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Wave Variables
At this point, one may assume that we have finished; indeed, we can derive a discretetime equivalent to any LTI port (graphically represented by a signal flow diagram involving shifts and arithmetic operations), and such elements can be connected using Kirchoff's Laws, which remain unchanged by the mapping (2.11). In particular, a network consisting of a collection of connected passive ports will possess a discrete equivalent of the passivity property, which has been called pseudopassivity [42]. The problem, however, is that a simple application of the bilinear transform to a given port usually leaves us with port variables which are not related to each other in a strictly causal way. For example, the difference equation (2.13) that results in the case of the inductor relates to at every time step so that if we try to connect such a discretetime oneport to another which has the same property (using Kirchoff's Laws, which are memoryless), we necessarily end up with nonrealizable delayfree loops [46] in our resulting signal flow diagram. In other words, we will not be able to explicitly update all the port variables in our algorithm using only past values stored in the delay registers.
The problem of these delayfree loops was solved by Fettweis [41] with the introduction of wave variables, a concept with a long history borrowed from microwave electronics [11,12]. For a port with voltage and current , voltage waves are defined by
and are referred to as wave variables, and in particular, is called an input wave and an output wave; the significance of these names will become clear in the examples of §2.3.4. This definition holds instantaneously, and will also be true for continuous and , though we will almost never have occasion to refer to analog wave variables in this thesis. The parameter is a free parameter known as the port resistanceits choice is governed by the character of the element itself. We also can define the port conductance by

(2.15) 
at a port with port resistance .
It is also possible to define powernormalized waves [46]
and
at any port with port resistance by
The two types of waves are simply related to each other by

(2.17a) 
but powernormalized quantities have certain advantages in cases for which a port resistance is timevarying or signal dependent (indeed, in these cases, powernormalized waves must be employed if passivity in the digital simulation is to be maintained). In general, however, in view of (2.17), it should be assumed that we are using voltage waves unless otherwise indicated.
The steady state quantities and are defined in a manner identical to (2.14), where we replace and by and .
Next: Pseudopower and Pseudopassivity
Up: Wave Digital Elements and
Previous: The Bilinear Transform
Stefan Bilbao
20020122