Digital Signal Processing Reference
In-Depth Information
In Fig.1 (d) the shifted filter response magnitude for
ω
01
=0.416
π
is shown. Another useful
analog prototype is the selective second-order (resonant) filter with central frequency
ω
0
:
(
)
2
2
0
H s
( )
=
a
s s
+
a w
s
+
(9)
The transfer function magnitude for such a filter with
α
=0.1 and
ω
0
=1.3 is shown in Fig. 1
(e). This will be further used as a prototype for two-directional filters.
(a)
(b)
)(b
b)
(e)
(c)
(d)
Figure 1.
Frequency response magnitudes of: (a) LP elliptic prototype of order 6; (b) LP elliptic prototype of order 4;
very selective first-order filter with central frequencies
ω
0
=0 (c) and
ω
0
=0.416
π
(d); (e) selective band-pass filter with
ω
0
=1.3
A useful zero-phase prototype can be obtained from the general function (1) by preserving
only the magnitude characteristics of the 1D filter; this prototype will be further used to design
2D zero-phase FIR filters of different types, specifically circular filters, with real-valued
transfer functions. In order to obtain a zero-phase filter, we consider the magnitude charac‐
teristics of
H
P
(
jω
), defined by the absolute value
|
H
P
(
jω
)
|
=
|
P
(
jω
)
|
/
|
Q
(
jω
)
|
. We look
for a series expansion of the magnitude
|
H
P
(
jω
)
|
that has to be an approximation as accurate
as possible on the frequency domain −
π
,
π
. The most convenient for our purpose is the
Chebyshev series expansion, because it yields an efficient approximation of a given function,
which is uniform along the desired interval. The Chebyshev series in powers of the frequency
variable
ω
for a given function on a specified interval can be easily found using a symbolic
computation software like MAPLE. However, we will finally need a trigonometric expansion
of
|
H
P
(
jω
)
|
, namely in cos(
nω
), rather than a polynomial expansion in powers of
ω
.
Therefore, prior to Chebyshev series calculation, we apply the change of variable:
