Digital Signal Processing Reference
In-Depth Information
Fig. P5.22. (a) RC circuit
system; (b) input signal.
R
+
y
(
t
)
v
(
t
)
v
(
t
)
C
−
1
t
T
T
0
LTIC system
−
2
2
(a)
(b)
Fig. P5.23. RC circuit with
sinusoidal input signal
considered in Problem 5.23.
R
y
(
t
)
x
(
t
)
C
−
5.21
Consider the LTIC systems with the following input-output pairs:
(a)
x
(
t
)
=
e
−
2
t
u
(
t
)
−
2
t
u
(
t
);
and
y
(
t
)
=
5e
−
2
t
u
(
t
)
−
2(
t
−
4)
u
(
t
−
4);
(b)
x
(
t
)
=
e
and
y
(
t
)
=
3e
−
2
t
u
(
t
)
y
(
t
)
=
t
3
e
−
2
t
u
(
t
);
(c)
x
(
t
)
=
e
and
−
3
t
u
(
t
)
.
For each of the above systems, determine (i) the transfer function, (ii) the
impulse response function, and (iii) the input-output relationship using
linear constant-coefficient differential equations.
−
2
t
u
(
t
)
−
t
u
(
t
)
+
e
(d)
x
(
t
)
=
e
and
y
(
t
)
=
e
5.22
Determine the transfer function of the system shown in Fig. P5.22(a).
Calculate the output of the system for the input signal shown in Fig.
P5.22(b).
5.23
Using the convolution property of the CTFT, calculate the output of the
system shown in Fig. P5.23 for the input signals (i)
x
1
(
t
)
=
cos(
ω
0
t
), and
(ii)
x
2
(
t
)
=
sin(
ω
0
t
).
5.24
Sketch
the
gain
and
phase
responses
for
the
LTIC
systems
in
Problem 5.20.
5.25
Sketch
the
gain
and
phase
responses
for
the
LTIC
systems
in
Problem 5.21.
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