Image Processing Reference
In-Depth Information
SFG, the coefficient multipliers can obviously be shared with all transfer functions H o (
z
)
,
o
; however, the respective outbound delay chains must essentially be duplicated.
Merging all of the above considerations, a signal separating DF requiring minimum
computation that, in addition, allows for simple channel selection or switching, respectively,
is readily developed as follows:
1. Multiply the incoming decimated polyphase signal samples concurrently and
consecutively by the complete set of all real coefficients (9) to allow for the exploitation of
coefficient symmetry (41) in compliance with Table 8.
2. Form a real and imaginary (R/I) sub-sequence of DF output signals being independent of the
selected channel transfer functions, i.e. o I , o II ∈{
∈{
0, 1, 2, 3
}
}
0, 1, 2, 3
,byusingall
R
-set coefficients of
Tabl e 8 .
3. Form an R and I sub-sequence of DF output signals being likewise independent of the
selected channels o I , o II by using all
o
I
(
)
-set coefficients of Table 8 multiplied by
1
to
eliminate channel dependency.
4. Form R/I sub-sequences of DF output signals being dependent of the selected channels
o I , o II that are derived from centre coefficients h 0, o .
5. Combine all of the above R/I sub-sequences considering the sign rules of Table 8 to select
the desired DF transfer functions H o i (
.
Based on the outlined DF implementation strategy, an illustrative example is presented in Fig.
24 with an underlying RHBF of length N
z
)
, o i ∈{
0, 1, 2, 3
}
, i
∈{
I, II
}
11. The front end for polyphase decomposition
and sample rate reduction by 2 is identical to that of the FDMUX approach of Fig. 22. Contrary
to the former approach, the delay chains for the odd-numbered coefficients are outbound and
duplicated (rather than interlaced) to allow for simple channel selection. As a result, channel
selection is performed by combining the respective sub-sequences that have passed the
=
R
-set
coefficients (cf. Table 8) with those having passed the corresponding
I
-set coefficients, where
o i ; o i ∈{
the latter sub-sequences are pre-multiplied by b i =(
)
}
∈{
}
.
Multipliers and delays for the centre coefficient h 0, o i signal processing are implemented
similarly to Fig. 22 without need for duplication of delays. However, the post-delay inner
lattice must be realized for each transfer function individually; its channel dependency follows
from Table 8 and (40):
1
0, 1, 2, 3
, i
I, II
2
) o i /2 ,
h 0
2 (
h 0
j o i
) o i /2 +
h 0, o i =
1
+
j
)
=
(
1
j
(
1
(49)
where o i ∈{
0, 1, 2, 3
}
, i
∈{
I, II
}
and h 0 =
1/2 according to (9). Rearranging (49) yields with
obvious abbreviations:
h 0
2 [(
h 0
2 [
o i
) o i /2 =
h 0, o i =
1
)
+
j
] (
1
b i
+
j
]
d i .
(50)
It is easily recognized that the inner lattices of Fig. 24 implement the operations within
the brackets of (50) with their results displayed at the respective inner nodes A, B, C, D. In
compliance with (50), these inner node sequences must be multiplied by the respective signs
d i =(
) o i /2 ; o i ∈{
1
0, 1, 2, 3
}
, i
∈{
I, II
}
, prior to their combination with the above R/I
sub-sequences.
To calculate a set of complex output samples at the two DF output ports, obviously the
minimum number of
(
+
)
N
5
/2 real multiplications must be carried out. Furthermore, for
Search WWH ::




Custom Search