Image Processing Reference
In-Depth Information
e
j
π
/2
1
2
,(b)
Fig. 21. DF operations: (a) Real HBF prototype centrosymmetric about
H
0
(
)=
Two selected DF filter functions, (c,d) Spectra of decimated DF output signals
by (38),
o
defined by (9). According to (39),
highest efficiency is obtained by additionally introducing a suitable complex scaling factor of
unity magnitude:
∈{
0, 1, 2, 3
}
, with the RHBF impulse response
h
(
k
)
e
j
a
4
h
e
j
4
[
k
(
2
o
+
1
)+
a
]
=
1
2
a
2
,
z
o
=
j
k
(
o
+
)+
h
k
,
o
,
a
=
(
k
)
h
(
k
)
h
(
k
)
(39)
−
−
N
1
N
1
where
. By directly equating (39), and relating the result
to (9) with a suitable choice of the constant
a
−
≤
k
≤
and
o
∈{
0, 1, 2, 3
}
2
2
=
2
o
+
1 compliant with (29), we get :
⎧
⎨
⎩
2
j
o
+
1
2
k
=
0
h
k
,
o
=
0
k
=
2
l
l
=
1,...,
(
N
−
3
)
/4
(40)
j
(
k
+
1
)(
o
+
1
2
)
h
k
k
=
2
l
−
1
l
=
1,...,
(
N
+
1
)
/4
with the symmetry property:
j
(
2
o
+
1
)
k
h
k
,
o
h
=
−
∀
k
>
0,
o
∈{
0, 1, 2, 3
}
.
(41)
−
k
,
o
The respective COHBF centre coefficient
)
4
]+
)
4
]
}
1
2
{
h
0,
o
=
[(
+
[(
+
∈{
}
cos
2
o
1
jsin
2
o
1
,
o
0, 1, 2, 3
,
(42)