Digital Signal Processing Reference
In-Depth Information
−ω
c
≤ ω ≤ ω
c
, the lowpass component
d
low
(
t
) is extracted. The information
signal
m
(
t
) is then obtained from
d
low
(
t
) using the following relationship:
m
(
t
)
=
2[
d
low
(
t
)
−
1]
.
(8.7)
8.1.2 Synchronous demodulation with non-zero epochs
In synchronous demodulation, the epoch
φ
c
of the modulating carrier is assumed
to be identical to the epoch of the demodulating carrier. In practice, perfect syn-
chronization between the carriers is not possible, which leads to distortion in
the signal reconstructed from demodulation. To illustrate the effect of distor-
tion introduced by unsynchronized carriers, consider the following modulated
signal:
s
(
t
)
=
A
cos(
ω
c
t
+ φ
c
)
+
Akm
(
t
) cos(
ω
c
t
+ φ
c
)
,
(8.8)
as derived in Eq. (8.2). Assume that the demodulator carrier is given by
c
2
(
t
)
=
A
cos(
ω
c
t
+ θ
c
(
t
))
,
(8.9)
which has a time-varying epoch
θ
c
(
t
)
= φ
c
. Using
c
2
(
t
), the demodulated signal
is given by
d
(
t
)
=
s
(
t
)
c
2
(
t
)
=
[
A
cos(
ω
c
t
+φ
c
)
+
Akm
(
t
) cos(
ω
c
t
+ φ
c
)] cos(
ω
c
t
+θ
c
(
t
))
,
(8.10)
which simplifies to
d
(
t
)
=
A
+
A
2
[1
+
km
(
t
)] cos(
φ
c
− θ
c
(
t
))
2
[1
+
km
(
t
)] cos(2
ω
c
t
+ φ
c
+ θ
c
(
t
))
.
d
low
(
t
)
d
high
(
t
)
(8.11)
Equation (8.11) illustrates that the demodulated signal contains a low-frequency
component
d
low
(
t
) and a higher-frequency component
d
high
(
t
). By passing the
demodulated signal through a lowpass filter, the higher-frequency component
is removed. The output of the lowpass filter is given by
y
(
t
)
=
A
(8.12)
2
[1
+
km
(
t
)] cos(
φ
c
− θ
c
(
t
))
.
Even after eliminating the dc component, the reconstructed signal has the fol-
lowing form:
y
(
t
)
=
A
(8.13)
2
km
(
t
) cos(
φ
c
− θ
c
(
t
))
,
where distortion is caused by the factor of cos(
φ
c
− θ
c
(
t
)). Since the epoch
θ
c
(
t
) is time-varying, it is difficult to eliminate the distortion. To reconstruct
x
(
t
) precisely, the phase difference between the carrier signals used at the mod-
ulator and demodulator must be kept equal to zero over time. In other words,
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