Digital Signal Processing Reference
In-Depth Information
frequency transformation is applied, which yields a 2D filter with the desired square shape in
the frequency plane. The proposed method combines the analytical approach with numerical
approximations.
3.1. Specification of diamond-type filters in the frequency plane
The standard diamond filter has the shape in the frequency plane as shown in Fig.2 (a). It is a
square with a side length of
π
2, while its axis is tilted by an angle of
φ
=
π
/4 radians about
the two frequency axes. Next we will consider the orientation angle
φ
about the
ω
2
− (vertical)
axis. In this chapter a more general case is approached, i.e. a 2D diamond-type filter with a
square shape in the frequency plane, but with arbitrary axis orientation angle, as shown in
Fig. 2(e). Next we refer to them as diamond-type filters, since they are more general than the
diamond filter from Fig. 2 (a).
The diamond-type filter in Fig.2 (e) is derived as the intersection of two oriented low-pass
filters whose axes are perpendicular to each other, for which the shape in the frequency plane
is given in Fig.2 (c), (d). Correspondingly, the diamond-type filter transfer function
H
D
(
z
1
,
z
2
)
results as a product of two partial transfer functions:
H z z H z z H z z
( , )
=
( , ) ( , )
×
(16)
1 2
1 1 2
2 1 2
(c)
(a)
(b)
(f)
(d)
(e)
Figure 2.
(a) diamond filter; (b) wide-band oriented filter; (c), (d) wide-band oriented filters with orientations forming
an angle
φ
=
π
/2; (e) square-shaped filter resulted as product of the above oriented filters; (f) rhomboidal filter
The frequency characteristic of
H
2
(
z
1
,
z
2
) is ideally identical to the frequency characteristic of
H
1
(
z
1
,
z
2
) rotated by an angle of
φ
=
π
/2. Since this rotation of axes implies the frequency
