Digital Signal Processing Reference
In-Depth Information
Fig. 11.14. DT sequence
x
[
k
]
used in Example 11.10.
x
[
k
]
1
0.75
0.5
0.2
0.125
k
−4
−2
0
4
1 0
12
14
16
18
with
N
=
3; and (ii) a time-shifted decaying exponential sequence, denoted by
x
3
[
k
] in Example 11.6, (iii), as follows:
p
k
u
[
k
]
,
x
3
[
k
]
=
with decay factor
p
=
0
.
5. In terms of
x
2
[
k
] and
x
3
[
k
], the expression for
x
[
k
]
is given by
x
[
k
]
=
0
.
75
x
2
[
k
−
6]
+
x
3
[
k
−
12]
.
Using the linearity and time-shifting properties, the DTFT
X
(
Ω
)of
x
[
k
]isgiven
by
−
j6
Ω
X
2
(
Ω
)
+
e
−
j12
Ω
X
3
(
Ω
)
.
X
(
Ω
)
=
0
.
75e
From the results in Example 11.6, the DTFTs for the sequences
x
2
[
k
] and
x
3
[
k
]
are given by
X
2
(
Ω
)
=
sin(3
.
5
Ω
)
sin(0
.
5
Ω
)
1
1
−
0
.
5e
X
3
(
Ω
)
=
−
j
Ω
.
and
Substituting the values of the DTFTs results in the following:
−
j6
Ω
sin(3
.
5
Ω
)
sin(0
.
5
Ω
)
1
1
−
0
.
5e
−
j
Ω
.
−
j12
Ω
X
(
Ω
)
=
0
.
75e
+
e
11.5.6 Frequency shifting
In the time-shifting property, we observed the change in the DTFT when the DT
sequence
x
[
k
] is shifted in the time domain. The frequency-shifting property
addresses the converse problem of how shifting the DTFT
X
(
Ω
) in the frequency
domain affects the sequence
x
[
k
] in the time domain.
If
DTFT
←−−→
x
[
k
]
X
(
Ω
)
then
DTFT
←−−→
x
[
k
]e
j
Ω
0
k
X
(
Ω
−
Ω
0
)
,
(11.43)
for 0
≤
Ω
0
<
2
π
.
Example 11.11
Using
the
frequency-shifting
property,
calculate
the
DTFT
of
x
[
k
]
=
cos(
Ω
0
k
) cos(
Ω
1
k
) with (
Ω
0
+
Ω
1
)
<π
.
Search WWH ::
Custom Search