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12
Most Efficient Digital Filter Structures:
The Potential of Halfband Filters
in Digital Signal Processing
Heinz G. Göckler
Digital Signal Processing Group, Ruhr-Universität Bochum
Germany
1. Introduction
A digital halfband filter (HBF) is, in its basic formwith real-valued coefficients, a lowpass filter
with one passband and one stopband region of unity or zero desired transfer characteristic,
respectively, where both specified bands have the same bandwidth. The zero-phase frequency
response of a nonrecursive (FIR) halfband filter with its symmetric impulse response exhibits
an odd symmetry about the quarter sample rate (
Ω = 2 ) and half magnitude ( 2 )point
[Schüssler & Steffen (1998)], where
Ω =
2
π
f / f n represents the normalised (radian) frequency
and f n =
1/ T the sampling rate. The same symmetry holds true for the squared magnitude
frequency response of minimum-phase (MP) recursive (IIR) halfband filters [Lutovac et al.
(2001); Schüssler & Steffen (2001)]. As a result of this symmetry property, the implementation
of a real HBF requires only a low computational load since, roughly, every other filter
coefficient is identical to zero [Bellanger (1989); Mitra & Kaiser (1993); Schüssler & Steffen
(2001)].
Due to their high efficiency, digital halfband filters are widely used as versatile building blocks
in digital signal processing applications. They are, for instance, encountered in front ends
of digital receivers and back ends of digital transmitters (software defined radio, modems,
CATV-systems, etc. [Göckler & Groth (2004); Göckler & Grotz (1994); Göckler & Eyssele
(1992); Renfors & Kupianen (1998)]), in decimators and interpolators for sample rate alteration
by a factor of two [Ansari & Liu (1983); Bellanger (1989); Bellanger et al. (1974); Gazsi
(1986); Valenzuela & Constantinides (1983)], in efficient multirate implementations of digital
filters [Bellanger et al. (1974); Fliege (1993); Göckler & Groth (2004)] (cf. Fig. 1), where
the input/output sampling rate f n is decimated by I cascaded HBF stages by a factor
of 2 I
z 2 n )
2 I
=
·
(
=
, in tree-structured filter banks for FDM de- and
remultiplexing (e.g. in satellite communications) according to Fig. 2 and [Danesfahani et al.
(1994); Göckler & Felbecker (2001); Göckler & Groth (2004); Göckler & Eyssele (1992)], etc. A
frequency-shifted (complex) halfband filter (CHBF), generally known as Hilbert-Transformer
(HT, cf. Fig. 3), is frequently used to derive an analytical bandpass signal from its real-valued
counterpart [Kollar et al. (1990); Kumar et al. (1994); Lutovac et al. (2001); Meerkötter & Ochs
(1998); Schüssler & Steffen (1998; 2001); Schüssler & Weith (1987)]. Finally, real IIR HBF or
spectral factors of real FIR HBF, respectively, are used in perfectly reconstructing sub-band
coder (cf. Fig. 4) and transmultiplexer filter banks [Fliege (1993); Göckler & Groth (2004);
to f d
f n
z d
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