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In dealing with networks and circuit elements in multiple dimensions, we must have a means of generalizing their energetic properties accordingly. In particular, the notion of passivity, which in the lumped case played an important role in developing stable digital filters directly from an analog network, must be expanded to include the distributed character of the system to be modeled. The definition of MD-passivity was given in [48], and more basic results are provided in [85] and [131]. The idea is nearly the same as in the lumped case--a passive $ N$-port cannot produce energy on its own, and hence a well-defined [12] network made up of Kirchoff connections of such passive $ N$-ports recirculates and possibly dissipates energy. The difference is that in MD, we would like to be able to take into account that for most physical systems, conservation of energy is a property holding with respect to time alone. We will need to make use of the coordinates defined in §3.3, so as to ensure that passivity holds with respect to all coordinates in the problem. In this section, we recap the main points of the definitions and derivations in [48].

Figure: $ k$-dimensional domain $ G$.
...rge {$dV$}}
\put(380,120){\large {$t$}}
\end{picture} \end{center} \end{figure}

We begin by defining a domain $ G$ in the vector space defined by the new coordinates $ {\bf t} = [t_{1},\hdots, t_{k}]^{T}$ under a transformation of the type (3.21) (which may be an embedding). Consider an $ N$-port defined over the domain $ G$, with port voltages $ v_{j}({\bf t})$ and currents $ i_{j}({\bf t})$, for $ j=1,\hdots,N$. The instantaneous absorbed power density, at any point in the interior of $ G$ is defined by

$\displaystyle w_{inst}({\bf t}) = \sum_{j=1}^{N}v_{j}i_{j}$    

and the stored energy flow as a vector field

$\displaystyle {\bf E} = [E_{1}, \hdots, E_{k}]^{T}$    

In addition, we can define the source and dissipated power densities within $ G$ to be $ w_{s}({\bf t})$ and $ w_{d}({\bf t})$. The energy balance of the $ N$-port can then be generalized directly from (2.3):

$\displaystyle \int_{G}\left(w_{inst}+w_{s}-w_{d}\right)dV = \int_{\partial G}{\bf n}_{G}{\bf E}d\sigma$ (3.26)

where $ {\bf n}_{G}$ is the $ k$-element row vector outward unit normal to the surface of $ G$, $ d\sigma$ is a surface element of $ G$, and $ dV$ is a volume element internal to $ G$. See Figure 3.3% latex2html id marker 80186
\setcounter{footnote}{2}\fnsymbol{footnote} for a graphical representation of some of the relevant quantities. The $ N$-port is called MD-passive if there is a stored energy vector field $ {\bf E}$, which is a positive semi-definite function of the state of the $ N$-port (i.e., all components of $ {\bf E}$ are non-negative, everywhere in $ G$) such that

$\displaystyle \int_{G}w_{inst}dV\geq \int_{\partial G}{\bf n}_{G} {\bf E}d\sigma$ (3.27)

and MD-lossless if (3.28) holds with equality. The total stored energy lost through the boundary of $ G$ must be less than the energy supplied through the ports in $ G$; this is equivalent, from (3.27) to saying that the energy dissipated in $ G$ must be greater than the energy coming from the source. The previous definition of MD-passivity has been more precisely called integral MD-passivity (with respect to a domain $ G$) [85]. A corresponding differential (pointwise) definition is

$\displaystyle w_{inst}\geq \nabla_{{\bf t}}\cdot{\bf E}$   $\displaystyle \mbox{{\rm in $G$}}$ (3.28)

An $ N$-port which is differentially MD-passive everywhere throughout a domain $ G$ will also be integrally MD-passive with respect to $ G$. The converse is not necessarily true.

It is also useful to define, for an $ N$-port, a scalar total energy [85] by

$\displaystyle \mathcal{E}(t) = \int_{G_{t}}{\bf e}_{t}^{T}{\bf E}dx_{1}dx_{2}\hdots dx_{n}$ (3.29)

Here $ G_{t}$ a spatial region defined as the cross-section of $ G$ at time $ t$, and $ {\bf e}_{t}$ is a column unit vector in the time direction; note that this definition is framed in terms of the untransformed coordinates $ {\bf u}$, and $ {\bf E}$ has been projected onto these coordinates under (3.21a). It can also be used as a measure of the total energy at time $ t$ in a given circuit, as we will see in §3.7.4.

Fettweis [44] looks at an extension of the idea of positive realness (see §2.2.2) to two dimensions, for the case of a real linear and shift-invariant $ N$-port. This idea generalizes easily to higher dimensions, as per some very early work in MD system theory [135]. Consider a real linear and shift-invariant (LSI) $ k$-dimensional $ N$-port, where the port quantities are in an exponential state of frequency $ {\bf s_{t}}$, where

$\displaystyle {\bf s_{t}} = [s_{1},\hdots, s_{k}]^{T}$    

are the frequency variables conjugate to $ {\bf t}$. Thus we have the real instantaneous voltages and currents

$\displaystyle v_{j}({\bf t}) = {\rm Re}\left(\hat{v}_{j}e^{{\bf s_{t}}^{T} {\bf...
...e}\left(\hat{i}_{j}e^{{\bf s_{t}}^{T} {\bf t}}\right)\hspace{0.3in}j=1,\hdots,N$    

where $ \hat{v}_{j}$ and $ \hat{i}_{j}$ are complex amplitudes. If there is an impedance relation between the voltages and currents, then we can write

$\displaystyle \hat{{\bf v}} = {\bf Z}({\bf s_{t}})\hat{{\bf i}}$    

where $ \hat{{\bf v}} = [\hat{v}_{1}, \hdots, \hat{v}_{N}]^{T}$ and $ \hat{{\bf i}} = [\hat{i}_{1}, \hdots, \hat{i}_{N}]^{T}$. The total complex MD power density at frequency $ {\bf s}_{t}$ can be defined as

$\displaystyle w({\bf s_{t}}) = \hat{{\bf i}}^{*}\hat{{\bf v}}$    

and the average or active power density as

$\displaystyle \bar{w}({\bf s_{t}}) = {\rm Re}\left(\hat{{\bf i}}^{*}\hat{{\bf v}}\right)$    

The positive realness condition on $ {\bf Z}$ for MD-passivity follows immediately, and is similar to (2.5), except that we now must have

$\displaystyle {\bf Z({\bf s_{t}})}+{\bf Z}^{*}({\bf s_{t}})\geq 0$   for$\displaystyle \hspace{0.3in}{\rm Re}(s_{j})\geq 0\hspace{0.2in}j=1,\hdots, k$ (3.30)

Thus the impedance must be positive real in all the new coordinates. The $ N$-port is MD-lossless if (3.31) holds with equality for Re$ (s_{j})=0$, $ j=1,\hdots,k$. It is important to note that because of (3.26) and (3.25), we have

$\displaystyle {\bf s_{u}} = {\bf V}{\bf H}^{-RT}{\bf s_{t}} \hspace{0.3in}{\bf s_{t}} = {\bf H}^{T}{\bf V}^{-1}{\bf s_{u}}$ (3.31)

where $ {\bf s}_{{\bf u}} = [s_{x_{1}},\hdots,s_{x_{n}},s_{t}]^{T}$ is the vector of frequencies in the untransformed coordinates $ {\bf u}$. Thus, due to the positivity condition on the elements of the last row of $ {\bf H}^{-RT}$ and the last column of $ {\bf H}^{T}$, we will have that

$\displaystyle {\rm Re}(s_{j})\geq 0$   for$\displaystyle \hspace{0.2in}j = 1,\hdots,k\hspace{0.2in} \Longleftrightarrow\hspace{0.2in} {\rm Re}(s_{t})\geq 0$    

so that for an MD-passive $ N$-port,

$\displaystyle {\bf Z}+{\bf Z}^{*}\geq 0$   for$\displaystyle \hspace{0.3in}{\rm Re}(s_{t})\geq 0$ (3.32)

It is thus seen that MD-passivity can be interpreted as passivity, but spread over a new system of coordinates (regardless of whether the new coordinates number more than the old).

next up previous
Next: MD Circuit Elements Up: Multidimensional Wave Digital Filters Previous: Embeddings
Stefan Bilbao 2002-01-22