Image Processing Reference
In-Depth Information
A
PPROACH
multiplications/sample
delays
FDMUX
N
+
3
(
3
N
−
5
)
/2
=
FDMUX ex.:
N
11
14
14
COHBF
(
N
+
5
)
/2
(
5
N
−
11
)
/2
COHBF ex.:
N
=
11
8
22
Table 9. Comparison of expenditure of FDMUX and COHBF DF approaches
at the expense of a higher count of delay elements. Finally, it should be noticed that the DF
group delay is independent of its (FDMUX or COHBF) implementation.
3.3 Linear-phase directional combination filter
Using transposition techniques, we subsequently derive DF being complementary (dual) to
those presented in Subsection 3.2: They combine two complex-valued signals of identical
sampling rate
f
d
that are likewise oversampled by at least 2 to an FDM signal, where different
oversampling factors allow for different bandwidths.
An example can be deduced from Fig.
21 by considering the signals
s
o
(
mT
d
)
←→
e
j
Ω
(
d
)
S
o
(
)
=
0, 2, of Figs.21(c,d) as input signals. The multiplexing DF increases the
sampling rates of both signals to
f
n
,
o
=
2
f
d
, and provides the filtering operations shown in Fig.
e
j
Ω
)
21(b),
h
o
(
)
←→
H
o
(
=
kT
n
,
c
0, 2, to form the FDM output spectrum being exclusively
e
j
Ω
)
composed of
S
o
(
=
,
o
0, 2.
3.3.1 Transposition of complex multirate systems
The goal of transposition is to derive a system that is complementary or dual to the original
one: The various filter transfer functions must be retained, demultiplexing and decimating
operations must be replaced with the dual operations of multiplexing and interpolation,
respectively [Göckler & Groth (2004)].
The types of systems we want to transpose, Figs.22 and 24, represent complex-valued 4
×
2 multiple-input multiple-output (MIMO) multirate systems. Obviously, these systems are
composed of
complex monorate
sub-systems (complex filtering of polyphase components) and
real multirate
sub-systems (down- and upsampler), cf. [Göckler & Groth (2004)].
While the transposition of real MIMO monorate systems is well-known and unique
[Göckler & Groth (2004); Mitra (1998)], in the context of
complex
MIMO monorate systems the
Invariant
(ITr) and the
Hermitian
(HTr) transposition must be distinguished, where the former
retains the original transfer functions,
H
o
(
o
, as desired in our application. As
detailed in [Göckler & Groth (2004)], the ITr is performed by applying the transposition rules
known for real MIMO monorate systems
provided that
all imaginary units “j”, both of the
complex input and output signals
and
of the complex coefficients, are conceptually considered
and treated as multipliers within the SFG
3
(denoted as truly complex implementation), as to
be seen from Figs.22 and 24.
The transposition of an
M
-downsampler, representing a real single-input single-output (SISO)
multirate system, uniquely leads to the corresponding
M
-upsampler, the complementary
(dual) multirate system, and vice versa [Göckler & Groth (2004)].
z
)=
H
o
(
z
)
∀
3
The imaginary units of the input signals and the coefficients
must not
be eliminated by simple
multiplication and consideration of the correct signs in subsequent adders; this approach would
transform the original complex MIMO SFG to a corresponding real SFG, where the direct transposition
of the latter would perform the HTr [Göckler & Groth (2004)].
