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 =
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)
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