Digital Signal Processing Reference
In-Depth Information
2.8.1.1 Compact Half-Band Coefficients
If the zero values were to be removed from the filter, we could represent the half-
band filter as a compact filter , of one half the length of the original low-pass
filter. Using the symmetry property as before, this yields a reduction in coefficient
count to one quarter of the original filter length to
L
/4+1 or [
L
/4]+2, depending on
whether
L
/4 is an integer, or not, as in the latter. The brackets [
a
/
b
] are interpreted
as the integer part when
a
is divided by
b
. This is the recommended approach for
×
ˆ
k
ˆ
2 time domain decimation using symmetric FIR filters. The filter
is obtained
k
from the original half-band filter
h
for two conditions of
L
/4, using:
k
L/4 = integer; L even:
ˆ
L
h
=
h
k
=
1
2
,
k
2
k
4
(2.29)
ˆ
h
=
h
L
L
+
1
+
1
4
2
L/4
integer; L even:
ˆ
L
h
=
h
k
=
1
2
,
[
]
+
1
k
2
k
1
4
(2.30)
ˆ
h
=
h
L
L
[
]
+
2
+
1
4
2
ˆ
The decimation process involving
when
L
is even becomes
k
L
1
ˆ
h
∑
ˆ
ˆ
y
=
1
h
x
+
h
(
x
+
x
)
m
=
1
2
,
1
(
N
P
)
m
L
P
+
1
k
2
k
+
m
1
P
2
k
+
m
+
1
+
m
1
K
2
ˆ
h
2
k
=
1
y
=
y
k
=
1
2
,
,
[
N
/
2
]
out
,
k
2
k
1
P
=
4
L
5
ˆ
h
(2.31)
where,
h
is the length of the compact half-band filter,
N
is the number of data
points and
y
L
ˆ
2
decimation operations where coefficient storage is limited. In order to implement
this filtering operation, two Matlab
out
is the decimated output. This approach is useful in dedicated
×
functions have been created for illustration
purposes:
compacLPF,
which compacts low-pass half-band coefficients, and
decx2
,
which carries out the
2 decimation in (2.31). Figures 2.24 and 2.25 present these
two functions. Note that in the implementation, we zero-padded
L
+1 zeros at the
beginning and end of the data set, so that there is no need to implement the shift
operation, as has been done previously. However, the consequence of this is the
×
