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# MD-passivity

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 -port cannot produce energy on its own, and hence a well-defined [12] network made up of Kirchoff connections of such passive -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].

We begin by defining a domain in the vector space defined by the new coordinates under a transformation of the type (3.21) (which may be an embedding). Consider an -port defined over the domain , with port voltages and currents , for . The instantaneous absorbed power density, at any point in the interior of is defined by

and the stored energy flow as a vector field

In addition, we can define the source and dissipated power densities within to be and . The energy balance of the -port can then be generalized directly from (2.3):

 (3.26)

where is the -element row vector outward unit normal to the surface of , is a surface element of , and is a volume element internal to . See Figure 3.3 for a graphical representation of some of the relevant quantities. The -port is called MD-passive if there is a stored energy vector field , which is a positive semi-definite function of the state of the -port (i.e., all components of are non-negative, everywhere in ) such that

 (3.27)

and MD-lossless if (3.28) holds with equality. The total stored energy lost through the boundary of must be less than the energy supplied through the ports in ; this is equivalent, from (3.27) to saying that the energy dissipated in 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 ) [85]. A corresponding differential (pointwise) definition is

 (3.28)

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

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

 (3.29)

Here a spatial region defined as the cross-section of at time , and is a column unit vector in the time direction; note that this definition is framed in terms of the untransformed coordinates , and has been projected onto these coordinates under (3.21a). It can also be used as a measure of the total energy at time 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 -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) -dimensional -port, where the port quantities are in an exponential state of frequency , where

are the frequency variables conjugate to . Thus we have the real instantaneous voltages and currents

where and are complex amplitudes. If there is an impedance relation between the voltages and currents, then we can write

where and . The total complex MD power density at frequency can be defined as

and the average or active power density as

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

 for (3.30)

Thus the impedance must be positive real in all the new coordinates. The -port is MD-lossless if (3.31) holds with equality for Re, . It is important to note that because of (3.26) and (3.25), we have

 (3.31)

where is the vector of frequencies in the untransformed coordinates . Thus, due to the positivity condition on the elements of the last row of and the last column of , we will have that

 for

so that for an MD-passive -port,

 for (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: MD Circuit Elements Up: Multidimensional Wave Digital Filters Previous: Embeddings
Stefan Bilbao 2002-01-22