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In-Depth Information
W
1
(e
j
ω
)
π
/2
πω
F I GU R E 14 . 14
Spectrum of the downsampled low-pass filter output.
Equation (
33
), we also get the odd indexed terms of
y
1
,
n
; however, as these terms are all zero
(see Equation (
26
)), they do not contribute to the Z-transform.
Substituting
z
e
j
ω
we get
=
1
2
Y
1
(
1
2
Y
1
(
−
e
j
2
e
j
2
e
j
ω
)
=
W
1
(
)
+
)
(36)
e
j
ω
)
Plotting this for the
Y
1
(
of Figure
14.13
, we get the spectral shape shown in Figure
14.14
;
that is, the spectral shape of the downsampled signal is a stretched version of the spectral shape
of the original signal. A similar situation exists for the downsampled signal
w
2
,
n
.
14.5.2 Upsampling
Let's take a look now at what happens after the upsampling. The upsampled sequence
v
1
,
n
can be written as
w
1
,
n
even
2
v
1
,
n
=
(37)
0
n
odd
The Z-transform
V
1
(
z
)
is thus
n
=−∞
v
1
,
n
z
−
n
V
1
(
z
)
=
(38)
n
=−∞
w
1
,
2
z
−
n
=
n
even
(39)
n
m
=−∞
w
1
,
m
z
−
2
m
=
(40)
z
2
=
W
1
(
)
(41)
The spectrum is sketched in Figure
14.15
. The “stretching” of the sequence in the time
domain has led to a compression in the frequency domain. This compression has also resulted
in a replication of the spectrum in the
interval. This replication effect is called
imaging
.
We remove the images by using an ideal low-pass filter in the top branch and an ideal high-pass
filter in the bottom branch.
Because the use of the filters prior to sampling reduces the bandwidth, which in turn allows
the downsampling operation to proceed without aliasing, these filters are called
anti-aliasing
[
0
,π
]
