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Multidimensional Unit Elements

As a first step towards reintroducing this port structure to a MDWD network (and hence towards relating the DWN to the MDWDF), we can extend the definition of the unit element (and recall from §4.2.1 that the unit element is a wave digital two-port which is equivalent to a single bidirectional delay line) to multi-D in the following way% latex2html id marker 85650
\setcounter{footnote}{2}\fnsymbol{footnote}. Suppose that we are dealing with a (1+1)D system, and new coordinates $ t_{1}$ and $ t_{2}$ are as defined by (3.18). We thus have two transform frequencies $ s_{1}$ and $ s_{2}$, as well as two frequency-domain shift-operators $ z_{1}^{-1}$ and $ z_{2}^{-1}$ in the two directions $ t_{1}$ and $ t_{2}$. The multidimensional two-port defined, at steady-state, by

$\displaystyle \begin{bmatrix}\hat{b}_{1}\\ \hat{b}_{2}\\ \end{bmatrix}= \begin{...
...{-1}&0\\ \end{bmatrix} \begin{bmatrix}\hat{a}_{1}\\ \hat{a}_{2}\\ \end{bmatrix}$ (4.121)

and shown in Figure 4.45(a) bears some resemblance to the lumped unit element discussed initially in §2.3.4; it is clearly lossless (because it merely implements a pair of shifts), but, unlike the unit element, it is no longer reciprocal. In this last respect, we remark that a multidimensional element so defined is perhaps closer in spirit to a generalization of the so-called quasi-reciprocal line (QUARL) proposed by Fettweis [46]. The two port resistances are assumed identical and equal to some positive constant $ R$. When the spatial dependence is expanded out, it appears as an entire array of unit elements, as in Figure 4.45(b), where we have assumed

$\displaystyle z_{1}^{-1} = z^{-\frac{1}{2}}w^{-\frac{1}{2}}\hspace{1.0in}z_{2}^{-1} = z^{-\frac{1}{2}}w^{\frac{1}{2}}$    

where $ z^{-\frac{1}{2}}$ and $ w^{-\frac{1}{2}}$ correspond, respectively, to unit shifts in time and space by $ T/2$ and $ \Delta/2$. We have thus chosen $ T_{1} = T_{2} = \Delta/\sqrt{2}$.

Figure 4.45: (a) Multidimensional unit element at steady state making use of shifts in directions $ t_{1}$ and $ t_{2}$ and (b) its steady state schematic when spatial dependence is expanded out.
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\end{picture} \end{center} \vspace{0.2in}

This element, like the standard unit element, is defined in the discrete (time and space) domain, and using wave variables. Rewriting the scattering relation in terms of steady-state discrete voltage and current amplitudes, (4.104) becomes the impedance relationship

$\displaystyle \begin{bmatrix}\hat{v}_{1}\\ \hat{v}_{2}\\ \end{bmatrix}= \frac{R...
...2}^{-1}\\ \end{bmatrix}\begin{bmatrix}\hat{i}_{1}\\ \hat{i}_{2}\\ \end{bmatrix}$ (4.122)

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Stefan Bilbao 2002-01-22