Digital Signal Processing Reference
In-Depth Information
13.16
Determine the output response of the following LTID systems with the
specified inputs and impulse responses:
(i)
x
[
k
]
=
u
[
k
+
2]
−
u
[
−
k
−
3]
and
h
[
k
]
=
u
[
k
−
5]
−
u
[
k
−
6];
−
k
u
[
k
−
4];
(ii)
x
[
k
]
=
u
[
k
]
=
u
[
k
−
9]
and
h
[
k
]
=
3
−
k
u
[
k
]
(iii)
x
[
k
]
=
2
and
h
[
k
]
=
k
(
u
[
k
]
−
u
[
k
−
4]);
−
k
(iv)
x
[
k
]
=
u
[
k
]
and
h
[
k
]
=
4
;
−
k
u
[
k
]
h
[
k
]
=
2
k
u
[
−
k
−
1].
(v)
x
[
k
]
=
2
and
13.17
When the DT sequence
x
[
k
]
=
(1
/
4)
k
u
[
k
]
+
(1
/
3)
k
u
[
k
]
is applied at the input of a causal LTID system, the output response is
given by
y
[
k
]
=
2 (1
/
4)
k
u
[
k
]
−
4 (3
/
4)
k
u
[
k
]
.
(i) Determine the z-transfer function
H
(
z
) of the LTID system.
(ii) Determine the impulse response
h
[
k
] of the LTID system.
(iii) Determine the difference-equation representation of the LTID sys-
tem.
13.18
Consider an LTIC system with the following transfer function:
e
sT
H
(
s
)
=
−
0
.
3
.
e
sT
Calculate the output response
y
(
t
) of the LTIC system for the following
input sequence:
∞
(0
.
2)
kT
δ
(
t
−
kT
)
.
f
(
t
)
=
k
=
0
13.19
Plot the poles and zeros of the following LTID systems. Assuming that
the systems are causal, determine if the systems are BIBO stable.
(i)
H
(
z
)
=
z
−
2
(
z
−
0
.
6
+
j0
.
8)(
z
2
+
0
.
25)
;
(ii)
H
(
z
)
=
(
z
−
2)(
z
−
1)
(
z
2
−
2
.
5
z
+
1)(
z
2
+
0
.
25)
;
(iii)
H
(
z
)
=
z
−
0
.
2
(
z
+
0
.
1)(
z
2
+
4)
;
(iv)
H
(
z
)
=
z
−
1
−
2
z
−
2
+
z
−
3
;
(
z
2
+
2
.
5
z
+
0
.
9
+
j0
.
15)
z
z
3
+
(1
.
8
+
j0
.
3)
z
2
+
(0
.
6
+
j0
.
6)
z
−
0
.
2
+
j0
.
3
;
(v)
H
(
z
)
=
z
3
−
1
.
2
z
2
+
2
.
5
z
+
0
.
8
z
6
+
0
.
3
z
5
+
0
.
23
z
4
+
0
.
209
z
3
+
0
.
1066
z
2
−
0
.
04162
z
−
0
.
07134
.
(vi)
H
(
z
)
=
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