Digital Signal Processing Reference
In-Depth Information
Problems
12.1
Calculate analytically the DFT of the following sequences, with length
0
≤
k
≤
N
−
1:
1
k
=
0
,
3
(i)
x
[
k
]
=
with length
N
=
4;
0
k
=
1
,
2
1
k
even
(ii)
x
[
k
]
=
with length
N
=
8;
−
1
k
odd
(iii)
x
[
k
]
=
0
.
6
k
with length
N
=
8;
(iv)
x
[
k
]
=
u
[
k
]
−
u
[
k
−
8] with length
N
=
8;
(v)
x
[
k
]
=
cos(
ω
0
k
) with
ω
0
=
2
π
m
/
N
,
m
∈
Z
.
12.2
Calculate
the
DFT
of
the
time-limited
sequences
specified
in
Examples 12.1(i)-(iv) using the matrix-vector approach.
12.3
Determine the time-limited sequence, with length 0
≤
k
≤
N
−
1, cor-
responding to the following DFTs
X
[
r
], which are defined for the DFT
index 0
≤
r
≤
N
−
1:
(i)
X
[
r
]
=
[1
+
j4
, −
2
−
j3
, −
2
+
j3
,
1
−
j4] with
N
=
4;
(ii)
X
[
r
]
=
[1
,
0
,
0
,
1] with
N
=
4;
(iii)
X
[
r
]
=
exp
−
j(2
π
k
0
r
/
N
), where
k
0
is a constant;
0
.
5
Nr
=
k
0
,
N
−
k
0
0
(iv)
X
[
r
]
=
where
k
0
is a constant;
elsewhere
(v)
X
[
r
]
=
e
−
j
π
r
(
m
−
1)
/
N
sin (
π
rm
/
N
)
sin(
π
r
/
N
)
where
m
∈ Z
and 0
<
m
<
N
;
r
N
(vi)
X
[
r
]
=
for
0
≤
r
≤
N
−
1
.
12.4
In Problem 11.1, we determined the DTFT representation for each of
the following DT periodic sequences using the DTFS. Using M
ATLAB
,
compute the DTFT representation based on the FFT algorithm. Plot the
frequency characteristics and compare the computed results with the ana-
lytical results derived in Chapter 11.
(i)
x
[
k
]
=
k
,
for 0
≤
k
≤
5
and
x
[
k
+
6]
=
x
[
k
];
10
≤
k
≤
2
(ii)
x
[
k
]
=
0
.
53
≤
k
≤
5
06
≤
k
≤
8
and
x
[
k
+
9]
=
x
[
k
];
2
π
7
π
4
(iii)
x
[
k
]
=
3 sin
k
+
;
j
5
π
3
π
4
(iv)
x
[
k
]
=
2exp
k
+
;
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