Digital Signal Processing Reference
In-Depth Information
variable change:
ω
1
→
ω
2
,
ω
2
→−
ω
1
, the transfer function
H
2
(
z
1
,
z
2
) can be derived from
H
1
(
z
1
,
z
2
) as
H
2
(
z
1
,
z
2
)=
H
1
(
z
2
,
z
1
−1
). A more general filter belonging to this class is a rhomboi‐
dal filter, as shown in Fig.2 (f). In this case the two oriented LP filters may have different
bandwidths and their axes are no longer perpendicular to each other.
3.2. Design method for diamond-type filters
The issue of this section is to find the transfer function
H
2
D
(
z
1
,
z
2
) of the desired 2D filter using
a complex frequency transformation
s
→
F
(
z
1
,
z
2
). From a prototype
H
P
(
s
)=
H
P
(
jω
) (which
varies on one axis only), a 2D oriented filter is obtained by rotating the axes of the plane (
ω
1
,
ω
2
)
by an angle
φ
. The rotation is defined by the following linear transformation, where
ω
1
,
ω
2
are
the original frequency variables and
¯
1
,
¯
2
the rotated ones:
é ù
w
é
cos sin
sin cos
j
j
ù
é ù
w
1
1
=
×
ê ú
ê
ú
ê ú
(17)
w
-
j
j
w
ë û
ë
û
ë û
2
2
The spatial orientation is specified by an angle
φ
with respect to
ω
1
−axis, defined by the 1D
to 2D frequency mapping
ω
→
ω
1
cos
φ
+
ω
2
sin
φ
. By substitution, we obtain the oriented filter
transfer function
H
φ
(
ω
1
,
ω
2
)=
H
P
(
ω
1
cos
φ
+
ω
2
sin
φ
). In the complex plane (
s
1
,
s
2
) the above
frequency transformation becomes:
s s
®
cos
j
+
s
sin
j
(18)
1
2
The oriented filter
H
φ
(
ω
1
,
ω
2
) has the frequency response magnitude section along the line
ω
1
cos
φ
+
ω
2
sin
φ
=0, identical with prototype
H
P
(
ω
), and constant along the perpendicular line
(filter longitudinal axis)
ω
1
sin
φ
−
ω
2
cos
φ
=0. The usual method to obtain a discrete filter from
an analog prototype is the bilinear transform. If the sample interval takes the value
T
=1, the
bilinear transform for
s
1
and
s
2
in the complex plane (
s
1
,
s
2
) has the form:
s
=
2( 1) ( 1)
z
-
z
+
s
=
2( 1) ( 1)
z
-
z
+
(19)
1
1
1
2
2
2
This method is straightforward, still the resulted 2D filter will present linearity distortions in
its shape, which increase towards the limits of the frequency plane as compared to the ideal
frequency response. This is mainly due to the so-called frequency warping effect of the bilinear
transform, expressed by the continuous to discrete frequency mapping:
(
)
w
=
(2 ) arctg
T
×
w
T
2
(20)
a
