Image Processing Reference
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2.1.2 Minimum-Phase (MP) IIR filters
In contrast to FIR HBF, we describe an MP IIR HBF always by its transfer function H
(
z
)
in the
z -domain.
Specification and properties
The magnitude response of an MP IIR lowpass HBF is specified in the frequency domain by
e j Ω )
, as shown in Fig. 8, again for a minimax or equiripple design. The constraints of
the designed magnitude response
D
(
e j Ω )
H
(
are characterized by the passband and stopband
deviations,
δ p and
δ s , according to [Lutovac et al. (2001); Schüssler & Steffen (1998)] related by
2
s
(
1
δ p )
+ δ
=
1.
(13)
The cut-off frequencies of the IIR HBF satisfy the symmetry condition (6), and the squared
magnitude response
is centrosymmetric about
2
2
2
1
2 .
We consider real MP IIR lowpass HBF of odd order n . The family of the MP
IIR HBF comprises Butterworth, Chebyshev, elliptic (Cauer-lowpass) and intermediate
designs [Vaidyananthan et al. (1987); Zhang & Yoshikawa (1999)].
e j Ω )
e j π /2
e j π /2
H
(
D
(
)
=
H
(
)
=
The MP IIR HBF is
doubly-complementary [Mitra & Kaiser (1993); Regalia et al.
(1988); Vaidyananthan et al.
(1987)], and satisfies the power-complementarity
2
2
e j Ω )
e j −π ) )
(
+
(
=
H
H
1
(14)
and the allpass-complementarity conditions
=
e j ( Ω π ) )
e j Ω )+
(
(
H
H
1.
(15)
/2 complex-conjugated
pole pairs on the imaginary axis within the unit circle, and all zeros on the unit circle
[Schüssler & Steffen (2001)]. Hence, the odd order MP IIR HBF is suitably realized by a
parallel connection of two allpass polyphase sections as expressed by
H
(
z
)
has a single pole at the origin of the z-plane, and
(
n
1
)
2 A 0 (
,
1
z 2
z 1 A 1 (
z 2
(
)=
)+
)
H
z
(16)
where the allpass polyphase components can be derived by alternating assignment of adjacent
complex-conjugated pole pairs of the IIR HBF to the polyphase components. The polyphase
components A l (
z 2
)
, l
=
0, 1 consist of cascade connections of second order allpass sections:
n
1
n
1
1
1
z 2
z 2
1
2
a i +
a i +
2
2
z 1
H
(
z
)=
+
,
(17)
+
a i z 2
+
a i z 2
1
1
i
=
0,2,...
i
=
1,3,...
A 0 (
z 2
)
A 1 (
z 2
)
n
1
where the coefficients a i , i
a i + 1 , denote the squared moduli
of the HBF complex-conjugated pole pairs in ascending order; the complete set of n poles is
given by 0,
=
0, 1, ...,
(
1
)
,with a i <
2
1 [Mitra (1998)].
j a 0 ,
j a 1 , ...,
j a n 1
2
±
±
±
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