Digital Signal Processing Reference
In-Depth Information
Substituting
t
by
α/
2 in Eq. (1.53), we obtain
α/
2
+
1
−
1
≤ α/
2
≤
0
1 0
≤ α/
2
≤
2
−α/
2
+
32
≤ α/
2
≤
3
0
x
(
α/
2)
=
elsewhere
.
By changing the independent variable from
α
to
t
and simplifying, we obtain
t
/
2
+
1
−
2
≤
t
≤
0
≤
t
≤
4
x
(
t
/
2)
=
−
t
/
2
+
34
≤
t
≤
6
0
elsewhere
,
which is plotted in Fig. 1.25(c). The waveform for
x
(0.5
t
) can also be obtained
directly by
expanding
the waveform for
x
(
t
) by a factor of 2. As for compression,
expansion is performed with respect to the
y
-axis such that the values
x
(
t
) and
x
(
t
/
2) at
t
=
0 are the same for both waveforms.
A CT signal
x
(
t
) can be scaled to
x
(
ct
) for any value of
c
. For the DTFT,
however, the time-scaling factor
c
is limited to integer values. We discuss the
time scaling of the DT sequence in the following.
1.3.2.1 Decimation
If a sequence
x
[
k
] is compressed by a factor
c
, some data samples of
x
[
k
] are
lost. For example, if we decimate
x
[
k
] by 2, the decimated function
y
[
k
]
=
x
[2
k
] retains only the alternate samples given by
x
[0],
x
[2],
x
[4], and so on.
Compression (referred to as decimation for DT sequences) is, therefore, an
irreversible process in the DT domain as the original sequence
x
[
k
] cannot be
recovered precisely from the decimated sequence
y
[
k
].
1.3.2.2 Interpolation
In the DT domain, expansion (also referred to as interpolation) is defined as
follows:
k
m
x
if
k
is a multiple of integer
m
x
(
m
)
[
k
]
=
(1.54)
0
otherwise.
The interpolated sequence
x
(
m
)
[
k
] inserts (
m
−
1) zeros in between adjacent
samples of the DT sequence
x
[
k
]. Interpolation of the DT sequence
x
[
k
]isa
reversible process as the original sequence
x
[
k
] can be recovered from
x
(
m
)
[
k
].
Example 1.17
Consider the DT sequence
x
[
k
] plotted in Fig. 1.26(a). Calculate and sketch
p
[
k
]
=
x
[2
k
] and
q
[
k
]
=
x
[
k
/
2].
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