Digital Signal Processing Reference
In-Depth Information
Table 1.2. Values of the signal
p
[
k
] for −3 ≤
k
≤ 3
k
−
3
−
2
−
1
p
[
k
]
x
[
−
6]
=
0
x
[
−
4]
=
0
.
2
x
[
−
2]
=
0
.
6
x
[0]
=
1
x
[2]
=
0
.
6
x
[4]
=
0
.
2
x
[6]
=
0
Table 1.3. Values of the signal
q
[
k
] for −10
≤
k
≤ 10
−
10
−
9
−
8
−
7
−
6
−
5
−
4
k
q
[
k
]
x
[
−
5]
=
00
x
[
−
4]
=
0
.
20
x
[
−
3]
=
0
.
40
x
[
−
2]
=
0
.
6
k
−
3
−
2
−
1
q
[
k
]
0
x
[
−
1]
=
0.8
0
x
[0]
=
10
x
[1]
=
0
.
80
k
1 0
q
[
k
]
x
[2]
=
0
.
60
x
[3]
=
0
.
4
x
[4]
=
0
.
2
x
[5]
=
0
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
k
k
0
0
−10
−8
−6
−4
−2
1 0
−10
−8
−6
−4
−2
1 0
(a)
(b)
1.2
1
0.8
Fig. 1.26. Time scaling of the DT
signal in Example 1.17.
(a) Original DT sequence
x
[
k
].
(b) Decimated version
x
[2
k
], of
x
[
k
]. (c) Interpolated version
x
[0.5
k
] of signal
x
[
k
].
0.6
0.4
0.2
k
0
−10
−8
−6
−4
−2
1 0
(c)
Solution
Since
x
[
k
] is non-zero for
−
5
≤
k
≤
5, the non-zero values of the decimated
sequence
p
[
k
]
=
x
[2
k
] lie in the range
−
3
≤
k
≤
3. The non-zero values of
p
[
k
] are shown in Table 1.2. The waveform for
p
[
k
] is plotted in Fig. 1.26(b).
The waveform for the decimated sequence
p
[
k
] can be obtained by directly
compressing
the waveform for
x
[
k
] by a factor of 2 about the
y
-axis. While
performing the compression, the value of
x
[
k
]at
k
=
0 is retained in
p
[
k
]. On
both sides of the
k
=
0 sample, every second sample of
x
[
k
] is retained in
p
[
k
].
To determine
q
[
k
]
=
x
[
k
/
2], we first determine the range over which
x
[
k
/
2]
is non-zero. The non-zero values of
q
[
k
]
=
x
[
k
/
2] lie in the range
−
10
≤
k
≤
10 and are shown in Table 1.3. The waveform for
q
[
k
] is plotted in Fig. 1.26(c).
The waveform for the decimated sequence
q
[
k
] can be obtained by directly
expanding
the waveform for
x
[
k
] by a factor of 2 about the
y
-axis. During
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