Wave Digital Elements

Under the bilinear transform (2.11), the steady-state equation for an inductor becomes

or, in the discrete-time domain,

Applying the definition of wave variables (2.14), we get, in the time domain,

If we make the choice

then (2.22) simplifies to

Thus the input wave must undergo a delay and sign-inversion before it is output as . In terms of steady-state quantities, we have

The reflectance is, as expected, LBR (see previous section). The resulting

The derivations of the wave digital one-ports corresponding to the resistor and capacitor are similar; their signal-flow graphs also appear in Figure 2.7. We note that the same choice of the port resistance should be made in the case of power-normalized wave variables. We also note in passing that we have used here the symbol to represent a unit delay in a wave digital filter^{}.

It is also possible to combine resistances and sources [46]; a resistive voltage source, shown in Figure 2.9(a), consists of a voltage source in series with a resistor of resistance . If the port resistance of the combined one-port is chosen to be , then the wave digital one-port [46] is as shown in Figure 2.9(b). A wave digital resistive current source can be similarly defined.

The classical transformer and gyrator two-ports can be treated in the same way. For example, the gyrator accepts two input waves and , and yields two output waves and . There are two port resistances, and . The instantaneous equations (2.10) relating the voltages and currents in a gyrator become, upon substitution of wave variables,

which simplifies, under the choice of and to

If we are using power-normalized wave variables, then the scattering equation for the gyrator becomes

In this case, any choice of the port resistances such that gives

The ideal transformer also can take on various forms, depending on the choices of the port resistances and on the type of wave variable employed. Under a choice of port resistances and such that , the equations (2.9) for the ideal transformer of turns ratio become

For the transformer and gyrator WD two-ports, we adopt general symbols that do not reflect a particular choice of the port resistances. If simplifying choices can be made in either case, than we can write the signal flow graph explicitly (see Figure 2.10). There may be occasions when it is not be possible to make these simplifying choices of the port resistances which yield (2.26) and (2.29). For example, when we approach the numerical integration of beam and plate systems in Chapter 5, as well as certain *balanced forms* (see §3.12) the WD networks contain gyrators whose port resistances are constrained, forcing us to use (2.25).

Numerous other wave digital elements have been proposed, namely circulators, quasi-reciprocal lines (QUARLS), as well as *unit elements* [46]. All have been applied fruitfully to filter design problems, but the unit element deserves a special treatment.