Image Processing Reference
In-Depth Information
Connecting all of the above considerations, the ITr transposition of a complex-valued MIMO
multirate system is performed as follows [Göckler & Groth (2004)]:
• The system SFG to be transposed must be given as truly complex implementation.
• Reverse all arrows of the given SFG, both the arrows representing signal flows and
those symbolic arrows of down- and upsamplers or rotating switches (commutators),
respectively.
As a result of transposition [Göckler & Groth (2004)]
• all input (output) nodes become output (input) nodes, a 4
×
2MIMO system is transformed
4MIMOsystem,
• the number of delays and multipliers is retained,
• the overall number of branching and summation nodes is retained, and
• the overall number of down- and upsamplers is retained.
Obviously, the original optimality (minimality) is transposition invariant .
to a 2
×
3.3.2 Transposition of the SFG of the COHBF approach to DF
As an example, we transpose the SFG of the COHBF approach to the implementation of
a separating DF, as depicted in Fig. 24. The application of the transposition rules of
the preceding Subsection 3.3.1 to the SFG of Fig. 24 results in the COHBF approach to a
multiplexing DF shown in Fig. 26. The invariant properties are easily confirmed by comparing
the original and the transposed SFG. Hence, the numbers of delays and multipliers required
by both DF systems being mutually dual are identical. As expected, the numbers of adders
required are different, since the overall number of branching and summation nodes is retained
only.
Moreover, it should be noted that also the simplicity of the channel selection procedure is
retained. To this end, we have shifted the channel-dependent sign-setting operators d i
=
) o i /2 , o i ∈{
(
, to more suitable positions in front of the summation
nodes G and H. Again, there is a total of 8 summation points, where the signs of the respective
input sequences must be adjusted: The 4 inner lattice output nodes A, B, C, and D, the 2 input
summation nodes E and F immediately fed by the imaginary parts of the input sequences,
and the 2 inner post-lattice summing nodes G and H. At all these summation nodes, the signs
of some or all input sequences must be set in compliance with the desired channel transfer
functions: H o (
1
0, 1, 2, 3
}
, i
∈{
I, II
}
z
)
, o i ∈{
0, 1, 2, 3
}
, i
∈{
I, II
}
, cf. Fig. 26. The sign selection is again most easily
performed, as shown in Fig. 27.
3.4 Conclusion: Halfband filter pair combined to directional filter
In this Section 3, we have derived and analyzed two different approaches to linear-phase
directional filters that separate from a complex-valued FDM input signal two complex user
signals, where the FDM signal may be composed of up to four independent user signals: The
FDMUX approach (Subsection 3.2.1) needs the least number of delays, whereas the synergetic
COHBF approach (Subsection 3.2.2) requires minimum computation. Signal extraction is
always combined with decimation by two.
While the four frequency slots of the user signals to be processed (corresponding to the four
potential DF transfer functions H o (
, centred according to (38);
cf. Fig. 21 ) are equally wide and uniformly allocated, as indicated in Fig. 28, the individual
z
)
, o i ∈{
0, 1, 2, 3
}
, i
∈{
I, II
}
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