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 $z_{1}^{-1}$ and $z_{2}^{-1}$. 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 ${\bf\hat{v}} = {\bf Z\hat{i}}$, or

$$\begin{bmatrix}\hat{v}_{1}\\ \hat{v}_{2}\\ \end{bmatrix}= \begin{... ...&Z_{22}\\ \end{bmatrix}\begin{bmatrix}\hat{i}_{1}\\ \hat{i}_{2}\\ \end{bmatrix}$$

This can be rewritten in a so-called hybrid form as

$$\begin{bmatrix}\hat{v}_{1}\\ \hat{i}_{2}\\ \end{bmatrix}= \frac{1... ..._{21}&1\\ \end{bmatrix}\begin{bmatrix}\hat{i}_{1}\\ \hat{v}_{2}\\ \end{bmatrix}$$

or as

$${\bf\hat{p}}\hspace{0.1in} =\hspace{0.98in} {\bf K}\hspace{0.8in}{\bf\hat{q}}$$

For the multidimensional unit element defined by (4.104), the hybrid matrix is, given the impedance relation (4.105),

$${\bf K}_{ue}(z_{1}^{-1},z_{2}^{-1}) = \frac{1}{1+z_{1}^{-1}z_{2}^... ...&2z_{2}^{-1}\\ -2z_{1}^{-1}&\frac{1}{R}(1-z_{1}^{-1}z_{2}^{-1})\\ \end{bmatrix}$$

It should be clear that the definition of the unit element holds regardless of which complex frequencies we choose. In particular, the delays $z_{1}^{-1}$ and $z_{2}^{-1}$ 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 ${\bf K}_{ue}$ so that, for example, ${\bf K}_{ue}(z_{1}^{-1},z_{2}^{-1})$ and ${\bf K}_{ue}(z_{2}^{-1},z_{1}^{-1})$ refer to unit elements of mirror-image orientation.

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

$${\bf\hat{v}}_{k} = {\bf Z}_{k}{\bf\hat{i}}_{k}\hspace{0.5in}{\bf\hat{p}}_{k} = {\bf K}_{k}{\bf\hat{q}}_{k}\hspace{0.5in}k=1,\hdots,N$$

Figure: Series/parallel connection of $N$ two-ports.

If we are interested in connecting the first ports of all $N$ 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

$$v_{1} = \sum_{k=1}^{N}v_{1k}\hspace{0.5in}i_{2} = \sum_{k=1}^{N}i_{2k}$$

and

$$i_{1k} = i_{1}\hspace{0.5in}v_{2k} = v_{2}\hspace{0.5in}k=1,\hdots,N$$

which hold instantaneously, and thus, in order to describe the two-port resulting from the connection, we may write

$${\bf\hat{p}} = \sum_{k=1}^{N}{\bf\hat{p}}_{k} = \sum_{k=1}^{N}{\bf K}_{k}{\bf\hat{q}}_{k} = \left(\sum_{k=1}^{N}{\bf K}_{k}\right){\bf\hat{q}}$$

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

$${\bf K} = \sum_{k=1}^{N}{\bf K}_{k} \mbox{{\rm Series/parallel combination of $N$\ two-ports}}$$