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Hybrid Form of the Multidimensional Unit Element
We now have an impedance relationship describing the multidimensional unit element in terms of the discrete frequency variables
and
. It is of interest, however, to introduce a particular type of hybrid form [12]. The reason for doing this ultimately has to do with the fact that in a DWN in interleaved form, such as that shown in Figure 4.14 or 4.21, a typical linking waveguide (or unit element) is connected in parallel at one port and in series at the other; it is somewhat easier to make the transition from wave digital filters to digital waveguide networks if we take account of this asymmetry.
Suppose we have a twoport which is defined, at steadystate, by the relationship
, or
This can be rewritten in a socalled hybrid form as
or as
For the multidimensional unit element defined by (4.104), the hybrid matrix is, given the impedance relation (4.105),
It should be clear that the definition of the unit element holds regardless of which complex frequencies we choose. In particular, the delays
and
could be replaced by delays in higher dimensional spaces (we will make use of this in §4.10.4, §4.10.5 and §4.10.6). We will enforce the order of the arguments of
so that, for example,
and
refer to unit elements of mirrorimage orientation.
Suppose now that we have twoports defined by their impedance and hybrid relationships
Figure:
Series/parallel connection of twoports.

If we are interested in connecting the first ports of all twoports to each other in series, and the second ports in parallel as in Figure 4.46, then we will have, for the total voltages and currents
and
which hold instantaneously, and thus, in order to describe the twoport resulting from the connection, we may write
Thus for such a series/parallel combination of twoports, the hybrid matrix of the connection is simply the sum of the hybrid matrices of the individual twoports, and thus
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Stefan Bilbao
20020122