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user signals may possess different bandwidths. However, each user signal must completely
be contained in one of the four frequency slots, as exemplified in Fig. 28.
Furthermore, by applying the transposition rules of [Göckler & Groth (2004)], the
corresponding complementary (dual) combining directional filters have been derived, where
the multiplication rates and the delay counts of the original structures are always retained.
Obviously, transposing a system allows for the derivation of an optimum dual system by
applying the simple transposition rules, provided that the original system is optimal. Thus,
a tedious re-derivation and optimization of the complementary system is circumvented.
Nevertheless, it should be noted that by transposition always just one particular structure
is obtained, rather than a variety of structures [Göckler & Groth (2004)].
Finally, to give an idea of the required filter lengths required, we recall the design result
reported in [Göckler & Eyssele (1992)] where, as depicted in the above Fig. 21(a,b), the
passband, stopband and transition bands were assumed equally wide: With an HBF prototype
filter length of N
=
>
11 and 10 bit coefficients, a stopband attenuation of
50dB was achieved.
4. Parallelisation of tree-structured filter banks composed of directional filters 4
In the subsequent Section 4 of this chapter we consider the combination of multiple
two-channel DF investigated in Section 3 to construct tree-structured filter banks. To this
end, we cascade separating DF in a hierarchical manner to demultiplex (split) a frequency
division multiplex (FDM) signal into its constituting user signals: this type of filter bank (FB)
is denoted by FDMUX FB; Fig. 2. Its transposed counterpart (cf. Subsection 3.3.1), the FMUX
FB, is a cascade connection of combining DF considered in Subsection 3.3 to form an FDM
signal of independent user signals. Finally, we call an FDMUX FB followed by an FMUX FB
an FDFMUX FB, which may contain a switching unit for channel routing between the two FB.
Subsequently, we consider an application of FDFMUX FB for on-board processing in satellite
communications. If the number of channels and/or the bandwidth requirements are high,
efficient implementation of the high-end DF is crucial, if they are operated at (extremely) high
sampling rates. To cope with this issue, we propose to parallelise the at least the front-end
(back-end) of the FDMUX (FMUX) filter bank. For this outlined application, we give the
following introduction and motivation.
Digital signal processing on-board communication satellites (OBP) is an active field of
research where, in conjunction with frequency division multiplex (FDMA) systems, presently
two trends and challenges are observed, respectively: i ) The need of an ever-increasing
number of user channels makes it necessary to digitally process, i.e. to demultiplex,
cross-connect and remultiplex, ultra-wideband FDM signals requiring high-end sampling
rates that range considerably beyond 1GHz [Arbesser-Rastburg et al. (2002); Maufroid et al.
(2004; 2003); Rio-Herrero & Maufroid (2003); Wittig (2000)], and ii ) the desire of flexibility
of channel bandwidth-to-user assignment calling for simply reconfigurable OBP systems
[Abdulazim & Göckler (2005); Göckler & Felbecker (2001); Johansson & Löwenborg (2005);
Kopmann et al. (2003)]. Yet, overall power consumption must be minimumdemanding highly
efficient FB for FDM demultiplexing (FDMUX) and remultiplexing (FMUX).
Two baseline approaches to most efficient uniform digital FB, as required for OBP, are
known: a ) The complex-modulated (DFT) polyphase (PP) FB applying single-step sample
rate alteration [Vaidyanathan (1993)], and b ) the multistage tree-structured FB as depicted
in Fig.
2, where its directional filters (DF) are either based on the DFT PP method
4 Underlying original publication: Göckler et al. (2006)
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