Wave Digital ElementsWhen modeling mechanical systems composed of masses, springs, and dashpots, it is best to begin with an electrical equivalent circuit. Equivalent circuits make clear the network-theoretic structure of the system, clearly indicating, for example, whether interacting elements should be connected in series or parallel. Each element of the equivalent circuit can then be replaced by a first-order wave digital element, and the elements are finally parallel or series connected by means of scattering-junction interfaces known as adaptors.
Wave digital elements may be derived from their describing differential equations (in continuous time) as follows:
- First express all physical quantities (such as force and velocity) in terms of traveling-wave components. The traveling wave components are called wave variables. For example, the force on a mass is decomposed as , where is regarded as a traveling wave propagating toward the mass, while is seen as the traveling component propagating away from the mass. A ``traveling wave'' view of force mediation is actually much closer to physical reality than any instantaneous model.
- Second, digitize the resulting traveling-wave system using the bilinear transform. The bilinear transform is equivalent in the time domain to the trapezoidal rule for numerical integration (see §7.3.2).
- Connect elementary units together by means of -port scattering junctions. There are two basic types of scattering junction, one for parallel, and one for series connection. (See §C.8 for the theory of scattering junctions.)
A Physical Derivation of Wave Digital ElementsThis section provides a ``physical'' derivation of Wave Digital Filters (WDF), which contrasts somewhat with the more formal derivation common in the literature. The derivation is presented as a numbered series of steps (some with rather long discussions):
- To each element, such as a capacitor or inductor, attach a
length of waveguide (electrical transmission line) having wave
impedance , and make it infinitesimally long. (Take the limit as
its length goes to zero.) A schematic depiction of this is shown in
Fig.F.1a. For consistency, all signals are Laplace transforms of
their respective time-domain signals. The length must approach zero
in order not to introduce propagation delays into the signal path.
Figure F.1: a) Physical schematic for the derivation of a wave digital model of driving-point impedance . The inserted waveguide impedance is real and positive, but otherwise arbitrary. b) Expanded view of the interior of the infinitesimal waveguide section, also representing the termination impedance as an impedance-step within the waveguide.
- The infinitesimal waveguide is terminated by the element. The element reflects waves as if it were a new waveguide section at impedance , as depicted in Fig.F.1b.
- The interface to the element is recast as traveling-wave components and at impedance . In terms of these components, the physical force on the element is obtained by adding them together: .
- The waveguide impedance is arbitrary because it has been physically introduced. We will need to know it when we connect this element to other elements. The element's interface to other elements is now a waveguide (transmission line) at real impedance .
- The junction is ``parallel'' (cf. §7.2):
- Force (voltage) must be continuous across the junction, since otherwise there would be a finite force across a zero mass, producing infinite acceleration.
- The sum of velocities (currents) into the junction must be zero by conservation of mass (charge).
reflectance of the terminated waveguide. That is, find the Laplace transform of the return wave divided by the Laplace transform of the input wave going into the waveguide. In general, the reflectance of an impedance step for force waves (voltage waves in the electrical case) is
This is easily derived from continuity constraints across the junction. Specifically, referring to Fig.F.1b, let denote the physical force and its traveling-wave components within the ``pseudo-infinitesimal-generalized-waveguide'' defined by the element impedance , with the `' superscript denoting a right-going wave.F.1 Similarly, let denote the velocity and its component wave variables on the side of the junction at impedance , and let denote the corresponding quantities on the element-side of the junction at impedance . Again, the `' superscript denotes travel to the right. Then the physical continuity constraints imply
Now that we've solved for the junction force , the outgoing waves are simply obtained from the force continuity constraint, :
Finally, the force-wave reflectance of an impedance step from to can be found by solving Eq.(F.3) and (F.2) for with set to zero:
reflectances of the elements used in LTI analog electric circuits, viz., the capacitor, inductor, and resistor.
impedance is (see §7.1.3)
inductor of Henrys, we have
Ohms, we get
Note that both the capacitor and inductor reflectances are stable allpass filters, as they must be. Also, the resistor reflectance is always less than 1, no matter what waveguide impedance we choose.
waveguide impedance which will give us a normalized, ``universal reflectance'' for each element:
- For the capacitor, setting gives
- For the inductor, setting gives
- And for the resistor, we set to obtain
bilinear transform means making the substitution
where is an arbitrary real constant, usually taken to be . Solving for gives us the inverse bilinear transform:
In this case, we see that setting further simplifies our universal reflectances in the digital domain:
- For the ``wave digital capacitor'' (or spring), Eq.(F.8) becomes
- For the ``wave digital inductor'' (or mass), Eq.(F.9) becomes
- And for the ``wave digital resistor'' (or dashpot), Eq.(F.10) becomes
F.1), we have that the general reflectance of impedance with respect to the reference impedance in the wave variable formulation is given by
In WDF construction, the free constant in the bilinear transform is taken to be . Thus we obtain . When is first order, it is possible to choose the reference impedance so as to eliminate the delay-free path in the digital reflectance , and so its value depends on the actual physical element being digitized.
mass , we have
spring with stiffness , we have the impedance
dashpot with coefficient , we have
force-wave reflectance of an infinite impedance (rigid wall or ``open circuit'') is
delay line with half a sample delay in each direction. As a result, it really belongs under the topic of distributed modeling. To avoid delay-free loops, Fettweis noted  that every pair of adaptors must be separated by at least one unit element. More recently, this objective is accomplished instead using ``reflection-free ports''  (see also §F.2.2).
Adaptors for Wave Digital Elements