Digital Signal Processing Reference
In-Depth Information
10.8
Using the sliding tape method and the following equation:
y
[
k
]
=
∞
x
[
m
]
h
[
k
−
m
]
,
m
=−∞
calculate the convolution of the sequences in Example 10.9 and show
that the convolution output is identical to that obtained in Example 10.9.
10.9
The linear convolution between two sequences
x
[
k
] and
h
[
k
] of lengths
K
1
and
K
2
, respectively, can be performed using periodic convolution by
considering periodic extensions of the two zero-padded sequences. Cal-
culate the linear convolution of the sequences defined in Example 10.8
using the periodic convolution approach with the fundamental period
K
0
set to 10. Repeat for
K
0
set to 13.
10.10
Repeat Example 10.7 using the periodic convolution approach with
K
set to 10.
10.11
Repeat Example 10.7 using the periodic convolution approach with
K
set to 15.
10.12
Repeat Example 10.12 with
K
set to 8.
10.13
Calculate the unit step response of the DT systems with the following
impulse responses:
(a)
h
[
k
]
=
u
[
k
+
7]
−
u
[
k
−
8];
(b)
h
[
k
]
=
0
.
4
k
u
[
k
];
(c)
h
[
k
]
=
2
k
u
[
−
k
];
(d)
h
[
k
]
=
0
.
6
k
;
∞
(
−
1)
m
δ
(
k
−
2
m
)
.
(e)
h
[
k
]
=
m
=−∞
10.14
Simplify the following expressions using the properties of discrete-time
convolution:
(a) (
x
[
k
]
+
2
δ
[
k
−
1])
∗ δ
[
k
−
2];
(b) (
x
[
k
]
+
2
δ
[
k
−
1])
∗
(
δ
[
k
+
1]
+ δ
[
k
−
2]);
(c) (
x
[
k
]
−
u
[
k
−
1])
∗ δ
[
k
−
2];
(d) (
x
[
k
]
−
x
[
k
−
1])
∗
u
[
k
],
where
x
[
k
] is an arbitrary function,
δ
[
k
] is the unit impulse function, and
u
[
k
] is the unit step function.
10.15
Prove Definition 10.3 by expanding the right-hand side of the periodic
convolution and showing it to be equal to the left-hand side.
10.16
Prove the time-shifting property stated in Eq. (10.24).
10.17
Show that the linear convolution
y
[
k
] of a time-limited DT sequence
x
1
[
k
] that is non-zero only within the range
k
ℓ
1
≤
k
≤
k
u
1
with another
time-limited DT sequence
x
2
[
k
] that is non-zero only within the range
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