Digital Signal Processing Reference
In-Depth Information
(
) (
)
B
=
3
a
×
Q
*
r
Q
+
ja
P
*
r
Q
+
ja
P
(43)
3
j
1
j
j
2
j
j
(
) (
) (
)
A
=
a
Q
+
ja
P
*
p
Q
+
ja
P
*
p
Q
+
ja
P
(44)
3
j
j
1
j
j
2
j
j
This implies the fact that the transfer function
H
3
(
z
1
,
z
2
) of the 2D three-directional filter with
templates
B
3
and
A
3
of size 7×7 results directly in a factorized form, which is an important
advantage in implementation. As a general remark on the method, using an analog prototype
instead of a digital one, as is currently done, simplifies the design in this case, as the frequency
mapping results simpler and leads to a 2D filter of lower complexity. The designed filters result
with complex coefficients, however such IIR filters can also be implemented (Nikolova et al., 2011).
6. Directional IIR filters designed in polar coordinates
We approach here a particular class of 2D filters, namely filters whose frequency response is
symmetric about the origin and has at the same time an angular periodicity. The contour plots
of their frequency response, resulted as sections with planes parallel with the frequency plane,
can be defined as closed curves which can be described in terms of a variable radius which is
a periodic function of the current angle formed with one of the axes.
It can be described in polar coordinates by
ρ
=
ρ
(
φ
), where
φ
is the angle formed by the radius
op with
ω
1
-axis, as shown in Fig.8(a) for a four-lobe filter. Therefore
ρ
(
φ
) is a periodic function
of the angle
φ
in the range
φ
∈ 0, 2
π
.
6.1. Spectral transformation for filters designed in polar coordinates
The main issue approached here is to find the transfer function of the desired 2D filter
H
2
D
(
z
1
,
z
2
) using appropriate frequency transformations of the form
ω
→
F
(
ω
1
,
ω
2
). The
elementary transfer functions (14) and (15) have the complex frequency responses:
(
) (
)
H j
( )
w
=
b b
+
cos
w
+
jb
sin
w
a
+
cos
w
+
j
sin
w
(45)
1
0
1
1
0
b b b
+
(
+
)cos
w
+
j b b
(
-
)sin
w
P
( )
w
1
2
0
2
0
H j
( )
w
=
=
(46)
2
a
+ +
(1 )cos
a
w
+
j a
(1 )sin
-
w
Q
( )
w
1
0
0
The proposed design method for these 2D filters is based on the frequency transformation:
2 2
F
:
¡
®
£
,
w
®
F z z B z z A z z
( , )
=
( , ) ( , )
(47)
1 2
f
1 2
f
1 2
