Image Processing Reference
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Fig. 33. Complete parallelisation of FDMUX front-end of SBC-FDFMUX filter bank (Fig. 32);
z d :
e j Ω ( d ) ,
Ω ( d ) =
=
2
π
f / f d , f d =
f n /8
. To this end, as required by Fig. 32, the unit delay z 1
0
H 1 (
z 1 )
is parallelised by P 0 =
8, as
shown in Fig. 31(a), while the subsequent down-sampler applies P 1 =
4, as described above
w.r.t. Fig. 31(b). Immediate cascading of parallelised unit delay
(
P 0 =
8
)
and down-sampling
(
(as induced by Fig. 31) shows that only those four PP components of the
parallelised delay with even time offset
P 1 =
4, M
=
2
)
(
p
=
0, 2, 4, 6
)
are transferred via the 4-branch SP-input
interface of down-sampling (2 P 1 =
8) to its PS-output interface with naturally ordered time
=
0, 1, 2, 3 w.r.t. P 1 =
offsets p
4. Hence, only those retained 4 out of 8 PP components
=
of odd time index p
7, 1, 3, 5, being provided by the unit delay's SP-input interface and
delayed by z 1
0
z 1/8
d
=
, are transferred (mapped) to the P 1
=
4 up-samplers with timing
offset p
0, 1, 2, 3 of the 4-branch PS-output interface of the down-sampler. Fig. 33 shows
the correspondingly rearranged signal flow graph representation of stage 1 input section
=
( ν =
λ =
.
As a result, the upper branch of stage 1, H 0 (
1
)
H 0 (
,isfedbytheeven-indexed
PP components of the high rate FDMUX input signal, whereas the lower branch H 1 (
z 1 )
z d )
z 1 )
H 1 (
is provided with the delayed versions of the PP components of odd index, as depicted
in Fig. 33. Hence, as in the original system Fig. 32, the input sequence is completely fed into
the parallelised system.
This procedure is repeated with the input branching and blocking sections of the subsequent
stages
z d )
H 0 (
ν =
2, 3: The PP branch filters H 0 (
z ν )
z d )
parallelised by P ν ,where P 2 =
2and
P 3 =
(
P 1 =
)
, are provided with the even-numbered PP components of the respective input
signals with timing offsets in natural order. Contrary, the set of PP components of odd index
is always delayed by z 1/ P ν− 1
d
1
4
H 1 (
and fed into filter blocks H 1 (
z
ν )
z d )
in crossed manner
(cf. input section
1).
3. P ν -fold Parallelisation of PP branch filters H
λ =
H λ (
λ (
z ν )
z d )
,
λ =
0, 1;
ν =
1, 2,
is
achieved by systematic application of the procedure condensed in Fig.
29 (for details
cf. Göckler & Groth (2004); Groth (2003)). To this end, H
is decomposed in P ν PP
components of correspondingly reduced order, which are arranged to a MIMO system by
λ (
z ν )
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