First we treat filter design for interpolation.
Consider an input signal
xn
x
n
that is
ω
0
ω
0
-bandlimited in the DTFT domain.
If we upsample by factor
L
L to get
vm
v
m
, the desired portion of
Vⅇⅈω
V
ω
is the spectrum in
-πLπL
L
L
,
while the undesired portion is the remainder of
-ππ
.
Noting from Figure 1 that
Vⅇⅈω
V
ω
has zero energy in the regions
2kπ+
ω
0
L2k+1π-
ω
0
L
,
k∈ℤ
2
k
ω
0
L
2
k
1
ω
0
L
,
k
(1)
the anti-imaging filter can be designed with transition bands in
these regions (rather than passbands or stopbands). For a given
number of taps, the additional degrees of freedom offered by
these transition bands allows for better responses in the
passbands and stopbands. The resulting filter design
specifications are shown in the
bottom subplot below.
Next we treat filter design for decimation. Say that the
desired spectral component of the input
signal is bandlimited to
ω
0
M<πM
ω
0
M
M
and we have decided to downsample by MM.
The goal is to minimally distort the input spectrum over
-
ω
0
M
ω
0
M
ω
0
M
ω
0
M
, i.e., the post-decimation
spectrum over
-
ω
0
ω
0
ω
0
ω
0
.
Thus, we must not allow any aliased signals to enter
-
ω
0
ω
0
ω
0
ω
0
.
To allow for extra degrees of freedom in the filter design,
we do allow aliasing to enter the
post-decimation spectrum outside of
-
ω
0
ω
0
ω
0
ω
0
within
-ππ
.
Since the input spectral regions which alias outside of
-
ω
0
ω
0
ω
0
ω
0
are given by
2kπ+
ω
0
L2k+1π-
ω
0
L
,
k∈ℤ
2
k
ω
0
L
2
k
1
ω
0
L
,
k
(2)
(as shown in
Figure 2), we can
treat these regions as transition bands in the filter design.
The resulting filter design specifications are illustrated in
the middle subplot (
Figure 2).
The Noble Identities
