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exploiting a multitude of multirate identities Groth (2003); Groth & Göckler (2001). The
resulting P
MIMO filter transfer matrix H λ (
ν ×
P
z d )
contains each PP component of H
λ (
z
ν )
ν
P
times whereas, as desired for
feasibility, the operational clock rate is concurrently reduced by P
times: Thus, the amount of hardware is increased P
ν
ν
. Hence, the overall
expenditure, i.e. the number of operations times the respective operational clock rate
Göckler & Groth (2004), is not changed.
4. Parallelisation of butterflies combining the output signals of associated PP filter blocks
is straightforward: For each (time-interleaved) PP component of the respective signals a
butterfly has to be foreseen, as shown in Fig. 33.
ν
4.3 Conclusion: Parallelisation of multirate systems
In this Section 4, a general and systematic procedure for parallelisation of multirate systems,
for instance as investigated in Sections 2 and 3, has been presented . Its application
to the high rate decimating FDMUX front end of the tree-structured SBC-FDFMUX FB
Abdulazim & Göckler (2005); Abdulazim et al. (2007) has been deployed in detail. The stage
ν
ν =
degree of parallelisation P ν ,
0, 1, 2, 3, is diminished proportionally to the operational
clock frequency f ν of stage
and is, thus, adapted to the actual sampling rate. As a result,
after suitable decomposition of the high rate front end input signal by an input commutator
in P 0
ν
=
=
8 in Fig. 33), all subsequent
processing units are likewise operated at the same operational clock rate f d =
P max polyphase components (as depicted for P max
=
f n / P 0
f 0 / P 0 .
Since inherent parallelism of the original tree-structured FDMUX (Fig.
32) has attained
=
P max
8 in the third stage, and the output signals of this stage represent the desired eight
demultiplexed FDM subsignals, interleaving PS-output commutators are no longer required,
as to be seen in Fig. 33. Finally, it should be noted that parallelisation does not change
overall expenditure; yet, by multiplying stage
hardware by P ν , the operational clock rates
are reduced by a factor of P ν to a feasible order of magnitude, as desired.
Applying the rules of multirate transposition (cf. Subsection 3.3.1 or Göckler & Groth
(2004)) to the parallelised FDMUX front end, the high rate interpolating back end of the
tree-structured SBC-FDFMUX FB is obtained likewise and exhibits the same properties as
to expenditure and feasibility Groth (2003). Hence, the versatile and efficient tree-structured
filter bank (FDMUX, FMUX, SBC, wavelet, or any combination thereof) can be used in any
(ultra) wide-band application without any restriction.
ν
5. Summary and conclusion
In Section 2 we have introduced and investigated a special class of real and complex FIR
and IIR halfband bandpass filters with the particular set of centre frequencies defined by
(1). As a result of the constraint (1), almost all filter coefficients are either real-valued or
purely imaginary-valued, as opposed to fully complex-valued coefficients. Hence, this class
of halfband filters requires only a small amount of computation.
In Section 3, two different options to combine two of the above FIR halfband filters with
different centre frequencies to forma directional filter (DF) have been investigated. As a result,
one of these DF approaches is optimum w.r.t. to computation (most efficient), whereas the
other requires the least number of delay elements (minimum McMillan degree). The relation
between separating DF and DF that combine two independent signals to an FDM signal via
multirate transposition rules has extensively been shown.
Finally, in Section 4, the above FIR directional filters (DF) have been combined to
tree-structured multiplexing and demultiplexing filter banks. While this procedure is
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