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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 two-port which is defined, at steady-state, by the relationship , or

This can be rewritten in a so-called 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 mirror-image orientation.

Suppose now that we have two-ports defined by their impedance and hybrid relationships

If we are interested in connecting the first ports of all two-ports 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 two-port resulting from the connection, we may write

Thus for such a series/parallel combination of two-ports, the hybrid matrix of the connection is simply the sum of the hybrid matrices of the individual two-ports, and thus

Next: Alternative MDKC for the Up: Incorporating the DWN into Previous: Multidimensional Unit Elements
Stefan Bilbao 2002-01-22