Digital Signal Processing Reference
In-Depth Information
R
1
node 1
i
R
1
(
t
)
+
−
+
−
−
−
v
(
t
)
i
(
t
)
R
2
y
(
t
)
L
y
(
t
)
C
R
C
i
C
(
t
)
i
R
2
(
t
)
Fig. P2.1. RC circuit consisting of
two resistors (
R
1
and
R
2
) and a
capacitor
C
.
Fig. P2.2. Resonator in an AM modulator.
non-linear
device
Fig. P2.3. AM demodulator. The
input signal is represented by
v
1
(
t
) =
A
c
cos(2π
f
c
t
) +
m
(
t
),
where
A
c
cos(2π
f
c
t
)isthe
carrier and
m
(
t
)isthe
modulating signal.
m
(
t
)
v
1
(
t
)
v
2
(
t
)
C
R
L
A
c
cos(2
p
f
c
t
)
Problems
2.1
The electrical circuit shown in Fig. P2.1 consists of two resistors
R
1
and
R
2
and a capacitor
C
.
(i) Determine the differential equation relating the input voltage
V
in
(
t
)to
the output voltage
V
out
(
t
).
(ii) Determine whether the system is (a) linear, (b) time-invariant;
(c) memoryless; (d) causal, (e) invertible, and (f ) stable.
2.2
The resonant circuit shown in Fig. P2.2 is generally used as a resonator in
an amplitude modulation (AM) system.
(i) Determine the relationship between the input
i
(
t
) and the output
v
(
t
)
of the AM modulator.
(ii) Determine whether the system is (a) linear, (b) time-invariant;
(c) memoryless; (d) causal, (e) invertible, and (f ) stable.
2.3
Figure P2.3 shows the schematic of a square-law demodulator used in the
demodulation of an AM signal. Demodulation is the process of extract-
ing the information-bearing signal from the modulated signal. The input-
output relationship of the non-linear device is approximated by (assuming
v
1
(
t
) is small)
v
2
(
t
)
=
c
1
v
1
(
t
)
+
c
2
v
1
(
t
)
,
where
c
1
and
c
2
are constants, and
v
1
(
t
) and
v
2
(
t
) are, respectively, the
input and output signals.
(i) Show that the demodulator is a non-linear device.
(ii) Determine
whether
the
non-linear
device
is
(a)
time-invariant,
(b) memoryless, (c) invertible, and (d) stable.
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