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Balanced Forms

Consider again the (1+1)D transmission line, with spatially-varying coefficients. It has been noted in the past [130,131] that the restriction on the time step, namely

with and is rather unsatisfying; the local group velocity at any point in our domain is given by , so we would hope that a more physically meaningful bound such as

 (3.93)

(which is obtained in using, for example, digital waveguide networks, which will be discussed in Chapter 4) could be attainable. Depending on the variation in and , the new bound can allow a substantially larger time step. We will show that this is in fact possible via a MDWD approach.

The transmission line equations given in (3.56) can be transformed in the following way: first introduce new dependent variables

where , the local line impedance is defined by

Such a transformation in fact changes to variables which both have units of root power. After a few elementary manipulations (namely scaling (3.56a) by and (3.56b) by ), we have
 (3.94a)

System (3.91) is still symmetric hyperbolic; referring to the general system from (3.1), for , we now have

 (3.95)

Note that because is now a multiple of the identity matrix, there is near complete symmetry between the variables and . We use the term balanced'' to describe such a system. Note also that new off-diagonal terms have appeared in (compare (3.92) with (3.57)), but they appear antisymmetrically, and thus do not give rise to loss--in other words these terms do not appear in , which determines the growth or decay of the solution, as per (3.5). In fact, these off-diagonal terms yield a lossless (but non-reciprocal) gyrator in the circuit setting.

In terms of the coordinates defined by (3.18), we can then rewrite (3.91) as

 0 0

where

(which should be compared with (3.63), for the standard form), and

As mentioned previously, in a MDKC setting, the terms with coefficient can be treated as a gyrator. The network and its wave digital counterpart are shown in Figure 3.23. The port resistances are given by

In order to accommodate the gyrator, we have been forced, in order to avoid delay-free loops, to set one of the ports to which it is connected to be reflection-free (see §2.3.5). In (1+1)D, we can choose either of these ports, but picking the bottom port in Figure 3.23(b) allows us to extend the idea to (2+1)D easily. This port resistance is then constrained to be

We have two simplifying choices for ; either we can choose it to be reflection-free as well, so that we will have a general gyrator described by (2.25), or we can choose

in which case the gyrator equations (2.25) reduce to a pair of throughs, scaled individually by and its inverse; this latter choice may be problematic if approaches zero, because one of the multipliers becomes unbounded. If is small over some part of the problem domain, however, it is probably wiser to remove the coupling from the network altogether over these regions (it can be replaced by a simple two-port short-circuit). We have assumed, throughout this development, that and (or rather, the local characteristic line impedance ) are differentiable. An offset-sampled version of this network is also possible, if we halve the port resistances and and double the delays at the same ports.

The stability bound, from a requirement on the positivity of and will be exactly (3.90). In an implementation, there will be of course the slight additional costs due to the extra gyrator and the rescaling of the new dependent variables and at every time step in order to obtain and . We note that this scaling can be fully incorporated into the MDKC by treating the scaling coefficients as transformer turns-ratios, though there is no advantage in doing so (other than putting one's mind at ease regarding whether such a scaling is a passive operation).

We will examine how this same technique can be applied to more complex systems when we approach the Timoshenko beam equations in §5.2; in that case, the maximum allowable time step can be radically increased for a system with only mild material parameter variation.

Subsections

Next: Extension to (2+1)D Up: Multidimensional Wave Digital Filters Previous: Note on Perfectly Matched
Stefan Bilbao 2002-01-22