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Homework Set #6: Discrete Fourier Transform
1.
Compute the
16-point DFTs of
(a)
x
1
[
n
]=[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]; (b)
x
2
[
n
]=[0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1];
(c)
x
3
[
n
]=[1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0];
(d)
x
4
[
n
]=[1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0]
(d)
x
5
[
n
]=[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
2.
An
8-point
sequence is given as
x
[
n
] = [3
3 -2
5 -4
1 -4
2], its 8-point DFT is expressed by
X
8
[
k
].
By
16-point
DFT program, we can compute
X
16
[
k
] = DFT
16
(
x
[
n
]), where we pad 8 extra zeros in the
end). Please compute the following results,
(a)
∑
=
−
7
0
8
]
[
)
1
(
k
k
k
X
;
(b)
DFT
16
(DFT
16
(
x
[
n
]));
(c)
X
8
[4];
(d)
X
16
[8];
(e)
11
)
8
/
2
(
7
0
8
]
[
=
=
∑
n
kn
j
k
e
k
X
π
.
3.
Followed by Problem 1, in terms of
X
16
(
k
) or
X
8
(
k
), please find the DFTs of the following sequences.
(a)

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