨n
A c
¨s
¨T
Real
Figure 7–22 Vector diagram for angle modulation, (S>N) in 1.