Image Processing Reference
In-Depth Information
Fig. 9. Optimum minimum-phase IIR HBF decimator block structure (a) and SFG of the 1st
(2nd) order allpass sections (b)
MoR:
f
Op
=
f
n
Dec:
f
Op
=
f
n
/2 Int:
f
Op
=
f
n
/2
(
+
)
n
mc
n
n
1
/2
N
M
(
n
−
1
)
/2
N
A
3
(
n
−
1
)
/2
+
1
3
(
n
−
1
)
/2
−
−
N
Op
2
n
1
2
n
2
Table 2. Expenditure of real minimum-phase IIR HBF;
n
:order,
n
mc
: McMillan degree,
N
M
(
N
A
)
: number of multipliers (adders),
f
Op
: operational clock frequency
z
2
Hence, the polyphase components
A
l
(
)
:
=
A
l
(
z
d
)
,
l
=
0, 1 of Fig. 9(a) operate at the
reduced output sampling rate
f
d
=
f
n
/2, and the McMillan degree
n
mc
is almost halved.
The optimum interpolating structure is readily derived from the decimator by applying the
multirate transposition rules (cf. Section 3 and [Göckler & Groth (2004)]). Computational
complexity is presented in Table 2, also indicating the respective operational rates
f
Op
for the
N
Op
arithmetical operations.
Elliptic filters also allow for multiplierless implementations with small quantization error,
or implementations with a reduced number of shift-and-add operations in multipliers
[Lutovac & Milic (1997; 2000); Milic (2009)].
2.1.3 Comparison of real FIR and IIR HBF
The comparison of the Tables 1 and 2 shows that
N
FIR
Op
N
IIR
<
Op
for the same filter order
n
,
where all operations are performed at the operational rate
f
Op
, as given in these Tables. Since,
however, the filter order
n
IIR
<
n
FIR
for any type of approximation, the
computational load of an MP IIR HBF is generally smaller than that of an LP FIR HBF, as it is
well known [Lutovac et al. (2001); Schüssler & Steffen (1998)].
The relative computational advantage of equiripple minimax designs of monorate IIR
halfband filters and polyphase decimators [Parks & Burrus (1987)], respectively, is depicted
in Fig. 10 where, in extension to [Lutovac et al. (2001)], the expenditure
N
Op
is indicated as a
parameter along with the filter order
n
. Note that the IIR and FIR curves of the lowest order
filters differ by just one operation despite the LP property of the FIR HBF.
A specification of a design example is deduced from Fig. 10:
n
IIR
=
n
FIR
or even
n
IIR
5and
n
FIR
=
14,
=
respectively, with a passband cut-off frequency of
f
p
0.1769
f
n
at the intersection point
of the associated expenditure curves: Fig. 11. As a result, the stopband attenuations of both
filters are the same (cf. Fig. 10). In addition, for both designs the typical pole-zero plots are
shown [Schüssler & Steffen (1998; 2001)]. From the point of view of expenditure, the MP IIR
HBF decimator (
N
Op
=
=
3) outperforms its LP FIR counterpart (
N
Op
=
=
9,
n
mc
12,
n
mc
8).