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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 $ f(n)$ on a mass is decomposed as $ f(n) = f^{{+}}(n)+f^{{-}}(n)$, where $ f^{{+}}(n)$ is regarded as a traveling wave propagating toward the mass, while $ f^{{-}}(n)$ 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 $ N$ elementary units together by means of $ N$-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.)

The next section will examine the above steps in greater detail.

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):

  1. To each element, such as a capacitor or inductor, attach a length of waveguide (electrical transmission line) having wave impedance $ R_0$, 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 $ R(s)$. The inserted waveguide impedance $ R_0$ is real and positive, but otherwise arbitrary. b) Expanded view of the interior of the infinitesimal waveguide section, also representing the termination impedance $ R(s)$ as an impedance-step within the waveguide.
    \includegraphics{eps/wdelt}

    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 $ R(s)$, as depicted in Fig.F.1b.

    • The interface to the element is recast as traveling-wave components $ F^{+}(s)$ and $ F^{-}(s)$ at impedance $ R_0$. In terms of these components, the physical force on the element is obtained by adding them together: $ F(s) = F^{+}(s)+F^{-}(s)$.

    • The waveguide impedance $ R_0$ 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 $ R_0$.

    • 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 a General Lumped Waveguide Termination

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

$\displaystyle \fbox{$\displaystyle \hat{\rho}_R(s) \isdef \frac{F^{-}(s)}{F^{+}(s)} = \frac{R(s)-R_0}{R(s)+R_0}$} \protect$ (F.1)

This is easily derived from continuity constraints across the junction. Specifically, referring to Fig.F.1b, let $ F_R(s) = F^{+}_R(s) + F^{-}_R(s)$ denote the physical force and its traveling-wave components within the ``pseudo-infinitesimal-generalized-waveguide'' defined by the element impedance $ R(s)$, with the `$ +$' superscript denoting a right-going wave.F.1 Similarly, let $ V(s) = V^{+}(s)+V^{-}(s)$ denote the velocity and its component wave variables on the side of the junction at impedance $ R_0$, and let $ V_R(s) = V^{+}_R(s)+V^{-}_R(s)$ denote the corresponding quantities on the element-side of the junction at impedance $ R(s)$. Again, the `$ +$' superscript denotes travel to the right. Then the physical continuity constraints imply

\begin{eqnarray*} F(s) &=& F_R(s)\\ 0 &=& V(s) + V_R(s) \end{eqnarray*}

By the definition of wave impedance in a waveguide, we have

\begin{eqnarray*} F^{+}(s) &=& \quad\! R_0 V^{+}(s)\\ F^{-}(s) &=& - R_0 V^{-}(... ... &=& \quad\! R(s) V^{+}_R(s)\\ F^{-}_R(s) &=& - R(s) V^{-}_R(s) \end{eqnarray*}

Thus,

\begin{eqnarray*} 0 &=& V(s) + V_R(s)\\ &=& \left[V^{+}(s)+V^{-}(s)\right] + ... ...s)}\right] &=& \frac{2}{R_0}F^{+}(s) + \frac{2}{R(s)}F^{+}_R(s) \end{eqnarray*}

Defining $ \Gamma_0 \isdef 1/R_0$ and $ \Gamma(s) \isdef 1/R(s)$, we have

$\displaystyle F(s) = \frac{2\Gamma_0}{\Gamma_0+\Gamma(s)} F^{+}(s) + \frac{2\Ga... ...a(s)} F^{+}_R(s) \isdef {\cal A}(s) F^{+}(s) + {\cal A}_R(s)F^{+}_R(s) \protect$ (F.2)

Now that we've solved for the junction force $ F(s)$, the outgoing waves are simply obtained from the force continuity constraint, $ F(s) = F^{+}(s)+F^{-}(s) = F^{+}_R(s)+F^{-}_R(s)$:
$\displaystyle F^{-}(s)$ $\displaystyle =$ $\displaystyle F(s) - F^{+}(s) \protect$ (F.3)
$\displaystyle F^{-}_R(s)$ $\displaystyle =$ $\displaystyle F(s) - F^{+}_R(s) \protect$ (F.4)

Finally, the force-wave reflectance of an impedance step from $ R_0$ to $ R(s)$ can be found by solving Eq.$ \,$(F.3) and (F.2) for $ F^{-}(s)/F^{+}(s)$ with $ F^{+}_R(s)$ set to zero:

\begin{eqnarray*} \hat{\rho}(s) &\isdef & \frac{F^{-}(s)}{F^{+}(s)} = \frac{F(s)... ...mma_0-\Gamma(s)}{\Gamma_0+\Gamma(s)} = \frac{R(s)-R_0}{R(s)+R_0} \end{eqnarray*}

as claimed.

Reflectances of Elementary Impedances

We now derive the reflectances of the elements used in LTI analog electric circuits, viz., the capacitor, inductor, and resistor.

Capacitor Reflectance

For a capacitor of $ C$ Farads, the driving-point impedance is (see §7.1.3)

$\displaystyle R_C(s)=\frac{1}{Cs} $

(or $ k/s$ for a spring with constant $ k$). Substituting into Eq.$ \,$(F.1) gives the reflectance

$\displaystyle \hat{\rho}_C(s) = \frac{R_C(s)-R_0}{R_C(s)+R_0} = \frac{1 - R_0 C s}{1 + R_0 C s} \protect$ (F.5)

Inductor Reflectance

For an inductor of $ L$ Henrys, we have

$\displaystyle R_L(s)$ $\displaystyle =$ $\displaystyle Ls$  
$\displaystyle \,\,\Rightarrow\,\,\hat{\rho}_L(s)$ $\displaystyle =$ $\displaystyle \frac{Ls-R_0}{Ls+R_0} = \frac{ s - R_0/L }{ s + R_0/L} \protect$ (F.6)

Resistor Reflectance

Finally, for a resistor of $ R$ Ohms, we get

$\displaystyle \hat{\rho}_R(s) = \frac{R-R_0}{R+R_0} = \frac{1 - R_0/R }{ 1 + R_0/R } \protect$ (F.7)

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 $ R_0>0$ we choose.

Choosing Impedance to Simplify Element Reflectance

Observe that there is a natural choice for each waveguide impedance which will give us a normalized, ``universal reflectance'' for each element:

  • For the capacitor, setting $ R_0 = 1/C$ gives

    $\displaystyle \fbox{$\displaystyle \hat{\rho}_C(s) = \frac{1 - s}{1 + s}$} \protect$ (F.8)

  • For the inductor, setting $ R_0=L$ gives

    $\displaystyle \fbox{$\displaystyle \hat{\rho}_L(s) = - \frac{1 - s}{1 + s}$} \protect$ (F.9)

  • And for the resistor, we set $ R_0 = R$ to obtain

    $\displaystyle \fbox{$\displaystyle \hat{\rho}_R(s) = 0$} \protect$ (F.10)

Digitizing Elementary Reflectances by Bilinear Transform

Going to discrete time via the bilinear transform means making the substitution

$\displaystyle s = c \frac{1-z^{-1}}{1+z^{-1}}$ (F.11)

where $ c>0$ is an arbitrary real constant, usually taken to be $ c=2/T$.

Solving for $ z^{-1}$ gives us the inverse bilinear transform:

$\displaystyle z^{-1}= \frac{1-s/c}{1+s/c} \protect$ (F.12)

In this case, we see that setting $ c=1$ further simplifies our universal reflectances in the digital domain:

Note that this choice of $ c$ 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 $ c=2/T$ (instead of $ c=1$) in the bilinear transform. From Eq.$ \,$(F.12), it is clear that changing $ c$ amounts to a linear frequency scaling of $ s=j\omega$. Such a scaling may be compensated by choosing the waveguide (port) impedances to be $ R_L = Lc = 2L/T$ (instead of $ R_L=L$) for the inductor, and $ R_C = T/(2C)$ (instead of $ 1/C$) for the capacitor.

Summary of Wave Digital Elements

From Eq.$ \,$(F.1), we have that the general reflectance of impedance $ R(s)$ with respect to the reference impedance $ R_0$ in the wave variable formulation is given by

$\displaystyle \fbox{$\displaystyle \hat{\rho}(s) \isdef \frac{R(s)-R_0}{R(s)+R_0}$} \protect$ (F.13)

In WDF construction, the free constant in the bilinear transform is taken to be $ c=1$. Thus we obtain $ \hat{\rho}_d(z) = \hat{\rho}[(1-z^{-1})/(1+z^{-1})]$. When $ R(s)$ is first order, it is possible to choose the reference impedance $ R_0$ so as to eliminate the delay-free path in the digital reflectance $ \hat{\rho}_d(z)$, and so its value depends on the actual physical element being digitized.

Wave Digital Mass

In the case of a mass $ m$, we have

$\displaystyle R(s) = ms $

which implies its reflectance is, from Eq.$ \,$(F.13),

$\displaystyle \hat{\rho}_m(s) = \frac{ms - R_0}{ms + R_0} $

Setting $ R_0= m$ gives

$\displaystyle \hat{\rho}_m(s) = \frac{s - 1 }{s + 1} $

and this choice also turns out to eliminate the delay-free path in the digital version. In view of the expression for the inverse bilinear transform in Eq.$ \,$(F.12), i.e., $ z=(1+s)/(1-s)$, the bilinear transform of $ \hat{\rho}(s)$ is immediately seen to be

$\displaystyle \fbox{$\displaystyle \hat{\tilde{\rho}}_m(z) = -z^{-1}$} $

where we defined $ \hat{\tilde{\rho}}_m(z) \isdef \hat{\rho}_m\left(\frac{z-1}{z+1}\right)$. The corresponding difference equation for the wave digital mass is

$\displaystyle f^{{-}}(n) = - f^{{+}}(n-1) $

and its wave flow diagram is drawn in Fig.F.2.

Figure F.2: Wave flow diagram for the wave digital mass. Note that the wave variables are written in the time domain as is customary in digital filter diagrams, while it would be more consistent (with the $ z^{-1}$ block) to keep them in the frequency domain as $ F^{+}(z)$ and $ F^{-}(z)$.
\includegraphics{eps/lWaveDigitalMass}

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.

Wave Digital Spring

In the case of a spring with stiffness $ k$, we have the impedance

$\displaystyle R(s) = k/s $

which gives the reflectance

$\displaystyle \hat{\rho}_k(s) = \frac{k/s - R_0}{k/s + R_0} $

As before, we may eliminate $ k$ by choosing $ R_0=k$ to get

$\displaystyle \hat{\rho}_k(s) = \frac{1 - s }{1 + s} = z^{-1} $

under the bilinear transform. So we have the digital reflectance

$\displaystyle \fbox{$\displaystyle \hat{\tilde{\rho}}_k(z) = z^{-1}$} \qquad\makebox[0pt][l]{(Wave Digital Spring)} $

and corresponding difference equation

$\displaystyle f^{{-}}(n) = f^{{+}}(n-1). $

Again the delay-free path has been eliminated. The wave flow diagram is shown in Fig.F.3.

Figure F.3: Wave flow diagram for the Wave Digital Spring.
\includegraphics{eps/lWaveDigitalSpring}

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.

Wave Digital Dashpot

Starting with a dashpot with coefficient $ \mu $, we have

$\displaystyle R(s) = \mu $

and reflectance

$\displaystyle \hat{\rho}_\mu(s) = \frac{\mu - R_0}{\mu + R_0} $

This time, choosing $ R_0$ equal to the element value gives

$\displaystyle \hat{\rho}_\mu(s) = 0 $

Conformally mapping the zero function yields the zero function so that

$\displaystyle \fbox{$\displaystyle \hat{\tilde{\rho}}_\mu(z) = 0$} $

as well. Thus, the WDF of a dashpot is a ``wave sink,'' as diagrammed in Fig.F.4.

Figure F.4: Wave flow diagram for the Wave Digital Dashpot.
\includegraphics{eps/lWaveDigitalDashpot}

In the context of waveguide theory, a zero reflectance corresponds to a matched impedance, i.e., the terminating transmission-line impedance equals the characteristic impedance of the line.

The difference equation for the wave digital dashpot is simply $ f^{{-}}(n)=0$. While this may appear overly degenerate at first, remember that the interface to the element is a port at impedance $ R_0=\mu$. Thus, in this particular case only, the infinitesimal waveguide interface is the element itself.

Limiting Cases

The force-wave reflectance of an infinite impedance (rigid wall or ``open circuit'') is

$\displaystyle \hat{\rho}(s) = \frac{R(s) - R_0}{R(s)+R_0} = \frac{\infty - R_0}{\infty +R_0} = 1 $

Similarly, the force-wave reflectance of a zero impedance (free termination, frictionless surface, or ``short circuit'') is

$\displaystyle \hat{\rho}(s) = \frac{0 - R_0}{0+R_0} = -1 $

For velocity waves, we obtain the opposite results: rigid terminations are inverting, and free terminations are non-inverting.

Unit Elements

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 [135] 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'' [136] (see also §F.2.2).

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Adaptors for Wave Digital Elements
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Acknowledgments