Wave Digital Elements
When 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
is regarded as a
traveling wave propagating toward the mass, while
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
- 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.)
An important benefit of introducing wave variables prior to bilinear transformation is the elimination of delay-free loops when connecting elementary building blocks. In other words, any number of elementary models can be interconnected, in series or in parallel, and the resulting finite-difference scheme remains explicit (free of delay-free loops).
A Physical Derivation of Wave Digital Elements
This 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.
Points to note:
- 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
- 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
- The sum of velocities (currents) into the junction must be zero
by conservation of mass (charge).
- Force (voltage) must be continuous across the junction, since otherwise there would be a finite force across a zero mass, producing infinite acceleration.
- 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.
Calculate the 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
By the definition of wave impedance in a waveguide, we have
Defining and , we have
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:
Finally, for a resistor of Ohms, we get
- For the capacitor, setting gives
- For the inductor, setting gives
- And for the resistor, we set to obtain
Going to discrete time via the 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
Note that this choice of is also the only one that eliminates delay-free paths in the fundamental elements. This allows them to be used as building blocks for explicit finite difference schemes.
We may still obtain the above results using the more typical value (instead of ) in the bilinear transform. From Eq.(F.12), it is clear that changing amounts to a linear frequency scaling of . Such a scaling may be compensated by choosing the waveguide (port) impedances to be (instead of ) for the inductor, and (instead of ) for the capacitor.
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.
In the case of a mass , we have
Thus, the wave digital mass is simply a unit-sample delay and a negation. The fact that the value of the mass has been canceled out will be addressed below in the subsection on ``adaptors,'' i.e., it only affects interconnection with other elements. For now, just remember that the reference impedance was chosen to be equal to the mass in order to get this simple wave flow diagram. Also note that the WDF mass simulator has no delay-free path from input to output.
Thus, the WDF of a spring is simply a unit-sample delay, which is just the negative of the WDF mass. If we were to switch to velocity waves instead of force waves, both masses and springs would again correspond to unit-sample delays, but the spring would become inverting and the mass non-inverting.
Starting with a dashpot with coefficient , we have
The difference equation for the wave digital dashpot is simply . While this may appear overly degenerate at first, remember that the interface to the element is a port at impedance . Thus, in this particular case only, the infinitesimal waveguide interface is the element itself.
The unit element two-port is simply a bidirectional 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