Image Processing Reference
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LP FIR MP IIR
N Op n mc Fig. N Op n mc Fig.
HBF Decimator
12
8
7
9
3
9
CHBF:
R C
11
11 12(a)
8
3
14
C C
CHBF:
24
16
13
18
6
15
COHBF:
R C
13
14 16(a)
17
5
18
C C
COHBF:
28
16
17
36
10
19
Table 7. Expenditures of real and complex HBF decimators based on the design examples of
Fig. 11; N Op : number of operations, n mc : McMillan degree; operational clock frequency:
f Op =
f n /2
contribution. This sectoral computational advantage of LP FIR COHBF is, despite n IIR <
n FIR ,
due to the fact that these FIR filters still allow for memory sharing in conjunction with the
exploitation of coefficient symmetry [Göckler (1996b)]. However, the amount of storage n mc
required for IIR HBF is always below that of their FIR counterparts.
3. Halfband filter pairs 2
In this Section 3, we address a particular class of efficient directional filters (DF). These DF
are composed of two real or complex HBF, respectively, of different centre frequencies out
of the set given by (1). To this end, we conceptually introduce and investigate two-channel
frequency demultiplexer filter banks (FDMUX) that extract from an incoming complex-valued
frequency division multiplex (FDM) signal, being composed of up to four uniformly allocated
independent user signals of identical bandwidth (cf. Fig. 20), two of its constituents by
concurrently reducing the sample rate by two Göckler & Groth (2004). Moreover, the DF shall
allow to select any pair of user signals out of the four constituents of the incoming FDM signal,
where the individual centre frequencies are to be selectable with minimum switching effort.
At first glance, there are two optional approaches: The selectable combination of two filter
functions out of a pool of i ) two RBF according to Subsection 2.1 and two CHBF (HT), as
described in Subsection 2.2, where the centre frequencies of this filter quadruple are given by
(1) with c
,or ii ) four COHBF, as described in Subsection 2.3, where the centre
frequencies of this filter quadruple are given by (1) with c
∈{
0, 2, 4, 6
}
. Since centre frequency
switching is more crucial in case one (switching between real and/or complex filters), we
subsequently restrict our investigations to case two, where the FDM input spectrum must be
allocated as shown in Fig. 20.
These DF with easily selectable centre frequencies are frequently used in receiver
front-ends to meet routing requirements [Göckler (1996c)], in tree-structured FDMUX
filter banks [Göckler & Felbecker (2001); Göckler & Groth (2004); Göckler & Eyssele (1992)],
and, in modified form, for frequency re-allocation to avoid hard-wired frequency-shifting
[Abdulazim & Göckler (2007); Eghbali et al. (2009)]. Efficient implementation is crucial, if
these DF are operated at high sampling rates at system input or output port. To cope with this
high rate challenge, we introduce a systematic approach to system parallelisation according
to [Groth (2003)] in Section 4 .
In continuation of the investigations reported in Section 2, we combine two linear-phase
(LP) FIR complex offset halfband filters (COHBF) with different centre frequencies, being
characterized by (1) with c
∈{
1, 3, 5, 7
}
∈{
1, 3, 5, 7
}
, to construct efficient directional filters for one input
2 Underlying original publication: Göckler & Alfsmann (2010)
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