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).
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