Image Processing Reference
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Fig. 23. I
=
4 point IDFT of FDMUX approach; N
=
11: (a) general (b) pruned for channels
=
o
0, 1
k
-5
-3
-1
0
1
3
5
h k , o
h k
j
(
1
j
2 j o
+
j
(
j
(
1
o 1
1
)
)
)
1
1
o
1
o
type
R
I
R C
I
R
I
Table 8. Properties of COHBF coefficients in dependence of channel index o
∈{
0, 1, 2, 3
}
;
I
:
C
with Re
{•} =
0
3.2.2 COHBF approach
For this novel approach, we combine two decimating COHBF of different centre frequencies
f o , o
∈{
}
, according to (38) in a synergetic manner to construct a DF for signal
separation that requires minimum computation. To this end, we first study the commonalities
of the impulse responses (40) of the four transfer functions H o (
0, 1, 2, 3
)
∈{
}
z
, o
0, 1, 2, 3
(underlying
=
+
constant in (39) subsequently: a
2 o
1). These impulse responses are presented in Table 8
∈{
}
as a function of the channel number o
0, 1, 2, 3
for the non-zero coefficients of (40), related
to the respective real RHBF coefficients.
Except for the centre coefficient exhibiting identical real and imaginary parts, one half of the
coefficientsisreal(
R
)and independent of the desired centre frequency represented by the channel
indices o
. Hence, these coefficients are common to all four transfer functions.
The other half of the coefficients is purely imaginary (
∈{
0, 1, 2, 3
}
: i.e., their real parts are zero) and
dependent of the selected centre frequency. However, this dependency on the channel number
is identical for all these coefficients and just requires a simple sign operation. Finally, the
repetitive pattern of the coefficients, as a result of coefficient symmetry (41), is reflected in
Tabl e 8 .
A COHBF implementation of a demultiplexing DF aiming at minimum computational load must
exploit the inherent coefficient symmetry (41), cf. Table 8. To this end, we consider the
COHBF as depicted in Fig. 17 of Subsection 2.3.1, applying input commutators for sample
rate reduction. In contrast to the FDMUX approach of Fig. 22, the SFG of Fig. 17 is based
on the transposed FIR direct form Bellanger (1989); Mitra (1998), where the incoming signal
samples are concurrently multiplied by the complete set of all coefficients, and the delay
chains are directly connected to the output ports. When combining two of these COHBF
I
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