Digital Signal Processing Reference
In-Depth Information
12.11
Without explicitly determining the DFT
X
[
r
] of the time-limited
sequence
x
[
k
]
=
[6
8
−
54 622789442
,
compute the following functions of the DFT
X
[
r
]:
10
(i)
X
[0];
(iv)
X
[
r
];
(ii)
X
[10];
r
=
0
10
(iii)
X
[6];
X
[
r
]
2
.
(v)
r
=
0
12.12
Without explicitly determining the the time-limited sequence
x
[
k
] for
the following DFT:
X
[
r
]
=
[12
,
8
+
j4
,
−
5
,
4
+
j1
,
16
,
16
,
4
−
j1
,
−
5
,
8
−
j4]
,
compute the following functions of the DFT
X
[
r
]:
9
(i)
x
[0];
(ii)
x
[9];
(iii)
x
[6];
(iv)
x
[
k
];
r
=
0
9
x
[
k
]
2
;
(v)
r
=
0
12.13
Given the DFT pair
DFT
←−−→
X
[
r
]
,
x
[
k
]
for a sequence of length
N
, express the DFT of the following sequences
as a function of
X
[
r
]:
(i)
y
[
k
]
=
x
[2
k
];
x
[0
.
5
k
]
k
even
0 elsewhere;
(iii)
y
[
k
]
=
x
[
N
−
1
−
k
]
(ii)
y
[
k
]
=
for
0
≤
k
≤
N
−
1;
x
[
k
]0
≤
k
≤
N
−
1
0
N
≤
k
≤
2
N
−
1;
(v)
y
[
k
]
=
(
x
[
k
]
−
x
[
k
−
2])e
j(10
π
k
/
N
)
.
(iv)
y
[
k
]
=
12.14
Compute the linear convolution of the following pair of time-limited
sequences using the DFT-based approach. Be careful with the time
indices of the result of the linear convolution.
k
0
≤
k
≤
3
2
−
1
≤
k
≤
2
(i)
x
1
[
k
]
=
and
x
2
[
k
]
=
0
otherwise
0
otherwise;
5
k
=
0
,
1
(ii)
x
1
[
k
]
=
k
for 0
≤
k
≤
3
and
x
2
[
k
]
=
0
otherwise;
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