Image Processing Reference
In-Depth Information
Dec:
R C
Int:
C R
Dec:
C C
Int:
C C
n mc
n
2 n
N M
n
2 n
N A
3
(
n
1
)
6
(
n
1
)+
2
6
(
n
1
)
N Op
4 n
3
8 n
4
8 n
6
Table 6. Expenditure of minimum-phase IIR COHBF; n :order, n mc : McMillan degree,
N M (
N A )
: number of multipliers (adders), operational clock frequency: f Op =
f n /2
Specification and properties
All properties of the real IIR HBF are basically retained except of those subjected to the
frequency shift operation of (28). This applies to the filter specification depicted in Fig. 8
and, hence, (6) is replaced with (32). Obviously, power (14) and allpass (15) complementarity
are retained as follows
e j c 4 ) ) |
2
e j −π ( 1 + c /4 )) ) |
2
|
H
(
+ |
H
(
=
1,
(35)
=
e j c 4 ) )+
e j −π ( 1 + c /4 )) )
H
(
H
(
1,
(36)
where (3) is applied in the frequency domain.
Efficient implementations
Introducing (34) in (16), the transfer function is frequency-shifted by f 1 =
(Ω = π
)
f n /8
/4
:
A 0 (
.
+
1
2
1
j
2 z 1 A 1 (
jz 2
jz 2
(
)=
)+
)
H
z
(37)
The optimal structure of an n
5th order MP IIR COHBF decimator for real input signals
is shown in Fig. 18(a) along with the elementary SFG of the allpass sections Fig. 18(b).
Doubling of the structure according to Fig. 19 allows for full-complex signal processing.
Multirate transposition [Göckler & Groth (2004)] is again applied to derive the corresponding
dual structure for interpolation.
The expenditure of the half- (
=
) COHBF decimators
and their transposes is listed in Table 6. A comparison of Tables 2 and 6 shows that the
half-complex IIR COHBF sample rate converter (cf. Fig. 18(a)) requires almost twice, whereas
the full-complex IIR COHBF (cf. Fig. 19) requires even four times the expenditure of that of
the real IIR HBF system depicted in Fig. 9.
R C
) and the full-complex (
C C
2.3.3 Comparison of FIR and IIR COHBF
LP FIR COHBF structures allow for implementations that utilize the coefficient symmetry
property. Hence, the required expenditure is just slightly higher than that needed for CHBF.
On the other hand, the expenditure of MP IIR COHBF is almost twice as high as that of
the corresponding CHBF, since it is not possible to exploit memory and coefficient sharing.
Almost the whole structure has to be doubled for a full-complex decimator (cf. Fig. 19).
2.4 Conclusion: Family of single real and complex halfband filters
We have recalled basic properties and design outlines of linear-phase FIR and minimum-phase
IIR halfband filters, predominantly for the purpose of sample rate alteration by a factor of
two, which have a passband centre frequency out of the specific set defined by (1). Our
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