Image Processing Reference
In-Depth Information
2.1 Real halfband filters (RHBF)
In this subsection we recall the essentials of LP FIR and MP IIR lowpass HBF with real-valued
impulse responses
h
represents the associated
z
-transform
transfer function. From such a lowpass (prototype) HBF a corresponding real highpass HBF
is readily derived by using the modulation property of the
z
-transform [Oppenheim & Schafer
(1989)]
(
k
)=
h
k
←→
H
(
z
)
,where
H
(
z
)
z
z
c
)
z
c
h
(
)
←→
(
k
H
(3)
by setting in accordance with (1)
e
j
2
π
f
4
/
f
n
e
j
π
=
−
=
z
4
=
=
z
c
1
(4)
resulting in a frequency shift by
f
4
=
(Ω
4
=
π
)
f
n
/2
.
2.1.1 Linear-Phase (LP) FIR filters
Throughout this Section 2 we describe a real LP FIR (lowpass) filter by its non-causal impulse
response with its centre of symmetry located at the time or sample index
k
=
0 according to
h
=
h
k
∀
k
(5)
−
k
e
j
Ω
)
∈
R
where the associated frequency response
H
(
is zero-phase [Mitra & Kaiser (1993);
Oppenheim & Schafer (1989)].
Specification and properties
A real zero-phase (LP) lowpass HBF, also called Nyquist(2)filter [Mitra & Kaiser (1993)],
is specified in the frequency domain as shown in Fig. 5, for instance, for an equiripple
or constrained least squares design, respectively, allowing for a don't care transition band
between passband and stopband [Mintzer (1982); Mitra & Kaiser (1993); Schüssler & Steffen
(1998)]. Passband and stopband constraints
δ
p
=
δ
s
=
δ
are identical, and for the cut-off
frequencies we have the relationship:
+ Ω
s
=
π
Ω
p
.
(6)
e
j
Ω
)
∈
R
As a result, the zero-phase desired function
D
(
as well as the frequency response
2
. From this frequency
e
j
Ω
)
∈
R
e
j
π
/2
e
j
π
/2
H
(
are centrosymmetric about
D
(
)=
H
(
)=
domain symmetry property immediately follows
e
j
(
Ω
−
π
)
)=
e
j
Ω
)+
(
(
H
H
1,
(7)
indicating that this type of halfband filter is strictly complementary [Schüssler & Steffen
(1998)].
According to (5), a real zero-phase FIR HBF has a symmetric impulse response of
odd
length
N
1 (denoted as type I filter in [Mitra & Kaiser (1993)]), where
n
represents the even
filter order. In case of a minimal (canonic) monorate filter implementation,
n
is identical to
the minimum number
n
mc
of delay elements required for realisation, where
n
mc
is known as
the McMillan degree [Vaidyanathan (1993)]. Due to the odd symmetry of the HBF zero-phase
frequency response about the transition region (don't care band according to Fig. 5), roughly
every other coefficient of the impulse response is zero [Mintzer (1982); Schüssler & Steffen
(1998)], resulting in the additional filter length constraint:
=
n
+
=
+
=
−
∈
N
N
n
1
4
i
1,
i
.
(8)