Image Processing Reference
In-Depth Information
Fig. 13. Optimum SFG of decimating linear-phase FIR CHBF
2.2.2 Minimum-Phase (MP) IIR filters
In the IIR CHBF case the frequency shift operation (3) is again applied in the
z
-domain. Using
(18), this is achieved by substituting the complex
z
-domain variable in the respective transfer
functions
H
(
)
z
and all corresponding SFG according to:
z
z
2
=
ze
−
j
2
z
:
=
=
−
jz
.
(24)
Specification and properties
All properties of the real IIR HBF are basically retained except of those subjected to the
frequency shift operation of (18). This applies to the filter specification depicted in Fig. 8
and, hence, (6) is replaced with (22). Obviously, power (14) and allpass (15) complementarity
are retained as follows
e
j
(Ω
∓
2
)
)
|
e
j
(Ω
±
2
)
)
|
2
2
|
H
(
+
|
H
(
=
1,
(25)
=
e
j
(Ω
∓
2
)
)+
e
j
(Ω
±
2
)
)
H
(
H
(
1,
(26)
where (3) is applied in the frequency domain.
Efficient implementations
Introducing (24) into (16) performs a frequency-shift of the transfer function
H
(
z
)
by
f
2
=
f
n
/4
(
Ω
2
=
π
/2
)
:
2
A
0
(
−
.
1
z
2
jz
−
1
A
1
(
−
z
2
H
(
z
)=
)+
)
(27)
The optimum general block structure of a decimating MP IIR HT, being up-scaled by 2, is
shown in Fig. 14(a) along with the SFG of the 1st (system theoretic 2nd) order allpass sections
(b), where the noble identities [Göckler & Groth (2004); Vaidyanathan (1993)] are exploited. By
